# 20: The Proof

Many parts of this chapter rely on the mathematical theory of ‘tensors’. It is therefore more mathematically oriented than the earlier ones. Please be sure to read ‘Before We Begin’ to gain some background, and to acquaint yourself with the general strategy we pursue here.

Figure 20.1

The principles we need to apply are very basic. We saw in the chapter ‘Before We Begin’ that it takes only a little imagination to adapt a framework of a line and two circles from a dodo to an ostrich; or a framework of two rectangles and an ellipse into a knuckle-walking gorilla. As Figure 20.1 shows, it also only takes a little imagination to adapt ordinary physical space into a “biological space” that we can then use to prove evolution.

It is invariably easier to picture simple points, lines, and shapes in ordinary two- and three-dimensional space. Figure 20.1.A shows three twirling batons. Each is made of two points with a strobe light that promenades from end to end as they twirl. They create a point in biological space where they intersect. Those promenading dots and batons are also the three intersecting curves in Figure 20.1.B. They oscillate between their minimum and maximum values to create the same point as in 20.1.A. Their twirls and oscillations also create the nested spheres (representing the minimums and maximums) and the point moving between them in Figure 20.1.C.

The force that makes those batons twirl is very similar to gravity, which is very familiar. Just like gravity, it passes through and pervades all objects. It can create orbits like planets, and conglomerations of objects like galaxies. If you can imagine all that, then you will have very little trouble indeed following the ideas we present in this chapter.

Refuting creationism and intelligent design, as we did in the last chapter, is one thing. But that only negates something. Building a positive proof that something else simply must be true is quite another. We shall positively prove Darwin's theory using a very simple method … one based on the common notion we met in the Introduction that “Things that are equal to the same thing are also equal to each other”. So if A is equal to B, and B is equal to C, then A should be equal to C. We use that as the basis for our Brassica rapa experiment which shows that creationism and intelligent design are deficient for they fail to meet this simple standard. We will use the identity tensor we met in the Refutation to prove this.

Figure 20.2

We shall also make good on the three claims we made in the Introduction:

1. We provide the definitive and deductive logical proof of Darwinian fitness, competition, and evolution.
2. We prove that every biological population can be uniquely characterized by the three properties of pitch, radius, and thickness we see in the helicoid in Figure 20.2. (A helicoid is the three-dimensional helix we can create by attaching a straight line to a central axis, and then rotating it while dragging it upwards). These create the (i) poloidal, (ii) meridional, and (iii) toroidal movements about the helicoid that define every population and species. The poloidal are the purely vertical, and so entirely timelike, from past to future. The meridional are purely horizontal, and so are spacelike, geographic, and structural. The toroidal are a time-and-distance measure containing elements of both.
3. We further prove that every biological population can be characterized by a single number—its evolutionary potential, η—that summarizes its entire evolutionary history.

If we are going to explain evolution and refute creationism and intelligent design, we must ultimately explain the molecular behaviour of biological organisms and the populations they form. That means explaining the internal energy that forms both individual biological entities and their populations. Our intention is to do exactly this and run an experiment (with the plant species Brassica rapa) to at last explain heredity and inheritance as the driving forces for evolution, in exactly the way Darwin suggested. However, all we are going to do is grow some plants. Since there is nothing special either about those plants or the conditions we grow them in, the trick must lie in how we propose to interpret that data and use it to explain the biomolecular behaviour and internal energy of all organisms.

Given the preparatory work we have done in the earlier chapters, we are obviously going to:

1. use the linear planimeter and the method of fluxes to analyse the absolute values of the plant population totals of internal energy; and then
2. use the polar planimeter and the method of tensors to analyse the relative values of the partitions of internal energy allocated to each individual plant.

We learned earlier that, thanks to James Maxwell, any difference between what the linear and polar planimeters measure is called a ‘curl’. Its symbol is ‘∇ ×’. We also learned that we can look on our three constraints of constant propagation, constant size, and constant equivalence as three separate orthogonal dimensions in the internal energy (i.e. mutually at 90°) that create our biological space. Those three are:

1. Number of entities in a population, n. This tells us the number of partitions we have at any time in the mass and energy aspects of internal energy we have across the population.
2. The mechanical chemical energy aspect of internal energy. This keeps a given number of moles of chemical components bound up within the population's internal energy. It is measured as m kilogrammes per each individual entity. Its average, including its distribution over the population, is . That average tells us the number of molecules allocated to each of the partitions into which internal energy is divided.
3. The nonmechanical chemical energy which energizes these same components. It services the partitionings in internal energy. It is measured as p joules per each individual entity. Its average, again including its distribution, is . This tells us how those molecules are configured within each partition, and so how the entities are using the internal energy by interacting both with each other, and with the surroundings.

When we combine (1) and (2) we produce the mass flux or Mendel pressure of the mechanical chemical energy, M, aspect of internal energy; when we combine (1) and (3) we produce the energy flux or the Wallace pressure and nonmechanical energy aspect, P, of internal energy; and when we combine (2) and (3) we produce the internal energy's current density and capabilities. This last can be expressed as either the dynamical work rate, W; or else as its inverse, which is the visible presence, V.

And … this is how we shall prove that creationism and intelligent design are false. We simply compare the individual to the population behaviours over time. When we combine (1) and (2) above, we produce nm̅ = M, which is a population behaviour. When we combine (1) and (3) we produce np̅ = P, which is another population behaviour. We are therefore going to compare independently to M at all points; and the same for and P. We also have two different ways of combining (2) and (3) to produce V (or W). We can use either the individual entity or the whole population values. We can use the individual values to make V = ; or we can use the population values to make V = MP. If creationism and intelligent design are true, those two should be the same so that = nm̅np̅ = MP. But these two can only be the same if the actual measured individual and population values for both mass and energy—i.e. and M, and and P—always change at the same rates. Numbers, n, can only have no effect if = MP at all times. Creationism and intelligent design mean that the individual and the population values must always match so they are both independent of everything to do with numbers. This requires that dm̅dt = dMdt and dp̅dt = dPdt, so that the effect of numbers is zero, and we have ∂n/∂t = 0. In other words, the linear and polar planimeters must always measure the same value so there is no curl.

We now have a very simple aim. We are going to show that the individual and the population values are not the same, which is that MP, that dm̅dtdMdt, that dp̅dtdPdt, and therefore that ∂n/∂t ≠ 0. We shall demonstrate, firstly, that these values cannot be the same in theory; and then demonstrate, secondly, that they are not the same in practice. That is the purpose of our Brassica rapa experiment. That experiment allows us to use our two planimeters to measure those differences between and MP, and to show that the magnitude of their difference is always equal to ∂n/∂t, which therefore acts as a force.

If creationism and intelligent design are true, then since M= nm̅, and P = np̅, we should of course have V = MP = nm̅np̅ = at all times. But that ignores differences in the rate of change between the individual and population values, which is the heart of Darwin's claim. Our linear and polar planimeters open up the way to measure precisely that difference. Any difference between those two values can only be caused by differences in n; and is caused entirely by differences in n. We must therefore find out whether or not M, P, , , and n do, or do not, change at the same rates; and if they do not, find out the causes.

A mathematical aside

We are going to build both a configuration space and a phase space for biological entities and populations. We already have the constraints to be imposed on that space, and that therefore describe all its possible configurations. Through our vector unit normals, we also have a system of generalized coordinates.

We are primarily interested in two things:

1. how a population's specific properties are manifested within any one generation; and
2. how heredity and traits are transmitted from generation to generation within the three-dimensional biological space of internal energy we have suggested.

We note that (1) and (2) immediately involve (A) and (B) above. They each require an analysis of both (a) absolute totals, and (b) relative individual values. The relative individual values are also statements of the distributions of the molecules that make up internal energy across both space and time.

Biological creatures may live in our three-dimensional biological space of n, , and V, but they also live in ordinary physical space with its standard x, y, and z dimensions. Indeed, ordinary physical space is the only place we can measure them. We therefore take perfectly ordinary measurements. We are just careful to pay special attention to the number of entities at each moment; their masses both individually and as a population; and the amount of energy they contain again individually, and as a population. We simply bring those values together and look on them as an addition to ordinary physical space. Biological populations then navigate both those spaces, each with three dimensions.

We are interested in heredity: the transmission of memes and genes we see in biology. Both our sets of spaces and dimensions need the extra fourth dimension of time, t, before we can use them to analyse that temporal transmission. Creationism and intelligent design fail precisely because they do not appreciate the implications of including this fourth dimension.

Heredity and traits require the transmission of properties over time, t. Since we are juxtaposing relative and absolute values, we clearly intend to use certain aspects of Einstein's special and general theories of relativity. The general theory in particular has a reputation for being complex, but we only need limited aspects. We shall therefore proceed largely pictorially.

Perhaps nobody sets out the creationist and intelligent design position regarding space and time more clearly than does Newton in the Scholium (or ‘comment’) to his Principia. He demonstrates, perhaps more convincingly, than anyone else could, why the position he describes continues to be so seductive in biology even though, thanks to Einstein, the rest of science has rejected it:

I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration ….

II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. …

III. Place is a part of space which a body takes up …. … The motion of the whole is the same with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the place of the whole is the same as the sum of the places as the parts, and for that reason, it is internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute place into another …. But real, absolute rest, is the continuance of the body in the same part of that immovable space …. …

V. … For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time …. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the flowing of absolute time is not liable to any change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow …. …

VI. As the order of the parts of time is immutable, so also is the order of the parts of space. …

VIII. But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes, and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. …

IX. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. …

Sir Isaac Newton, 1689. Scholium to “Definitions” in Philosophiae Naturalis Principia Mathematica (Book 1). Translated by Andrew Motte, 1729. Revised by Florian Cajori, 1934, Berkeley: University of California Press.

Newton's declarations are certainly very endearing, but we now know that although his master work, Principia, as a whole, stands inviolate over science, nothing he says here in his Scholium explains the universe around us. We shall use the term ‘firmament’ to describe Newton's now outmoded vision of time and space.

We may be joining modern science in rejecting the biological implications of Newton's firmament, but the rest of his physics stands. It still only needs one or two modifications to incorporate the quantum theory that accounts for the molecules that constitute both biological and non-biological entities throughout the cosmos. It is the firm edifice on which all of science rests.

Figure 20.3

We can use Figure 20.3 to express the difference between the firmament views that creationism and intelligent design advocate on the one hand, and the non-firmament views that Darwinian evolution advocates on the other. We can observe it in the realization that everything that changes—including every planet—is always hovering between two different forms of perfection: the perfectly linear and the perfectly curved.

There is a widespread belief that, as in Figure 20.3.A, Newton took science into the modern era by refuting Aristotle's ideas of perfectly circular motion. But that is not quite the reality. Aristotle's circular motion did not become irrelevant.

Kepler laid the foundations for Newton by proving, as in Figure 20.3.B, that if we measure a planet's motions from some moment t-1 in the past to some moment t1 in the future, as at Locations I and II, then those two black areas are equal. Newton then successfully combined Kepler's laws of planetary motion with Galileo's observation that all bodies would move in a straight line—and so at the tangents drawn in Figures 20.3.A, B, and C—unless acted on by a force. The planet's orbit is an ellipse … which is a combination of the straight line and the circle.

The circular orbits we see in Figure 20.3.A, and that are now so firmly associated with Aristotle, are entirely possible in Newton's theory. But by Galileo's dictum, the earth is always trying to move straight ahead on its tangent. So it keeps moving, in its straight line, infinitesimally further away from the sun. But as it moves fractionally further away, it slows down microscopically, because it has to fight against the sun's constant pull. At a certain point it is so far away, and so close to being stationary, that it can no longer avoid the sun's pull, the sun's gravitational force yanks it back, and so it accelerates back in. It then swings past the sun, and the same happens the other side. Its orbital path therefore changes continuously between the straight line and the circle.

A circular orbit is possible in these conditions, but it requires a perfect exchange between the tangential-going-ahead and circularly-pulling-around energies concerned. This is possible in theory, but impossible to realize in practice. We only need to introduce one other planet, and we will immediately have a set of perturbations that make the circular orbit impossible.

Since the earth and other planets swing back and forth around the sun, they feel a variable force, and therefore have variable accelerations. Figure 20.3.C describes this effect as we measure between the times t-1 and t1.

The tangent measures how close to a straight line the moving body is. Leibniz also realized, however, that every elliptical path also has a curvature. That curvature can be measured with the lightly drawn circule we see in Figure 20.32.C that he called the circulus osculans or ‘kissing circle’, and now known as the “osculating circle”. The osculating circle tells us exactly how curved the body is at that point.

The osculating circle at any given point to a curve is the circle that just touches the tangent at that same point. While the tangent now tells us how straight the curve is, i.e. in terms of the slope of the straight line that just touches it; the osculating circle similarly tells us how curvy the same curve is, in terms of the osculating circle that just fits it.

We can perhaps begin to see the advantage of having both a linear and a polar planimeter to work with. The straight line, or tangent, is one form of perfection. It has zero curvature. The tangent measures linearity: how close to a straight line the moving body is.

But although the tangent and linear planimeter might tell us how fast ahead the object might be moving, and so reveals the velocity; it tells us nothing about how much the object might be curving as it is yanked back around. Leibniz realized that every elliptical path also has a curvature, which needs to be measured. His osculating circle does this. A circle, the other kind of perfection, can reveal this. A circle is completely symmetric. It has exactly the same degree of curvature everywhere about it.

As in Figure 20.3.C we can determine how curvy something is from its “degree of aberrancy” which is the extent of its departure from being a circle. We can measure any curve with an osculating circle.

The osculating circle has zero aberrancy because it is perfectly, and symmetrically, curvy. We can see this by drawing a line at 90° to the tangent, which is then the ‘normal’ at that point. We next draw a series of chords or lines within both the curve and the osculating circle, all of them parallel to the tangent. Since the osculating circle is a perfect shape, it bisects all its chords. All their midpoints lie on its radius. One of its chords will pass directly through its centre and act as its diameter.

The curve that describes our planet does not have the osculating circle's perfect symmetry. The midpoints of its chords do not lie on their normal. They instead fall off the normal, and create the “axis of aberrancy” at that point. The greater is the curve's aberrancy, then the more do the normal and the axis of aberrancy differ. The degree of deviation tells us the degree of aberrancy.

Aristotle's perfect circular motion has zero aberrancy at all points. It is the template for perfect circular motion. In the same way, creationism and intelligent design claim that biological entities always follow the template for their species. They also have zero aberrancy at all points.

The planets, however, do not move in circles. Kepler's elliptical motion is not circular. And since it is not circular, it has some measure of aberrancy. It is always either pulling a planet so it moves a little further out away from a circle, or else reeling it back in behind where a circle would be. If a planet remains in regular elliptical motion, then its axis of aberrancy will first lean one way; lean increasingly that way; return; then lean another way; lean increasingly in that other direction; and then return. The ellipse can now sometimes be more curvy than a circle, and sometimes less. The sum of all those aberrancies, all around its orbit, can now be zero … but without being constantly zero in the way Aristotle's perfectly curvy circle demands.

Creationism and intelligent design are now the biological equivalent of Aristotelian circular motion. They insist that all biological entities follow some abstract template without deviation and therefore without variations … and so without aberrancies.

Darwin disagrees with this proposal. He instead asserts that it is perfectly possible for the creatures in a species to deviate a little this way, and a little that way, over a period of time, and so that they can maintain the equilibrium that is a species by summing their aberrancies to zero.

However, the aberrancies are not obliged to point to any one specific spot in the firmament. When the sum of the aberrances changes slightly, we have evolution.

We can reframe this debate as that of determining whether or not biological populations have aberrancies. All we then have to do—as we did in our Brassica rapa experiment—is measure a difference between our linear and polar planimeters. That is exactly the difference between linearity and curviness, and is an aberrancy. If we measure that, then we will have achieved our purpose. It will be a direct refutation of creationism and intelligent design, and a proof of Darwin's thesis.

If we wish to understand biological creatures as they move about the circulation of the generations, then we need the extra dimension of time, t. Creationism and intelligent design do not appreciate that this four-dimensional approach has proven that a population free from Darwinian fitness, competition, and evolution is impossible. They instead prefer to perpetuate the erroneous firmament views that Newton describes so clearly in his Scholium. They thus embody a near-fanatical refusal to take Einstein's discoveries about matter and energy, and space and time into account.

It is important to understand, clearly, that although Newton's views on the firmament might have been refuted, his physics still stands. It has, in fact, been greatly extended by incorporating that fourth dimension. A particularly apposite extension to Newton's physics came on September 21, 1908, when Einstein's former college mathematics lecturer Hermann Minkowski spoke to the 80th Assembly of German Natural Scientists and Physicians in Cologne, Germany. He helped demolish the scientific relevance of Newton's firmament by presenting his somewhat easier to understand, and more geometric, interpretation of Einstein's recently published special theory.

According to Abraham Pais's biography Subtle Is the Lord: The Science and the Life of Albert Einstein, Einstein initially dismissed Minkowski's four-dimensional interpretation of his theory as ‘superfluous learnedness’. He changed his mind very quickly when he saw its simplicity and utility:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. ….

A point of space at a point of time, that is, a system of values, x, y, z, t, I will call a world-point. The multiplicity of all thinkable x, y, z, t systems of values we will christen the world.

Hermann Minkowski, 1908. Quoted in: Volkert, K., From Legendre to Minkowski–the History of Mathematical Space in the 19th century.

Figure 20.4

We will eventually prove that every biological population must evolve precisely because the parallel lines and definite angles and ratios—i.e. free from all aberrancies—we see in Figure 20.4, and that creationism and intelligent design require, are impossible once we incorporate the fourth dimension of time into biological affairs. We shall prove that a population free from Darwinian fitness, competition, and evolution is impossible because the “right helicoid” shown in profile in Figure 20.4.A, and end on in 20.4.B, is impossible.

The figure is called a right helicoid because as it spirals upwards, it describes a cylinder whose sides are parallel all around it, its base also being at right angles to those sides. The pitch and the thickness are also constant. When viewed on its end, it describes a circle of constant radius. It is therefore free from aberrancy. We are going to prove that this is impossible both macroscopically and microscopically, given the quartet of values x, y, z and t, that Minkowski provides, and the innate characteristics of the internal energy they describe.

Put very simply, the requirement that aberrancy be zero tries to stipulate the behaviour of molecules. This, however, breaches the famous Heisenberg uncertainty principle: that it is impossible to predict both of any molecule's position, q, and momentum, p, to a sufficient degree of accuracy. This makes it impossible to establish the regular shapes we see in Figure 20.4.

Although it seems very obvious that the shape in Figure 20.4.C is a square, reasoning in four dimensions—which is the easiest way to achieve our purposes—is often counter-intuitive. ‘Straight’ and ‘parallel’ becomes a little less obvious. The square's corners are not just potentially at different points in space. They are also at different moments in time. This means that if two things are parallel, then they must also change at the same rates so they do the same things at the same point in time.

This is all an attempt to specify the molecules that make up internal energy. There is a certain sense in which the two circles in Figure 20.4.D are parallel. They have the same centre and can go round and round indefinitely without ever meeting. The two shapes in Figure 20.4.E and 20.4.F could also be considered parallel because they are made from the same curves, and have the same areas and perimeters, even though they look so different. They could easily be moving at complementary rates, just like two planets in matching elliptical orbits about the same sun. Figure 20.4.G could also be considered parallel to both 20.4.E and 20.4.F because it again has the same area and perimeter. We must take very great care with differences and similarities when looking at things in four dimensions. This extends to the molecular behaviours and patterns of the objects concerned. Two things can be parallel if one starts at an earlier moment but finishes at a later one, and simply through having the same rates. We will refute creationism and intelligent design not only by showing that they do not respect these kinds of parallels, but also by measuring the exact amounts by which the doctrines fail, which is what we did in our experiment.

A mathematical aside

A helicoid is a ruled surface. It can be generated by a straight line. It is uniquely defined by the three properties (1) x = ρsinθ; (2) y = ρcosθ and (3) z = tan α + kθ when expressed in Cartesian coordinates. The angle θ takes the population around each circulation, while -ρ to ρ measure the range from minus to plus infinity for the number of generations, and k and α are constants that establish the helicoid's size and orientation. In cylindrical coordinates this is simply z = kθ. This confirms that every biological population is uniquely defined with three, and only three, properties.

We must begin by dealing with three important misconceptions about Einstein's theory. They lie at the heart of our third and last anomaly. It joins our previous two.

The first misconception arises because we are going to completely reassess the traditional ideas of what “forces” a given biological organism to be a “member of a species”, and/or to be “in a population”. It is therefore vital to appreciate that, in the Einstein theory, gravity is not a “force”. Gravity is instead a means of cajoling objects to behave in particular ways … which is what the “species force” does to biological entities in the biological space of internal energy we have constructed.

Figure 20.5

As in Figure 20.5, the sun and the earth do not exercise forces on each other. Newton's firmament of absolute space and absolute time differs, very greatly, from Einstein's views on this very point. The sun, earth, and other celestial bodies instead cause distortions or warpings in the field Minkowski called “spacetime”. So according to Einstein's theory of general relativity, every collection of mass-energy distorts the local spacetime geometry. A ball doesn't “fall” to the earth's surface because some Newtonian-style attraction, acting within the firmament, exists between it and the earth. It instead feels, and causes, the effects of a “propagating current-element” in the fabric of spacetime as it slides along the “geodesic” or curved path of shortest length that the earth creates in the spacetime field surrounding both of them, courtesy of its own mass. Every body warps its local spacetime, and then seeks to induce all others to slide towards it. These cosmos-wide collections of warpings establish the same kind of overall spacetime geometry that species fashion within their surroundings.

In general relativity the gravity field lines associated with a given body always appear to originate out at infinity, relative to that body. That is the spacetime fabric to which every body with mass and energy contributes. But while all field lines might originate elsewhere, from the perspective of each body, they all terminate on that body, and so on some specific mass or other. No individual body may “know” where any field lines begin, but they all “know” where they end: i.e. right on themselves. All masses act as localized gravity sinks, of a greater or a lesser magnitude, within their local spacetime geometry. Other bodies can then fall towards them, along those lines they provide. The bodies set up fields of potential energy complete with propagating current-elements, which then register as the kinetic energy of those bodies that respond and that accelerate through their established fields.

A body apparently falling under gravity is simply responding to the local geometry. It is spontaneously gliding along its local field lines towards the end point of some field line. It is converting the potential energy of its position, relative to some body, into kinetic energy. It does so by rolling on those field lines it finds around it. It is then apparently getting closer to whatever larger body is responsible for creating that localized warp, which therefore—and in its turn—seems to be exerting a force; and which is seemingly reeling the smaller body down into the gravity well it has provided. It thus consistently gives that smaller body less far to fall towards it at each passing moment, again relative to itself. But the falling body is simply following the field lines established in that local geometry, and as a propagating current-element.

So as again in the Figure 20.5, we in our turn remain glued to the earth's surface not because of an attractive gravitational force, but rather because our feet constantly slide along the earth's lines of curvature; while the earth similarly and simultaneously slides along the sun's lines. We thus “fall” because the intensity of our curvatures is minuscule. It creates propagating current-elements of negligible power when compared to either the sun's, or even the earth's. This all appears as us succumbing to, and feeling, the force of gravity that Newton successfully explained, albeit using his now-antiquated concept of a firmament. Gravity is not, however, a force. It is the localized warping and gravity well formed by the surrounding celestial bodies. Space is not a featureless firmament abstractly waiting for bodies to be placed in it. Space is instead formed directly by bodies acting gravitationally, and creating potential energy wells around themselves. There is therefore no firmament holding some form of template for absolute space and time that pre-exists them, and that they can all move about in. Despite all claims to the contrary that creationism and intelligent design make, the same is true in biology. Our biological space is composed of an internal energy that permeates all points, and behaves very similarly to gravity.

As for the second important misconception about relativity, many people believe that Einstein's theory says that nothing can move faster than the speed of light. As we saw earlier, this is only partly correct. The speed of light has many ramifications in relativity theory. In this context it more accurately insists that nothing can travel faster than the speed of light … IN FREE SPACE. That addendum is important because the speed of light varies with the medium, immediately invoking microscopic behaviour.

Quite apart from the tachyon issue we have already discussed, it is perfectly possible for an object with inertia to move faster than the speed of light, but within some specific medium. This is because light, as an electromagnetic radiation, is transported by photons which are quantum and microscopic particles that, like all others, interact with the molecules contained in their transporting medium. Those collisions then refract the light. But its speed of propagation, across that medium, must then change. Glass, for example, has a refractive index of 1.49, meaning that light passes through it with a speed of c/1.49 (where c is the speed of light). That differential is caused by the interaction light has with the particles in the glass. But although the speed of light in glass is different from its speed in free space, an individual photon's “speed” never deviates from c. The interference the medium spectroscopically imposes simply means that light's journey through that medium takes a little longer than it would take to cover a similar distance in free space. The glass has warped and changed the nature of the space that light has to pass through. The photon's “speed” is always the same, but the ultimate result depends upon the medium. Light in some ways has “further to travel” within a medium, because it “runs into many more obstacles”, which then “take up its time”. This produces the apparent change in light's phase velocity, and that we quantify as its refractive index. It is just as accurate to say that glass warps the space inside it to give light a greater—relative—distance to travel, internally to it, than it looks like externally to it and relative to ourselves out here in (relative) free space. If we see a piece of glass 1 centimetre thick then since it is filled with obstacles, we should think of it as a passage of free space 1.49 centimetres long.

When light travels in some medium, it has stopped being merely a disturbance in the electromagnetic field. It has instead become a disturbance of both (a) the field, and (b) the positions and the velocities of whatever charged particles reside within that medium. But light's “true speed”—which it would and does adopt when in a vacuum, and so when free and solely a disturbance of the underlying electromagnetic field—never changes.

Since the amount by which light is seemingly slowed depends entirely upon the medium, it is perfectly feasible that the atoms or charged particles within some medium could be moving so energetically that they move faster than light's photons can within that same medium … which is exactly what happens with the Cherenkov radiation found inside, for example, nuclear power reactors. Largish nuclear particles can be ejected from nuclei with such vehemence that their high momentum allows them to ram their way through their surroundings at a speed greater than the light they emit. Those emitted light photons, being less massive, now have considerably less forwards momentum than do the particles that emit them, for they are much delayed and detained by the many obstacles they run into in the highly energetic medium that the nuclear reactor imposes. The large particles that have emitted those photons are now of higher momentum, and so can charge ahead, outpacing the light they have emitted, so causing the distinctive bluish glow of Cherenkov radiation within that nuclear medium. We cannot, therefore, discuss biological behaviour without taking careful account of differences in medium between different entities and populations. Biological activities within internal energy can sometimes slow down, and sometimes speed up.

And finally, the third misconception about gravity is partly a product of the Newton firmament proposal. It is associated with the apparent “redshift” of galaxies beyond the Milky Way, which is an apparent change in the frequency of their emitted light.

Some of the apparent galactic redshift motions are caused by the Doppler effect that is similar to the apparent change we hear in a siren's pitch as a police car or similar passes by. This happens because we hear and can easily measure the changes in the relative velocity between ourselves and the wailing police car.

Einstein's general theory proposes something rather different. It instead proposes that space itself is expanding, and quite independently of any relative galactic motions. The distance between galaxies can appear to increase simply because space itself is growing as the universe expands and changes its energy density. The lengths of gravitational field lines also appear to change. It now makes no real sense to talk of galaxies as moving and receding, because they are not. It is the expanding space that gives the illusion of their motion. The effect of space expanding is very similar to the Doppler effect, which is therefore a useful explanatory device. The cause, however, is radically different, and so that explanation should only be taken so far.

Propagating photons, in an expanding universe, seem to be “stretched” into a cosmological redshift in the sense that they now take longer to arrive—because the distance is greater—than they otherwise would have done. The effect is perceivable in the energy change in the arriving photons we detect over time. But this global or cosmic redshift due to the expansion of space is again quite distinct from the more localized firmament-based Doppler redshift seemingly caused by relative velocities. One belongs to the fabric or medium of spacetime; the other to the objects embedded in that fabric. This is an important distinction.

The net consequence is that no two planets at different points in the cosmos have a clear and well-defined relative velocity. Velocity in that sense requires a Newton-based firmament approach, which belies the known reality. It may seem useful to talk in terms of relative velocities, Doppler effects, and redshifts, but that is not the reality. Galaxies can appear to be moving relative to each other, when they are not. In the same way, it is entirely possible that two species or configuration of biological internal energy in biological space could appear to be moving—i.e. evolving—when they are not. They could also appear to be not evolving, when they are.

Figure 20.6

It is important to bear the above three points in mind because:

1. We are going to show you that biological organisms appear to us to be found in species and populations because they “fall” into such categories, projected there by the localized lines and configurations of their immediate biological-ecological spacetime geography and mass-energy potentials and landscapes, as we see in the helicoid of internal energy in Figure 20.6. Time moves from pole to pole, which is poloidally from the past of minus infinity to a future of plus infinity, and upwards on the vertical axis from some t-1, onwards towards some t1. Those are placed either side of some present moment, t0. Biological populations then go toroidally round the circulation of the generations, with the generation length, T, being the circumference. The entire circulation depends on a combination of (I) the pitch, (II) the radius, and (III) the medium.
2. We are also going to show you that (i) there is a definite “speed of reproduction” that is a property of biological space, and that is therefore a characteristic of any given biological population, when seen as a medium of propagation; (ii) that there is also a universal speed of reproduction, τ, held in common by all populations, and so that is a universal constant of nature; (iii) that this universal speed is as unattainable and/or unsustainable as is the speed of light in free space; and (iv) that the Darwinian fitness, competition, and evolution we observe is entirely due to variations about that universal and constant speed of reproduction and propagation and caused by biological space itself.

We can glean the bulk of what we need about relativity, and so prove all the above, directly from Einstein, who presents the essentials very simply indeed. He states that he based his entire theory on the common notion “things that are equal to the same thing are also equal to each other”. We shall in our turn prove that “things that are equal to the same thing are also equal to each other” both within and between populations and their constellations of internal energy, and therefore that Darwinian competition and evolution exist.

Einstein's initial proposal was:

… we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A… :—

1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Albert Einstein, On the Electrodynamics of Moving Bodies, 1905.

This reads like a tautology, but he developed it into the far-ranging theories that made him world-famous. We shall soon present a proposal—it involves our third anomaly—that will, in its turn, also appear like a tautology, but that is the sole cause of Darwinian competition and evolution.

A mathematical aside

We now have the principal characteristics for our biological configuration and phase spaces.

Now we understand that light travelling through some medium has both (a) a “pure element” that is solely the disturbance of the electromagnetic field; and (b) an “impure element” that varies with the medium; we can use our tensor to (i) find out what the speed of reproduction might be in our proposed three-dimensional biological medium, as well as (ii) to measure the transformations that biological entities undertake. We can then place absolute values along the rows and/or columns in our tensor as appropriate; place relative ones in whichever other; and then see how they interact.

Our biological potential, μ, sets the timetable for reproduction. It is the confluence of (A) the mechanical chemical energy of a mass of chemical components acting as genes and genomes within internal energy; and (B) the nonmechanical chemical energy of the chemical pathways that heat, light, and biochemical transformations make available through the Gibbs and the Helmholtz energies active within the same internal energy. The “speed” at which reproduction passes through a biological medium, or population, is then a question of the mass, the numbers, and the energy that any given population carries both absolutely and relatively, and at the direction of its biological potential. This is intrinsically molecular. We must again account for the potentials that drive such behaviours, which means clarifying those Gibbs and Helmholtz energies.

As suggested by our third maxim of ecology, which is the maxim of succession, heredity is the question of the potentialities embedded in a medium of propagation, which is a population in our three-dimensional biological space. Creationism and intelligent design suggest that heredity proceeds via an invariant template independently of any and all media, spaces, and even of the entities themselves. Darwinian evolution denies this but is then left with the well-known problem of properly defining fitness, competition, and evolution, and then accounting for their transmissions. We are going to resolve all this by approaching it geometrically.

We are going to show that each of these biological proposals has its “shape”. Creationism and intelligent design fail because they demand regular shapes within the geometry of four dimensions. Once we recognize that biological organisms cannot survive without the suites of electrical and biochemical reactions that drive their molecules, then it becomes impossible for them to sustain the rates of activity that will make things equal to the same thing and so equal to each other, over succeeding generations, in the way creationism and intelligent design demand. We shall show that they demand the four-dimensional consequences of the square, the circle, and the right helicoid all of which are impossible.

Biological entities are all maintained by (1) electrical fields which radiate from one definite point-centre to another; (2) magnetic fields which instead loop continuously; and (3) the electromagnetic field which is their combination, and which attempts to propagate linearly in time and space, but that must instead oscillate about itself as it does so. All three have important biological consequences. Biological organisms are therefore subject to the rates and limitations inherent in electromagnetic theory, which creationism and intelligent design breach. We also have the beginnings of the four-dimensional macroscopic and microscopic shapes we are looking for.

The French physicist Charles Augustin de Coulomb was the first to realize that all electrical forces, including the biological, obey a form of the inverse square law Newton discovered when he proved Kepler's theory. The electrostatic force between any two objects is dependent on the strength of their charges, and their distances. Coulomb also discovered that all charged objects also behave exactly as if their charge is concentrated at a single point-centre within them. They then do work while (a) moving about a boundary; and (b) simultaneously covering an area. They also always act as if they share, and emerge from, a common centre. This is a definite shape.

Coulomb’s new law was very similar to Newton’s in proposing that all the electrical charge in a body is concentrated at a single point-centre within it. But the Frenchman Siméon-Denis Poisson next extended Coulomb’s original laws and equations to show that electrical charge could simultaneously be thought of in a different way.

Poisson showed that while charge was always concentrated, as Coulomb had observed, at the point-centre within, it could still be regarded as residing upon the object's surface. So according to Coulomb, electrical charge acts—relative to a distant object—as if it emerges from that centre. But Poisson showed that we need to examine the object more carefully … and that when we do, we will observe that its charge resides exclusively upon its surface.

A charged body distributes its charge upon its surface in such a way that if we pierce the surface and approach the centre, the apparent force we thought we detected inside it vanishes. The force acts as if it is not there … even though it continues to appear, from outside it, as if it continues to emerge, undisturbed, from that same centre. The charge can in other words be regarded as being distributed all around the object's surface, and as if we can get no further towards that charge-source than its surface. The charge therefore, on the one hand vanishes behind the surface; but, on the other hand, still acts, to all intents and purposes, as if at the centre.

If we now test a known conducting object for its charge and bring a positive point-charge close, then the overall electrical potential is maintained. Negative charges are attracted to its near side, while positive ones are repelled to its far side. But the centre responsible for the activity seems inactive overall, because even though external electrical behaviours are maintained relative to all such point-centres, nothing seems to happen behind the surface.

Poisson thus proved that internal charge distributions adjust themselves as if they are all around the surface, but always relative to the centre. All molecules are therefore jointly responsible for electrical effects, no matter what may be the object’s size or composition. There is an inherent and measurable charge distribution. But relative to external objects, all charges associated with an object can be allocated to its point-centre to produce the surface forces.

Potential energy exists whenever an object with either charge or mass occupies some position in an electrical or gravitational field. And, equivalently, an electric or gravitational field can be defined by its potential. Poisson’s equation, which makes all this measurable, can be derived from both Coulomb’s law and a theorem from the famous German mathematician and mathematical physicist Carl Friedrich Gauss, sometimes called the “prince of mathematicians”, and widely regarded as the greatest mathematician in history. Thanks to Coulomb, Poisson, and Gauss, we can calculate the force a first body exerts upon a second simply by regarding them each as concentrated point-centres, relative to each other, and in spite of their seemingly contrary behaviour when observed closely and separately. So we not only have a shape, we have rates of change for that shape in given directions, as well as between objects of similar molecular composition. And … these are all things it is very easy to measure. As we shall soon see, biological entities and their internal energies behave in exactly this same way. That is to say, every biological entity and population acts as if it is a surface manifestation. Every entity and population acts as if it is emerging from some definite point centre, with itself as the surface.

A mathematical aside

Our configuration and phase spaces ensure that biological organisms and their populations are subject to these laws.

Figure 20.7

Since we are dealing with movements in both space and time—i.e. meridionally and poloidally—we have to be very careful what we are measuring. Figure 20.7 takes us back to Figure 0.6 when we first exercised our imagination with circles and other simple shapes. It is also a close-up view of the helicoid of internal energy from Figure 20.6. It shows that reproduction and inheritance are ultimately circular motions achieved through the electrochemical and molecular Coulomb forces of chemical activity that allow a set of progenitors to leave progeny behind them before they dissipate and die. The progenitor is then a four-dimensional shape with a line or curve that leads to the progeny, establishing further shapes and rates of change. Those paths are the toroidal circulations of the generations.

The Poisson distribution suggests that our progenitors and their progeny act exactly as if they are located on the helicoid's four-dimensional spacetime surface. We will not find them anywhere else. But the Poisson distribution also says that they will act—to all external observers—as if they are centrally located at the four-dimensional spacetime point-centre indicated by the polar planimeter. We can therefore measure them with both the linear and the polar planimeter. And since both progenitors and their progeny are composed of molecules with charge, they will obey the laws of both Coulomb and Poisson as if they are distributed about a surface focused on that point-centre.

We shall prove that it is indeed always possible to place a polar planimeter at the centre of the arc inevitably formed by any propagating biological current-element moving between progenitor and progeny. It is simply a specific point in the generation. Since those molecules are moving both through space and across time, then they immediately have (a) electrical or energetic fields, (b) magnetic or looping components, and (c) oscillatory phenomena and that transform between their minimum and maximum values. That is the four-dimensional shape we shall prove. It is part of our new biological geometry.

As biological populations propagate from progenitor to progeny, and so across the span of the generations, their mechanical and nonmechanical energies are obliged to obey, amongst other things, the important Biot-Savart law. This is a product of the Maxwell laws of the electromagnetic field, which also governs all chemical and thermal behaviour.

The Biot-Savart law relates a magnetic field to the magnitude, direction, length, and proximity of whatever electric current generates it. It in essence states, as the polar planimeter shows, that the intensity of any ongoing biological activity has:

1. a central point of generation;
2. a clear line of action and a direction to and from that point; and
3. a magnitude distributed throughout the entire neighbourhood of associated points through which it passes.

Figure 20.7 shows the four-dimensional consequences of Coulomb's, Poisson's, and Maxwell's discoveries as also enshrined in the Biot-Savart law: it is the torus-like or toroidal circulation of the generations. Each set of linked propagating current-elements all about that circulation represents the activities of some biological population busy undertaking the chemical and electromagnetic transformations that carry it about its biological cycle, in our space of internal energy, from progenitors to progeny. Each population is always moving through our three-dimensional spaces, but taking time to do so, which is the fourth dimension. Its configurations move through the ordinary time and space measured by Minkowski's x, y, z and t, and as the world-points he described. They move at some specified speed that depends upon their medium. As we shall prove very shortly, that medium is their composition, and their internal energy.

A mathematical aside

We now have the forces that cause the movements in our configuration space. We therefore now have a complete phase space. And, furthermore, both of our configuration and phase spaces are capable of being either positive definite, negative definite, or indefinite, entirely depending on the generalized coordinate system we choose to impose, and therefore on how we choose to measure them. They can also be either Riemannian or pseudo-Riemannian, again depending on how we choose to measure. The metric tensor can be made nondegenerate. These spaces are therefore completely ready to impose a Lorentzian metric upon them. A Minkowski space exists, and we can both parallel transport within it, and be torsion free. The fundamental theorem of Riemannian geometry holds throughout, for the space has a Levi-Civita connection. This configuration space and its associated phase space are therefore smooth Riemannian manifolds that are also capable of being symplectic and supporting hamiltonians.

The polar planimeter that stands at the centre of all the populations it is measuring demonstrates that, as the Biot-Savart law requires; and in spite of their varied compositions and behaviours; every current-element propagating throughout this three-dimensional and biological spacetime has both (a) a sense of direction, and (b) a rate at which it moves as it oscillates between its maximum and its minimum values. It is the conjoining of the three constraints of constant propagation, φ, constant size, κ, and constant equivalence, χ, that establish the configuration space and medium of three dimensions for all biological activity. When we add time, we have the four dimensions Minkowski proposed. That is our complete biological field of internal energy.

Every biological entity is a propagating current-element. Every one is an offshoot of the transformations wrought by some preceding and also propagating biological current-element within our biological space. This is a biological geometry.

When the first maxim of ecology, which is the maxim of dissipation, is conjoined to the second maxim of ecology, the maxim of number, they make it clear that all possible biological organisms emerge from others. That is a behaviour that extends into the fourth dimension. Every organism is born at a specific moment in time, and so creates a four-dimensional arc.

By the third maxim of ecology, which is the maxim of succession, progenitors and their progeny together form loops and circuits in space and time, which are their generating circulations of the generations. And since all biological organisms appear from others, then they all follow these curving paths of regular increase and decrease across space and time, in their important values. Those are now the surface-type activities of propagating current-elements that have a dedicated centre. By the Biot-Savart law, as indicated in Figure 20.7, it is always possible to determine a specific point, and a specific value, responsible for any observed biological activity, and so to determine that rate and curvature relative to that centre. By the laws of electromagnetism, that activity is then the rate of biological activity in space and time for that increase and decrease, and as if originating from that point of curvature. Its value can be stated, along with its distribution, which is the fourth maxim of ecology, the maxim of apportionment. This is again the workings and the import of the linear and the polar planimeters in our four-dimensional biological spacetime of internal energy.

This all means that if we see, for example, a deer, then we straight away know its initial conditions; the kind of metabolic and physiological events it has already been through; the kind it will be going through; and its general life trajectory. We immediately suspect the fawn from which that deer came, for both that prior fawn and the deer we now observe are a part of the overall neighbourhood of points summarized by the biological propagating current-element that is this deer. The deer is right now curving in four-dimensional biological spacetime about its known centre in the field of internal energy. And we know these same things about the buck or the doe it will become. We further also know, straight away, that the deer had a progenitor with similar properties of fawn and buck and doe; and that that prior deer also had progenitors as its own fawns, bucks, and does; and so forth; and we even have a shrewd idea of the masses, energies, and times concerned for all. Those are all the results of propagating current-elements within the four-dimensional space. They all, without exception, have a suitable generating centre in the field of internal energy; with specific values; and which are a measurable set for that same centre and set of centres.

Upon seeing any biological entity or a propagating current-element, in spacetime, we can always do as we did for the above deer. We can immediately position a polar planimeter, which then declares the characteristic values of , , n, and τ, all at a given moment, t; and all along with fluxes that move that surrounding population all about itself as their centre. This is the doctrine of propagating current-elements and the Biot-Savart law. It forms what we shall now term an allowed set of molecular configurations, and changes in configurations in internal energy.

A mathematical aside

By the Biot-Savart law any biological activity occurring over any stretch of time, Δt, immediately points to some proposed centre of generation. Any field of activity induced by such a flowing current element will also be strongest at right angles, which is then the generating centre. It always exists.

And since a flowing current element of this kind cannot exist only in a single point for it must propagate, then its derivative must be taken. Integrals must also be taken all around its boundary to determine it. The implied curvature, and its rate, is the current element that in its turn implies some generating length which is its boundary. An ∫M dT and so forth are also promptly implied. That implied generation length is then the complete circulation, along with a curvature in biological potential; and all as implicitly expressed over that observed Δt, and which is a part of the allowed set that defines the size and breadth of the configuration space.

Our allowed set gives us the beginnings of something to measure. We have to find values to put in our tensor's rows and columns to represent it. The allowed set is the collection of linked values that tell us a population's behaviour.

We can find the set of linked values we are looking for by turning to Einstein's theory of relativity … and … it is surely a brave person who takes issue with him over some topic in his theory: but that is exactly what the German physicist and mathematician Hermann Weyl did. He greatly improved Einstein's theory with his discovery of the “Weyl tensor” which is the equivalent, in tensors, of the Maxwell electromagnetic field.

The debate here concerns the repetition of values symptomatic of both planetary and galactic orbits, and circulations of the generations. The Einstein general theory had recently been experimentally validated, but since it used tensors, Weyl's view was that Einstein had left it deficient. Riemann's original geometry, on which tensor calculus is based, proved that twenty equations were necessary for a complete explanation of the repetitions of values, and Einstein had only used ten. Since Einstein had “thrown away” half the supporting reasoning and validation, Weyl pointed out that Einstein could not, and did not, use tensors in his theory to account for the transmission of values and properties across spacetime, which is gravitation's specific medium of propagation. We can think of this as the distinction between our allowed set, and a further required set of molecular configurations and changes in configurations in internal energy. We can think of the required sets as the promenading points and baton twirls in Figure 20.1.A. Each required set generates the allowed sets, which we saw in the various shapes, waves, circles, and spheres in Figures 20.1.B and C, all by interacting with each other.

In Weyl's opinion the issue was describing the appropriate behaviour correctly, using tensors. He insisted that Einstein had not adequately explained how an object embedded in spacetime “knows” how it should behave in a given situation. How does any given body know what values or properties to repeat for its orbit? Since Einstein has eliminated forces, how does each one get its information on what to do?

The earth, for example, is a definite distance from the sun. It pursues a definite orbit. Quite independently of each other and of their own distinct characteristics, all other bodies of that mass and at that same distance from the sun will all feel exactly the same solar influence. How do they all “know” how to behave at that set distance from the sun or any other body?

The earth's, or any other body's, correct gravitational behaviour, relative to the sun, is now just as much a property of the sun, and the field the sun establishes, as it is of the earth or other body itself. Since the Einstein theory no longer uses forces to explain events; and since all we have is the fabric of spacetime; then how does any body “know” that the spacetime field around it is warped; how does it know the amount; and how does it know which way it must roll, and the speed at which it should do so for it only strictly knows about its own self and properties? This is again the distinction between the allowed and the required set.

If we now remove the earth from its orbit and replace it with one twice the mass, how does that new earth know how to conduct itself simply as a consequence of its location, and in accordance with some required set? This required set of information is again put out by the sun, and not the earth.

More generally: all celestial bodies are recipients of outwardly propagating—and required—gravitational information from all other bodies, all of them at locations unknown to each other. That information in each case concerns the generalized character of spacetime, and states their required behaviour under those conditions. Therefore, argued Weyl, a specific and required set information, about all time and all space, must be transmitted from location to location, and quite independently of any and all objects … and all simply as a property of gravitational spacetime. In Weyl's view Einstein's theory completely failed to account for this required set that told every distinct body how to behave, meaning that it could not properly explain the propagation and warping that was gravity.

Figure 20.8

Weyl was pointing not just to a problem with Einstein's theory, but to a more general capability that tensors need to acquire. Figure 20.8 is a circle formed by collapsing the helicoid onto itself so it is all on one level. The circulation's beginnings and ends coincide. It is still, however, a four-dimensional structure. Time is still embedded along its length, and the helicoid still carries internal energy upwards through time with its pitch. We have simply set those concerns aside for the present. We are highlighting the repetition of values.

Figure 20.8 clarifies the issues of transmission that heredity raises. How does a biological entity in some successor, and remote, generation “know” how to conduct itself simply through being located in that succeeding generation, and quite independently of either its own specific construction, or that of what might have preceded it? How was that behaviour transmitted? How does any successor entity know what either of its allowed or required sets of informations and behaviours should be or are? That is the implication of the circulation lying upon itself. It is the issue we must resolve. Weyl shows us how to do this: how to repeat and transmit values and properties.

Figure 20.8 represents the conundrum of the transmission and propagation of inherited traits. It displays a range of locations, in biological space, all at equivalent distances from a generating centre, and courtesy of our polar planimeter. They are different quantities and intensities of internal energy. Those embedded traits are defined as a linked surface on the internal energy helicoid. They are linked by their shared electrochemical and reproductive activity through the Biot-Savart law and its circulation of the generations. Like all the locations at a given distance from the sun, all the locations in the three-dimensional medium of internal energy defined by that circulation are in the same region on a helicoid surface. Since they are the same region, they share a definite potential in their allowed and/or required sets. They share these simply through being at equivalent locations, and so through sharing a joint curvature of generational energy, all relative to that central location, and all in four-dimensional biological spacetime.

We still, however, face a problem. How does any entity found in any location receive the information it needs on how to conduct itself, and whether a set be allowed or required? And how does it know what to do to transmit its information to others? This is the problem of heredity. We are now approaching it as the issue of a four-dimensional loop lying upon itself in the internal energy field.

Weyl resolved the issues he raised by turning them into a discussion of gradients and potential energy, and so of rates and changes in tensor values, their bases, and their components. More specifically, the sun and the moon will each establish gradients across the earth. Since their masses and distances are different, these are different gradients and rates of change. They therefore produce different behaviours and tides in the oceans, each according to its own mass and distance. The spring tides are when earth, moon, and sun align at the new and full moons, and when their gravities combine. Tides are higher. Neap tides are when the moon is in the first or last quarter, and sun and moon are pulling at right angles. Tides are then lower. The earth experiences every cosmic body as a gradient and potential across itself. It “knows” the distances of the sun and moon, and all other bodies, from the differences it feels at each moment in those energies and their gradients falling upon it from the surroundings.

Thanks to Weyl, we only now need to measure a gradient, or set of rates of change, across space and time locally. The solar and lunar masses and distances are easy enough to measure locally. So also are their distinct tides. Heredity and traits are becoming similar local tidal forces across populations. Each entity and population only needs to measure its surroundings.

The Weyl tensor, as the tensor equivalent of the Maxwell electromagnetic field, measures the immediately surrounding gradients and rates of change of whatever description; then extends those values outwards; and all as an intrinsic property of the tensor's components, and so ultimately of spacetime itself. It has the same properties as the Maxwell electromagnetic field, and achieves the same purposes: it explains how transformations can propagate into given locations; affect any objects embedded there; and then propagate onwards to yet other locations, all with a given rate and intensity; and simply due to that space's gradients and characteristics, independently of the embedded objects. This is again the issue with heredity and inheritance.

In Weyl's hands, therefore, Einstein's general theory came to recognize all forms of potentials and gradients. His tensor describes the curvature, or rate of change, at any point, but not as directly caused by whatever object might be embedded or located there. Thanks to the Weyl tensor, any object embedded at any given location will experience definite properties, gradients, and potentials simply because of that location and its gradients, which is a feature of being at a specified distance from those given sources. It is the potential that handles any changes in the energy intensity or gradient experienced in any space, and that is not directly attributable to, or caused by, whatever mass or energy might be found embedded at that location at that time. Just as the moon and the sun cause tides which the earth experiences as a set of gradients transmitted to it simply by being in that space, so also is a biological entity impelled to behave by the gradients and behaviours in its internal energy. This is again the issue with heredity and traits.

We should remember the sometimes confusing anomaly that redshifting galaxies can cause. This means that both populations and galaxies can look as if they are moving when they are not; or else they can look stationary when they are not. Their merely apparent changes in position, imposed by changes in space itself, are easily confused and taken as the motions of those same galaxies relative to ourselves.

Figure 20.9

We can understand the implications of the Weyl tensor for our macroscopic and microscopic biological space through Figure 20.9. It is a set of propagating biological gradients in internal energy. Those gradients form the required sets of our twirling batons and their promenades.

Just as the sun and the moon impose tides, a required set of biological gradients impels a biological entity to undertake a set of both macroscopic and microscopic transformations simply by being born to, or appearing in, its particular four-dimensional spaces. The Weyl tensor therefore describes exactly how one set of properties can succeed to another simply because of the local and measurable propagating gradients and current-elements, complete with their inbuilt potentials. Just as with tides or with changes in an underlying space, whatever appears in a region governed by the Weyl tensor is going to encounter certain kinds of gradients and rates of change across itself, all over space and time. This is a description of the seeming inevitability of heredity.

Although Figure 20.9 tries to show the Weyl tensor's general properties by presenting two identical curves that differ only in length … the idea should be treated with great caution. The issue is the philosophy of the firmament as against the real fabric of both spacetime and the quantum universe. The stretched out tensor underneath is implying an earlier start point and a later end point in the field of internal energy, and so a difference in rate, range, or both. However, what the Figure depicts is not really possible … at least … not in four dimensions; and not with the degree of accuracy the diagram implies. We will eventually see that since this is impossible, creationism and intelligent design are also impossible.

Figure 20.4. showed a variety of shapes which could nevertheless be considered parallel when we considered their areas and perimeter lengths as different points in space and time. In the same way, while the Weyl tensor certainly transmits mass-energy information, it does not specifically associate that mass-energy information with any one curvature, in any region, any more than the perimeter and area equivalences of the various shapes in Figure 20.4 demand any particular shape. Since mass and energy are equivalent, the resulting Weyl tensor curvature depends upon both the forms of energy, and how they are transported, for those define the space.

As an example, an electrical company that sells power by the kilowatt-hour is not concerned with whether the consumer used a low powered gadget for a long time, or a high powered one for a short time. A home can be allocated more kilowatt-hours, which is more energy. In the same way, the Weyl tensor similarly indicates a more general mass-energy “pressure” in the internal energy that it makes available in a given region of spacetime. It does not stipulate, as Figure 20.9 tries to do, the niceties of the allocation. There need be no further specifications, for just as all gadgets that consume or produce a set number of kilowatt-hours are equivalent, so also are all the activities that produce a given Weyl tensor equivalent.

As another important point, the energy that the Weyl tensor transports in the internal energy field depends upon all the objects, combined, throughout space and time, for it operates in four dimensions and defines a fabric. Since, for example, the sun's light takes eight minutes to arrive on earth, then objects on earth are always responding to the sun in a different state from its current one. But the earth also responds to the moon which is 1.3 light seconds away, relative to it; just as it responds to Alpha Centauri 4.37 light years away. The Weyl tensor the earth experiences is unique, for it occupies a unique point relative to all possible cosmic bodies. In the same way, heredity is always a unique response to a prior and propagating set of conditions.

Two propagating Weyl tensors can only be similar, in the way Figure 20.9 tries to suggest, if they originate in universes containing identical objects differing only in their mass-energies, and in the exact proportion the length difference suggests. The differences propagated would then differ only in the amount of time they take to reach, and influence, our object of interest through travelling through a presumably denser space, for the speed of light is constant.

Figure 20.9 is trying to suggest that two populations of biological entities can be identical, differing only in the speed and intensity with which they grow and develop. But that growth and development is their interaction with the surrounding universe. The universes that support them are proposed as identical, excepting only that everything is taking proportionately longer in one. But this is not possible. If those two universes indeed contained identical objects, then they would have identical intensities, and their Weyl tensors would be identical, including in their proposed lengths. The same things would be experienced at each location in each. To express it astronomically, it is not possible to be closer only to Alpha Centauri, while everything else in the cosmos remains the same. Our distance from the sun, the moon, and everything else also changes immediately.

Seeing one person jump six metres while another jumps two-and-a-half certainly makes the six-metre jumper look impressive, until we learn that the six metre jumper did it on the moon … which is equivalent to one metre on earth. That is not nearly so impressive. Elements within four-dimensional space can only be compared, in the way Figure 20.9 suggests, if they are elements in the same local space. We can therefore only make comparisons when objects are in the same space.

It is not really possible to know whether two galaxies are approaching each other or receding, relative to each other, until we have examined the space. If their velocities can indeed be measured, locally, then they could easily be approaching each other, relatively, with only the cosmic expansion making it look as if they are moving apart, for their relative approach velocities are less than that imposed by the expansion of space; or conversely; or they might even appear static because the two velocities balance. Once again, the discussion of galaxy recession does not really make sense for it makes unwarranted assumptions about the nature of space. Thus a true Weyl tensor comparison requires what is called “parallel transport”, which is a very important difference between relativity's spacetime and the absolute space and time of Newton's firmament. The circumstances and conditions in which the objects are embedded must be made equivalent before they can be compared. Two objects at two different points, even on the same Weyl tensor, are like objects on two different planets, and do not have the well defined relationships the Figure suggests.

Figure 20.9 can only suggest the Weyl tensor's generalized biological capabilities. The Weyl tensor can certainly “extend” spacetime, but since this occurs over four dimensions, it does not happen in the precise and clinical way Figure 20.9 suggests. There would certainly be an extension of mass-energy pressures and transformations. A major effect could also be an increase in the temporal intervals into which those pressures and transformations are extruded. The transformations would be, broadly, of the general type shown. However, the biological entities would undertake transformations that would be more than just an extension of time. The precise circumstances depicted to convey the idea are impossible, but the underlying impression holds: biological mass-energy and configurations and potentials can certainly extend, with one of the manifestations being over time.

The Weyl tensor is our biological potential, μ. This also has its inbuilt rates of change—and so gradients—for the three constraints of constant propagation, size, and equivalence. They are the proportionate changes dφ, dκ, and dχ. Exactly like the Weyl tensor, our biological potential thereby causes transformations and oscillations throughout its medium which will be experienced by any object in that space.

As per the Maxwell electromagnetic theory and the Biot-Savart law, the biological potential is a transmission mechanism constantly busy propagating through the medium as an ongoing sequence of current-elements courtesy of the polar planimeter and its potentials at the centre. Biology is now therefore defined by the space that biological potential creates, and by the medium that supports it. Heredity is a propagating potential of allowed and required sets of configuration changes along our helicoid surface, all in four-dimensional biological spacetime. Creationism and intelligent design are now impossible because the Weyl tensor behaviour they demand in internal energy is impossible.

A mathematical aside

The Weyl tensor is technically defined so its “contraction” between its indices is zero:
Cλμλκ = 0.

It is given, in the general theory, by:
Cμναβ = Rμναβ + ([gμβRνα - gμαRνβ + gναRμβ - gνβRμα]/(n-2)) + ([gμαgβν - gμβgαν]R/(n - 1)(n - 2)),
where gαβ = gβα is the metric tensor that defines how that space is measured. It therefore incorporates any ongoing changes in the basis. It can always be calculated over all the time and space available to the mass density and energy intensity into which any body is being inserted irrespective of how much that space, or the basis, changes. It therefore gives the body concerned the information it needs on the behaviour of the objects surrounding it so it can respond to the influences and gradients or rates of change that they propagate, and quite irrespective of any relative and ongoing changes in spaces and basis.

The Weyl tensor defines the transmissibility of the configurations in the phase space, and thus the principle equations of motion.

Now we have confirmed that our biological space of internal energy can create curves, gradients, and potentials which it can propagate across its entire extent as its allowed and required sets, we can attend to the embedded objects themselves: our biological entities.

There are two aspects to every gravitational body: attracting, and being attracted. The moon may experience the attractive effects of the sun, the earth, and all other bodies, but it also makes a distinctive attracting contribution to spacetime. All other bodies, and the earth in particular, experience its actions. In the same way, all biological entities influence the four-dimensional texture and characteristics of biological spacetime with their distinct ecologies and behaviours. They are not merely passive residents.

Figure 20.10

As an example of what we need, Figure 20.10 is a more detailed, more internal, “embeddings” view of Figures 20.8 and 20.9, our helicoid surface of internal energy. It is the medium's thickness. It states how the objects that receive an ongoing biological potential of allowed and required sets of information will behave. It tells us what general properties they will exhibit.

Figure 20.10 tells us that biological entities will enter the population through birth and reproduction with a given amount of internal energy as defined by their entry point. They enter through the symbolic helix shaped tunnel that represents their environment and ecology. Ours is shown constant or “flat” in the number dimension. The tunnel's beginning and end are on the same level. As many entities enter as leave at every point so there is no overall change in numbers. (This is simply to aid comprehension. If the tunnel was perhaps shown e.g. changing colours across itself (or perhaps across its breadth?), then it could represent a population size change, with the numbers entering and leaving being different. It is difficult to represent the four dimensions of number, mass, energy, and time in the only two we have available on this flat monitor or piece of paper).

The entities in the helicoid tunnel will now use their metabolisms and physiologies to absorb energy and chemical resources as their mechanical and nonmechanical chemical energy fluxes, and so they can satisfy the allowed and required sets they find there. This is their set of varied and varying points in the space. It is their meridional interaction. They will thus go through the cycles of growth and development. The time this takes is their pole-to-pole or poloidal movement. The reproduction that is the circulation of the generations is the net toroidal activity.

The overall tunnel size changes with their activities. They will then depart at the other end of the environment-ecology tunnel as progenitors. Their progeny will simultaneously enter the next piece of tunneling to repeat the poloidal, meridional, and toroidal processes, and so on and so forth, smoothly and uninterruptedly along the helicoid surface of their molecular composition.

Where Figures 20.8 and 20.9 and their Weyl tensor tell us (a) the overall distance from the centre of curvature a given population inhabits, and so also (b) its overall circulation or generation length; Figure 20.10, and its associated “Ricci tensor”, tell us the overall sizing or volume around each entity and/or population, at each location as it, and they, feel the effects that the Weyl tensor simultaneously imposes through those allowed and required sets. Their Ricci tensor tells us how much energy is trapped or embedded at each location. By the Ricci tensor, the actual entity and population sizes or volume elements at any time depend on their own masses, numbers, energies, and activities, and as influenced by the characteristics of the four-dimensional mass-energy medium passing through them and over them.

Figure 20.11

Slightly more formally, the Ricci tensor, shown in Figure 20.11, governs the evolution of the sizes of the volume elements of internal energy within Figure 20.10's “tunnel of availability”. The Ricci tensor tells us the mass-energy “pressure” of internal energy that both an individual entity and an entire population feel or exhibit, and therefore the ease or otherwise with which they move and grow in biological space as they propagate through spacetime, and along the tunnel of ecology and environment. As again in Figure 20.11, when the surrounding environment and ecology exert a pressure, then the volume elements allocated to them tend to shrink; and when the surroundings relent and retreat, then the Ricci tensor encourages the volume elements to bounce back and to increase in both size and numbers. These are effects in space and over time.

The Ricci tensor governs the helicoid's radius and thickness: the degree to which matter and internal energy will converge or diverge, in time. It describes the scale of activity that both biological entities and populations engage in. It states the overall expansion and contraction of their masses, numbers, and energies, and describes the fluxing; the dynamical size; and the proportionate rates of growth of their overall material and energetic manifestation at each time point and spatial location, all as their ongoing divergences and convergences. In so far as they are the comings and goings of mass, energy and numbers, then the Ricci tensor governs the instantaneous metabolisms and physiologies of entities … and since number is a dimension, then it dictates not only the size of the entities, but also the sizes of entire populations—as in their number density—at each point on the helicoid of internal energy. When the Ricci tensor contracts, less is possible; and when it expands, more can be achieved.

Figure 20.11 for the Ricci tensor should be treated with as much care as Figure 20.9 for the Weyl one. The difference between them is that this Ricci tensor has more influence on the meridional molecular quantities, partitioning, and behaviour, while the Weyl tensor can influence entities and populations more poloidally by changing the helicoid's pitch; and so eventually the radius and thickness. They together govern the toroidal allocations and dispositions of internal energy, that in their turn govern biological entities and populations.

The Weyl tensor causes the internal energy helicoid to stretch or contract more longitudinally and toroidally, and to establish volumes and densities in energy and resources tidally along the symbolic tunnel. But although a change in a helicoid's pitch or radius might change the overall circulation, and so generation time; it does not necessarily change the total volume of energy or resources used, in total, over that changed time span. Should the Weyl tensor indeed cause a toroidal and generational stretching, then the entities and/or population can easily maintain that same overall volume of materials and resources. But they must then contract meridionally, largely through the Ricci tensor, at each time point and location. This creates an instantaneous convergence in mass and/or energy. And since number is a dimension, these events affect population size.

The need for a contraction immediately enters the province of the Ricci tensor. Under the conditions outlined above, there will now be a smaller number of smaller entities at each point mapped out by the Weyl tensor. But they could alternatively stretch out through the Weyl tensor, yet maintain the same overall mass-energy volumes at each moment through the Ricci one. But since they are now maintaining the same instantaneous energies and resources for a longer period of time, then the overall totals over that increased span must increase. These two work together.

The Ricci tensor more directly affects the helicoid's radius and thickness. It congregates or disburses internal energy. It mainly affects quantities in absolute time … and so therefore eventually affects rates. It therefore and unavoidably affects the pitch, which ultimately reaches across the generations. All this can be measured by the Ricci tensor.

While the Weyl tensor governs the more longitudinal, wave-like, toroidal and so also poloidal and timelike extensions and contractions—i.e. in t and τ—throughout the biological space of internal energy, the Ricci tensor describes the more meridional and spacelike movements: its relative sizes and its densities in mass and energy within discrete volume elements, and so in both entities and their populations—i.e. in , M , P …and so also in n. Since the poloidal and meridional movements coincide in the toroidal ones, we must obviously find a way to quantify them and their influences on molecular behaviour. We only have to describe what creationism and intelligent design look like under this proposal. We can then run an experiment. We have our right helicoid waiting in the wings.

A mathematical aside

A fuller version of the Einstein field equation incorporating the Ricci tensor, Rμν, is:
Rμν + Λgμν - (½ • Rμν) = (8πG/c4) Tμν
where gμν is again the metric tensor giving information about the basis, and Λ is the cosmological constant Einstein introduced that states an intrinsic energy density throughout the space. We have Ω from the previous chapter, the refutation, to substitute for this, and therefore acting as an intrinsic biological energy density for a medium that is equally characteristic of this biological space. It is the biological analogue of the Einstein cosmological constant.

The Ricci tensor defines the number and density of the configurations in the configuration space, and that are then transported, in the phase space, by the Weyl tensor.

It is now time to tackle our third anomaly. We can highlight it by emulating Einstein. We propose, in the manner of Einstein, that no matter what their absolute values might be, if two populations are each, for example, half-way through their respective circulations, then they are “synchronized” in the important way he suggests. Since we have already signified the temporal distribution as τ, we are then proposing that all other things being equal, all populations with, for example, τ = 0.5 are all similar through being half way through their respective circulations of the generations.

The four populations in Figure 20.7 demonstrate these issues. They have vastly different generation lengths, but are each nevertheless a single revolution in biological potential. As a revolution, they link the perimeter and the areas, both of which are infinitesimal increments:

1. By the linear planimeter, the generation length is the sum of all the time intervals, t, about the circulation which is an arc length or line integral, ∫C, that gives us an absolute number, T, in seconds, and is similar to kilowatt hours used.
2. However, by the polar planimeter, that same circulation is also the sum of all the angles, which is the sum of all the areas and distributions, and so is a spanning and an area integral, R, which is τ, and a quantity of mass energy. As a measure of distance, it is more like a light-year. This at first sounds like a measure of time, but is actually a measure of distance. This biological τ is more properly a distance travelled into a generation.
3. A mathematical aside

And we can also now add, since the Weyl and Ricci tensors are in play, that those areas will also move in time. Their propagating surfaces therefore describe an enclosed volume of resources and energy, which is a triple integral,  ∭W. They together determine the rate of configuration change propagation in the phase space of internal energy.

We have been here before. It is the same anomaly between totals and average individual values; between the population and its individuals; and between the infinitesimal increments of these two distinct properties. Though the line and the area integrals must ultimately form the same totals, their infinitesimal spans or increments need not be the same at every point, meaning dt dτ.

We must take care what this means. In the most simple of terms, the Weyl and Ricci tensors do not simply scale, for energy does not simply scale. That is an illusion that belongs to the Newtonian-style firmament of absolute times and spaces. If we, for example, fling more and more mass onto a planet, we do not just get a bigger planet. We instead get a range of circumstances that are the life cycle of a star. We will eventually end up with a black hole. In the same way, neither absolute time, t, nor biological-generational time, τ, increment in the same ways, and so are most unlikely to maintain anything other than a temporary quantitative equivalence.

We can take our three real populations in Figure 20.7 as an example. If we take the day as our basis and reference, then:

1. our representative mosquito has the generation length T = 4 days (to give 4 : 1);
2. we measured Brassica rapa at T = 36 days (to give 36 : 1); and
3. a blue whale has T = 31 years (to give 11,315 : 1).

These are very different measures … yet … we have a single statement, for the entire generation, of τ = 1. This τ = 1 is a statement of the total number of configurations we require to navigate all the biological events that span a generation. And meanwhile, the clock time, t, is a statement of how long it will take to work through all those configurations. Those measures therefore cover the same rate of spanning the generation, dτ, for the polar planimeter for all of them; but the absolute infinitesimal spans of time, dt, they each cover all over the helicoid differ all over, for all of them. These infinitesimal increments are not the same, and dtdτ anywhere, even though values for each dt and dτ exist everywhere. The issue is whether or not the number of configurations per unit of clock time in each population remains constant. That is what creationism and intelligent design aver, but which Darwinian evolution denies.

A mathematical aside

1. If we also now take the mosquito as our basis, and use it to establish a set of T’ values relative to it, we have 9 : 1 for Brassica rapa, and 2,828.75 : 1 for the blue whale.
2. If we instead take B. rapa as a basis, we have 0.111 : 1 for the mosquito, and 314.3 : 1 for the blue whale.
3. And, finally, if we take the blue whale as our basis, we have 0.00035 : 1 for the mosquito, and 0.0032 : 1 for B. rapa.

Taking each of these in turn as our basis, which the tensor allows, gives us all the relative tensor values we need. We then have three different T’ values, which are relative generation lengths; but we always have the same τ = 1, for it is the relative value common to all. It is the total number of configuration changes needed to span a generation. But the biological events are unlikely to differ simply by scale. They are likely to differ in type and intensity.

There is also, of course, the complementary issue. If we take an arbitrary absolute time period of t = 4 days, then the mosquito can amass everything it needs for its generation, which is again T = 4 days. Since it has τ = 1, we again have 4 : 1. Brassica rapa has less of a gradient and curvature, and so can only amass enough mass to cover τ = 0.111 of its generational needs to give 4 : 0.111. The blue whale has a yet flatter curvature or rate of change, meaning it can only access enough to travel τ = 0.00035 of its own generation to give 4 : 0.00035. So we now have the same absolute time span of t = 4 days, but three different values for the relative τ for each. We again have dtdτ expressed in relative terms … and we are still left with the conundrum of overall distributions. This is the problem of variations, heredity, and generation length. The time spans each population needs to complete a set of biological events is different; and is also subject to change.

The Maxwell electromagnetic theory and the Biot-Savart law tell us that since biological potential is heat, light, and electrochemical behaviours, it cannot maintain constant and linear values. The relevant magnetic and electric fields cannot remain stationary. Biological organisms therefore change constantly as heat, light, and biochemical activity and configurations are incessantly propagated along the helicoid. Their populations will be transported over time and space by the ever-changing biological potential and as a propagation through the internal energy of biological space.

We cannot discuss evolution, nor prove anything either way, with no systematic way to discuss overall population behaviours. We cannot quantify comparisons from one species to another, or isolate relevant differences without correcting this.

We learned during the refutation that we can measure a generation length through the angles, areas, and tangents it presents all along our helicoid. But this always requires that we bring two out of our possible four dimensions together. If we bring number together with mass or energy, the population's time to span a generation changes with the ever-changing ratios in /n and /n. The generation length in other words changes. However, bringing mass and energy together as / creates the visible presence, V, which is a measure of the entropy and energy density. Therefore, by measuring those ratios, we immediately measure some sector or other of the generation; and by measuring the entirety of those changes, we have measured a generation for those two-by-two values, which is a circuit of the helicoid and a circulation in internal energy.

A mathematical aside

We should note that since we are working with tensors, we are quite at liberty to invert the two axes. We would then be using n/m̅ to measure the angle and the generation, which would be the cotangent relative to our current orientation, rather than the tangent. We can measure the helicoid equally well as x = ρsinθ, y = ρcosθ, and z = tan α + kθ or as x = ρcosθ, y = ρsinθ, and z = cotan α + kθ . More comprehensively, we can have either y = x tan z/k, or y = x tan k/z. And if we measured both the total population mass flux, M, and the numbers, n, and then used those two to calculate the individual mass values, then that could be either a sine, a cosine, a secant or a cosecant, or the arc equivalents, and all entirely according to the orientation we chose, and when we discuss them relative to some other measure. For the sake of simplicity and consistency we will remain in the specific orientation between mass and number we have selected and continue to refer to it as a tangent, but our tensors will easily accommodate any of the other possibilities.

Since biological entities are composed of molecules, the Weyl and Ricci tensors ultimately dictate molecular potentials and constructions. Since we want to assess both the sequencing and the timing of their various configuration events, it is very important to keep both the similarities and the differences between the Weyl and the Ricci tensors clear or we will not be able to quantify them, and will never unmask evolution. Our helicoids have different sizes and so different measures to and from each pole; different measures across their meridians; and different lengths around their circulations and toruses.

Figure 20.12

The two people in Figure 20.12 illustrate the importance of separating the sequencing and the timing of events. They are each going to stop that large moving shuttlecock; and then return it to the same location on the other side of the net. Since the displacements or distances moved through are the same for both, the work they must each do, and the energy they must each expend, on the shuttlecock, is also the same, because work = force times displacement: W = Fd, where displacement is the distance a force moves in a specific direction. The same force over the same distance always produces the same energy.

More technically, work is the integral, ∫, of force, F, with respect to displacement, d, and so is written as W = ∫F dd. Work and energy are always determined solely by displacement: i.e. the distance a force pushes in a specific direction. They are both independent of the time taken to cover that distance. A concrete block has the same mass and must move through the same distance whether we push it slowly or quickly, and whether we do so by ourselves, or call on some friends to assist. The type and intensity of work done might differ, but the overall total is the same.

The two people in Figure 20.12 are confronted with the same sequence of events. The man on the left, who is catching the shuttlecock by hand and then throwing it back, is clearly going to take a longer amount of time over the whole process than is the woman on the right, who has the benefit of the racquet. The woman can potentially give the shuttlecock a bigger impulse over a shorter period of time, and thwack it firmly back. The man, however, has to be in physical contact with the shuttlecock for a much longer period of time, and also over a longer distance. The sequence of events is the same, but the impulse they each give to the shuttlecock is different. Impulse is force times time: J = Ft. More technically, impulse is the integral of force with respect to time, and so is written as J= ∫F dt. We notice carefully that one of these measures is related to distance, the other to time.

Even though the man has to spend more time in contact with the shuttlecock by gently buckling the knees, absorbing its momentum, and then throwing it back while the woman simply strikes it; if the woman nevertheless stands in exactly the right place and moves appropriately, then the total distance the shuttlecock travels will be the same for both. If we now assume that the woman uses her racquet to lob the shuttlecock relatively gently back so that it ends up in exactly the same spot and with the same speed as it does when the man throws it, then the shuttlecock receives exactly the same amount of work and energy, although the impulses are still different. While the work and the energy depend only on the distance and sequencing of events and are both properties independent of the time taken, the impulse depends entirely on the time, and is independent of the distance. The shuttlecock is admittedly in motion in both cases, and so the one does not tend to accumulate without the other. The two properties of impulse and energy still focus, nevertheless, on the very different dimensional attributes of distance and time. The forces have simply been differently distributed over the times and the distances involved in each case in Figure 20.12. In the same way, the Weyl and Ricci tensors govern the total distributions of mass-energy by making adjustments over different paths, times, and possibilities.

Darwin asserts that the entities in a population can, for example, take longer to undertake some activities than do others, and that this is a variation. The Weyl and Ricci tensors might govern some form of four-dimensional mass-energy behaviour, but we still have to account for how they distribute that mass and energy within the three dimensions around us, and especially at the molecular level. The first and second equations we derived in the refutation show that biological entities are composed of molecules that raise the same kinds of issues. Both those equations have a numerator, on the top, of dm̅. The entity uses itself as its own basis for measurement. It undertakes a proportionate rate of change in mass, which is dm̅. How much it changes at any point depends upon how much of it there is … which is ultimately its molecular count. We have an infinitesimal increment in mechanical chemical energy flux all about the generation but expressed, as the Weyl tensor requests, as a gradient relative to the entity. The total change is independent of the time needed.

If the entity we are dealing with is a blue whale, then dm̅/ involves a large number of molecules; but if it is only a mosquito then it only involves a small number. But since it is expressed as a proportion, the values can be the same even though the absolute amounts are different. We can measure the Weyl and Ricci tensors, and also the numbers of molecules involved. We get this latter from the masses in conjunction with the Avogadro constant.

The second term in the two equations we derived earlier is a little different. The one for mechanical chemical energy multiplies the incremental proportional change in mass, dm̅/m̅ by 1/(n + dn). Since that n + dn is in the denominator, then if population numbers decrease, mechanical chemical energy and numbers of molecules will increase. The increased dm̅ on the top has the same effect as the decreased n + dn on the bottom. The smaller the entities in a population, or the greater the numbers, then the faster that population traverses its generation length. We proved this initially surprising dependency of generation length on mass in our experiment.

The term dp̅/p̅(n + dn)—which has its equivalent through the constraint of constant equivalence as dχ/χ(φ + dφ)—does exactly the same for proportionate changes in the Wallace pressure aspect of internal energy. This is again the effect we wanted. The two between them stipulate the four dimensions of the Weyl and the Ricci tensors, but also allow us to separate the various effects so we can run experiments. When numbers decline, generation length increases, and conversely. We again confirmed this with our experiment.

Since biological space is composed entirely of internal energy, all populations need to do work to consume enough energy to complete the generation. They do not complete their generations at the same absolute speeds. As the contrast between the blue whale and the mosquito makes clear, every population also has its minimum and its maximum values for mass and energy.

Figure 20.13

Figure 20.13 highlights the same issues of time and distance as Figure 20.12 with the man and the woman and the shuttlecock, but from a different perspective. Every population has a lower and an upper bound. This is the inner and outer peripheries of the spiral staircase.

Biological entities must always pass through time by ascending this symbolic spiral staircase from the past pole to the future one. They are also obliged to go toroidally around the circulation of the generations, undertaking their necessary configuration changes in internal energy. Those entities and populations whose masses and energies behave more like the man in Figure 20.12 will be placed, relatively, closer to the periphery. They will have greater masses and energies, but smaller numbers. Those behaving more like the woman will be closer to the central axis with smaller masses and energies, but in greater numbers.

There is always some relationship between the absolute clock time moving from one pole to the other, and the journey of biological activity about the helicoid, and as a torus. The central axis on the spiral staircase, and so on the helicoid, represents the flow of absolute clock time through the space. It is Sir Arthur Stanley Eddington's famous arrow of time, always ascending as the helicoid's pitch, which is forwards through time. Every generation then has some absolute time interval, T. Every such time interval, T, is composed of a set of distinct moments, t, which can be measured with a clock.

The generation time of biological events that also passes is a sequence of biological events. Those events are the helical corkscrew motion that stretches out over the time, τ. Every generation is determined by its relationship between T and t, which is between its absolute clock time, and the portion of generation that elapses.

A spiral staircase always feels steeper if we ascend it closer to the centre. So also, the closer to the central axis a given entity remains, the greater is its rate of change in its electromagnetic activity, and the greater is its rate of curvature. So if a first biological entity is smaller than a second, then just like there is a difference between the times the man and the woman take to return a shuttlecock, there is immediately a difference in their proportionate rates of change, stated as dm/. The smaller entity is closer to any central axis, and will always process its materials all about the circulation more rapidly. It ascends timelike up the staircase more rapidly, and simply because it has less materials to process. This is something we can measure, and that we confirmed in our Brassica rapa experiment.

The converse also holds. If one entity is larger at every point than another, then that larger one must travel larger spacelike, horizontal, distances, and so must also maintain the lower relative timelike rate of ascent, in just the same way as anyone ascending a spiral staircase further away from the centre must maintain a higher horizontal and spacelike rate, but a smaller vertical and timelike one. The larger entity has a larger set of configuration changes it must undertake. We again confirmed this in our Brassica rapa experiment.

The smaller an entity is, then the faster is its timelike rate of configuration propagation for its internal energy all around the generation; the greater is the amount of instantaneous change at any moment; the greater is the rate of timelike change; and the closer to the centre of the helicoid or spiral staircase that entity is. However, since we have in the denominator of the same expressions, then the larger the entities are, the further away from the central axis they also are, and the lower is the poloidal and timelike and proportionate rate of change, while the greater is the meridional or spacelike one. In other words, larger entities will always take longer to go about the circulation of the generations. They simply have more configurations of internal energy to navigate. These are variations and deviations within a population. That is what we measured in our Brassica rapa experiment.

The Weyl and the Ricci tensors that govern biological events in biological space are more concerned with a general pressure of internal energy that works itself through four dimensions. The expressions for proportionate rates of change in both mass and energy have an n + dn factor which, since it is again in the denominator, inversely affect population sizes. An instantaneous decrease in the numbers, dn, therefore has the same effect as an instantaneous increase in population mass and/or energy, dm̅ and dp̅. A decrease in numbers institutes the same spacelike change of decrease in processing speed as an increase in mass and/or energy. Both induce a move to the outside of a helicoid or spiral staircase. These are warpings in biological spacetime that are independent of any distinct entity, and which immediately extend the circulation. They are effects that are extremely easy to measure … which is precisely what we did in our Brassica rapa experiment. We indeed saw there that when the population had its smallest average individual mass and/or greatest numbers, it took only 28 days to complete the cycle; but that when the average individual mass was at its largest, and/or numbers were at their smallest, the same cycle took 44 days … exactly according to these four-dimensional expectations.

We should now note two pieces of information we shall be needing shortly:

1. The special theory of relativity tells us that nothing can move faster than the speed of light in free space. This is another way of saying that every population has its maximum of resources and energy it can use. It can therefore support a maximum number. There is, correspondingly, a minimum amount of resources and energy per individual member that the population, or any of its members, can use, and as is consistent with the medium or chemical components. This is the inner bound on the helicoid of internal energy at any time. It establishes an upper bound for the intensity of configuration activity; a minimum for the basis in mass; and so a lowest possible interval for generation length, T, for any viable population. Every population therefore has a closest possible approach for its internal energy to the central axis on that spiral staircase.
2. Since the mass-energy required per each time interval increases as the curvature decreases, then the flattest possible curvature will seek to embrace all possible mass, and all possible energy, in the entire universe as its alternative basis in for its configurations and genome. The amount of internal energy increases. There is therefore a minimum number that the population can support. This means that there is a theoretical upper bound to any generation length. This is the biological entity whose potential size, and whose circulation of mass and energy, is the birth and death of the entire cosmos. This is all possible internal energy. There is therefore a lower bound to the actual rate, and an upper one to the possible basis in mass; as well as a longest possible circulation or time interval for any generation length. This is the greatest possible amount of resources and energy the population, or any of its members, can use, again consistent with the medium or chemical components. This is the outer bound on the helicoid. Every population therefore has a furthest possible location away from the central axis on the helicoid.

The above two potentialities in maxima and minima are always part of any allowed set. The actual lower and upper absolute bounds in any given case will, of course, depend upon a population's specific rates of change in its internal energy. We nevertheless have an intrinsic rate of activity for one bound, that depends completely upon the medium; and an extrinsic bound for the other, that depends upon the available resources. These two limits form a part of a population's required and allowed sets of mechanical and nonmechanical chemical energies, and its molecular interactions in its internal energy.

Since we are now using entirely proportionate rates of change, each population is left free to determine its own absolute times and amounts, again the effect we need. So as the polar planimeter turns about itself in our four dimensions to measure a helicoid from its centre, the mosquitos close to that centre in Figure 20.7 will have small absolute values, but high relative rates of change; while the blue whales out on the periphery will instead have large absolute values, but small relative rates of change. They each thus produce their different generation times, yet still retain the comparability in τ and all other values that we seek. We can now use the four dimensions to compare transformations across all populations, and quite irrespective of type.

All biological populations are now given by our biological potential of μ = dS = dφ + dκ + dχ, which is a complete set of relative proportional changes in internal energy everywhere and every when. It is therefore, and in principle, possible to stack all possible generation lengths beside and/or on top of each other, so they form one helicoid, and to measure all their internal energies with but a single turn of a universal polar planimeter. There is then—and immediately—the one equally universal speed of reproduction. Populations now differ only in their relative rates of curvature, and in the relative distances and gradients they each report to each other, as well as in the quantities of mass and energy they each scale at their respective locations in our three-dimensional biological space of numbers, n, mass, M, and energy, P.

All this granted, we can then follow Einstein and establish a cosmological truth for all biological populations albeit—and as his did—it initially looks like a tautology:

We establish, by definition, that every biological population
takes exactly one generation, T, to circulate the events of a generation, τ
.

If we now fill our tensor with values that transmit heredity and potentials from point to point, we can study the molecular materials and energies that make up the surfaces and tunnels of our four-dimensional helicoid. We shall soon see why the above is no more of a tautology than Einstein's original quote near the beginning of this chapter.

A mathematical aside

We can again refer to Green's theorem of the vector calculus to express all possible increments in mechanical chemical energy as the equalityC m̅ dn + n dm̅ = R ((∂/n) - (∂n/))  dτ between a line and an area integral. Any increment in either mass or number now has an immediate double effect on (1) the length of time spent about the boundary, which is T for the generation; and (2) the area enclosed, which latter is also the spanning of a generation, τ. That latter area or generation span, which is the term representing the area integral on the right, is again measured by the relative rates of change in both mass and number through the partial differentials it contains. In particular … the area enclosed increases if numbers decrease, which manifests itself in the radial stretching out of our polar planimeter … and as an increase in the Ricci tensor.

We can of course derive a similar equality for the nonmechanical chemical energy: C  p̅ dn + n dp̅ = R ((∂/n) - (∂n/))  dτ.

And since we have the Weyl and Ricci tensors, then as long as we take appropriate orientations across the surfaces and manifolds so that we have a viable unit normal (it is simply our generational weighted average), we will have a further equality between our volume and surface integrals. By Gauss' divergence theorem (of the vector calculus), then since we have specified beginning and end points which are the movements in time and so that are our bounding surfaces; and if we use m̅’ as the weighted average and unit normal; then the total mechanical chemical energy or Mendel flux passing through the surfaces, and so across those boundaries, is equal to the total flow within the contained volume over the time period as in R ndm̅’ = W ∇ • nm̅dt. The same holds for the nonmechanical energy of the Wallace pressure: R ndp̅’ = W ∇ • np̅dt.

These are a part of the forces that determine the overall rate at which configurations propagate in the phase space, and both establish the Weyl tensor while confirming the inverse dependency on numbers over time.

Figure 20.14

The issue is now how heredity is transmitted across time and space, and through our internal energy. Since the reproductive cycle demands a variety of behaviours, it is time to look more closely at the molecules that compose all biological entities and populations. We can show that the above declaration is not a tautology by paying closer attention to the molecular implications of Figure 20.14 which define all possible molecular behaviours and interactions for the internal energies of all biological entities.

Figure 20.14 is a ‘triple point of water cell’. It depicts ice, water, and water vapour held at their ‘Gibbs triple point equilibrium’. It declares the molecular driving forces at every point on the helicoid, and so all around the circulations of the generations as transmitted to biological entities by the Weyl and Ricci tensors.

Since all biological entities are composed of the molecules that enshrine their internal energy, they all have what the University of Berkeley-based US chemist Gilbert Norton Lewis called a “fugacity” or “escaping tendency”, which is defined by the Gibbs triple point equilibrium. As shown in Figure 20.14, the escaping tendency defines all the molecular potentials available to biological entities by specifying their unique molecular driving forces and interactions. It does so by stating the energy and the entropy of the diverse states its molecules can take up due to the internal energy they enshrine.

A mathematical aside

By the first law of thermodynamics, a given biological entity is a discrete thermodynamic system with a given number of moles, q, of chemical components. It will therefore have a set of molecular transformations given by (∂S/∂U)q. If the number of its constraints, β, and its moles of components or amount of substance, q, hold constant, while its internal energy, U, changes, then its entropy, S, must also change. So if the molecules in a first System A are more energetic than those in a second System B, then molecular energy and momentum will transfer over to B as their molecules interact, for A has the greater escaping tendency. This is a one-sided and irreversible effect upon all systems whose escaping tendency is lower, for a lower escaping tendency is immediately a greater capturing tendency. If System A is a biological entity while System B is the surroundings, then it is an irreversible surrender to the surroundings as their joint entropy increases. Either System A, or a population containing such a system, must now find a way to increase its capturing tendency above the surroundings so that replacement molecules can enter and the circulation can repeat. This happens upon the helicoid as biological populations first, in the net, lose molecules; and then regain them by adjusting their escaping and capturing tendencies.

By the second law of thermodynamics, each of the three phases of gas(-eous vapour), solid (ice) and liquid (water) in the Gibbs system displays its measurable fugacity or escaping tendency. This is its irrevocable and irreversible tendency to surrender molecules. Each of the phases interacts both with the surroundings and with each other across their various boundaries. But they all interact with the surroundings. The system as a whole therefore maintains an equilibrium with the environment. But because each part or phase must make a distinct contribution, we can measure those and allocate values to the molecules involved. We can therefore measure the differences between the inner and outer bounds on the spiral staircase in Figure 20.13.

Since the second law of thermodynamics guarantees that molecules must always escape from any state or condition, the most those in each distinct phase can hope for is to create a countervailing ‘capturing tendency’ to pull molecules back in to themselves, using the conditions, so they offset any being lost to the other two phases via the escaping one. Each phase's tendency to expand, through its molecular capturing tendency, is then exerted under these imposed conditions of pressure and temperature. Each phase's tendency to expand through its capturing tendency matches its tendency to contract as its molecules dissipate into the other two through its escaping tendency. These are now exerted by all phases under the same set of conditions. When all three phases have attained a balance both with each other and with the surroundings, the energies and the entropies of all the molecules concerned are evenly distributed across all three. All molecules are then equally free to enter and to exit all the arrays or configurations available to them. They exert mutual equilibrium pressures. The three macroscopically evident phases are in balance, at those imposed conditions, on a microscopic level. They have a chemical equivalency. The Gibbs triple point equilibrium is the balancing of all molecular driving forces. The system and all its molecules can then be measured; and all the molecules will have definite values. Once we have found those values, we can use them to measure our biological entities all around the circulation of the generations.

By the ‘Gibbs phase rule’ and the chemical equivalency of the Gibbs triple point equilibrium, every substance capable of existing as solid, liquid, and gas, has a unique combination of pressure and temperature that allows all three of its phases to coexist, equivalently, in this manner. Water is now the only substance that can exist as vapour, solid ice, and liquid water at a temperature of 273.16 kelvins, and a pressure of 611.3 pascals or newtons per square metre. The molecules concerned are exerting equilibrium forces upon each other, and must also have the chemical formula H2O. No other is possible. The process is completely general, and can be extended to all molecules, of whatever type.

Scientists such as Carnot, Clausius, Hermann von Helmholtz, Julius Mayer, James Joule, Josiah Gibbs, Lord Kelvin and numerous others struggled to understand a vast array of anomalous and initially incomprehensible interactions under energy, and such as are collected together in this triple point equilibrium. They of course wanted to express all such interactions in the same kind of language that had proved so immensely successful for Newtonian mechanics.

Figure 20.15

We are looking to follow the above scientists and give concrete values to the biological Weyl and Ricci tensors that govern heredity and biological behaviours. Figure 20.15.A shows the various nonmechanical chemical energy stages in our prototype biological cycle. The apertures are open. Masses do not cross boundaries. Since the stages in Figure 20.15.A are the conversion of ice into steam, they happen without exchanges of mass. They are nonmechanical chemical work and force. The changes in energy density happen at constant volume, and again only involve changes in cell configuration. They do not involve changes in mass or in mechanical chemical energy. Nonmechanical chemical energy may enter and leave; and work might thus be done, but this is entirely within the system.

The equivalent to this process in biology could be something like a chrysalis transforming into a butterfly, or any of the sets of processes we depicted in Figure 20.9. Since no further mass needs to enter the system, no piston goes up and down in the surroundings to exchange work as ongoing exchanges in mass. It is again, therefore, nonmechanical.

The significance of Figure 20.15.A is that it is one of Clausius' closed system interactions. The enclosed internal energy is exchanging heat energy with the surroundings, but is not exchanging mechanical work. There is only an exchange of nonmechanical energy over the boundary. No pressure or volume changes therefore occur in the surroundings, for no piston moves. There is again merely an exchange of nonmechanical energy, in this case as solar radiation, which is heat. All mass remains internal, and simply undergoes a configuration change, which nevertheless increases the internal energy. It is also measurable as work, for it is an energy conversion.

The prototype cell's entropy remains constant. The nonmechanical energy entering or leaving simply fuels the configuration changes, all at a constant cellular volume and entropy. It is a display of molecular force and a change in energy density and in which energy is exchanged with the surroundings, with no change in the quantity of chemical components. The specific interactions could just as well arise from a set of ongoing metabolic and physiological activities such as proteins folding, but they are still all internal to the cell. All the work being done is strictly contained within these molecular and biological boundaries. It is all redolent of the Weyl tensor. We can therefore measure the changes in energy imposed on each volume element all around the circulation of the generations.

The various stages in Figure 20.15.B are very different. They instead show the two mechanical chemical energy stages in our prototype biological cell's internal energy, where its orifices have opened. These are Clausius' open system interactions where a mass of chemical components does now cross the cell boundary. Molecules are exchanged with the surroundings. The entropy also changes as that volume of molecules and cellular components changes. Mechanical work is now being done in and by the surroundings. It is obligatory. Mass enters and/or leaves the cell, which is the definition of mechanical work. The passage of that molecular mass of chemical components is work done against a force that the cell must either exert against the surroundings, or else resist against as the surroundings reclaim those components. This affects the amount of internal energy.

These two stages require that the rock depicting them in Figure 9.1 moves mechanically either up or down. Those exchanges in mass are therefore all done against a force. And since the system must bear the weight of that rock—which is the force of the components exchanged—to compensate for the entry or exist of material mass, then that work is being done under the constant pressure, imposed by that rock, which is representative of the force of and against the surroundings, and so as against the piston. There are now definite and measurable pressure and volume changes in the surroundings courtesy of the force and work in the exchanges of mass. It is measurable as the mass exchanged. This is a molecular-based “constant pressure” interaction. It is all redolent of the Ricci tensor. We can therefore measure all changes in the sizes and quantities of our volume elements all around the same circulation.

In the prior two stages, nonmechanical energy can enter or leave without any need for external mechanical work or changes in pressure or volume, and no matter what the scale of the internal ones. This is the molecular-based constant volume and configuration-changing, work of Figure 20.15.A which is also a way of measuring internal energy. It is very different in kind from the constant pressure, mass- and molecule-exchanging, work of Figure 20.15.B, which also increases internal energy. The latter has the added factor of work done by molecules against an external force. This difference is exactly what we need to quantify before we can explain evolution, or conduct any experiments to unequivocally prove evolution's necessary existence. We must also quantify it all in both absolute and relative terms, which our four equations from the previous refutation chapter certainly allow us to do.

We can measure what we need, and we can distinguish between the Weyl and Ricci tensors using Figures 20.15.C and 20.15.D which depict one of the major triumphs of Newton’s system: the way it equates the potential energy of the gravitational attraction caused by a position in space, to the kinetic energy caused by that potential energy, and by its attraction, as objects then move through space because of that potential. Figure 20.15.D adds the stresses and strains of some elastic medium, also at work in a gravitational field, and so with the same mix of potential and kinetic energies. The combination of these with absolute clock and relative generational time is the force that drives the molecules across the generation. We must find a way to measure this.

Newton solved the problem of how two bodies in space affect each other, but he could not resolve ‘the three-body problem’, such as with the sun and two earths in Figure 20.15.C. We now know that that third body introduces the perturbations and aberrancies that drive any mass-energy system and its rates of change away from its expressions of perfect motion. It can maintain neither a constant velocity nor a constant acceleration.

The mathematicians Ernst Bruns and Henri Poincaré eventually showed, in 1887, that there is no general solution to the three-body problem—at least when using algebra and integrals—because while the three bodies might each repeat their individual motions, they never precisely repeat their joint relative motions, except in very special situations. However, that unpredictable non-repeating of n bodies does not affect the complete predictability of any two, nor its extension into the elastic situation of Figure 20.15.D. They will always obey some general rules and establish the inner and outer bounds we want to measure.

Since we want to track all molecules and their energies, we want to summarize the differences between the man and the woman as they return the shuttlecock in Figure 20.12. We can do this using the technique the French mathematician Jopseph-Louis Lagrange developed when he tackled the three-body problem. He saw that he needed a clear way to describe the gravitational attraction that could be directly attributed to points in space. Those are the differences we want to measure in our biological internal energy space.

Lagrange wanted to calculate the force that an infinitesimally small test body must be experiencing at some given Location A, so that by the time it got to Location B, it had acquired its values for kinetic energy because of previously having been at A. He observed that if he subtracted a given amount of kinetic energy from the body when it was at B, and then backtracked to A, he could assign the value he had just subtracted to Location A itself as its inherent ability to impose exactly that quantity of force, upon the body, so it would acquire the kinetic energy it would have by the time it was at B. He therefore devised a “potential function” for this purpose, and so he could more accurately describe planetary behaviour. The heredity we want to describe is also a potential function.

Lagrange could now calculate the force impressed upon a body when it was in a specific position and so that, entirely because of being there, it would gradually acquire, through its ensuing movements in space, the kinetic energy it would then exhibit by the time it was at some other point. Since this was now a force inherent in space; and since it was caused by position and the potential function; the Cambridge University mathematician and physicist George Green (of Green's theorem fame) later gave Lagrange's discovery the term “potential energy”. Thanks to Lagrange, therefore, potential energy, of whatever kind, is an object’s stored capacity to do work due to its position, but with respect to some reference. Potential energy is the work some body will undertake because of where it is; or else because of what it is; but always when compared to—or relative to—some other body or state. Our two references, in biology, are progenitor and progeny.

The Irish physicist, astronomer, and mathematician, Sir William Rowan Hamilton later realized that if a planet was maintaining a steady orbit, then the sum of its potential and kinetic energies was always constant. There is a constancy because the one is always being exchanged for the other. When a planet moves, it is either moving closer to or further away from some body, and so is either creating or realizing a potential with respect to that body. Its current velocity is always a function of both (1) the positions it has recently been in relative to some body, and (2) the positions it will shortly enter again relative to that body. This is true all around the cycle or orbit. When a planet is, for example, far away from the sun, it will shortly be closer; and when it is close, it will soon be far away. As a more general principle, when a body falls in a gravitational field, then it is gradually exchanging the potential energy of its previous capacity-for-falling into the kinetic energy of actually-falling. The sum of those potential and kinetic energies is again constant, and is now called the ‘hamiltonian’, H, in his honour.

We should note carefully that the energies making up the hamiltonian are related additively. So if two different populations have the same hamiltonian, but it is composed of different different quantities of the kinetic and the potential energies, then they will always transform bertween them at different rates. Their positions and speeds will always be different, even though they keep their hamiltonians the same.

The three-body problem may be unsolvable, but all three bodies are still individually predictable. They all still have both a position and a momentum. Where Hamilton applied his idea largely to single particles and systems, the French mathematician Joseph Liouville proved, with the “Liouville theorem”, that any large system or ensemble's conjoined hamiltonian will remain constant, no matter what permutations that increased number or “ensemble” of bodies might go through. Liouville therefore explained the behaviour of a whole population of biological entities.

Figure 20.16

Since we are now considering an ensemble of bodies, we more generally represent their momentum with p, and their position with q. As in Figure 20.16, we can now represent Body A's position in the x-dimension with qxA, and its momentum at that position with pxA. Since we have just given ourselves two new coordinates in x—one of px in momentum, and one of qx in position, both in x—then Body A's momentum at every point in the x dimension is now a point on a new px-qx plane. We can do the same for some Body B. We now have a complete record of all those motions in x for every body in the ensemble. We can repeat this for the y and z dimensions to give ourselves the six-dimensional “phase space” px-qx-py-qy-pz-qz to record this entire collection or ensemble of bodies everywhere it goes.

Since every distinct body in the ensemble has its hamiltonian, through which it exchanges its potential and kinetic energies, then each one will follow a determinate path in this new six-dimensional phase space. This holds for each member in the ensemble as they each switch between their expressions of potential and kinetic energies. The whole ensemble of bodies, points, and particles is now something like a gas or cloud of such points in this space, all moving together, again through this phase space, and as a statement of their energies and motions. The sum of all their hamiltonians is constant and they have an overall “phase volume” within the space.

The Liouville theorem now states that the overall phase volume that our ensemble of bodies or points occupies will be constant. If one body within the cloud acquires a high momentum in, say, the z direction, then the others will all gradually follow suit, or else have done so in the very recent past, and the whole cloud keeps together. And if one particle or body slows down, then the others will also eventually do so, and the cloud again remains coherent relative to its total energy. The range of positions the cloud and ensemble members occupy is narrow enough, and definite enough, to preserve the total ensemble energy, which is their joint hamiltonian. There is therefore an overall density of presence of these bodies in the ensemble in every energy neighbourhood that is consistently maintained. That density will be preserved as the overall phase and energy volume cloud changes its shape with the changes in position and momentum that the individual bodies in the ensemble undertake. Since the bodies within the ensemble follow similar paths, their density in any neighbourhood in phase space remains constant no matter where they might go, and how they might move. The “Liouville constant” is a scalar value that measures that constancy in energy.

A mathematical aside

As we shall see very shortly, the Liouville constant also acts as the “Ricci scalar”, R, for biology, making it the determining characteristic of populations and species.

The Liouville theorem only applies to whole ensembles of bodies. It establishes the behaviour of entire neighborhoods in the phase space. It deals with the existence and the behaviour of systems of ensembles of bodies and points, and embraces both (A) macroscopic bodies; and (B) their molecules. Anything violating the Liouville theorem immediately violates the second law of thermodynamics, which is soon derived from it through the fluctuation theorem concerning the relative probability of a system's entropy.

A mathematical aside

The Liouville theorem and its hamiltonians are principal determinants of the configuration and phase spaces.

The Einstein tensor, Gμν, that describes the behaviour of all matter can be much more compactly written, using the Ricci scalar, R, as:
Gμν = Rμν - (½ • Rgμν ).

We can now call Gμν the “Einstein-Linnaeus tensor”. In conjunction with (a) the Ricci tensor, Rμν, (b) the Ricci scalar, R, and (c) the metric tensor, gμν, which establishes the basis for all measures and values; the Einstein-Linnaeus tensor predicts the numbers, forms, and masses of all the entities in any population.

Hamiltonians and the Liouville constant can easily be extended to include the spring with its stresses and strains shown in Figure 20.15.D, and therefore transformations in media of all kinds. As the spring bounces up and down, the weight at the bottom first stores, and then releases, energy within the spring's materials. Those molecules are therefore undergoing the transformations of potentials and their kinetic energies. Two things are therefore storing its potential energy: (A) the weight's physical location as a distance in space; and (B) the positions of the molecules within its medium. This is a juxtaposition of mechanical and nonmechanical energies. Both are now the source of the spring's conjoined potential and kinetic energies.

The French mathematician Baron Augustin-Louis Cauchy showed how to analyse those spring effects, behaviours, and changes in energies and intensities with his “Cauchy tensor” of Table 20:1, which we should instantly recognize as the inner 3 × 3 or Owen tensor subset of our larger 4 × 4 Haeckel tensor. We create it by removing the left column and top row:

 x (length) y (breadth) z (height) x (length) xx xy xz y (length) yx yy yz z (height) zx zy zz

The term xy in the Cauchy tensor now refers to the force directed in the x direction that has produced a movement or result in the y direction, whereas the term yx refers to a force intended for the y-direction that instead produced a change or movement in x. So if a soccer ball bounces, a cushion merely falls, or an object of any kind transforms in space, we can carefully analyse the forces, stresses, and strains in each direction, and because of each component. General relativity takes this same Cauchy tensor and adds the fourth dimension of time to analyse the time rate of change of forces and energies across all four dimensions, including their rates of change over time. So the difference between the soccer ball and the cushion is that in both cases forces in the z-dimension of height cause changes in both x and y. As the soccer ball and soft cushion flex, distort, and transform, they do so in different ways and at different rates over time. But as we observe them both over time, previous values for x, y, and z are eventually restored in the soccer ball so it regains its shape. This is why it bounces. They are not restored in the soft cushion, which is why it does not bounce. The energy instead either remains as its changed shape, or else dissipates into the surroundings as heat.

We should note the special status of the components on the diagonal. These are xx, yy, and zz where the forces intended for x, y, and z respectively actually produced movements in those intended directions. Those are called the “normal pressures”, and tell us about the object's normal and most characteristic responses.

The six off-diagonal components in the Cauchy tensor are called “shear stresses”. They summarize the distortions imposed upon the object. We can now quantify all internal and potentially nonmechanical, constant volume, Weyl tensor-like transformations due to mechanical, constant pressure, Ricci tensor-like external forces. The Ricci scalar is the value that summarizes both the three normal pressures and the six shear stresses. It is unique to every tensor.

The molecules in the Gibbs triple point equilibrium constitute a Gibbs “ensemble”. We can soon find out what this ensemble of molecules is equalizing; how they are jointly doing it; and then similarly quantify it all within biology by first turning to Gabriel Daniel Fahrenheit, who investigated heat and temperature.

Fahrenheit—famous for the quality of his thermometers, and for the temperature scale named after him—was the first to note that water can be chilled below its freezing point without turning to ice; but that when the vessel containing it is shaken, it turns immediately to ice as its temperature rises back to freezing point. This phenomenon greatly interested the eighteenth century Scottish physicist Joseph Black. He was the first to note that biology is impossible without the absorption and emission of heat energy; the first to think of directing his own breath at a sample of lime water, or calcium carbonate, CaCO3, to quantify the ensuing gaseous reaction; and the first to observe that biological organisms survive by the systematic exchange of gases we know as respiration, although the ultimate proof for that insight came from the French chemist Antoine Lavoisier.

Black was the first to appreciate the consequences of a very common observation such as we can see with icebergs, or ice cubes in any iced drink: that if equal amounts of water and ice, both initially at 0° C, are exposed to the air, then although the water around the ice will gradually warm … the ice itself will remain steadfastly at 0° C throughout, and even as it melts. He gave the first accounts of his observations concerning diversity, internal constant volume transformations, and their hamiltonians on April 23rd. 1762, at the University of Glasgow:

If the complete change of ice and snow into water required only the further addition of a very small quantity of heat, the mass, though of considerable size, ought all to be melted in a few minutes or seconds or more. Were this the case, the consequences would be dreadful. Even as things are at present, the melting of great quantities of snow and ice occasions violent torrents. But were the ice and snow to melt … suddenly … the torrents would be incomparably dreadful.

Joseph Black, quoted in Harvard Case Studies in Experimental Science, Vol I, James B. Conant, 1957

The Weyl and the Ricci tensors might be inextricably interlocked in four dimensions, but we can nevertheless untangle their effects in this, the three-dimensional world we inhabit, by observing Black's seminal discoveries. He believed that heat was an important factor in determining chemical affinities, or the diversities and different behaviours of substances. Based on his understanding of Newton’s third law of actio-reactio—“to every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts”—Black established, in chemistry, the idea that if a first agent acts upon a second, then that second agent will exhibit a reaction to the first. This was his concept of a “chemical reaction”. He was the pioneer of modern chemistry and biochemistry.

If we want to understand biological diversity, then we must understand the scale of Black's achievements. He paved the way for modern energy and biology by showing that if we add energy to a substance, it can change its energy density, its configuration, or both.

Black took the first big step towards the discovery of this nondynamical, constant volume, nonmechanical equivalent, in heat energy, to Newton’s mechanical and dynamical, constant pressure, potential energy. He was therefore the first to properly notice what he then called “latent heat”, and which we now call “phase change enthalpy”, where enthalpy is taken from the Greek enthalpein when en means in, and thalpein means warm or heat.

Enthalpy is internal to a system. It is a measure of the energy content within a body or substance's boundary. It tells us the total internal molecular force attributable to our collection of internal energy. It is not just the system's proposed heat content, in terms simply of its current stock of vibrating molecular energy, but also incorporates the work that those molecules had to do to establish that system within that environment. As Black observed, enthalpy is also the measure of a potential for diversity through being the measure of the stored or latent energy: i.e. energy contained which can still express itself in the surroundings via a heat transfer or interaction which will then change both that system and the environment's configuration and/or behaviour. It tells us the full scale of the energies that the Weyl and Ricci tensors distribute over time and space.

The Gibbs phase equilibrium in Figure 20.14 exists between the different varieties of chemical and molecular reactions that Black and others discovered. We now know, thanks to them, that no matter how diverse they might be, the reagents in a reaction always have a measurable potential with respect to the products, and irrespective of type. The basis of the atomic theory is that all atoms behave the same way everywhere.

Black was again the first to realize that heat and temperature are quite distinct; and so he was the first to carefully separate the quantity of heat in a body, which we now call its enthalpy, from its intensity, which we now call its temperature. The significance is not generally noted, for although temperature plays an important part in daily life, there is a less frequent need to note differences in energy content. And since there is little need to note differences in energy content, there is even less need to note any differences in the rate at which energy content changes. This last is a critical factor in evolution, which is predicated on exactly such changes over time.

The Weyl and the Ricci tensors, as four-dimensional constructs, have different ways of commanding the quantities and intensities of the energy they deliver into three-dimensional space at each moment. Blue whales are not just bigger than mosquitos, they distribute their masses and energies very differently, and so are qualitatively very different indeed. We must, however, measure this.

As an example, a hot plate might contain only relatively small quantities of heat energy at any time, but it attempts to create its Prévost style radiative equilibrium by transferring that heat into us at a very fast rate, relative to us. We therefore notice its temperature. That is far more important, to us, than the total amount of heat transferred … although we intuitively realize that if something has not yet cooked properly, then it needs more time to transfer more of that heat, which is an energy content. The cooked food depends on both the intensity, or temperature, and energy content, which is a function of the cooking time.

And similarly, while we intuitively realize that an ice cube has a relatively small total heat content, the cold temperature we feel is more significant to us. That is the relatively slow rate at which it transfers our heat out of us. That rate, however, is distinct from its size or net content. A large frozen lake has a much greater heat content than either the hot plate or ice cube … although it certainly shares the same temperature, and so rate of heat transfer, as the latter. A small hot object might have less total heat to transfer than a large cold one, but we more readily notice the intensity or temperature of the transfer, and not so much the amount.

A large frozen lake has much more total energy to transfer than a small hot plate. The large frozen lake could certainly become problematic if we are forced to remain in it for lengthy periods. The steady transfer of our heat out of us into that lake is just as damaging, in the long run, as is the heat pushed so rapidly into us by the hot plate; but we identify more with the temperatures and the rates than we do with the amounts or enthalpies. We also identify more with the traits, memes, and behaviours of biological organisms than we do with their net energy contents.

Black was again the first to carefully measure the masses involved in chemical reactions, and to note the significance. He also proved that a gas is an independent chemical substance which can react with solids; that heat tends towards an equilibrium; and that biology depends on a cyclical series of chemical reactions. That cyclical series ultimately results in heredity and reproduction under the Biot-Savart law.

The Gibbs triple point of Figure 20.14 might be holding the molecules in the ice, water, and vapour in equilibrium, but there is clearly something different about their diverse configurations. They have very different energy densities, which is a measure of their diversity. The three phases are thus in very different states and conditions. Black initiated the process of understanding that diversity by extending Leibniz’s doctrine of the conservation of momentum out to heat.

In 1792 Black first showed how we can link the Weyl tensor's expansions and contractions in Figure 20.9 to the Ricci tensor's broadening and narrowing in Figure 20.11 and thus explain all biodiversity. He showed how to separate the quantity of heat, or enthalpy from its rate or intensity or temperature. We would now say that he used a hamiltonian. It is the method to help unlock evolution and evolutionary potentials.

Black's method was both very simple, and very general. He placed some ice-cold water into two pails, first immersing them both in a surrounding ice bath. He placed a drop of alcohol into one pail to prevent it from freezing. When he was sure, from his thermometer, that they were at the same temperature as the ice bath, he removed them from it and positioned them both in front of the same open fire. The frozen pail took ten hours to melt, while the other rose to 140°F (78°C). Therefore, he concluded, the quantity of heat absorbed by the frozen pail, in ten hours, was equal to the quantity of heat needed to raise the intensity of the same quantity of water by 140 °F (78 °C). He in other words used the yet-to-be-derived first law of thermodynamics to state both their diversity and their hamiltonian equivalence.

Black had realized, in his simple-seeming experiment, that the quantity of heat absorbed by his first pail's internal energy, per unit of its mass, which is its energy density, could be measured relatively. It could be measured as the second pail’s rise in temperature over that same time interval, and also per unit mass, which is its heat capacity or its ability to change its intensity to heat. So the difference or diversity between the ice and the water lies in their different energy density responses to the same quantity of heat energy. It is a very definite amount of heat energy that he could measure using his simple hamiltonian method of equating their potentials and their actuals.

To put it in modern terms, Black had exploited his realization that the quantity of heat he put into the ice causes it to change its molecular configuration while it maintains its same temperature, or heat intensity. And since the heat energy absorbed was hidden from his thermometer, Black called it a “latent form of heat”. It is a phase change, and we now call the heat needed for this transition from solid to liquid the “enthalpy of fusion”. It is a change in energy density and molecular configuration and so a cause of diversity, but at a specific heat intensity or temperature. Temperature is therefore a measure of what was formerly called “heat power”.

In another experiment, using this same hamiltonian idea, Black put a known quantity of water at 50 °F (10 °C) in a flat-bottomed, tin-plated pan. It took four minutes to increase its temperature or heat intensity and reach boiling point. He then noted that that known mass of water took twenty minutes to boil away, and so to be fully converted into vapour, which has a different energy density and heat content, all at that same temperature or heat intensity. He therefore concluded that water’s latent heat of vapourization was 810 °F (432 °C) … or the temperature that mass of water would have risen to, over that same time span, if it had instead simply kept rising in its temperature and so maintaining a tendency to change its intensity response to heat, rather than staying at the same temperature and boiling away. Therefore, the difference between water and its vapour is entirely due to the enthalpy or heat content absorbed. That is also a phase change enthalpy. It is a chemical behaviour funded by another very definite amount of heat energy, and which caused a change in energy density at a given heat intensity or temperature. It is the enthalpy of vapourization.

Black's hamiltonian method is again completely general. We now have a way of stating any change in diversity or energy density, both absolutely and relatively. Using his system we can always calculate the potential energy for any change in state, even those that are intrinsically and intricately biological. We simply do as Black did and use each entity or population as its own basis, and compare it to its own self over time.

Temperature measures a substance's tendency to spontaneously surrender energy to the surroundings. Entropy plays a role by helping to establish the amount of energy that that specific configuration surrenders, and under those conditions. The temperature multiplied by the entropy, TS, states both the quantity and the intensity of energy currently available—or unavailable—for undertaking work-and-heat involved transformations.

An object with lower entropy, S, has a greater amount of potential transformations and available energy. This is what entropy measures. If two objects contain the same amount of energy but are at different temperatures, then other things being equal, the one at the lower temperature is undertaking less transformations and has the lower entropy. Its molecules are moving more slowly. It therefore has more energy available for further interactions because there is a greater number of interactions it could undertake, relative to the other. Those other interactions will gradually appear as its temperature is raised.

If two objects are at the same temperature, then their molecules are moving at the same speeds. If one has a higher entropy, then it is undertaking more transformations. But just as something cool can be heated so it undertakes more transformations, something with a lower entropy has the capacity to undertake further transformations. In both cases, the object must absorb energy. While the cool one absorbs that energy by raising its temperature, the other one absorbs it by changing in its state.

Black observed that the temperature remains constant as ice converts to water. But since heat energy is constantly absorbed, there must be an enthalpy or heat content change. There must be some difference in the way ice and water are each configured at that same temperature. We have the chemical and energetic interaction TiceSiceTliquidSliquid, where S refers to their unit configuration difference or entropy. The double-sided arrow implies that there is some equivalence between the two. It means that the two stocks of internal energy are undertaking the same numbers of vibrations in their molecular movements, for they are at the same temperature; but they are somehow vibrating in different modes and ways. They must have different configurations, and therefore different energy contents and enthalpies; even though vibrating at the same speeds. In those established conditions, there is an equivalence in their two rates and amounts of energy and transformations, and as contained in their masses. This is a statement of the chemical reactions between substances and their surroundings. This is the same kind of interaction that occurs between the Weyl and the Ricci tensors as biological entities transform and reconfigure over time. That time is a movement in our fourth dimension.

The amount of heat energy, H, that we add to the ice to produce the change in energy density that transforms it into water—which is a change in the amount of energy surrendered to the surroundings at that temperature—is the heart of their chemical equivalence, albeit their only difference is in their energy content. Their molecules vibrate differently, while maintaining the same rate. That energy content difference is the heat or enthalpy of fusion, Hfusion, which we can measure using Black's hamiltonian method. It is a transformation under energy quantity but temperature identity. Ice and water contain different amounts of internal energy, by vibrating with different modes, all at the same mass and temperature, which is their rate of vibration.

Once the ice has completely converted to water, we can add yet more heat. There is a different effect. The ice is now saturated. If we add that more heat to the liquid water we procured by adding heat to the original ice, we persuade the water to increase its rate of vibrations. It now gets warmer. We can eventually carry the water right up to boiling point. This rise in temperature is a transformation of internal energy under continuing quantities of energy, but it is a very different response.

We can then keep adding yet more heat, and transform the liquid water into vapour. We are back to a transformation under energy quantity but temperature identity. This is the enthalpy of vapourization, Hvapourization, which we can again measure as a Black equivalence or hamiltonian. And since the temperature once more holds constant, we have another change in the energy density, and we get the similar energy equivalence TliquidSliquidTvapourSvapour … another transformation and reconfiguration under energy quantity. This is another change in modes of vibration, but all at the same rates and speeds.

The Gibbs triple point equilibrium is where diverse substances hold each other in equilibrium under certain given conditions. All three phases are present. Some molecules will therefore move directly from the solid to the gas phase, bypassing the liquid one altogether. The well-known “dry ice” of carbon dioxide, seen in the swirl of fog in discotheques, is an example. This gives the equivalence TsolidSsolidTvapourSvapour. This enthalpy of sublimation, Hsublimation, is always greater than that of fusion and vapourization. It is another form of transformation under energy quantity, but temperature identity.

The three reverse processes of removing heat from each phase are (1) freezing or solidification (liquid → solid), (2) condensation (vapour → liquid), and (3) deposition (vapour → solid) will decrease energy densities and quantities. They change entropies and configurations in the reverse directions, as the substances release the energies that they previously took on. We can measure those same energies of releasing, and they are the precise converse.

We are interested in molecular behaviour. We must quantify it in biological entities. As Black proved, a biological cycle is a similar sequence of reversing and reversible chemical reactions where energy absorbed in one stage is released in another, albeit at different rates. This is the source of all biological diversity in both space and time.

We can now measure the Gibbs energy under standard conditions for any chemical or biochemical reaction whatever, and then predict the energies those substances will either take on or give off when they undertake those same reactions, under those same conditions. We know their molecular potentials. The energy that molecular substances of any kind take on or give off in chemical reactions is now known as the ‘Gibbs energy’ after the Harvard scientist Josiah Willard Gibbs who made what Black first discovered far more rigorous, and who established chemical thermodynamics.

Gibbs proved that all possible chemical reactions—including these phase changes we are using to understand them—are caused by the different speeds, movements, arrays, and configurations of molecules, all of which can be measured through their energies. The challenge he met was to express such diversities as a potential, which is the Gibbs energy. Like all energy potentials, it is measured in joules, but in this case as joules per mole per kelvin.

Since “things that are equal to the same thing are also equal to each other”, then the various changes in energy density, as heat is added or removed, between the three phases at the Gibbs triple point are entirely responsible for all their changes in configuration and energy density. Since these form one system, we now have TiceSiceTliquidSliquidTvapourSvapour. But since each phase has both its escaping and its capturing tendency, then something measurable causes the molecules to flip from one to another and to change back and forth in energy. That measurable cause is the quantity of energy they take on and release as their internal energy changes.

Black revolutionized heat and energy studies by meticulously clarifying the difference between enthalpy and temperature, which he did by contrasting ice, water, and steam. His very careful work inspired his friend James Watt to invent the steam engine, which Carnot then analysed, eventually leading to Clausius, Maxwell, temperature, and entropy. In other words, if we hold both the volume, V, and number of particles in a system constant, N, which we can write as {V, N}, then any changes in a system's entropy, S, with changes in its internal energy, U, must now be due to its increased molecular motion. There is no other possibility. The net result is a rise in its temperature, T. This is how temperature is defined. It measures the rate at which a system changes in its internal energy, through its molecular motions, when its volume and its number of particles are considered as constant.

A mathematical aside

If we have a system's volume and numbers of molecules held constant as {V, N}, then temperature is defined as the inverse partial derivative T = 1/(∂S/∂U)V,N.

The Gibbs triple point equilibrium does not have any changes in temperature. All three phases hold themselves in a steady equilibrium. Therefore, something else must he happening to cause those observable changes in their internal energy. Since all three phases in that equilibrium are at the same temperature, then their heat powers or abilities to transform are all the same, and in spite of their diverse conditions. This is again TiceSiceTliquidSliquidTvapourSvapour. Since they all have the same intensity or heat power, they must therefore be expressing some other ability to create transformations. This can only be the quantities of energy they take on and give off. This is the Gibbs energy, G.

The Gibbs energy measures a substance's ability to change in its internal energy from one condition to another by instituting specific chemical reactions—which is changes in vibrational modes and in chemical bonding—when energy is introduced or removed. It is defined as the ability to cause changes in entropy and configuration, S, which is to cause chemical reactions at a given temperature and pressure, and in a given number of particles. It is therefore a constant volume and nonmechanical interaction. It is similar to temperature because it measures the ability to transform by causing diversities in behaviour … but it is distinct from temperature because it is solely concerned with changes in state, and changes in modes of vibration, rather than in rates of vibration. And … biology is all about balancing modes of vibration.

The Gibbs energy of formation for liquid water, H2O, which is its potential, is -237.14 kilojoules per mole; while that for carbon dioxide, CO2, is -394.39 kilojoules per mole, each in standard conditions. The negative sign means those two reactions are “endothermic” or energy demanding. The similar energies of formation for benzene, C6H6, and nitrogen dioxide, NO2, both in their gaseous states, are +129.7 and +51.3 kilojoules per mole respectively, both again in standard conditions, and both “exothermic” or energy releasing. These are, again, potentials in energy. It means they are energy changes we can confidently predict for any molecules held in those conditions. This includes within biological entities. If, for example, they use their internal energies to undertake any kind of water-creation transformations, we know exactly how much energy they have used.

The Gibbs energy is defined as GH - TS where H is the enthalpy or energy content, T is the temperature, and S is the entropy. This simply means that every specific chemical reaction either needs or releases a given quantity of energy. The equation is telling us that the three phases in the Gibbs triple point equilibrium differ only in their energy content. They differ in their ability to induce chemical reactions due to that energy content: due to its intensity at that temperature, which affects their configuration potential. The Gibbs energy measures configuration potential at that energy quantity and intensity, or enthalpy and temperature.

The Gibbs energy is the potential assigned to the components involved in any interaction, relative to each other. As with the above four examples of water, benzene, and carbon and nitrogen dioxides, the Gibbs energy is always a precise and known value in joules per mole, which is a specific number of molecules. It states the force or potential that those specified molecules carry as they interact with each other in those conditions. It is their power to transform … and so to carry biological entities about their circulation. Every one of their biochemical configurations has its energy, its entropy, its potential, and its power to transform at that given point in the cycle or circulation; and we can now use the information from the Gibbs triple point to account for all the nonmechanical work that biological entities engage in.

Now we have fully accounted for the nonmechanical and constant volume chemical energy and work that largely drives the Weyl tensor, we turn to the mechanical and constant pressure variety that biological entities and populations also engage in, so we can express those molecular behaviours, also, as a potential. This, obviously, largely drives the Ricci tensor.

The first person to realize that biological organisms are heat intensity pumps; that they follow the laws of heat; and that they cannot live without incurring a cost; was the German physician and physicist Julius Robert van Mayer who, in 1840, volunteered to be a ship's doctor on a tour to the Dutch East Indies … and then produced the entire energy concept from his observations.

Mayer observed that the blood of some of his patients was far lighter—a bright red—than it was when they were back in Germany. It was so bright that it was almost the same colour as arterial blood, and he initially thought that he had missed a vein and struck an artery when he had tried, as was the medical convention of the time, to let blood. He concluded that the higher temperatures in the East Indies meant that his patients needed to burn less food to survive in those conditions. This led him to the first ever version of the law of the conservation of energy, for since they were using less oxygen then some of it must be conserved, or not used, otherwise they could not return to their previous physiologies on their return to Germany. Those reactions would not be available to them under those changed conditions if they were not conserved. They must, therefore, always have had both a potential and a hamiltonian and remained equivalent.

Mayer also insisted that when biological entities interact with the environment, they have to either push the surroundings aside, or else draw it into themselves. This, he said, required more of the energy that he conjectured, and whose existence he proved, than did the simpler act of just using substances that had already been acquired. He had in other words separated the mechanical and the external from the nonmechanical and the internal.

From such simple observations Mayer did what no-one had done before him, and deduced the necessary existence of energy. His first paper, On the Quantitative and Qualitative Determination of Forces, 1841, even, and for the first time, calculated a value for the universal gas constant, R, and the mechanical equivalent of heat, J. The relation CP = R + CV links the constant pressure, CP, and constant volume, CV, specific heats of any substance to the universal gas constant, R, which states the number of molecules involved. This is called ‘Mayer's relation’ in his honour. He realized, long before anyone else, that the defining property of an ideal gas is that its internal energy is independent of its volume in an isothermal process; and that its enthalpy is also independent of its pressure. The corrected value he gave for R in 1845, when improved experimental data was available, is within 0.5% of today's value of R = 8.3143 joules per kelvin per mole. It states the energy involved in any molecular interaction of any type, at any temperature, biological or non-biological.

Mayer was the first to publish. He was also far more comprehensive in his vision of energy, and what it could do. But although he had by far the broader insight, he was not as accomplished a mathematician as the German physicist Hermann von Helmholtz who, working independently, had seen the implications of Clapeyron's discovery of PdV. Helmholtz might not have had such clarity of vision, but he could do something Mayer could not do, which was explain the mechanism. Helmholtz saw that a pressure pushing through an infinitesimal volume is the Carnot cycle's driving force. That driving force therefore achieved all energy effects.

Since Helmholtz saw the theoretical implications, he was coming to the same overall conclusions as Mayer, who was working more practically and experimentally. Helmholtz, therefore, did not initially have as comprehensive an insight as Mayer. Although Helmholtz ceded priority to Mayer, his explanations are the more widely accepted. It was Helmholtz who gave the now accepted explanation for the interaction between pressure and volume in mechanical work, and who gave clear definitions of internal energy, U, available work, A (taken from the German Arbheit), and other such concepts that Mayer struggled considerably harder to explain:

What do we understand by “Forces”? and how are different forces related to each other? Whereas the term matter implies the possession, by the object to which it is applied, of very definite properties, such as weight and extension; the term force conveys for the most part the idea of something unknown, unsearchable, and hypothetical. … Forces are causes: accordingly, we may in relation to them make full application of the principle—causa aequat effectum. If the cause c has the effect e, then c = e; if, in its turn, e is the cause of a second effect f, we have e = f … = c. In a chain of causes and effects, a term or a part of a term can never, as plainly appears from the nature of an equation, become equal to nothing. This first property of all causes we call their indestructibility. … Two classes of causes occur in nature, which, so far as experience goes, never pass one into another. The first class consists of such causes as possess the properties of weight and impenetrability; these are kinds of Matter: the other class is made up of causes which are wanting in the properties just mentioned, namely Forces, called also Imponderables, from the negative property that has been indicated. Forces are therefore indestructible, convertible imponderable objects (Mayer, 1841).

Julius Mayer. 1841. Remarks on the Forces of Nature. Quoted in Lehninger, A, Bioenergetics—the Molecular Basis of Biological Energy Transformations, 2nd. Ed.; The Benjamin/Cummings Publishing Company; London; 1971.

Mayer certainly expressed the idea of the conservation of energy: i.e. that in spite of all the changes all material stuff constantly goes through as energy passes through it, the energy itself remains constant. However, the appropriate potentials and transformations are now measured according to the much clearer definitions Helmholtz provided. Joule knew of Helmholtz, whose lead he followed, but he did not at that time know of Mayer.

The “Helmholtz energy” is a thermodynamic potential that measures that part of any system's energy that is capable of affecting the surroundings through doing external, materially sensible, mechanical work. This is to move masses within the surroundings in the constant pressure way that Mayer first described.

We can now begin unmasking the effect, in biology, of these Weyl and Ricci tensors. Let us for simplicity assume that all three phases in our Gibbs triple point equilibrium of Figure 20.14 each have the same internal energy, U. So the total energy is 3U.

We immediately know, from their different configurations, that the three phases in our system have different Gibbs energies. They have different heat contents and enthalpies, which is their energy densities. They have different internal effects and potentials. We are now concerned with the effects these have on the work done, and on the total amounts of energy that can cross the boundaries, from each phase, and so potentially affect the external world. They cannot have identical effects either on each other, or potentially on their surroundings.

If we attach a piston to the entire Gibbs triple point arrangement, it could certainly do some external mechanical work. We simply heat it so it goes through the changes of the Carnot cycle. It acts as a steam engine. However, since each of the three phases is at a different stage each will act very differently, which is the root of all diversity both biological and non-biological. They are the heart of the Weyl and Ricci tensors and—as in Figure 20.12—they are the causes of all differences between the internal and the external.

1. The vapour phase, as an example of internal energy, has already been fully converted for maximum external impact. It has the highest enthalpy or energy content, when seen as a combination of both the heat contained and the effect on the surroundings in terms of the energy molecules must exert to make space for themselves. The potential for extra mechanical work is given by the Helmholtz energy, which is defined as AU - TS. It tells us that the higher is the entropy, S, for a given quantity of internal energy, U, and temperature, T, then the less is the available work, A, because the less capable those configurations are of pushing out into the surroundings. The less, therefore, is the effect we will get by pushing further out into the surroundings with that internal energy. Since this phase is already a gas, it is already the most “externally oriented”. It is already doing work by moving a piston, or else has just done such work. It has the highest entropy, and so has the relative disadvantage of having the least options available to it for further pressure-volume transformations over the cycle. It is already where it needs to be, which is as vapour or steam. Enthalpy is given by H = U + PV, and this gas has both a higher entropy and a higher enthalpy because it has already taken on energy to spread its molecules out, which increases its volume … the process that would allow it to drive a rocket. Its high entropy, low pressure, and high volume therefore give it the lowest measure for available, and future, work out into the surroundings of the three. This vapour phase therefore has the the highest enthalpy, H, of the three; and the lowest ‘push-out-ability’ potential and Helmholtz energy, A.
2. The liquid water expression of internal energy has a lower enthalpy and a lower entropy than the vapour phase. Its molecules are more tightly bound than is the gas or water vapour, and it has a less open configuration. There is now the option to increase the injections of heat, and to take it through a phase transition, and so to increase the total amount of vapour at our disposal for running the engine. We can boil this water. This phase's higher inpulling internal pressure plus its lower consequent volume mean that it has both a lower entropy and a greater potential than has the vapour, and so more by way of pushing-out, constant pressure, available work in its future. Enthalpy indicates how far into the surroundings a given system has spread and, in this liquid state, the system is less spread out than is the vapour, meaning it has some further expanding in its future. The entropy in this liquid phase is lower than it is in the vapour one because there are fewer molecular collisions in each unit of time. The molecules are more tightly bound. The enthalpy is also lower and so there is, overall, less energy in the PV term, for the system has pushed itself less far out into the surroundings than has the vapour. It takes up less space and its internally directed pressure is greater. But since the entropy is lower, the volume lower, and the pressure higher, it is more ready—i.e. has more potential—than is the vapour to undertake some future transformations. This is its Helmholtz energy or available work. Therefore, less mechanical work has so far been done in and by this phase; but more is available to be done in its future available work, because the entropy is lower at this value for internal energy. Once again, the Helmholtz energy measures that potential for external mechanical work, which is greater in this phase. Out of the three phases, it has the middling values for both enthalpy, H, and Helmholtz energy, A.
3. The ice, as a further expression of internal energy, has the lowest enthalpy or heat content of these three phases. Its low entropy means that its molecules are extremely close together, having a low number of mutual collisions per unit time. It has pushed out the least. If the ice is to expand yet maintain its same temperature, then the molecules must transport their energies across greater distances, which requires heat energy injections. The ice is, however, currently confined to a small volume, V, by being so tightly configured. Its internal pressure, P, is high, but its volume,V , is low, thus giving us the H = U + PV of its low enthalpy value in this case. This phase has the lowest current energy content, but the most available work in its future, through being the least spread out, of the three, into the surroundings. This ice can transition through an extra phase and so go through a greater total number of pressure and volume transformations, which will together produce its increased amount of external mechanical work, although we will have to input the most heat energy to extricate that available work. It therefore registers the highest of all the Helmholtz energy potentials, because since A = U - TS, that lowest value for entropy means there is the least to subtract in respect of the configurations, and so from the internal energy. Out of the three phases the ice has the least value for its enthalpy, H, and the greatest value for its Helmholtz energy, A.

It is extremely important to appreciate that, like gravitational potential which must be stated in terms of height above a specific surface with a specific mass, the Gibbs and Helmholtz energies must always be measured with respect to some basis or reference. We can invariably provide that basis or reference through a tensor. We can then compare different populations, deriving accurate measures for each.

A mathematical aside

The Gibbs and the Helmholtz energies join with the constraints of constant propagation, constant size, and constant equivalence to be the principal forces acting upon the phase and configuration spaces that biological entities inhabit.

Now we are clear on these most important differences in internal energy, which are attributes of the molecules that compose biological organisms, we are at last in a position to use our tensors to see both (A) why a population free from evolution is impossible; and (B) why one with fitness, competition, and evolution is not just obligatory, but completely inevitable for them all. Darwinian evolution is, as we can now see, a simple consequence of our four dimensions in conjunction with the second law of thermodynamics.

#### The law of diversity

We now have everything we need to explain how the molecules that create the internal energy of our biological space can band together and be partitioned so it produces all possible biological diversity. We can derive another law of biology (our third) to cover it.

It is easier to think and work in physical space, and to adapt it as we need for the specifics of our biological space with its field of internal energy. We can therefore conceive of our biological space as being very much like physical space. Every point is defined with coordinates stretching in three dimensions. Objects are regions created by the Ricci tensor. They contain specified quantities of internal energy aggregated around their given point-centres. Biological space is defined, and objects are created, by the twirlings of our three batons and their accompanying curves and spheres from Figure 20.1.

We already know we want to either prove or disprove the link between evolution and diversity. We shall therefore begin exactly where we know we already want to end, which is with the ability to either prove or disprove it. We must therefore take a somewhat different approach from usual. We need a way to measure both longitudinally along the Weyl tensor, and transversally across the Ricci one … insofar as such notions make sense in four dimensions.

If we want to measure evolution, then we need some way of assessing when two biological entities are alike; and when two biological entities are different. We need a clear measure of how near to each other they might be. But that measure must also incorporate some means of portraying how far along in the circulation of the generations they each are. This has to involve their internal energy, which is whatever numbers and configurations of molecules define the sector of internal energy they currently reside in. But as we have seen with the Gibbs and the Helmholtz energies, this must always be done relative to something.

We are interested in the circulation of the generations: i.e. the reproductive cycle. This is a departure from, and then return to, a point in our internal energy field. We therefore go right back to where we first started, which was setting the scene, and we exercise a little bit of imagination.

We picture a traditional African hunter-gatherer community leaving a camp. The group will eventually return to that camp. The returning hunter-gatherers will want to find everything the same. No matter how far they might travel in either time or space, they want to return to the same point. They also want to find everything in their surroundings the same, so they can sustain themselves in the same way. We need some way to indicate this return to sameness.

‘Displacement’, d, has a property we need. It is a vector quantity that measures distance. It tells us the distance between two objects by using one as a basis or reference for the other. If, for example, an object is in circular motion when measured from a specific location, then its total distance travelled around the circumference can keep on increasing with each orbit travelled; but its current displacement from the centre, or any other point, first increases and then decreases in regular fashion. The circumference measures the distance travelled round and around, whereas the displacement measures the direct radius. That direct radius measure is the displacement away from that centre point, currently taken as the reference or basis.

If our hunter-gatherers go off trekking in, say, a series of large circles and eventually end up back at the same camp, then although they might have travelled a distance of 1,000 miles on their perambulations, their net displacement, when they return, is zero because they have ended up in the same place. While the 1,000 mile distance tells us how much ground the hunter-gatherers have covered in total during their trek, their displacement tells us how far ‘out of place’ they were, at any time, in terms of their change in position relative to their start point. So our hunter-gatherers could have travelled 1,000 miles and more in their trek, but might only ever had a maximum displacement of, say, 25 miles in the sense that the group was never more than that distance away from their initial camp. They just toured round and round the same basic region.

If four different trees are located five kilometres north, south, east and west of their base camp respectively, then they have four different displacements through being in four different directions, even though their distances are all the same at five kilometres. Displacement is distance with a direction. The four trees might all be equal in distance, but they are not equal in displacement. The north-south, east-west, and up-down pairings are opposites in displacements. They between them create the overall distances, areas, and volumes of space and internal energy through which the band of hunter-gatherers is wont to roam.

Linked with the concept of displacement is ‘placement’, p, which is the reciprocal of displacement: i.e. p = 1/d. Thus where displacement gives a measure of ‘farness’, or separation, placement instead gives a sense of ‘nearness’. Or in other words, where displacement measures absence, placement measures degree of presence, whether it is in physical space, or in degrees of internal energy. And since they are reciprocals, we always have pd = 1.

We should note that although placement and displacement are multiplicatively related as inverses, two populations with different circulation lengths will not interchange between them at the same rates. The seemingly trivial fact that pd = 1 must always be true, for all populations, but that they will exchange at different rates over different distances is going to be very important to us in the near future.

We also already know that every population will maintain its hamiltonian. We know that the hamiltonian is additive. We therefore have two different constants or invariants, one additive and the other multiplicative. We can use these to discuss any and all populations.

If an object is in circular motion and we measure it from a specific point upon the orbit, then its nearness keeps decreasing for the first part of its orbit as it circles away. It reaches its minimum nearness—or maximum farness—at the far end of the orbit. So when our hunter-gatherer group is 5 miles away from camp, its placement or nearness is measured as p = 1/5 per mile. This means that for every mile of placement measured between the group's current location and the original camp, we only produce 1/5th of the distance needed to get to the target. And when the group is 10 miles away, then its placement is 1/10th per mile, meaning that for every mile measured, only 1/10th usefully covers the distance towards the target. The nearness therefore keeps increasing as the object returns to its target or home position in the latter part of an orbit, but keeps decreasing in the first part when it is moving away. Placement therefore allows us to express the location of a specified object in terms of the separation, relative to another.

When we substitute the beginning of a generation for our base camp, placement becomes a useful concept for we have both (a) a basis; and (b) a measurement using that basis that is therefore relative to the beginning of a generation. We can say how far into its generation—or into its internal energy—any entity is.

Biological entities also go about their circulations of the generations at specific speeds. There is therefore another important aspect to placement and displacement, which is both time spent and potential rapidity. It is obviously useful to know how much time objects have spent away from base camp, and how rapidly things are moving relative to it at any time … especially if our point of origin is the beginning of a generation.

Velocity is a displacement changing over time. It is a specific movement, measured in a specific direction, and per each unit of time. It is also always the result of a transfer between a potential and a kinetic energy. There is, however, another property we are interested in.

We also want to know how long our group of hunger-gatherers has been away from base camp. In biological terms, we want to know how long it has been since a specific cohort of biological entities was born. We also want to know how much of it has expended, given that its hamiltonian is constant. We can use all this to tell us how far away it is from the point at which it is likely to start reproducing.

The little known scientific properties of “absement” and “presement”, respectively, measure this amount of time away from a given location. They are, in another sense, a measure of the ‘speeds’ at which farness and nearness are devoured across a given unit of time. Something might be travelling with a given velocity, but how long has it been travelling, overall? How long has it been away? And how much energy has it used?

A batsman in say a game of cricket, who has been playing for five hours might have the same score as one who has been batting for only fifty minutes, but they are not in the same condition. One has been out in the middle, batting, for far longer. They will still not be in the same condition if the first batsman, who has been in longer, suddenly ups his game so that the two score evenly from that point on. Absement adds this idea of how long a body has been in some given energy condition to that of the simple motion that velocity measures.

“Absement” is constructed from absence-displacement; while presement is constructed from presence-displacement. [We are grateful to Steve Mann—father of the wearable computer, and inventor of the hydraulaphone and much else besides—for clarification on these various terms]:

To understand presement/absement in terms of a metaphor from real life, consider absemence [sic] as when spending some time away from a loved one. With absement being a product of time and displacement, the more time, and the farther away from home, the more absement one feels. Borrowing from a common saying, we might say that “absement makes the heart grow fonder”. Being two hundred miles from home for one day, produces approximately the same feeling of absement as being one hundred miles from home for two days. As another example, consider a long-distance phone bill based on a time-distance product. Units of absement are the product of time and distance. Whereas velocity is measured in metres per second, absement is measured in “metre seconds”.

Steve Mann, Ryan Janzen, Mark Post, “Hydraulophone Design Considerations: Absement, Displacement, and Velocity-Sensitive Music Keyboard in which each key is a Water Jet”, Proceedings of the 2006 ACM International Conference on Multimedia (ACM MM), October 23-27, Santa Barbara, USA.

The absement that an object has at any point is its current displacement from our measurement location, multiplied by the time it has spent there. More correctly, it is the integral of displacement with respect to time: A = ∫ d dt. We can now measure how long any biological entity takes to reproduce, which is to return to the same state and internal energy condition as its progenitor.

We are looking for a way to describe variations, and so for a form of measurement we can use across the reproductive cycle, to state its equalities and differences. Absement is also linked with speed in the sense that when, for example, a piano key is played loudly, the pianist strikes it so that it displaces itself rapidly from its home location. The piano action transmits the actions to the hammers supported on its brackets. The hammer strikes the string and falls back with its escapement or double escapement. The key will very soon return to its home position. When the piano is played quietly, however, the pianist plays more lingeringly, and the key moves correspondingly more slowly. It will take longer to return.

Another way of understanding absement is to imagine two people who regularly walk together to their respective offices, always leaving their home together. Since our two people start at home, both their absements start at zero. Their absements each get a value as soon as they step out the door. They both have a displacement, relative to home. When they have been away from home for 1 second, and are also 1 metre away, then they have an absement of 1 metre-seconds. When they are two metres away, their rates of acquiring absement will increase to 2 metre-seconds for each second that passes, and so they will both accumulate absement at an increased rate in each unit of time.

Now suppose these two people have arrived where one of them works. The second person walks on, while the first enters the office. This first one now acquires absement at a steady rate through being stationary at the office, while the second continues to increase the rate at which it is acquired, by continuing to walk on. The amount of displacement in each unit of time keeps increasing because the displacement keeps increasing.

When the second person at last arrives at his or her office, he or she stops increasing the rate at which he or she is acquiring absement, and they both now take it on at continuous rates. However, the second person acquires more absement with every passing moment, through being further away and so having the greater displacement at every moment.

Both these people now acquire absement at their steady rates, the one greater than the other, until the end of the day when the journey home begins. The second person walks back to pick up the first. He or she continues to increase absement, but now at a diminishing rate. The first person, who is waiting, continues to acquire it at his or her slower but steady rate. When the two meet up, they again accumulate absement at the same rates, which now declines for both of them, at the same rate, as they walk home together. They continue to acquire absement because time keeps moving along; but the rate keeps diminishing because the displacement is diminishing. And when they re-enter their front door, absement no longer increases for either of them because their displacement returns to zero. They both now have a total; but they also have different values for their absement.

We can now state the total absement values for each of these two people in metre-seconds. And since the second person spent more time further away, and so held the greater displacements, he or she will have the greater value for absement, even though they spent the same amounts of time away from home. There is now something measurably different about them. They are exhibiting variations. We now have a way to measure important differences in internal energy.

Presement is the inverse of absement. So on the journey away from home to work, the presement or nearness value for the two above workers diminishes as soon as they step out of their door. When they are 1 metre away, their presement is 1 seconds per metre. If they keep walking away, then their placement continues to decrease at a steadily increasing rate, for the time increases as their nearness also decreases. When each gets to where he or she works, his or her presement maintains a steady increase in its value because the distance is no longer being incremented in each unit of time. Their presements then begin to reverse and to increase at a steadily increasing rate as they return home. When they re-enter their front door, presement no longer increases because their placement is again at a maximum and they have the maximum nearness. The one who went further away will now record the lesser value of nearness. That total value on their return is again their presement for their respective journey in seconds away per metres travelled (i.e. metres seconds-1).

We must also, of course, take care to account for the time and displacement spent increasing and decreasing the rate of acquiring absement, such as when moving from one metre away to two. The absements steadily increased all across that interval both because (a) displacements were increasing; and also (b) the time away from home was increasing. Therefore, absements are steadily increasing at a constantly increasing rate, as the two people continue on their journeys to work, and away from home; and then equally steadily decrease on the return journey.

Our two people acquired different amounts of absement and presement. Since they spent the same amount of time away, they must have done so at different rates. If something is one metre away from its home position; and if it moves so that it takes one second to return home from that away location; then we have 1 × 1 = 1 metre second seconds, or metre-seconds2, of “absity” as a way of devouring that displacement, constructed from absence-velocity. This is now equal in its absity or absence-displacement-acquisition speed to a second object that is two metres away, but that takes half a second to return, for that also gives an absity of 2 × ½ = 1 metre-seconds2. And on the principle that “things that are equal to the same thing are also equal to each other”, then these two are in their turn equal in their absity to a third object that is half a metre away, but that takes two seconds to return, as in ½ × 2 = 1 metre-seconds2. So even though they are all at different displacements; and even though the objects at the different displacements are moving at different velocities to return to where they are being measured; there is something common to them all. The velocity needed to return within a given time, which is their absity, is in direct proportion to their displacement or farness. Distance or absence from a location is thus linked to the time taken to get or return there, and with a given time to cover that distance, and so velocity. The further away something is the more absement it can acquire or consume, with a greater absity, just as a cell phone or mobile telecommunications device can also consume more power in poor reception areas by being further away. These are again measurable variations.

Just as displacement and absement have their converses in placement and presement, absity also has its converse. The above three objects all also enjoy the same “presity”, which measures the velocity with which nearness is being devoured, relative to some specific location (of course constructed from presence-velocity). Since the first object is the furthest away at 2 metres, it has the least amount of nearness of the three. However, if the object located there moves the fastest of the three, it will devour that complete nearness in only ½-a-second, which is with the greatest (absolute) velocity of the three (… but …not the greatest presity). The middle object has the middle amount of nearness at 1 metre, and the object located there can devour it by moving at the middle velocity. Its velocity might be lower, but it ends up with the same presity of 1 seconds2 per metre. The third object has the maximum amount of nearness at ½-metre, but since the object located there can appear to have the slowest velocity, it can take 2 seconds to devour that larger amount of nearness. It is devouring that largest amount of nearness the most slowly of the three, and therefore ends up with the same value for its presity, or nearness-devouring-propensity, as the others. We have therefore linked nearness to the rate of devouring it. So we can now not only measure and compare relative closenesses and separations, but also the times taken to attain and relinquish them. This is surely necessary when discussing evolution, which certainly invokes debates of time and activity.

Absity and presity measure any actual velocities with which farness and nearness are being devoured at any time, but relative to some specific location. An object can be moving with a given absolute velocity; but that need not be in the direction of, or relative to, the place or state we are actually interested in, and from which our measurements for absement and/or presement, absity and/or presity, are better taken for our given purposes. Each distinct state of location can have its own absities and presities relative to itself. An object, therefore, need not have the same absities or presities when measured from two different locations, and relative to those distinct locations; although it could easily have the same (absolute) velocity when compared to some standard held in common to both.

And since we have absity and presity, “abseleration” and “preseleration” measure any rates of change in absity and presity again relative to specific locations for measuring displacement and placement, respectively. And with these abilities to measure nearness, farness, and their various changes in hand, we shall surely soon unravel evolution for we can now measure biological entities relative to each other, to the surroundings, and to their reproductive point of origin.

A mathematical aside

Absement, also sometimes called “absition”, A, is, more technically, the -1th derivative of position with respect to time, or else the first integral of position with respect to time: A = ∫ d dt. It is the area under a displacement-time graph. And where absement can be regarded as the time interval of farness, presement, P, its reciprocal, is the time interval of nearness, with placement as the reciprocal of distance then being the measure of nearness. And just as velocity is the double integral of position with respect to time, so also is absity the double integral of displacement, or the integral of absement, and also the -2nd derivative of position with presity being the inverse; and with abseleration then being the triple integral of displacement, the integral of absity, and the -3rd derivative of position, all with respect to time, and preseleration again being reciprocals.

Reproduction is always a relative, and not an absolute, affair involving closeness and separation. Much like the Helmholtz energy, which must always be measured relative to some specific system and environment, every biological entity always reproduces in some specific environment involving relative closeness and separation to others. Reproduction always involves specific entities, that must then “measure” each other. They must measure their likenesses and unlikelihoods, as well as their surroundings; and all relative to themselves. That biological form of relativity is what we want to measure.

Darwin helped promote his theories by discussing the role that natural selection played in, for example, the evolution of dogs:

But man himself cannot express love and humility by external signs so plainly as does a dog, when with drooping ears, hanging lips, flexuous body, and wagging tail, he meets his beloved master … Nor can these movements in the dog be explained by acts of volition or necessary instincts, any more than the beaming eyes and smiling cheeks of a man when he meets an old friend. … With mankind some expressions, such as the bristling of the hair under the influence of extreme terror, or the uncovering of the teeth under that of furious rage, can hardly be understood, except on the belief that man once existed in a much lower and animal-like condition.

Charles Darwin, The expression of the emotions in man and animals, John Murray, London, 1872. (First edition).

The central issue is not how dogs—or horses, donkeys, or anything else—might look to us, as human observers. What matters is how they all look relative to each other, and how close or separate they are—from each others' perspectives—as they are each carried around their independent circulations of the generations. The diverse breeds of dogs are generally regarded as being reproductively accessible to each other, and thus as one breeding community. They are close enough to each other, all measuring each other relative to each other. But the possibilities of reproductive success become a little more dubious when we consider a chihuahua and a great dane, or a Pekingese and a St. Bernard, or any other such combination. Small breed dog females often have foetuses that are too large for them, and that cause considerable problems. Such puppies often cannot be delivered naturally, requiring c-sections. It is far simpler for a chihuahua or Pekingese male dog to impregnate a great dane or St. Bernard female, than the reverse. There are many dog breeds and other animals where a male can impregnate a female and be successful, but where the opposite fails. This is obviously a matter of quantifying this closeness and/or separation, and then expressing them relatively to each other … and in terms of our linear and polar planimeters.

We have to find some way of quantifying the configuration changes the Weyl and the Ricci tensors impose with their four-dimensional pressures and transformations. There is also the question of nearness and farness in space, time, and configurations. We know from our study of memes and genes in Before We Begin, and the ellipsoids we considered in Figure 0.5, that we must reckon various kinds of interactions. Each population's genetic and memetic space depends on how far it stretches in each dimension relative to all the others, forming the various areas and volumes. We must be able to judge the nearness and farness of say a mule from the horse and the donkey that produce it, as well as their nearness or farness from each other. We have prepared ourselves for this.

We must accommodate the realization that, for example, a mule, i.e. a male donkey–female horse coupling, is much easier to obtain than a hinny, which is a male horse–female donkey coupling. Our internal energy space must somehow reflect the biological fact that a horse has 64 chromosomes, a donkey 62, while both mules and hinnies have 63. This incompatibility prevents the chromosomes from pairing up correctly, and most mules and hinnies are therefore infertile. Indeed, except for one recorded case, all male mules seem to be infertile. A very few female mules, less than eighty, seem to have had reproductive success when mated with either a purebred horse or donkey. Virtually all hinnies, however, seem to be infertile because the donkey male now has the lower chromosome count as against the horse female, so making breeding more of a hit and miss affair. The reproductive system is inevitably incomplete. However, mules and hinnies are both extensively used in China, and both fertile female mules and hinnies seem to have been verified by chromosomal investigations. In each case involving a female hinny, the hinny had mated with a donkey—which has the lower chromosome count—to produce an apparently viable hybrid foal.

While we as human observers classify lions and tigers, and grizzly bears and polar bears by their farnesses from each other, and so as different species, they have enough nearness in each others' eyes to successfully mutually interbreed, and to produce viable offspring; and even though we as observers continue to believe that they are distinct species. The sole issue, however is their own reproductive accessibility relative to each other and their given surroundings and circumstances, and not what standards of farness we wish to impose from outside by lumping them as species.

The liger is a male lion–female tiger cross, while a tigon is male tiger–female lion one. The surroundings and environment become important, because these hybrids only exist in captivity. Lions and tigers live in noncontiguous habitats, and so do not meet naturally in the wild. Ligers, which are easier to produce, usually grow to be bigger than either of their parents, whereas tigons, bred from male tigers, tend to be about as large as the female tiger, and so larger than the female lion that bears them. Both ligers and tigons were originally thought to be infertile, but have recently been successfully mated, although not consistently.

Grizzly bears and polar bears are both being affected by global warming and habitat destruction. Those conditions are again important, because their habitats now intersect. Mixes have been produced both in the wild, and in captivity. The hybrid offspring go by a variety of names. As with ligers and tigons, a convention is growing that the male name should come first. The offspring have been variously called pizzlies, grolars, polizzlies and grizolars. The Inuit names nanuk, for the polar bear, and aklak for the grizzly have also been used to form nanulak for the male polar–female grizzly mix, and aknuk for the male grizzly–female polar one. Grizzlies and polar bears consider each other to have enough nearness to breed, although we as humans tend to think in terms of their farness as species.

The problem in all these hybrid reproductions is measuring the nearnesses and farnesses. We seem, however, to have found a system using our three dimensions with its distances, areas, and volumes. We are very nearly ready to do exactly this.

Every entity and population contains a stock of internal energy, and is therefore a region in our biological space. Since reproduction is a relative affair of accessibility and of given environmental conditions, we now establish, as a general point of principle, that two populations cannot be reproductively accessible to each other unless their generation lengths, T, and their temporal distributions, τ, allow them to overlap at each point in (A) their numbers and number densities, n; (B) their average individual masses and therefore genes, ; and (C) their chemical types and rates of activity, ; which is in both their mechanical and their nonmechanical chemical energies that measures their nearness. They must have similar values for absement and presement, for absity and presity, and for abseleration and preseleration. And that is now what we set about to prove using these concepts of nearness and farness, which are the integrals of displacement with respect to time (i.e. ∫ d dt). We place them in our tensor.

Figure 20.17

We can see an exercise in nearness, farness, and reproduction in Figure 20.17, which is a tensor that represents a study in the population dynamics of the snowshoe hare, Lepus americanus; its immediate predator, the Canadian lynx, Lynx canadensis; and the snowshoe hare's varied sources of vegetation which L. B. Keith, the researcher, referred to as ‘woody browse’ (Begon and Mortimer, 1986; Smith, 1986). Each of the three populations views the others relatively from its own perspective, and from within its own surroundings, as each still tries to reproduce and to recreate its internal energy.

The biological-ecological backstory for these mutual measurements is that the snowshoe hare inhabits a core, foundation woodland area. Since it is relatively protected there, it gradually increases its consumption of woody browse. Its sizes and population numbers increase. The rising population pressure forces succeeding members into increasingly sparser areas where there is not only less vegetation for them to eat, but where the lynx find them much easier to hunt. The lynx now flourish. The hares that are now forced to live, by peer population pressure, outside the favourable woodland areas suffer two-fold as the browse is also gradually over-utilized. As the lynx predation increases, the snowshoe hare population begins to decline, which in its turn begins to stress the lynx. The woody browse eventually collapses from its over-utilization, and this combines with the increased lynx predation to force a collapse in the snowshoe hare population. As it retreats back to its core areas, which has maintained its relatively high-density lush vegetation, the lynx population suffers yet further because that high-vegetation area is relatively immune from their predations, so making hunting harder. The Lepus americanus retreat now gives the woody browse time to recover … whereupon the snowshoe hare sets off on another increase, and the approximately ten-year cycle repeats. This is a very common set of biological-ecological interactions that we can use to prove our case.

Each population first takes it and its internal energy, at some specific time point, as a basis for making measurements. It is then going to measure both itself and the other two all around their joint ten year cycle to reconstruct the same internal energies. We thus have the beginning and end points t-1 and t1 at either end of the generation. The points of measurement are always the present moment, t0. We therefore range from t-1 to t1, right across each t0.

• Each component in Figure 20.17 has a population icon facing leftwards. This represents each population being measured. We can also think of it as time t-1.
• Each component also has an icon facing rightwards. This represents each one doing the measuring. We can think of it as time t1.

We will adopt the convention that the population in the left icon at each position in the tensor has the basis, and does the measuring, while the one on the right is the one currently being measured. Each population uses itself as its basis to take its measurements of placement and displacement, absement and presement, across some interval. Each therefore reports all values for internal energy, both of itself and of others, in its own units. Each population therefore measures both itself and the other two all across their respective circulations, all in both absolute and relative terms. The three diagonal positions match the normal pressures of xx, yy, and zz from the Cauchy tensor. They are where each population measures itself across some interval, and so all around the generations. But although we adopt that left-to-right convention, we must be able to reverse it at any time and adopt the opposite convention, and as tensors demand.

Whichever way round we measure, each population uses itself and its internal energy as its own basis. Each compares entities Einitial and Efinal at whatever time to determine any changes in state, and in internal energy, by taking relative measures. A first dog, for example, can measure the mass of a second by referring that other dog to its own mass to establish a scale. It then has a sense of the Biot-Savart laws as at work in them both at any time t0 between t-1 and t1, and so at either end of a generation. It has a sense of the kinds of curvatures they each have; their relative abselerations and preselerations; and so knows how to proceed biologically and ecologically relative to that other. We then get the story of a generation within those conditions with our values for absement, presement, and the rest, and slowly build up the picture of their joint interaction.

Figure 20.18

We do not know how any biological entity or population measures or assesses any other, but a combination of both quantities and rates in the three dimensions in biological space is clearly important. If a lynx is going to hunt a snowshoe hare, it has a rough idea how much smaller than itself the hare needs to be; and also knows how fast it needs to run. All such activities are combinations of quantities and rates.

Since every population and interaction must involve quantities, rates, and/or some combination of both, we can therefore suggest to any population that it uses one or more of the three systems of measuring shown in Figure 20.18. They are methods for fixing quantities and rates, which are locations and regions in internal energy. Our tensors can cope with them all.

1. Figure 20.18.A uses a standard rectilinear coordinate system to measure internal energy, and fixes every position in biological space with three separate quantities relative to the basis. This is like the x, y, and z of physical space. Since we have three dimensions, this is a one-to-one correspondence between quantities and dimensions in our space.
2. Figure 20.18.B shows a more cylindrical system. The relative measures between any two dimensions, such as we see on the lower disc, fix an angle or distribution which also sweeps out the generation. We thus know where we are within, and relative to, a fixed point in the generation. One quantity fixes both (a) the angle to rotate around the disc relative to the other, and (b) the exact distance to be travelled in that direction. This is again the Biot-Savart law at work. It is an absement. We then have an exact amount for both quantities, even though one was stated proportionately, and relative to the other. A second quantity then moves upwards into the third dimension, giving the effective distance between the two proposed discs or planes of measures and distributions. Each distinct point in our configuration or phase space is now pinpointed with two quantities and a rate, with that rate being effectively a curvature. There is now only a one-to-one correspondence between two of the quantities and dimensions. This is a combination of two quantities and a rate.
3. Figure 20.18.C shows a spherical system. This uses two angles, or rates, or distributions, to measure pertinent transformations all about the generation. These cover the entire universe of biological possibilities. They do not, however, fix any quantities. We must therefore have one quantity that fixes the distance outwards at whatever angles. But that one quantity still participates in both the others to create their distributions, and its own. So these two rates or curvatures and one quantity similarly pin down all specific locations in biological space. But we now have only one one-to-one correspondence between one quantity and one dimension. The other two are stated proportionately, as rates. This is again a combination of quantities and rates.

Now we have our systems of measure and can fix any point in biological space, we have to focus on the things we would like to measure. It is intuitively obvious that all biological populations seek for some form of equilibrium, with that equilibrium being some combination of quantities and rates.

It may be obvious that biological entities and populations seek for equilibrium, but biologists are unfortunately completely at odds in trying to describe what that equilibrium is; how it is attained; how it is maintained; and almost everything else about it. Most agree that biological populations show some form of environmentally-based self-regulation within an interacting community, all of whose members contend with the same environment. Most of the parameters involved change over time, but there is nevertheless a self-sustaining state around which deviations occur, so that there is no net change in certain important variables. Those important variables fall into three large groups which give three general categories of equilibrium.

Nothing could make our difficulties in explaining evolution clearer than the well-known, and well-accepted, ‘Hardy-Weinberg equilibrium’. It tries to handle diversity by focusing on the genes and the biochemistry that must be an intrinsic part of any biological internal energy.

The Hardy-Weinberg equilibrium assumes a large population; random mating; and no disturbing factors. It claims that if matings are based on the chance meetings of alleles and genes, which are evenly and thinly spread over the population, then the frequencies and percentages of dominant and recessive alleles and their genotypes will remain constant. It therefore assesses genetic variation, and states that populations will maintain an overall and constant genetic state.

A first difficulty with the Hardy-Weinberg equilibrium is biochemical and genetic. Mutations can introduce new alleles at any time. These immediately disrupt the equilibrium. Natural selection and nonrandom mating also disrupt gene frequencies by helping or hindering the reproductive success of specific organisms. Another problem with the Hardy-Weinberg law is that it also requires that all members can breed, for otherwise, of course, the genes of barren members leak out of the population and disturb the equilibrium. Genetic drift, particularly in small populations, is another factor that can change allele frequencies away from the Hardy-Weinberg formulation, which asks that populations be infinite in size. Gene flows that transfer new alleles into previously distinct breeding populations through geographic migrations or other factors can also be disruptive. There are other factors, all of which occur freely in nature, and all of which can cause disruptions. So although the Hardy-Weinberg formula can tell us the frequencies of alleles, such as the rate of albinism in a given human population, it does not account for changes in absolute population sizes, or in the distinct characteristics and overall masses of the individuals or populations. It is therefore an idealized state that is rarely realized, but that nevertheless gives a very useful base line that allows genetic variations to be measured. From our perspective, this one focuses mostly on rates. It focuses mostly on configurations of molecules as various alleles and genes, and therefore on the Gibbs energy. While it does highlight the first of the three important classes of variables and equilibria, it does not help us establish quantities.

The Lotka-Volterra is another well-known model. It highlights a second important class of variables. It tries to discuss predator-prey interactions in general, but within an equilibrium context. It produces a differential equation incorporating the density of prey, the density of their predators, a predation rate coefficient, an intrinsic rate of prey population increase, a predator mortality rate, and a reproduction rate for the predators per prey eaten. It is therefore also rate-based.

The Lotka-Volterra model also has its problems. It is not, for example, strictly an equilibrium model, because it shows predators and prey cycling endlessly. Unless other information is introduced into the model, they never reach a stable point. They oscillate constantly about what would otherwise appear to be a stable point, but one that the model cannot confirm.

The Lotka-Volterra model is an example of the more general ‘Kolmogorov models’ which the Russian mathematician Andrey Kolmogorov developed out of his dissatisfaction with the Lotka-Volterra. Kolmogorov models are more sophisticated than Lotka-Volterra ones, and try, more generally, to model competition, disease, and other ecological factors. They work by treating population flows as large eddies, which are then considered intrinsically unstable. Population flows transfer energy to smaller eddies which in their turn break up and transfer to yet smaller eddies in an energy cascade. This continues until an eddy motion results that is sufficiently small to be without turbulences and so stable. Molecular criteria then kick in, and a molecular viscosity associated with the eddy dissipates the molecular kinetic energy, and remaining turbulences, as heat.

Amongst their other difficulties, Kolmogorov models are not valid in small populations. Those remain too consistently close to their boundaries in the phase space. The models also have difficulty explaining how large eddies form; why they should be regarded as intrinsically unstable; and the scales and stages at which the dissipations are supposed to occur. But from our perspective, this one focuses mostly on rates and on mass fluxes, as the molecules enshrined in the entities seek their equilibria relative to each other. It is an example of the second broad form of equilibrium.

Population biology uses another quite common equilibrium that focuses upon life cycles. It calculates an equilibrium age distribution population using a BIDE model involving matrix algebra. The BIDE stands for number of Births within the population over a given time interval, the numbers Immigrating, the number of Deaths, and the number Emigrating. When the population of interest is at its equilibrium, then its intrinsic rate of natural increase is zero so that it is neither increasing nor decreasing in numbers. The age distribution is stable, with the numbers leaving each age class being exactly replaced by those arriving at that same age class. All of those dying, at all ages, are being exactly replaced by those being born so there is no net increase or decrease in population numbers. The numbers may now be balanced, but the equilibrium says nothing about genes or individual entity sizes and capabilities. These equilibrium age distribution populations, using the BIDE model, therefore have the advantage of incorporating numbers, and rates of change of numbers, but the disadvantage of ignoring both masses and chemical and genetic conditions.

There are numerous other models and equilibria. As a general principle, they all work the same way, although none quantify involved parameters adequately.

All the equilibria struggle with the concept of an interacting natural community that exhibits some form of unspecified equilibrium. The fundamental idea is that an initial state exists. It has some property of either nearness or farness. Either that initial state is not at equilibrium, or some force acts to move the system sufficiently far away to elicit a response. The population or system seeks to return. The equilibrium values becomes increasingly easy to discern as the observation time tends to infinity. Their unifying idea is that once at equilibrium, a population can recreate itself indefinitely, without any overall changes in numbers or essential characteristics, and over increasingly long periods of time.

We take a somewhat different approach to biological diversity. We focus on nearness and farness, and their measures and movements in the internal energy of our biological space.

As an example of the possibilities for farness and nearness measures, the comet-chasing probe Rosetta was at one time on earth. It was near to all of us. Launched in 2004, it chased after the comet 67P/Churyumov–Gerasimenko, eventually having a farness, relative to all of us still here on earth, that put it somewhere just past the orbit of Jupiter some 500 million miles, 800 million kilometres, away from the sun. But it did at one time share the same “inertial frame of reference” as we whom it left behind back here on earth.

Although Rosetta is now far away from us all, any human on board it would still be sharing an inertial frame of reference with it. He or she would have gone through the same diverse accelerations relative to us, here. But both he or she and Rosetta would still be linked to all of us, still here on earth, through those movements in space and time. We could travel backwards from Rosetta, throughout its entire journey, all the way back here to its origin in space and time to this earthbound inertial frame.

As another example, when a car is driven skillfully we, as passengers, accelerate and decelerate smoothly with it, and there are no “jerks”. When it is driven badly, we suffer many jerks, judders, and jolts. The ideal in the elevator industry is to build elevators that are completely smooth so that passengers do not even know they have accelerated and decelerated at the beginnings and ends of their journeys: i.e. without jerks. So a passenger sitting in a car that is accelerating with zero jerk will feel a constant force. An increase in acceleration is a positive jerk, while a decrease in acceleration is a negative one. Both those jerks cause changes in force in their respective directions. A change in acceleration always requires a change in force, which is a jerk. And if a ball is dropped from high up in the atmosphere, then all the time it is falling freely through space, and undertaking a constant acceleration, it is free from jerks. If we suddenly strike it at some point then the change in force makes it suddenly change in its acceleration, which is a jerk or jolt. All ball games depend on the successful exploitation of jerks and jolts. A jerk, therefore, is the scientific name for a change in an acceleration. If we want to replicate journeys in the style of Rosetta, but within biology, then we need to examine jerks, along with abselerations and preselerations. Those jerks are changes in force, and create and produce changes in internal energy. The most usual source for a jerk is some event in the surroundings.

A mathematical aside

More technically, jerk is the derivative of acceleration with respect to time: da/dt. It is the second derivative of velocity with respect to time, d2v/dt2; or the third derivative of displacement again with respect to time, d3d/dt3. Newton notated it as ȧ.

Jerks are the source of diversity in initially comparable frames of reference. If we follow Newton's convention, we can write it as  ̇ḋ ̇. We can now measure the smoothness of any given trajectory between times t-1 and t1 with the integral of jerk over that time span.

We can, of course, link jerk to placement/displacement, d. It is the fourth integral of displacement, and the integral of abseleration, and so the -4th derivative of position with respect to time. It tells us the smoothness or lack of smoothness with which a first object changes its state of farness relative to another, over time. This is called its “abserk”. Its complement of the rate of change in an acceleration towards nearness also immediately exists and is called “preserk”.

We shall now build a biological inertial frame of reference. It will allow for the same kinds of journeys in the internal energy of biological space that Rosetta and other celestial objects take in physical space. The frame will allow us to take measurements directly along; across; and into the Weyl and Ricci tensors. In particular, a backwards journey in our biological inertia frame of reference will be a journey into an organism or population's evolutionary past.

Just as Einstein took it as literally true that all physical laws take the same form in all inertial frames, so are we going to make it literally true that our four laws of biology (with two still waiting to be stated), our four maxims of ecology, and our three constraints of constant propagation, constant size, and constant equivalence are all literally true in all possible biological inertial frames, which is for all forms of biological internal energy.

Now we have a target, we must first be clear what an inertial frame of reference is. It hangs on Newton's first law: “Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon”. All inertial frames are in constant and rectilinear motion with respect to one another. They are not accelerating, again with respect to each other. So there was a time when Rosetta was not accelerating with respect to us here on earth. It was going around the sun at the same pace as us; it was in the same relative locations; and we were each enjoying the same accelerations as it. But that ceased to be true at its lift-off. That is when it suffered a jerk and started to accelerate away from us, giving itself different values. A jerk allows a first object to accelerate away from another. We can then define evolution as some kind of departing acceleration: a jerk or jolt in biological space that begins to separate one set of entities from another by giving them motions relative to each other.

Non-inertial reference frames are somewhat different from inertial ones, because they are accelerating. They contain a jerk—and so an abserk and a preserk—relative to whichever frame is observing them. Einstein's theory of special relativity refers to non-accelerating frames as ‘special’, precisely because they are not jerking or accelerating relative to the observer.

Einstein's general theory is different from his special one. It applies to all frames and viewpoints generally, whether they are jerking and accelerating or not. In the same kind of way, we want to take a measurement such as the proportionate rate of increase, dm̅/m̅, in say a Lepus americanus frame, and then find the equivalent value in the Lynx canadensis one to see what that species is up to. We then want to take some other measurement in the L. canadensis frame, go back, and find its equivalent in the L. americanus one. We especially want to compare one generation of a species to another, and determine if they are moving smoothly relative to each other, which is to keep the same values for internal energy.

We now need a general idea for what a biological form of an inertial frame of reference might be. Nobody yet has defined “species”, but we understand it here to be a group of organisms that have a shared set of values: i.e. without jerks, abserks and preserks relative to each other in their internal energies and its forces.

We now say that any biological entities sharing a biological inertial frame of reference do not evolve with respect to each other because they create the frame for each other through their joint biological activities, especially including reproduction. These internal energies are equivalent. As Newton's first law requires, they are all stationary with respect to each other. Evolution then becomes the question of what constant motion might be within this space; what sustains it; and what form jerks or relative accelerations might take.

Before we can discuss what constant motion in biological space is, we have to determine what a ‘point’ in that space is, for all motion occurs relative to given points and locations. Along with ‘line’, ‘plane’, and ‘set’, ‘point’ is one of the undetermined terms in any space or geometry, that each geometry is free to define. If biological entities are going to evolve, then they must accelerate away not just from others, but from points occupied by others.

We can define the points in our biological space that biological entities occupy by understanding that it brings together the two concepts of presence and absence. We always want to know how far along any Weyl or Ricci tensor we are, relative to its beginning, or other known point, so we know how far into the circulation we are.

Our biological inertial frame of reference with its internal energy must abide by the five conditions necessary for all metric spaces. The distance, d, between two arbitrary points q and r must always satisfy:

1. the identity condition: if q = r, then d(q, r) = 0;
2. the symmetry condition: d(q, r) = d(r, q);
3. the separation (or non-negative) condition: d(q, r) ≥ 0;
4. the triangle inequality condition: d(q, r) + d(r, s) ≥ d(q, s); and
5. the identity of indiscernibles condition: if d(q, r) = 0 then q = r.

All five are needed for a metric space … but … we have a particular interest in the identity of indiscernibles condition, which is not quite the same as the simple identity condition: i.e. the identity of indiscernibles is not the same as the indiscernibility of identicals. The identity of indiscernibles is the way to unlock evolution. We simply mean by this that if two entities occupy the same relative point in our biological space, then they are indiscernible.

Figure 20.19

We can see the consequences of the identity of indiscernibles condition in the right helicoid in Figure 20.19. As we follow the arrow of time and move upwards in generation time, every point q on a lower circulation has an indiscernible point r above it; and then another, s, above that; and so on. Points q, r, s and the rest directly above it differ only by pitch, which is the amount ρ. There is an infinite number of other indiscernible points both below q and above s.

If we look at the right helicoid sideways on, as at bottom right in the Figure, we will only see a circle of given radius. We see q, r and s indiscernibly coincident upon each other, with q = r = s, thus satisfying the identity condition.

If one generation is to be indiscernible from another, then the values for number of partitions in internal energy, and the values for the mechanical and the nonmechanical aspects of internal energy, must each remain constant from generation to generation. The values for the helicoid's pitch, radius, and medium or thickness must remain invariant.

Every point in our biological space is a point and a value for internal energy. It has two important aspects. Those aspects start introducing a sense of direction in this space. That direction is the force in internal energy that helps complete the circulation.

When we look at ordinary physical space, all objects exist at some distance from some mass that has created the gravitational field that surrounds them, and in which they are inserted. We look to the internal energy of biological space to create the same kinds of relationships.

A first aspect in biological space is the biological nearness or presence that one indiscernible point has relative to another. Where a metric space simply asserts a notion of distance between any two points that satisfy the requirements, a “vector space” states the pre-defined way in which the points must interact relative to some frame. That interaction then has a magnitude established by the points; as well as a direction determined by wherever the frame is defined. Those definitions declare the force that completes an orbit … or, in our case, a circulation of the generations.

Just as every point in physical space experiences some measure of gravitational attraction, from all other points and objects, so does every point in our space have some sphere of influence in internal energy. That sphere of influence is determined by both distance and direction. When a given point is taken as a reference or zero point, it defines all locations accessible to it, and that are a set distance from it. Since q and r are indiscernible, then every point a set distance from q is also some set distance from r, and relative to their indiscernibility … and so that when q and r are brought together, the net result is the identity of the indiscernible point. Every point thus has its range of influence immediately before and after it. That range helps it to institute its indiscernible properties both for itself, and for all others. This is its unit sphere of points. They are mutually indiscernible. What we define for any point p, we also define for all the qs, rs, ss and others that are indiscernible from it.

Just as every mass in physical space has a reach for the gravitational influence it can exert, so does every point in biological space have its similar “size”, which is its unit sphere of reach. The solar system may have no boundaries, but it has a size of influence which is defined by its force. Voyager I recently breached its heliosphere. A biological point's size depends upon the internal energy or medium used to define it in terms of its pitch, radius, and thickness.

The second part of a point, in this geometry, is its absence or remove from a point, measured by its displacement. This combines with the time to produce its absement. The helicoid of internal energy always looks like a screw being threaded, and so as if there is a constant upward or downward slope.

Position is measured by (a) the distance from the axis, and (b) by the angle, θ, through which we have rotated. But there is also the complement angle, 1 - θ, through which we must further rotate to get back to the next indiscernible point. These together measure how far away one indiscernible point is from another poloidally, meridionally, and toroidally on the helicoid. This is the difference in internal energy, and so in the behaviour of molecules which are also forces. Molecules of internal energy are always measured by how they are configured at that time, which includes how they must act in order to return to or arrive at some given state.

Since we have both position and absence, then there are always two intersecting ways of measuring time in this spatiotemporal geometry of biology which creates and recreates indiscernible points. They arise from poloidal movements vertically in time, meridional ones horizontally in space, with the toroidal being a mixture of both about the helicoid.

Figure 20.20

The tensor we see in Figure 20.20 brings these different aspects together. It shows both (a) reproduction along the rows, and (b) the substitution of one indiscernible for another up the columns. It has the two measures of time: the poloidal, which reflects the arrow of time; and the toroidal, which is about the circulation. One measure for time, the toroidal, moves constantly upwards through Columns I, II, and III. It is a form of energy density. It is the journey from indiscernible point to indiscernible point. It is therefore a circulation and an orbit, but which is also a stationary state. The three columns that create this structure are the three constraints of constant propagation, size, and equivalence:

1. The constraint of constant propagation has a rate of P’ joules per generation upwards through this column. It can easily handle all number-based BIDE equilibrium distribution populations as its set of indiscernible objects which is how the molecules that produce biological internal energy are partitioned.
2. The same quantity of biological energy also drives the mechanical chemical energy as the constraint of constant size over the generation. It states the constant pressure and external interactions with the surroundings. This column can handle Lotka-Volterra or Kolmogorov style equilibria as its set of indiscernible objects.
3. The same internal energy also has a nonmechanical aspect which is the constraint of constant equivalence over the generation. It tells us the constant volume internal interactions over the generation, which is how the molecules and partitions are configured. This one can handle the Hardy-Weinberg equilibrium as its set of indiscernible objects.

These three constraints always express themselves as the distribution over the entire generation length as τ, and so as the biological potential, μ. All measurements upwards along the columns are therefore per the generation, which is the time period t-1t1 in which one set of indiscernible objects is manufactured and then substituted for another. That is a rule in this geometry.

The indiscernible points and structures must not just express themselves vertically, and so over a generation. They must also express themselves horizontally in time. The tensor tells us, through the rows, that the same internal energy moving upwards from indiscernible point to indiscernible point also has an impact, or momentum, within the surroundings. We can simultaneously measure it in clock time. We can measure it in absolute temporal units at a sequence of times. So as well as by the generation, we also always measure, in this geometry, in absolute time units. Both are required. Every population must always state both its absolute clock time, and its current location in its generation. That is a rule in this geometry.

Our second measure for time, at each present moment t0, therefore moves constantly along Rows I, II and III as a material, rather than largely energy, expression. The three rows that express themselves in time are energetic and material fluxes:

1. The biological energy expresses itself in the top row against a clock, stating a definite average number density of observable biological entities as partitionings of internal energy maintained per second. This is always at some rate and over some interval. Since it interacts with a column, any present moment t0 is always at some point in a circulation and generation, and so is always between some t-1 and t1.
2. That same energy density is also apportioned, in the second row, into the mechanical chemical energy of the Mendel flux. It has a population mass density of M kilogrammes of chemical components maintained per second. Since it also interacts with a column, it is always at some point in a circulation and generation.
3. And, finally, the internal energy is also apportioned as an energy density which is the Wallace pressure or energy flux measured in watts per kilogramme, again over the same interval. The same proviso about columns holds.

While we always measure along the rows in real clock time, we must always measure up the columns as proportions of a generation. That is an expression for nearness or farness from one end of a generation or another; along with a rate for approaching or departing them. The rows and columns taken together are always some distance along the helicoid of internal energy.

A mathematical aside

We can make good use of the general principle that (a) time integrals of position measure farness, while (b) time integrals of the reciprocal of position measure nearness; and that (c) time derivatives of position measure swiftness; while (d) time derivatives of momentum measure forcefulness. These hold good for both absolute linear clock time and relative biological generation time. So our mass flux is measuring the effects of the forcefulness contained in mechanical chemical energy; while the energy flux is measuring the simultaneous power inherent in the nonmechanical variety. The time-integral of instantaneous power as applied along a trajectory of the point of application produces the total work done and energy used on that path, which is the essence of Green's theorem.

The constraint of constant propagation is one way to find indiscernible points. It describes a circle or orbit. As we learned in Before We Begin, it does so by always satisfying the description  T0 dP = 0. This simply means that if we go all around the generation, the infinitesimal increases and decreases sum to zero. We will end up right back where started.

The balancing and coming together of all the infinitesimal changes a population can go through means that every population influences, and is influenced by, a definite time period. That time period is a generation. Over that time period, the energy flux, P, first increases, absolutely, and then decreases, between limits, and so that the sum of all those changes is zero to create an indiscernible point. There is then a definite average of P’ belonging to that point, and which must be satisfied over the reach of time, T, all about, and relative to, that point. That is its sphere of influence all around it. It is where its placement and displacement, its presement and absement combine. That is its unit sphere as a point … while simultaneously summarizing the rows stretching across it.

Our biological space composed of internal energy must incorporate all the properties we need. We must preserve both the hamiltonians in energy which are additive, and the absements and presements which are multiplicative. A movement in one will ultimately induce its complement and a restoration.

Figure 20.21

As in Figure 20.21, the rows and the columns in this geometry cannot be separated from each other. They are both involved in preserving both invariants: the hamiltonian additive and the displacement-placement multiplicative.

Every temporal expression along a row is a manifestation of some indiscernible point-structure that stretches across a generation. Every indiscernible point must reach out to another like itself across some span of clock time.

Just as the Weyl and Ricci tensors interpenetrate, every point stretching upwards as a column also sits in a well or vortex constructed for it by the rows stretching across it, with the same holding for the rows relative to the columns. Every point therefore has its balancing cascade of energy, which is its set of increases and decreases to and from itself. That cascade forms a unit sphere of rows and columns communicating with each other that defines the points around it. This helps it satisfy the condition d(p, q) = 0; and so that p = q are indiscernible. That is a generation and a return to a biological point. If we do not measure a balancing cascade as an orbit, then the two points p and q we are comparing are not equilibrium ones. There is a net force in some direction, and towards the one or the other. That is a rule in this geometry.

The same conditions hold for the constraints of constant size and equivalence: T0 dM = 0, and T0 dW = 0, respectively. Those two work with  T0 dP = 0 to help define the indiscernible point that is a generation. These again mean that no matter what happens in each moment, the sums of all these various increments and decrements is zero all about a generation.

A point in this geometry is a collection of internal energy that establishes a circulation about itself. A point in this geometry is a constant attractor for the mass, energy, and number that establishes the internal energy for a generation. Each point sustains a very definite biological dance all the way about it. Each has these very strict properties we have stated. We can always determine them by measuring. All such cascades are at rest relative to the point from which they are measured. There are always both the lines and the circles of Figure 20.21. Every point that is not zero tends to some point, close by, that is, for it remains indiscernible. That is the medium of propagation and the definition of a point.

Now we have elucidated the properties of a single point in our biological space, we can turn to the properties of a line. A line is the reach from one indiscernible point to another. It generates an orbit. It preserves both the additive and multiplicative aspects. These are the required and allowed sets for molecules and their configurations.

Figure 20.22

To understand a line we must first contemplate Figure 20.22, which shows the pre-established way in which points interact from one indiscernible one amongst them to another. They thereby create their notions of magnitude and direction relative to each other. (The lines are shown straight for ease of comprehension, but they are actually curved around our helices and helicoid, potentially completing circulations).

Absolute clock time, t, is generated and imposed by the materials and energies in the surrounding physical universe. Biological entities try to maintain themselves and their populations against its incessant activities. That externally imposed incessant march causes displacements in biological internal energy. Therefore: no point can exist, in this geometry, without its set of placements and displacements. No point can exist without a sum of efforts that preserve the nearness of each location as its set of indiscernible points and activities. No point can exist without its effort to find its indiscernibles both above and below it in time, and so to return itself to its zero reference state relative to those. Those journeys in time form that point's spheres of influence which are its areas and its volumes. These are its forcefulness in the rows and its power in its columns. Those are the rules.

Lines and points can be straight relative to each other where they intersect … but they can also be curved relative to each other where they do not. Clock time drives onwards along lines … while biological time strives to stay the same in its indiscernible points for a generation. The two together produce the circulation of the generations. Hamiltonians use their constancy in energy to maintain their integrity, while their placements and displacements balance them with dp = 1. Those are rules in this geometry.

We return, briefly, to Mayer to clarify these possibilities. He was the first to suspect that energy existed, and that it could hold constant through all interactions and situations. He was the first to point to that constancy as something expressed in the constant volume, Cv, and constant pressure, Cp, interactions of systems. Unlike others around him, he knew how to make the most of the limited information available. He built on the known fact that the constant volume specific heat capacity, Cv, of any substance is slightly smaller than its constant pressure one, Cp. He gave a reason. He reasoned, correctly, that a gas does external work against the surroundings when it expands itself out with its constant, Cp, into those surroundings. We now call that additional work that must be done the Helmholtz energy.

Biological populations have similar considerations. They must sometimes do work internally, and sometimes do it externally. The two sets of forward and backward walks in Figure 20.22 mean that if, for example, a population can manufacture and exploit a protein, then it must be possible for that same population to produce a complementary set of rates and activities in which it survives while the same protein is surrendered; and then all the while that it is lacking. If this is not so, then there is no equilibrium.

In the same way, Mayer also reasoned that the heat a gas gave off when it was kept at the same temperature while being compressed, isothermally, was simply the surroundings recovering the heat it had lost when the gas had previously expanded, and perhaps even cooled. His argument, then, was that although no real gas or substance is perfect, they all tend towards the same behaviours as a limit. They all express their internal and external behaviours in different ways. But there is still an ideal gas law. Each real gas deviates from that in its characteristic way. But the ideal gas was therefore something more than a theoretical convenience. It could be extrapolated from the many measurements of real gases, and then used as the foundation to accurately describe them all. We are searching for a similar biological limit point explanation for the Weyl and the Ricci tensors, and for the internal and external expressions that biological organisms and populations must also have.

Clock time can keep marching forwards, but if biological populations are to maintain their indiscernible points, then their biological processes must be reversible. They must expand out and then collapse back into points. Clock time may move straight, but it must also lead around in a circle, relatively, so the population gets back to the same indiscernible point. That is the property of a line in this geometry. Individual biological entities can form line segments and are statements of absolute time, in this geometry. But their populations form circles and are statements of indiscernibles and points.

We now have to bring the lines, areas, and volumes of our biological space and its internal energy together into a coherent whole. We have to reconcile lines with curves, rectangles with circles, and cubes with spheres. We also have to reconcile the additive constancy of the energies in the hamiltonian with the multiplicative constancy of placements and displacements. Those reconciliations must also all be measurable.

Figure 20.23

Since the columns in Figure 20.21 intersect the rows, while the rows simultaneously intersect the columns, then every point in this biological space is in fact a pair of coincident and indiscernible points. Every point is both the beginning and end of a generation. They are coalesced indiscernibly into one, even as a span of absolute and linear clock time stretches between them.

Since this biological space consists of three dimensions, then there are at least three such sets of indiscernible points and spans. As in Figure 20.23, a point therefore loops a line and surface about itself, to form a ‘0-sphere’.

Every point in this biological space of internal energy has a minimum construction. Since each point is indiscernible, that minimum structure is a pair of points. But what looks like a pair of points to a column, is a single point to a row; and what looks like a single point to a row looks like a pair of points to a column. And, of course, what looks like a pair of points to a row is but a single point to a column, while what looks like a single point to a column looks like a pair of points to a row.

The same holds for the different aspects of time. While clock time stretches linearly out from the beginning to the end of a generation, they are the same indiscernible point relative to a generation which is a column. That pair of time points is connected and made the same by the journey through the generation that restores their indiscernible natures and brings them to the same state. A point therefore has a line stretching across it to another like itself. The two points thus form the boundary of a line-segment. As shown in the middle of Figure 20.23, and in Figure 20.21, that line-segment is a portion of the internal energy for a helix.

In accordance with the majority mathematical convention, we call the line-segment that is the twirling baton that links two such indiscernible points in time a ‘1-ball’ because it appears to go out and back somewhat like a bouncing ball. (NB: The literature is a little inconsistent in its usage. We follow the convention, here upon this site, of using ‘ball’ for the interiors of objects (hint: there are no e's in ball); and ‘sphere’ for the exteriors of objects (hint: there are two e's in exterior, and also two in sphere to match them)).

Every point is a 0-sphere. Every point is formed from the exterior that is traced out by the twirling batons we first met in Figure 20.1, each of which is a 1-ball in one or another of the dimensions. They twirl together create a sphere of 0 dimensions. That 0-sphere is a generation. It is a point. It is a population. That is so by definition, and in this geometry.

In the same way that a generation is a surface that surrounds and embraces a collection of entities, the 0-sphere or point contains within it—and defines—the one-dimensional lines that are the twirling batons. It is indiscernible and the description of entities going about their generation. That line stretches both across and around the point to another one of the same description; and with similar points on either side; covering its stretch of absolute time; transporting individual biological entities from indiscernible to indiscernible. Those end points are identical; coincident; and make a 0-sphere. These are the two parts of Figure 20.21. In one orientation, seen as columns, those line segment ends come together as the ends of line segments, and as one indiscernible point. But in another orientation, seen as rows, those same 0-sphere points stretch those same two points indiscernibly together.

Identical points, when approached, can open up and offer their contained 1-ball line segment interiors. They then form the line-segments of absolute time that transport individual entities along the rows and from one indiscernible point to another for a generation. Every such line of action is then a 1-ball whose points form the interior of some 0-sphere which regards those points as indiscernible. That 0-sphere is then a point that has very definite properties taken from those indiscernible points and as a journey on a helicoid of internal energy. Those points, which are indiscernible to the 0-sphere, both determine and are determined by their 1-ball interiors and conversely. Each point's properties also depend upon the properties of its neighbouring 0-spheres or points, and the intrinsic 1-ball lines that they in their turn stretch between and contain. Generations can then have properties independently of the absolute times, and entities, within them, and which flow up the columns.

A mathematical aside

Another way of expressing the same property is to say that since every point is a 0-sphere; and since every line is a 1-ball; then every point is the closed set [-r, +r], and so the sum of a placement and a displacement; of an absement and a presement; of an absity and a presity; of an abseleration and a preseleration; and of an abserk and a preserk. Those two complements always cancel out to produce the same value right across any proposed line segment, so creating their 0-sphere of indiscernible points as a 1-ball line. That is a rule of this geometry.

Since all entities and/or their components are subject to loss, then their surrounding populations must be able to reverse both those kinds of losses. The populations must therefore pass through periods of reversing sizes, and of reversing losses in entities and configurations, so that others can then be created that can later do the same. And so, in this geometry and in this space, there is always a reversing rate of change associated with each point, with a net zero effect overall over the span of a line from point to point that is the length of a generation; and so therefore this reversibility must be over some specified reach per each point, or 0-sphere and its contained interior of a 1-ball. A 0-sphere is an entire generation as defined by ∫dN = ∫dM = ∫dP = 0. This surrounds the 1-ball, of that length, for a point-to-point helical motion.

In order to make the zero for a point and a 0-sphere, then a positive and a negative must always come indiscernibly together, in every dimension. Since we must be able, about the entire 0-sphere, to eventually reverse everything happening along the 1-ball or line segment, which is absolute time, then one of the three differentials of dN, dM, and dP—which are the changes in number-energy, mechanical-energy, and nonmechanical-energy—must always be moving in opposition to the others to produce a contrary derivative or rate of change. We can see this in Figure 20.1.B where the curves for the three twirling batons always have two moving in opposition. Each of those three infinitesimals must decrease and reduce its overall quantity when the other two each increase theirs, and conversely, to preserve the overall equilibrium that is the 0-sphere and the point. We will also produce the three integrals or infinitesimals sums of ∫dN = ∫dM = ∫dP = 0 all about the circulation that produces the various equilibria of P’, M’, and N’. They must all at some point do this.

Our biological configuration space must now have, as a part of its definition, the possibility for an equilibrium between forward and backward rates along each axis, and relative to each point, and that involves locations either side, and so from point to point on a line. Real physical and clock time cannot, of course, go backwards, but biological and generational time can … and must … for individual biological organisms can change their sizes and configurations quite dramatically, and then dissipate, to be replaced by others. Populations endure and can adhere to points and line segments. While clock time drives ever forwards, biological points seek constantly to reverse its effects and to remain themselves. This means that they or their progeny must be able to go backwards through whatever is currently being imposed upon them. For every point on a 0-sphere there is a point with opposite properties symmetrically located, so they are either side of their midpoint, which then enjoys the same description.

As in Figure 20.22, every forwards reach and rate for a population must now be balanced by some backwards reach and rate for that same population to create a unit sphere and a 0-sphere; and to populate the 1-ball line segment for and within that point. That is the power of a point in this geometry. So just as ice, liquid water, and vapour can all exert different pressures on the surroundings yet create a mutual equilibrium, so also all the different possible configurations of biological entities will also have their different energies, quantities, and balances. Every point in this space, in other words, has the potential for a Gibbs-style triple point equilibrium embedded in it. It is measurable as rates around each point.

Our biological space achieves its purposes for internal energy by following the right-hand rule of vector spaces, again shown in Figure 20.22. Our unit ball, in other words, both exists in and creates a normed vector space. So there will always be one aspect or side of any reaction or configuration change that is the more energetically favoured relative to some point. That aspect surrenders mass and/or energy and/or number to the surroundings when measured in clock time, again relative to some specific point. But there is another rate and configuration to match it that is not energetically favoured, and so that absorbs mass, and/or energy, and/or number from the surroundings. One of these rates will therefore occur with less energy and/or smaller numbers and/or at a smaller mass relative to a point, and so that the two can create the average of each of these that belongs to that point. But, once again, one of those must act counter to the others, and increase when they decrease and conversely and so that we can preserve the indiscernibles of points, which is ∫dN = ∫dM = ∫dP = 0 all about themselves for the 0-sphere.

A mathematical aside

If a, b and c are specified vectors within the field we have established; and if c and d are scalars applied to them then:

(1) a + b = b + a; (2) a + (b + c) = (a + b) + c; (3) a + 0 = a; (4) + (-a) = 0; (5) c(a + b) = ca + c ;
(6) (c + d)a = ca + da; (7) (cd)a = c(da); (8) 1a = a (Stewart, 2007, p. 810).

These are the complete rules for well-behaved vectors and they apply to every 0-sphere and 1-ball. All therefore abide by (1) the gradient theorem of line integrals, (2) Green’s theorem, (3) Stokes’ theorem, and (4) Gauss’ theorem or the divergence theorem, which is the complete set for the vector calculus. Every vector therefore has its standard basis vectors or unit normal where:

basis of a vector space. A set of linearly independent vectors such that every vector of the space is equal to some finite linear combination of vectors or the basis (James and James, 1992, pp. 27, 194).

Those are part of the rules. We therefore have separable two-dimensional vector spaces with orthogonal components. Each is linearly independent, so allowing the dot products to vanish, making them orthogonal in the Euclidean sense, and with everything that implies for the mass and energy fluxes. These are all true by definition.

And since biological entities are made from molecules, then it is a part of the definition of this space that the energetic-demanding rate is a multiple of the Boltzmann constant, kB, relative to the energetic-favouring one. Every point in this space therefore allows biological systems to create equilibria by balancing rates in molecules in both forwards and backwards directions all about it. These molecules will also do so by distributing themselves, throughout time and space, in accordance with the Maxwell-Boltzmann distribution. We can then make predictions for this space and see if anything in the real world follows them. That was the purpose of our Brassica rapa experiment.

The rates of activity in any one instant need not be equivalent in each direction, but their overall time rates of distribution, across the entire generation, must match each other to create such an equilibrium centred around a viable point in this space. Therefore, for every property X in a collection of components and/or entities using mass-energy in this space, there exists a relevant average of = X/N, however that average and its distribution may be computed, along with a characteristic distribution for that property amongst those N components or entities of whatever size and type, and which can also always be stated by the mole or other such relevant chemical or other unit, and about that point. Those are the rules.

But since the above distribution results from a set of dynamic activities tending from point to point, then it also represents a path for the line-segment that is the 1-ball for a 0-sphere. No point or configuration is isolated. Every one is either a 0-sphere; a part of a 1-ball for some 0-sphere; or both. Every configuration is therefore on the path for others. So there is, around every point, a time-independent stationary state whose length is τ = 1 when measured by the generation, but T seconds when measured absolutely. The distribution about the point is a complete record of all values belonging to that point, and it is a collection of the stationary states or configurations needed to create the phase space for that collection of points. Each state can only be maintained by the rates specified by that stationary state configuration, and those around it. The values of the stationary states are the instantaneous values for the path leading to and from the point, and so for the phase space, at each moment. There is therefore a numerical identity between the path and the state values that produce the indiscernibles.

More generally … for every variable X at a point, there is immediately implied an associated Y lying upon that same point where X is the path in some units per time span, and Y is the stationary state in units for that point, and such that, numerically, X = Y. So whatever X might be, Y states how many of them there are per second or other time interval. Additionally, of course, a exists that can be taken over whatever N are found, be it molecules or entities, in that distribution. The X and the Y are always related such that Y = ∫ X dt; X = dY/dt; a Y’ exists over any interval; and also such that T0 dY = 0 over a point-set or collection. One is in other words the rate of the other.

We have now defined a set in this geometry. A set is everything gathered in the unit sphere or vortex around an indiscernible point and its companion being a complete collection of rates and states in X and Y, and and , all linked by n in rates and states thereof.

The lines and the points in this space are also defined to have the orientations ABCD found in Figure 20.22, which is the right-hand rule. Each point (i.e. such that ∫dN = ∫dM = ∫dP = 0) is therefore capable of acting as the average value for a generation of activity surrounding it, for that is its zero value. Each point can thus be the weighted average N’, M’, or P’, for a generation, and so for some range about it whose absolute value is determined parallel to each dimensional axis. That range of values about that point for each dimensional axis then forms a required set for that axis and back and forth from that point; across that range of values; and for that line. The required set is the 1-ball for a dimension, with all three dimensions being the 0-sphere of the point they together define, and for which they are the joint interior. The 0-sphere surface and 1-ball interiors follow Coulomb, Poisson, Biot-Savart, and Maxwell rules. Each point then always sits in such a specific range about it, and from one point exactly like it to another, which is the length called a generation. It goes from zero to zero, and that is a definition.

As in Figure 20.22, we can imagine that the population ranges in values, over the generation, back and forth with that small walking figure in the centre of the Figure, and on that required set line and its range … which is the 1-ball of a single dimension for a conjoined indiscernible 0-sphere. It is a single baton whose strobe light goes from end to end on its required set walk. The line value first increases away from a point towards others, but will then return and decrease back to the point; and then continue further behind; and then again turn about and return; to produce the net zero effect in incremental changes for that point. It is the required set walk across the entire breadth of the 1-ball for that dimension in the 0-sphere.

The Weyl and Ricci tensors create entire biological communities around us by working in their four-dimensional spacetime. They create more than points and lines. Lines give only one dimension in the real physical space around us. Our twirling batons intersect and act in a variety of ways for two and three dimensions.

Our biological configuration space and its internal energy have three dimensions, which are their three ‘degrees of freedom’. All movements are defined by changes in (1) mass, (2) energy, and (3) number. Each of these is precisely equivalent to the x, y, and z dimensions of ordinary physical space.

Figure 20.24

The 0-sphere we started with has a set of 1-ball interiors, each of which is a twirling baton. Since each twirls, then as in Figure 20.24, it must twirl into at least one other dimension to make it an area. The twirling baton's 0-point boundary describes an arc, with respect to that other dimension. It immediately creates a ‘1-sphere’.

Since 0-spheres have 1-balls inside them, then if this new object is officially termed a ‘sphere’, it must logically be some form of exterior. And if a 0-sphere exterior contains a 1-ball interior, then a 1-sphere exterior must in its turn have a 2-ball interior.

As again in Figure 20.24, where a 1-ball is a more direct and straight line, a 1-sphere is what we generally think of as a circle. The arc a twirling baton describes is a 1-sphere. And since it is a sphere, then it is only the boundary or exterior bound for the 2-ball or ‘disk’ that it surrounds. Thus a 1-sphere is the boundary for a 2-disc area. And conversely a disk—being the 2-ball interior for some circle or 1-sphere busy circumscribing it—is the entirety of that 1-sphere's interior.

Since the 2-ball that defines an area and a distribution in Figure 20.22 follows the right-hand rule with respect to both its dimensions, it is the larger walking figure in and for the interior energy it bounds. It does so in conjunction with one other dimension.

All three of our dimensions can walk. They can all interact with each other. Since there are three and they can all take it in turns walking, then there are six 2-ball walks. One is when n walks against , another is when walks against n. The same goes for n and , and and . We already know those last two— and —as the work rate and the visible presence, respectively.

We see the effects of the right-hand rule at work in the way the batons twirl as they create the four separate parts of each required set walk—AB, BC, CD, DA—shown in Figure 20.22. For every step towards the generation average on the required set for a generation that a point represents, and that is therefore also an increase, there is an accompanying increase in some other property. It is relative to the positive and the increasing direction in that dimension. It agrees with the right hand rule for this point, and is relative to that. It is therefore also an increase with respect to that second axis and dimension, and for at least part of the range. But there must eventually be a decrease. The two dimensions therefore produce the four parts of the walk because they can (1) increase together; (2) decrease together; (3) one can increase while the other decreases; and (4) conversely. We thus have both displacements and placements. For every nearness there is a farness. The 1-ball line segment (within the 0-sphere point) is the 1-sphere, or bounding circle, for the 2-ball or area it surrounds. So every point, in this geometry, also has an intrinsic area or 2-ball disk which supports the walk contained inside it, and also outside it.

We now have the out-and-reverse walks that is the required set of walks for the small figure shown in Figure 20.22, traversing the 1-ball of the 0-sphere. So to accompany the ‘outwalk’ BC there is the ‘inwalk’ DA back towards the point; and which is an integral part of this point's definition. The inwalk is exactly the same range as the outwalk … but traversed in the reverse direction. They are the nearnesses and farnesses interchanging about the zero. This gives the complete 2-ball disk interior, and the circular 1-sphere line segment exterior that is the circulation of the generations. Those are the 0-sphere points. Those are the rules.

There are three twirling batons all interacting in similar ways to create this biological space of internal energy. We know from Figure 20.21 that all three columns circle the rows, and that all three rows circle the columns. The reverse walk or inwalk that defines a line ranging about a point is matched by some equivalent range of values in its second property. It follows the right-hand rule. So its area and its rate distribution is also defined for it by the third dimensions which also moves through time, along the helicoid axis, to create a surrounding volume and sphere, which is the entire 0-sphere point.

Each of the three dimensions now twirls with the two others to create a complete 0-sphere volume with its set of linear 1-balls. They bring together both absolute and generational time. The areas any two batons create between them each form an allowed set; as also the volume they jointly create, which is also an allowed set. But since this is an indiscernible point, then as fast as something stretches away, just so fast does something else approach to create the 0-sphere surface. The inverses of approaches and retreats therefore hold for the third dimension, which makes its separate 1-sphere and 2-ball contribution to that same allowed set which is the entire volume for the 0-sphere.

One part of a required set walk now results in a boundary that is at a higher energy and/or value. The other part provides the lower boundary for the 1-sphere that surrounds the 2-ball interior area. As in the diagram, the combination of out- and in-walks is therefore a complete domain or area of ABCD. It creates an allowed set of values in a 2-ball, and within its bounding 1-sphere for a contained area; and all within the 1-ball that is a line segment within its 0-sphere volume that is a point. These in- and out-regions match the required set range about and within the point; and also relative to a second dimension's similar required set walk. The required set is the range demanded about the point, while the allowed set is the method of satisfying that required range in concert with each other dimension, and which is always affected by that third. And since time also moves onward then the same happens to this dimension from the third dimension, and so that that dimension also provides an area to match a required set in that third dimension. There is, then, an entire three-dimensional energy volume, all about that point, and which is also part of the allowed set; and all within the point. This is a 0-sphere.

A mathematical aside

We have now defined Stokes' law of the vector calculus as an intrinsic part of this geometry. The required set defines both an area and a circulation. Every point is therefore capable of acting as a part of the vector curl for some other dimension, and so can define a curl as a circulation per unit area over an entire generation. We have therefore also immediately added Stokes' law to Gauss', which is the divergence theorem; and to Green's theorem; and to the gradient theorem of line integrals, which is the fundamental theorem of calculus. So we have successfully linked points, to lines, to areas, to surfaces, and to volumes, and we can operate in zero, one, two, and three dimensions. We have a series of vector fields complete with standard basis vectors and unit normals: a set of linearly independent vectors such that every vector of the space is equal to some finite linear combination of vectors or the basis (James and James, 1992, pp. 27, 194). Each is therefore independently a separable two-dimensional vector space, with an orthonormal basis, and that each follow all tensor rules, while still being orthogonal in the Euclidean sense.

And now we have defined a point, a line, an area, a volume and a set, we can define an entire biological plane of activity. A plane in this geometry means that every point has infinitely many indiscernible points lying upon it and around it, and ranged on either side of it. Since  T0 dP = 0 then it is possible to travel both backwards and forwards for an infinite number of generations—and always find another point, which can easily support a biological entity, with exactly the same description, for that dimension; and also to move through exactly the same range of values to find it. This is the journey up and down the helicoid from -ρ to ρ; from the pole of minus infinity to the pole of plus infinity. This holds separately for each specific dimension. All of mass, energy, and number can be infinitely repeated.

The plane in this geometry is defined as flat, relative to a given dimension, when the absolute amount of time, T, needed for those repeating points is constantly the same, and so that τ always has the same length. The helicoid's spirals are then always constant in pitch, radius, and thickness. The ratio T:τ must be consistently the same all along the axis for that dimension for it to be flat or even. If not so, then the plane is curved for that dimension. This holds, separately, for all three dimensions. If there is no such thing as evolution, then we should be able to measure a flat plane in every direction because nothing is changing.

A completely flat biological space, i.e. for all three dimensions, will produce a flat biological medium. This is one that keeps its generation length the same everywhere in all three dimensions. Otherwise, the space is curved, and the curvature can soon be determined. We can also easily find if there are any jerks. We conducted our Brassica rapa experiment to determine its net curvature in biological space, and to locate its jerks, if any, which are its aberrancies.

Now we have a plane, which is a surface, we can cut and divide the whole of our biological space to create the regions that are biological entities and populations. We have a sense of direction, along with a sense of near and far, bigger and smaller, before and after and so forth that we can use to describe any population.

If we amended the earth and removed its mountains and valleys, then we could treat its surface, just as well, as a plane. It would have the same effect. We could walk around and around it as often as we wanted in any direction, and never come to the end of it. It would therefore stretch just as infinitely far in any direction as any supposedly flat plane. We can therefore substitute the one for the other without immediate fear of detection.

Figure 20.25

Since the surface of a sphere is also a plane, then as in Figure 20.25.A, we have two ways to conceive of creating our regions, entities, and populations. With a first method, which we see in the upper part of 20.25.A, we use a straight line of infinite extent to cut the plane. One plane, or dimension, placed in space, can then divide all of space into two sectors. We have an origin upon the plane. This can be zero or unity. Everything then lies either one side or the other. Everything is either less than, or greater than, positive or negative, relative to the plane, and is now near or far. Absolute or clock time can behave like this. Everything is either past or future, either side of ‘now’. We have our t0 as the present, with t-1 being the past, and t1 being the future.

We see the second method of creating the plane in the lower part of Figure 20.25.A. This time, we go up a dimension and use a sphere. We take up ‘3-ball’.

We can think of a 3-ball as an ordinary sphere (almost!!). However, a 3-ball as we generally observe it is actually the interior of a 2-sphere. That 2-sphere is the infinitesimally thin boundary surrounding the 3-ball interior. That 2-sphere is the 3-ball's entire surface.

The 2-sphere or infinitesimally thin surround for a 3-ball is now our infinite plane. We can go about on it indefinitely. Everything in space is then either to the inside or the outside of that 2-sphere boundary. We can place positive or greater than, or nearness, values inside; and negative or less-than, or farness values outside (or vice versa). The surface is then the origin or the unit, as appropriate. That is still one side of the surface or the other, for our 3-ball can be any size, with the 2-sphere it provides, as the surface, then being infinitely large in the sense that we can go round and round it without restriction for endless generations. Those are all the same. Our polar planimeter can now sit at the centre of all populations and generations, and measure them all as the same 2-sphere in terms of near and far, absement and presement, and so forth.

If we have two planes that do not intersect, and therefore that are parallel in the traditional sense, as in Figure 20.25.B, then all of space is divided into three regions. We have (a) one side of both planes; (b) in between the planes; and (c) the other side of both planes. The same goes for two 2-spheres, as the surfaces of two “ordinary” 3-balls, that also do not intersect. The upper and lower parts of Figure 20.25.B are therefore identical.

Since the two 2-spheres could stand apart from each other and intersect nowhere, then they would also always be parallel everywhere (by definition); and similarly give us three non-coincident regions, exactly the same as the two planes. They are as if in constant motion relative to each other, never affecting each other. We now have two entities or populations that do not interact in any way, and so that are parallel and enjoy special inertial frames of reference, and we can measure the farness that causes this. We also have a definition for species that do not evolve relative to each other. They are in constant motion in special inertial frames of reference. This sense of parallel and not evolving is now something we can measure.

If the planes or spheres are non-parallel, as in Figure 20.25.C, then whether they are planes or spheres, they will intersect. Space is then divided into four quadrants, again in each case. We can call one plane or 2-sphere the x- dimension, and the other the y. Whichever way we approach it, by plane or by sphere, we then have the four quadrants −x−y, −x+y, +x−y and +x+y. Those are our placements and displacements, and our nearness and farness.

If we have three planes—or three dimensions—to our space, then as in Figures 20.25.D and E, space is divided into the eight regions generally called ‘octants’. The plane case in Figure 20.25.D makes it clear why. If we now call each of the three 2-spheres or planes in either of these situations x, y, and z, then we have the eight octants: −x−y−z, −x−y+z, −x+y−z, −x+y+z, +x−y−z, +x−y+z, +x+y−z, +x+y+z.

If there is no such thing as evolution, then we should be able to prove either:

1. That all biological entities live completely flat, as in Figure 20.25.C, and without intersecting with each other. We should be able to prove that they live only within the four quadrants of a two-dimensional flat plane surface, never moving into a third dimension; or else we should be able to prove
2. That they live completely upon the surface of one of the three 3-spheres, never either entering into it or departing it, so that the radius in Figure 20.18.C is constant. This would then be the spherical equivalent of constantly remaining within the same four quadrants, through always being on the surface of one sphere.

Either of these is now trivial to predict and to prove, because we have some very definite values (n = 1 at all times) and rates of change (dn = 0 at all times) that we should be able to measure in the real world (by setting n + dn = 1 or invariant at all times).

There is, however, another important factor. We have our Owen and Haeckel tensors. One is a 3 × 3, the other a 4 × 4. All biological entities live not just in space, but also in the fourth dimension of time. Evolution then becomes the problem of establishing what is possible, and not possible, in that fourth dimension. Creationism and intelligent design insist that there are never any long-term changes of mass, energy, or number in time. If so, we should be able to confirm that with our measurements. The 4 × 4 will help us determine what we should measure.

Figure 20.26

As in Figure 20.26, that fourth dimension also introduces a fourth line as a plane, and/or a fourth sphere. This 3-sphere case is now a little more tricky. We have to have a 4-ball for its interior. The closest we have, in this real world, to a 3-sphere, to act as the surface is a little inadequate. We have only the ordinary 3-balls we have in this reality. That is to say, every 3-ball we see is not the interior of a 2-sphere. It is instead the 3-sphere surface of a four-dimensional 4-ball hypersphere that we cannot see properly. We cannot draw that 4-ball interior … but the 3-spheres we have that are their surfaces are there (as best as possible) depicted in the two-dimensional diagram of 20.26. The diagram does its best to show, with the four 3-spheres of different colours, that this gives 15 different 4-ball intersection regions. But those four 3-spheres are again supposed to be showing the 4-ball interiors of a 3-sphere exterior, which we cannot show correctly. It is impossible in three dimensions, so we can do no more than “count the ways”. Those ways are the possibilities open to biological entities.

Since nothing can move faster than the speed of light, then we can make it a rule of this 4 × 3-spheres geometry that one of the four 4-balls, which are the interiors, must contain all of the time that passes in this entire universe. That specific 4-ball's 3-sphere boundary defines particles such as photons which travel constantly at the speed of light. Those are the surface which can never be crossed. And since nothing in this universe can be outside that boundary, everything is always inside that fourth sphere. Everything is positive and/or moving with respect to time. That then accounts for time; how it flows in this universe; and also makes sure that nothing in biology breaches the speed of light. That is another rule in this geometry.

Nothing can go literally backwards in real and physical clock time, but biological populations can certainly move backwards through biological time. Adults can recreate their young which then become adults. So instead of keeping the four 4-balls the same size as we have in the upper part of the Figure, we can observe, as in the lower part, that:

1. One 3-sphere, and its 4-ball interior, can be completely included in the union of the other three.
2. The intersection of two can be completely contained inside the union of two others.
3. The intersection of three can be contained inside one.
4. None of the above are true.

This gives us four constantly changing spherical hypertriangles, separated by six hyperarcs of hypercircles, which intersect at four very definite contact points. Those four spherical contact points give us the ‘4-simplex’ or ‘5-cell’ which is the tetrahedron figure—figure with four faces—also shown in Figure 20.25. It is sometimes called a ‘pentachoron’ or ‘pentatope’.This 4-simplex has five points, or vertices, or ‘0-faces’; ten ‘1-faces’ or edges; ten ‘2-faces’ that we in this reality might call areas, and which are equilateral triangles; five 3-faces or volumes, which are pyramids and which are also the 4-simplex or tetrahedron's maximum contact with this ordinary reality; and one ‘5-face’ of four dimensions. Each of those four points P1,2,3, P1,2,4, P1,3,4, and P2,3,4 cover all possibilities in time and space.

Every two 3-spheres have a tangent or ‘radical hyperplane’ stretching between them, making six in all. They are the tangencies or contacts between the spheres: S12, S13, S14, S23, S24, S34. They will always intersect at the unique point shown inside that tetrahedron. Since each of the six hyperplanes is a tangent, each is effectively an alleyway or passage transferring values from one 3-sphere and 4-ball to another. Each of those six tangents is therefore a perfectly normal three-dimensional space just like this one, complete with its rates and flows of time. Each such tangential reality simply has different rates of change of its properties as each tends between one 3-sphere surface and the other, thus producing the rates of change around us. We are building a curved biological space with distances, masses, and energies, that can all change their rates over both time and the generation, and so that can build us different kinds of helicoids, and therefore populations.

The central point in the tetrahedron now divides biological spacetime into four trihedra, each contained within. That inner point therefore uses the four contact points P1,2,3, P1,2,4, P1,3,4, and P2,3,4 to create the volumes and the surfaces that define their joint interaction. Each partial volume is therefore the intersection between three 3-spheres and the 4-simplex or tetrahedron, and again helps define our surrounding space. We must also take care to impose the vector right-hand rule we need. We can eventually calculate all those volumes, surfaces, lines and points and determine the entire biology, plus its rates of change over time, from all those various perspectives.

But we also, of course, have generation length to consider, which could be a separate dimension. Navigating through biological space then becomes a matter of navigating through those five 4-spheres and 5-balls, or hyperspheres and hyperballs, and so forth. This becomes somewhat more complex, but the essentials are still approachable through Helly's theorem. At its very simplest:

1. A first 5-ball and 4-sphere contains the intersection of four others; and simultaneously, the union of those four others contains the first. We again gradually end up with a tetrahedron like above, but the descriptions of all parts are a little more involved. However, since our latest sphere and ball for generation time is still allowed both positive and negative values, we perhaps need not pay too much attention to its specific description; except that (i) its values in either direction must not allow the events it governs, in absolute clock time, to breach c, the speed of light in that specific sphere; and (ii) it may freely swing back and forth and so have positives and negatives for its own values, but its oscillations must also not breach the maximum available in rates and resources in that relevant sphere, and so that it creates the inner and outer bounds for our helicoid. We must therefore be careful to select configurations that respect these requirements.
2. The union of two contains the intersection of three others; and simultaneously the union of those three contains the intersection of the first two. This ends up as a triangular hyper-prism-like figure bounded by two triangles and three tetragons, nine arcs and six vertices. We then again have to take care to ensure that (a) one is always positive for the fourth dimension of time; and (b) that a second always respects rates and quantities of mass-energy and resources.

We can now move perfectly safely, in our three-dimensional biological geometry or internal energy. We can move from indiscernible point to indiscernible point on our helicoid. We can have reversing rates and events; still always move forwards in clock time; and conduct tightly controlled experiments, whose values we can predict.

Figure 20.27
The third law of biology: the law of diversity (1)

Our three biological dimensions emulate the standard x, y, and z dimensions of space. The right helicoid case may well prove to be impossible, but it remains the easiest to describe. And the fourth dimension may be hard to visualize, but we can determine it by observing the transformations that biological entities and populations go through over time.

The right helicoid cross-section in Figure 20.27 shows movements in the mass dimension. We see the AB meridional stretch from minimum to maximum, or from inner to outer bound on the right. That is the required set walk that increases mass. Its reverse is CD on the left where mass declines. The values hold constant on the BC and DA walks at top and bottom.

The other two dimensions—one for energy and the other for number—are similar. The energy dimension is located above mass, and is rotated through 90° so that it is orthogonal. The number dimension is similar; is underneath; and is again orthogonal to both. The underneath number dimension is initially stationary. It does not walk. This gives us the n + dn = 1 constancy we need so we can investigate to see if creationism and intelligent design are even remotely possible.

The cross-section of a right helicoid in Figure 20.27 helps define the configuration space, or set of quantities and variables, that characterize our three-dimensional biological space at any given time. It is easiest to first consider the walk BC.

Figure 20.28
The third law of biology: the law of diversity (2)

As the population walks from B to C, in Figure 20.28, energy keeps entering. But the number of chemical components has just moved from A to B and so is at its maximum and holding steady. If energy keeps entering, but mechanical chemical energy has reached a maximum, then the population must be undertaking some other kind of transformation.

Figure 20.28 presents the Weyl tensor we first met in Figures 20.8 and 20.9. Since no more mechanical chemical energy is entering, the mass and the number of components remain constant. All incoming energy must therefore divert to effecting a set of internal changes in state. That incoming energy institutes a set of biological interactions and reconfigurations. Since mass holds constant and numbers also hold constant underneath, then the observed transformations are a manifestation of the required set walk currently taking place in the energy dimension that is above this mass one. The configuration changes along BC mean that the energy and the pressure of the constant volume and nonmechanical chemical form is steadily increasing. Since this induces energy to keep entering the population; but since numbers remain constant; and since mass or mechanical chemical energy is also constant and holding at its maximum; then only constant volume and entirely internal changes in configuration are possible for this population. This means an increase in the Weyl tensor's activities.

All these ongoing configuration changes we observe as we stand still in the mass dimension, and look above to the energy one, are happening at the behest of the positive part of the required set walk currently ongoing in that energy dimension. More energy is entering, but its effects are entirely of the nonmechanical variety. The mass dimension has provided all the ingredients; and holds them at the current value. No further mechanical chemical energy enters. The incoming energy helps to drive the above walk by providing the mechanical mass and inertia the energy dimension above needs so it can operate. And since we can “see” that walk happening above us; and since this mass dimension is responsible for providing the materials for those effects; then that movement is interpreted, by this dimension, as a curl in its own mass flux. The formal mathematical designation is: ∇ × M.

And that is another rule in this geometry. Whenever we see a dimension, above or below, undertaking its intrinsic required set walk, then the dimension which holds the value institutes a curl relative to its own dimension. That curl projects into that upper or lower dimension. It feeds the ongoing required set 1-sphere walk in that other dimension. The holding dimension therefore sees a rate of change of the activity it provides occurring at right angles to itself in that other dimension. Those are the rules.

A mathematical aside

We have already calculated that the complete curl in mass, as transmitted to energy, is: ∇ × M = ∂/∂t - ∂n/∂t; while the complete curl in energy, as received by energy, is ∇ × P = ∂/∂t - ∂W/∂t - ∂n/∂t. These form a part of the Weyl tensor.

We are currently holding two things in our developing geometry of biology constant: (a) the total number of chemical components contained in the population; and (b) the total number of individual entities, which is the number of partitions. Since those two are constant, then the mass flux stays constant over the entire interval.

A mathematical aside

We already know that any molecular or thermodynamic system's escaping tendency or fugacity is (∂S/∂U)q. We also already know that when we hold both the volume, V, and the number of particles, N, in an ordinary system constant; add energy; and see its entropy, S, change with its internal energy, U; that we will then have 1/T = (∂S/∂U)V,N. The escaping tendency is an increase in molecular motion, which is a rise in temperature.

If we call the integral of the mass flux over any given time a ‘mendel’ of so many kilogrammes of chemical components, then our biological system currently has both (a) a total mendel of U kilogrammes; and (b) a given number of ways, N, in which those components have been partitioned. That partitioning is the number of biological entities, N. This is {U,N}. We must now determine the molecular effect and escaping tendency upon our helicoid, subject to that constraint.

We only have three dimensions in our biological space. We are presently holding two—the number of components, and the number of partitions for those components, which is the number of entities—constant. So as in Figure 20.9, which illustrates the Weyl tensor's effects, the only way the Wallace pressure can increase is for the biological entities to increase their overall activity rates. This is to increase the average individual Wallace pressure, . And the only way this can change, also under these conditions, is for the entities to change their internal cellular configurations in a more energy intensive direction. They must increase the observed diversity, which is the observed changes in configuration we see in Figure 20.28. This is set of constant volume internally directed transactions and transformations. We have successfully isolated all physiology, and all diversity, in biology.

‘Average individual Wallace pressure’, , is a very important part of this geometry. It produces a change in helicoid pitch and/or radius and./or thickness. It changes the Gibbs energy and the internal configuration of biological entities. But it is fairly convoluted to keep on repeating that phrase. So we shall from now on refer to it as the population's biopressure. The biopressure is any population's average individual pressure, .

A mathematical aside

We therefore now have the formal definition of biopressure: it is what increases, over a population, through the curl in mass, and at constant population mass and numbers.

A change in a population's biopressure is a change in its energy intensity while both the numbers of entities and numbers and types of molecules remain the same. It is defined as ≡ (∂S/∂U)U,N. And since these are chemical reactions, then they are also changes in the population's Gibbs energy, which is the potential to engage in these very chemical reactions.

We now consider the walk AB that preceded the BC walk we have just investigated. As Figures 20.11, 20.27 and 20.28 make clear, the AB required set walk happens because of the Ricci tensor. It increases the mechanical chemical energy from its minimum to its maximum, and from its inner bound to its outer one. Since the energy dimension above is now not walking; and since numbers underneath are also not walking; then we now have {V,N} to show that it is now the visible presence and the number of partitions of biological entities that hold constant.

If energy density and numbers hold constant, then the only possibility for change is an increase in the number of chemical components held across that population. Since number is holding constant, then each individual biological entity must increase the quantity of chemical components apportioned to it. That is to increase in its mass; but all while holding its biopressure constant. Mass is now enjoying a two dimensional interaction in partnership with number. We are now creating an increasing area as the radius we work with increases.

The required set walk in this dimension is the divergence in the mass flux of ∇ • M, which is also the flux density, . It is therefore a rule in this geometry that a walk in a given dimension, that produces a change within that dimension, in conjunction with an underneath dimension, is the divergence for that walking dimension. It is a rate of change in the direction of, and along, the field itself. It is an expression of the Ricci tensor for that dimension. It is manifest, in this case, as the increase in the average individual mass pressure of genetic and chemical components, .

‘Average individual Mendel pressure’ is now also an important part of this geometry, and also changes the helicoid's pitch and/or radius and./or thickness. Since it is equally convoluted to keep on repeating, we from now on refer to it as the population's mendelity. A change in a population's mendelity is a change in the number of moles, q, that a biological entity holds, while its biopressure or energy intensity, , remains the same. The mendelity is the point at the centre of a population that describes the components that create its current internal energy. Its value can be attributed to the entire population, which then acts as the surface, taking that value for mendelity as its centre. The mendelity is the average individual mass, .

A mathematical aside

We also have the formal definition of mendelity: it is what increases, over a population, when the divergence in mass changes at a constant number and energy density or diversity. It is therefore defined as ≡ (∂S/∂U)V,N.

The Weyl and Ricci tensors therefore carry rates and potentials. They are four-dimensional. We can quantify their potential for diversity, and for biological transformations, using Black's and Lagrange's hamiltonian method.

When Lagrange studied planetary motion, he wanted to calculate the force that moved a test body from Location A to Location B. He calculated the future kinetic energy it would have when it got to B, and assigned it to A as a potential. Its position at A was then a statement of its energy. Its potential was a measure of the work it would undertake because of that position, which was a displacement relative to another. It was a stored capacity to do work with respect to some reference, having the combined properties of nearness–farness, and the forcefulness of and for each. Therefore, the potential any biological entity has when at the beginning of any interval in our geometry is simply—as Lagrange taught—the energy difference between the two points.

A potential simply needs a reference; a target towards which it develops; and something by which it may be increased. Black used the same kind of reasoning for his ice and water, and water and steam experiments. He used one as a basis and a measure for the other.

On the same basis as Black's experiment, we find a heat source and heat up two samples of water. We hypothesize a first sample, Winvariant that never changes in its phase or configuration. It will always respond by rising in temperature. It never changes in its state. It maintains the same heat capacity, and always responds by increasing in its temperature. It therefore keeps the same molecular configuration throughout.

We now also take up a sample of ordinary water, Wnormal. This one will change in state. It is currently frozen as ice. It will respond, as normal water always does, by undergoing all the usual changes in phase as we input heat energy.

Now we have our two samples of water, we can calculate a potential final temperature, F, to which both Winvariant and Wnormal can potentially rise when we heat them. This is F = 628 °C. We have simply summed the two temperatures from the two experiments Black conducted. But Wnormal will not of course rise to that temperature. It will instead do what Black observed and change its configuration. At the completion of any such experiment, our potential heat source will be exhausted while Winvariant rises to some high temperature, with Wnormal instead rising to some lesser and final measured temperature, M, because it has diverted some of its heat energy into its changes in state. We can then compare that final temperature, M, to the F we calculated for Winvariant. The difference between the two will be the heat energy that was needed for those configuration changes. We will have a measure of energy. That was Black's realization. The difference between the calculated and actual temperatures, F and M, is the hamiltonian that states the total energy difference, and so the diversity in their configurations. It is a measure of diversity in configurations.

We begin our experiment with our reservoir of heat energy containing A joules. It is ready to place in our two samples. If we remove the correct amount of heat from our reservoir and direct it at our two samples, Winvariant will rise by exactly 1 °C while Wnormal melts. It has not used that heat for a change in temperature. It has instead diverted the heat to a change in state. We must therefore reduce Wnormal's potential final temperature by 1 °C to (F - 1) °C. Our reservoir has tended to zero, A → 0, while Wnormal has tended to its final measured temperature, FM. That heat absorbed and that descent of a potential final temperature measure their equivalence.

If we remove more heat from our reservoir and inject it into our two samples, Winvariant will rise by a further 1 °C while Wnormal again melts instead of rising. Its potential final temperature F again declines by yet another degree, so that both A → 0 and FM.

For every further injection of heat energy that causes a phase change in Wnormal, our Winvariant will rise ever closer to F by 1 °C while we will have to reduce the potential final temperature to which Wnormal could yet rise by 1 °C, and so closer to our final M. As Black showed, the difference between F and M will always quantify the diversity and changes in configuration we see. It will be all the variations in configurations that Wnormal has undergone when compared to F and the original Winvariant. The heat that would have taken the temperature to 628°C is what Black called its ‘latent heat’.

We now have a way of stating all diversities, and any changes in energy densities both absolutely and relatively. There always exists an initial condition of some Winvariant that is a potential match, with its proposed specific changes in state, for any molecular reconfiguration, or expression of diversity, that any second body will go through, no matter what the type. We simply measure the energy we put into some second body which changes its state, and determine how the first body would act if it absorbed that same energy, but refused to change its own state. Therefore: whenever we have changes in configuration to consider, we can keep the one invariant relative to some initial condition, using that as a basis for an initial value; and measure the other and final value relative to some specified change the initial one would have gone through with that same quantity of energy, and relative to it.

We now use Black's hamiltonian method on our biological population. We select an entity at each end of our two outwalks which is at A and at C. This gives us the full range of states across which a 0-sphere changes from its minimum to its maximum. We call the two entities Einitial and Efinal. They represent the complete set of changes in both mass and energy.

We can copy Lagrange and measure any biological population across any chosen interval in our geometry, and assign the energy we measure at either end of any interval to a potential held at the beginning. So Einitial holds all the potential eventually expressed in Efinal, and conversely. This is also the Black and hamiltonian method, and it is a further rule in our geometry. We now call the energy that effects that transformation, and that we would place in our reservoir, the reproductive potential, A. It is a property of every point through its required set walks, and the absement and presement of a generation, τ.

The energy we inject will institute a set of transformations. They also have a potential. We define it as the “Franklin factor”, K. It is named after Rosalind Franklin who did the essential crystallographic work that enabled Watson and Crick to unravel DNA.

The Franklin factor indicates the scale of the transformations allowable to—and so that are a potential for—the population in its current state. We simply measure the visible presence or energy density at the end of our measurement period or walk, and use it as a basis. We can then determine the Franklin factor at any time of interest from K = (V/Vfinal) - 1. So if a given biological organism has completed all its configurations over the two walks, then its Franklin factor is Kfinal = (Vfinal/Vfinal) - 1 = 0. This simply means that the potential is exhausted. The Franklin factor therefore quantifies the potential configurations or transformations in the relative fashion we need by comparing the energy density at any point to the value it has at the end of our selected interval. It gives us the dV/V or proportionate incremental values we need for the equations and tensor we wish to use.

We can now repeat Black's hamiltonian thought experiment and place a known number of joules of potential energy into a reservoir for our two walks. We then measure the population's current energy density. We use that to hypothesize a terminal mass, based on the quantity of energy in our reservoir, for an Einvariant that will not change its energy density across that outwards two-walk interval. It will keep the same energy density and eschew all configuration changes. The hypothetical final mass is the mass that the entity will reach, if it does not reconfigure itself and increase in its energy density. That hypothetical mass is now the Franklin energy, F, stated in kilogrammes of mechanical chemical energy. It is similar to Black's 628°C temperature that the water would have reached if it had not instead change in phase.

If the entity reconfigures its chemical bonds and changes in state, which is to change its energy density, then it will instead tend towards a final measured mass, M, that is less than F. When Brassica rapa is a seed, for example, its initial mass is 1.171 x 10-3 grams while its Franklin energy—i.e. the mass it would attain if it did not change in state and maintained the same energy content as an adult that it had when it was a seed—is 2.142 x 10-3 grams. This gives it a Franklin factor of K = 0.829. That is a measure of its transformation in that transition.

When we remove one joule of energy from our reservoir, it is tending to zero, A → 0. We then place that one joule into our entity. We now have the following possibilities:

1. It can increase its mechanical chemical energy while maintaining the same state and configuration. It then thrusts itself out further into the surroundings. The mendelity, and the population mass flux, M, both increase while the energy density remains invariant. As with the Ricci tensor in Figure 20.11, we simply have a larger expression of the entity.
2. The entity can instead increase its biopressure and diversify, as in the Weyl tensor of Figure 20.9. Chemical components are reconfigured with no changes in mass. It maintains the same mass, but has a denser energy configuration. It goes through its cycle and acts more like a bonsai tree that never increases in size.
3. The entity can respond with a combination of both the mechanical and the nonmechanical forms of chemical energy. It gradually gives us a range of values for both mendelity and biopressure, as well as in the population mass and energy fluxes, M and P.

If the population responds through either (B) or (C) above, then its Franklin factor decreases, and its Franklin energy—i.e. the additional mass it could support if it did not change in this way—necessarily and also decreases. The amount of diversity is being precisely measured by the decrease in the Franklin energy which is a decrease in the size it could maintain, thanks to the ongoing reconfigurations. It is thus a measure of the Gibbs energy.

We now have a proportionate value, and a rate of change, between F and M. And since the potential for further increases in energy density declines, then the projected final value for attainable mass over the population declines. The Franklin energy, F, approaches the actual and measured mass for that population to give FM. This simply means that all potential configurations in chemical components available to a given sample of biological matter are gradually being exhausted. And since the potential energy driving those changes is also gradually being exhausted, we also have A → 0. If two populations start with the same values, but then differ in M, then they will also differ in F which is a statement of their mechanical chemical energy and their power for thrusting out into the surroundings. They must also differ, and increasingly so, in their energy fluxes and biopressures.

We have now successfully replicated the Black experiment and measured all possible biological configurations and diversity within any population or biological inertial frame of reference. For every joule of energy we input that elicits a reconfiguration or change in visible presence, then the Franklin energy declines, and F—the hypothetical mass that could be attained—will ever more steadily approach M, the actual mechanical chemical energy observed, and as the population diversifies. This is a statement of potentialities and actualities. The greater the variations and diversity, the greater the difference between F and M, and also in all the measured energy densities, rates, and masses over the population. We can measure the energy that goes in, as well as the masses and the energy densities which are the exact numbers of molecules and how they are bound.

Since we now have both A → 0 and FM, we have suitably quantified all possible variations and configurations, and so all possible biological diversity. The A → 0 tells us how many joules are being taken on as we approach reproduction, while the FM tells us the constant pressure to constant volume conversion rate, or the difference between the internal and external energy expressions of populations and entities.

We have now quantified the complete allowed set of interactions, which is a resultant of both the curl and the divergence about the circulation. Our geometry has at last given biological populations a potential to match any in any other science. We can therefore state:

## The sum of all the paths that satisfy Law 2 constitutes the allowed set for the entity and its equivalents; while that which permits them to satisfy Law 1 constitutes the required set.

#### The law of reproduction

We must now deal with reproduction: biology's most distinguishing aspect. If we cannot incorporate it into our biological space of indiscernible points and internal energy, this exercise is fruitless.

Figure 20.29

We are still trying to prove that if some Body A is equal to some B, and that if B is equal to C, then we have A = B = C. Although it at first sight seems complex, we are basing our approach on the Riemannian geometry that Mirzakhani studied. Since it is a four-dimensional geometry, we must also picture it in a variety of ways. We need only count from zero to three, with the zero being the 0-sphere we see in Figure 20.29.

We can unravel evolution if we appreciate that our 0-sphere point also contains time. All possible circumstances are contained in a single 0-sphere, which is both a point and a state. It ranges from indiscernible point to indiscernible point and wraps up an entire population, incorporating both its beginning and its end, so we can compare them directly. That structure encloses the entire fourth dimension, and holds a dynamic population in a timeless state of reproducing and being reproduced. It gives us one of some objects, and three of others. We have altogether::

• a 0-sphere time-space envelopement which contains …
• three 1-ball line segment twirling batons,
• three 1-sphere arcing circular boundaries,
• three 2-ball disc interiors,
• three 2-sphere regional covering exteriors, and
• one 3-ball of a volume and spatial interior.

Our three twirling batons are our diameters and 1-balls. Their moving points describe the three dimensions. Their twirling outsides form the areas that are the 2-spheres that cover our biological space of internal energy. Their 3-ball volume interior is the sum of all the biological events for a circulation. They allow progenitors to produce their progeny. The whole taken together is the 0-sphere and point.

Figure 20.30

Figure 20.30 shows the impossible creationist and intelligent design template of the right helicoid that the twirling batons can describe for mass, energy, and numbers in a perfect Newton-style firmament that has no aberrancy, no evolution, and no Darwinian competition. The values for number, mendelity, and biopressure—n’, m̅’ and p̅’—are rigidly imposed. All the helices are even and parallel. They neither approach nor leave each other. Both absolute clock and relative generational time flow in a completely regular and predictable manner. No entities or populations vary away from the firmament's ideal templates. Since modern science began with the rejection of Aristotle's similar theories of motion, we shall now call any population that meets this firmament description an “Aristotelian population”. We shall eventually prove that it is impossible.

A mathematical aside

A right helicoid shares the same general formula x = ρsinθ, y = ρcosθ, z = ρtan α + kθ as all other helicoids, but always has α = π/2. Since α can never vary, then evolution is impossible. Aristotelian populations can therefore be represented by x = βsinθ, y = βcosθ, z = ρθ where β is the right cylinder's radius for that population, and ρ is its resulting pitch. As the populations go about the central axis and θ increases, then just as they would upon a spiral staircase, each population rises, but out at its appropriate distance, β, upon the helicoid; and so that z, the pitch, gradually increases from 0 to 2πρ for all of them. They are all therefore measured with the same turn of our polar planimeter and differ only in: (1) dρ/dt, which is the absolute time they each take to traverse the generation upwards on the vertical axis which is the pitch; (2) dθ/dt, which is the (lateral) speed at which they rotate about the central axis, and which is therefore the spacetime quantity of moles of molecules that form that medium and that is their genes, genomes, and overall sizes; and (3) dβ/dt which is the gradient of their rise, and so their felt differences in the medium's density and thickness, which is their energy intensity, or how they configure those molecules at each moment to formulate their appearances and behaviours.

Mechanical chemical energy does work externally. Nonmechanical chemical energy does work internally. The mechanical variety is due to motion or position, and is always measured relative to some environment or reference point. It is stated as the Helmholtz energy. The nonmechanical energy is, at its simplest, all energy not due to either position or movement. It is electrical, thermal, magnetic, electromagnetic, chemical and the like. It also requires that the environmental conditions under which it is being measured be specified. It is stated as the Gibbs energy.

Figure 20.31

Mechanical energy shows itself in Figure 20.31 in the soccer ball's ability to use its Helmholtz energy. It can convert its impact into the drive that keeps it going through the surroundings. The cushion, by contrast, expresses itself through its greater amount of internal and nonmechanical energy, which is its Gibbs energy. The Helmholtz energy is equivalent to a biological population's conversion of its Franklin energy, F, into its Mendel pressure, or mass flux, M, which helps it maintain its physical-material expression in the environment. The Gibbs energy features more heavily in any population with a greater degree of configuration energy and biopressure, , and that manifests itself more through a set of internalized transformations in its internal energy. Nonmechanical energy can nevertheless always be converted into, and equated with, the mechanical variety via a hamiltonian. So just as some balls can bounce higher than others, then if some Population A takes on mass at a faster rate than some Population B, then A must have the greater Helmholtz energy for it is doing more mechanical work into the surroundings. Its increased use of its internal energy manifests itself as a mendelity of , and is the force that populations use to acquire genes, genomes, and biochemical resources. The Gibbs energy potentials induce the soft cushion to do a greater amount of interior work than does the soccer ball, and so to absorb the impact into its reconfigurations. It manifests itself, in biology, as the constant volume interior work of configuring and reconfiguring. But this interiorized work eventually leads to the thrusting out into the world—through Helmholtz energy—of the resulting progeny. This conversion from Gibbs to Helmholtz over time is reproduction … which we must now account for.

Figure 20.32
Photograph of Chorthippus brunneus (common field grasshopper) courtesy of Alex Hyde Photography

Figure 20.32 displays the results of a five-year case study by the British field researchers Richards and Waloff undertaken in the early 1950s in fields around Ascot, Berkshire, United Kingdom (Richards & Waloff, 1954). They studied the common field grasshopper, or cricket, Chorthippus brunneus, a member of Acrididae, and a semelparous annual. They saw the overall population increase; then collapse under environmental stresses due largely to onsets of damp weather; and then recover to previous levels.

Chorthippus brunneus shows an example of the thrusting out of internal energy that we must account for. Its primary focus is the the BIDE equilibrium which is the constraint of constant propagation, φ, and the method of number, n. It is the essence of reproduction. It is the first of the three attributes we need for an indiscernible point.

As Figure 20.32 indicates, the Chorthippus brunneus reproduction is the corporate property of an entire collection of adult-plus-22-fertilized-eggs set circulating about a definite spacetime centre in biological space. The crickets oscillate around their equilibrium age distribution. Each fertilized Chorthippus brunneus egg starts out with only a 4.6% probability of lasting the course. This tells us that if they are to survive their journey through space and time, then each adult must produce somewhere around 22 fertilized eggs; and, conversely, each surviving C. brunneus adult requires that 22 eggs pre-exist it.

When the first egg in a 22-egg-set fails, the remaining 21 improve their odds slightly and have a 5% chance for success. When a second is lost, the probability increases to 5.3% over the others. The individual probabilities for survivor entities continue to increase until the sole progenitor in each 22-egg-set remains. Each individual cricket is thus the property of a set of joint interactions and probabilities involving mass and energy in interaction with the environment over historical time. Every population thus oscillates around some similar centre. Every biological entity is the physical manifestation of a probable path through a biological system or inertial frame of reference.

Figure 20.33

Every dimension in our biological space must have both a forwards and a backwards rate or the circulation cannot be completed. Figure 20.33 shows this as a complete helicoid or circulation collapsed onto itself. It is another view of our developing biological inertial frame of reference and its three methods of creating an indiscernible point. As also did Figure 20.8, it incorporates the Weyl and Ricci tensors. It confirms that there must always be some rate running counter to the others, for otherwise the circulation of the generations cannot curve about itself and complete.

The Weyl tensor is the length, and everything that leads to it, all about the circulation in Figure 20.33. The Ricci tensor is then the breadth, thickness, and volume, and all that leads to that. These together establish the gradients and the potentials in internal energy. Nonmechanical energy is then being absorbed and emitted on the vertical axis; mechanical energy is on the horizontal; and numbers are on the saggital axis. Time is longitudinally about the circulation. Those are the four dimensions. The “reproduction zone”—i.e. where progeny is produced and the indiscernible point is created—is visible at the front.

Figure 20.34

Figure 20.34 is yet another view of the same circulation of the generations. The twirling batons have traced out those contour lines. They are the forces in our biological internal energy. They show the interaction between the 0-, 1-, and 2-sphere exteriors, and their respective 1-, 2-, and 3-ball interiors. They are also measured by the linear planimeter. They are the results of the three constraints of constant propagation, size, and equivalence.

The contour lines represent the biological forces that create six varied populations. They also result from a single turn of their common polar planimeter. Four populations share that same centre. Two smaller ones are offset. That centre is a point in internal energy that we can use to make measurements so we can express any population in terms of any other.

Populations with larger circulations have large, overall, horizontal movements. They shift large quantities of resources and energy. If we, for example, take a 2-day time span and observe a blue whale calf, it will maintain larger overall values with smaller rates of change than will populations closer to the centre. Those instead face a much larger temporal gradient and will exhibit greater relative rates of change in time, working on smaller magnitudes. The blue whale's relative lack of change in its absolute values is in spite of it gorging on enough mother's milk to gain 91 kilogrammes, 200 pounds, in weight every day for its first year of life. However, its initial mass is also very high, relative to the mosquito. It weighs in at some six to eight tons. This is why its proportionate rate of increase is so small. Even at that high rate of consumption, the blue whale has a smaller relative rate of change, and takes about ten years to ascend, timelike, about the generation to reach maturity. Its generation length is 31 years as opposed to 4 days. So in spite of those huge absolute amounts of resources, this is a very small amount of abseleration and absity over that two day period. It is a mere τ = 0.000177 about the circulation. We do not see much in the way of relative change, because our chosen two day observation is a very small proportion of the blue whale's overall generation length.

A mosquito stands in stark contrast to the blue whale. It has τ = 0.5 over that same two day period. Where the blue whale maintains a high placement, the mosquito instead has a high displacement, and accelerates itself half-way around its far smaller circulation. Its overall consumption might be small, but its relative changes are large.

Any biological entity with a smaller mass or energy relative to any other immediately has greater values for its absement, absity, abseleration, and abserk in mechanical and or nonmechanical chemical energy. But in all cases, the placement times the displacement is unity for they are inverses.

Since placement and displacement, absement and presement, and all other such couplings are all multiplicative inverses, then it is a truism that they must hold constant—pd = 1—over all possible populations. But … their hamiltonians, which are additive inverses, also always hold constant. These two sets now combine to define each population. One defines the quantities, while the other defines the rates and changes.

A mathematical aside

Each circulation tells us how much work a population must do to carry itself about its generation length. Our placements, displacements, presements, absements and presities, absities and the like give us the sets of relative measurements directly along the 1-ball for each 0-sphere. We can measure those proportionate increments of mechanical and nonmechanical chemical energy and numbers for a directional derivative using the relative values we derived in the refutation as µ = dS = χ(dm̅/) + ΩT’(dV/V) = κ(dm̅/) + ΩT’(dP/P).

Whether a population is a mosquito positioned close to the centre, or a blue whale positioned far away, we can measure all generations in both absolute units t, and in proportionate generational time segments, τ. But since every population completes a circuit, then one dimension must always be moving in opposition to the others to circumscribe each generation.

Since we can describe our population with μ(nV), it will have the gradient μ = (μ/n, μ/, μ/V) which is the directional directive. But … this is just another vector, for it can be written as μ(P(nm̅, V)). An Aristotelian population is now not permitted to respond to μ/n. That is our potential aberrancy. We can measure it at every point to see if the Aristotelian variant is true.

We already know that the various population hamiltonians are the sums of their Gibbs and Helmholtz energies. Those hamiltonians, being statements of energy, record all potentials. They are conserved. They are the sums of any and all changes in energetic activities undertaken, and/or in work done; heat emitted; and so forth. It is therefore also always true that the sum of the hamiltonians for all possible populations will remain constant over their respective generations. We also measure those energy activities over the generations with our placements, displacements, absements, presements and the like. The rates of change between potential and actual, and between Gibbs and Helmholtz, will therefore be determined by these coupled inverses, the one additive, and the other multiplicative.

We can now bring these two different sets of constant values together to describe our indiscernible points and the surfaces and interiors they form. This leads to the Liouville theorem. This states that any population's hamiltonian will remain constant, no matter what its permutations. The forces exerted about the circulation boundaries—which form the Liouville phase areas and volumes—will therefore be preserved. This is the Liouville constant, L. It is the combination of (a) the hamiltonian sums, and (b) the multiplicative inverses that determine positions. And since all such pairs of corresponding values for absement, presement, and the like must also hold constant, then these two sets can be taken together. Since each is constant, then that constancy becomes the biological equivalent for the constancy of the speed of light, c. As with the speed of light in the Einstein theory, things can move faster or slower than the speed of light, but nothing can be brought to move constantly at the speed of light, for that requires infinite energy. This describes the right helicoid in internal energy, which is similarly impossible, and for the same reasons.

Figure 20.35
Original diagram courtesy of John Denker.

Figure 20.35 shows an ensemble of nine populations moving through the internal energy of our biological space under the influence of such constants. It presents only two dimensions, number and mass, and is yet another view of our four-dimensional biology.

There are nine populations going around the circulation. The number of entities or partitions in internal energy, n, is on the vertical axis of both graphs. The mechanical chemical energy contained in each of those entities is expressed as the number of moles of components per entity, , on the horizontal axis. We therefore have population size, and the current mechanical and chemical inertia. We have coloured two of the nine populations white to make their interactions in force and energy—and thus the behaviour of their internal energies—easier to follow.

The nine populations we are following in Figure 20.35 could be either (a) the same population over nine successive generations; or (b) nine different subpopulations, of the same species being observed simultaneously; or else (c) any combination in between. The populations seek the same equilibrium within the same 0-sphere indiscernible point. The four-dimensional surface is the circular arrow tending from A to B. It is the mean for the Biot-Savart law that the polar planimeter describes for every population sharing the same helicoid in internal energy. It is the polar planimeter measuring them all in Figures 20.7 and 20.34.

The left-hand graph in Figure 20.35 shows the populations' internal energy states and configurations at any time. The right-hand one shows the forces that lead to those states. The two graphs together give the values and rates for both generational and absolute time. They are the two-dimensional workings of the Weyl and Ricci tensors. I.e. they tell us all interactions between number and mass, n and m.

The two populations we have marked white in the complementary graphs are on opposite sides of the curved arrow that marks their mean population values for energy and force. Since the left-hand graph is the quantities of molecules that currently compose internal energy, while the right-hand one is the forces derived from them, it is easy to see that all populations that have below-the-mean and right-of-the-axis (or positive) values, in the quantities graph upon the left, have above-the-mean and left-of-the-axis (or negative) forces acting upon them on the right and vice versa. The one therefore pushes the other towards the mean; while the other reciprocates by pulling the other in towards the mean. These therefore counteract each other to produce the overall equilibrium.

The ensemble of nine populations on the left are all increasing in numbers of partitions in internal energy from A to B and towards their maximum. The number of entities in the population is increasing.

Simultaneously, however, the number of chemical components each entity holds , was at its maximum, but is now decreasing towards the generation average which is the origin and the axis. The entities' average individual masses or mendelities, , are therefore decreasing.

Population numbers will be a maximum when the populations arrive at B. However, entity sizes, or , will have decreased from a maximum towards the generation average. They will continue to decrease, beyond B, as mendelity continues to fall. We thus see, all the way from A to B, larger numbers of increasingly smaller progeny. This is both (a) reproduction, and (b) the constancy in phase volumes that is the Liouville theorem.

The graph on the right, for forces, shows that the force to increase in numbers of partitions in internal energy is at its maximum at A, but beginning to decrease, reaching its lowest impetus at the end of this A-to-B period, when it has the average for the generation. We also see that the impetus in was at the generation average at A, but was switching to the negative to cause the decreases we see in the q-state of moles of components held, with its minimizing efforts reaching their own peak at B.

The overall ensemble shape changes through t1, t2, and t3. Everything at bottom left in the graph upon the left is at top right in the one upon the right. Since the populations are inverted about the mean, those displaced from the mean so they enjoy either higher or lower values, and so that are furthest away irrespective of direction, will suffer the greatest forces to change and to move closer to their ensemble average. Populations of different sizes will exhibit forces of different sizes. They will use different quantities of mechanical chemical energy to compose themselves of different moles of chemical components within the same dynamical and environmental conditions. Their oscillations around those mean values are now measurable variations both (a) per population; and (b) per entity. This is the exchange in information that creates reproduction. Those are Darwinian variations.

Figure 20.36
Original diagram courtesy of John Denker

We can see the constancy that the Liouville theorem refers to in the two graphs of Figure 20.36. We see that the ensembles at A, when they are at t1 in the graph on the left, have a more open configuration than in the one on the right. We can see that the graphs on both left and right vary. The Liouville theorem is that we can add and/or subtract the areas on each side, and they will be constant.

The graph on the left in Figure 20.36 is the coming together of (a) the mechanical chemical energy or number of chemical components, and (b) the scale of the force of acquisition those components feel. This is both the row and the column marked II in our tensor brought together. The graph on the right in Figure 20.36 is the energy devoted to numbers at that same moment. It establishes the number of partitions for the molecules, which is ultimately the number of biological entities in the population. This is the row and the column marked I in the tensor. (There will of course be similar graphs at A for I and III, and II and III).

The graph for the ensemble number density upon the right in Figure 20.36 is more closed, but its two white marker populations are over on the opposite side from those on the left. They are forces at work that are sensitive to direction. They are always either increasing or decreasing placements and displacements with specific velocities.

Figure 20.37

We can grasp what all this implies from Figure 20.37, where we see the significance of the right-hand rule that this biological geometry imposes. It means our three twirling batons will only ever interact so they drive the population and generation in the one direction that already accords with absolute clock times.

As we saw in Figure 20.22, each required set simply walks across its own range, and so from its minimum to its maximum and back again. But that walk produces an area in some plane because each dimension walks in interaction with another to create an allowed set which is both an area and a force in internal energy. It is a complete area of interaction that is a number and type of molecules in interaction. It is also a 1-sphere surrounding a 2-ball.

By the right-hand rule of vector forces, the area formed by a required set walk always has its unit normal, or generation average, upon its left, as in Figure 20.37. The outwalk therefore ranges around higher values while the inwalk ranges through lower ones, relative to the unit. But as we again see in Figure 20.37, the inwalk is not only at a lower magnitude, it contributes negative values to the resulting divergence and area it establishes with that other dimension. The force is in the opposite direction. It is contributing a net negative force and momentum to the population relative to the point of measure. Since all its values are now negative—relative to the unit normal—they must be subtracted from the whole.

Since the two movements on each side of the required set walk are vectors, then their impulse, J, on one side is opposed to the impulse on the other. And since the impulses are opposed, then so also are the placements, displacements, absements, presements, and the like. When one is close to the mean the other is far; when one is approaching the mean the other is departing from it; when one is moving quickly the other is moving slowly; and when one is accelerating the other is decelerating. Therefore, the area on the right graph for the work being done through the force on number density, at t1 in Figure 20.36 must be subtracted from the similar area on the left graph that is busy establishing the total moles of molecular components. These are both vector forces, but working oppositely on their joint surface which is a step on the helicoid of internal energy.

The force for number density is currently increasing the number of partitions into which the components must be divided. However, the same net force in our biological internal energy is acting to drive the actual number of those components out of each partition. We therefore end up with a greater number of smaller biological entities for that collection of internal energy.

We can immediately identify one of the conditions we need so we can define reproduction in this biological geometry of internal energy. It is a declining mechanical constant pressure thrusting out into its environment, combined with an increasing number of partitions. This is a net decrease in the Helmholtz energy … which is in fact the “reproductive potential”, A.

We can closely identify this stage in the usage of reproductive potential, which is the Helmholtz energy:

1. It is when dn/dt ≥ 0. Numbers are either increasing or holding steady. If numbers are decreasing, we do not identify this stage.
2. It is when dm̅/dt ≤ 0. Mendelity is either decreasing or holding steady. Or alternatively, the moles of chemical components held per each entity, over the population, is either holding steady or declining. If mendelity is increasing, we do not identify this stage.
3. It is also when > 0 throughout. Law 1 of biology, the law of existence, must hold throughout. Mechanical chemical energy must always be positive so that there are always biologically active chemical components bound together within internal energy. If this condition is not met, we do not have biology.

We can see that these definitions are correct because when we get to t2, in Figure 20.36, the qs on the left are almost completely closed. The ensemble is crossing the axis. It is at the mean position; and its forces are also at the mean. There is therefore virtually nothing to add or subtract. Its net forces are either zero, or else cancelling. The moles of components held is at the generation average; and so also is the force to acquire them … but which is now about to increase. The two white marker populations are almost in line as they switch orientations. This again means that they are approaching, or are at, the unit normal or generation average. The entities may still be getting smaller, but the force impelling them to do so is gradually reversing itself to induce them to become larger.

Since a hamiltonian is involved, then a previous potential, held all the time properties were below average, is now about to be realized. The sum is constant. If one commodity increases, then the other must diminish, and conversely. And since the area on the left at this point is zero or very nearly so, then its matching area on the right must contain the balance of the potential and the actual—and the positive and the negative—set of values that is the most representative of the Liouville ones for the whole ensemble. But while the force over on the right for number of partitions into which the population's molecules are to be divided, may also be approaching the unit normal, it is doing so from further away, and also over on the other side. Where numbers were previously increasing, they are either now increasing at a very much lower rate, or else are slowly reversing. They will soon pick up speed in their rate of number losses, rather than number gains. They will surrender to the environment. But the hamiltonian is being preserved for the potential to eventually reverse this—which is to increase in their Helmholtz energy and their reproductive potential, A—is also increasing. The reproductive potential is the biological analogue of the Helmholtz energy. We now have dA/dt > 0.

We now continue to t3 where the ns have become even more open. The force acting is now at its minimum and switching, for numbers are just below or at the generation average, and again about to reverse direction. They have continued their decline in numbers, but simultaneously increased in the potential force that will eventually make them change direction, which is again the Helmholtz energy and reproductive potential, A. And while, relative to where they were at t2, the qs at t3 seem to have opened out, that more open area must now be subtracted from the much more open number area over on the right. Mechanical chemical energy is now about to realize a potential to produce larger biological entities. But the right-hand graph, for the number of partitions, tells us that this is via a decreased, and decreasing, number of entities. That number partitioning density has its own potentials and its own required set inwalks and outwalks, but the two conjoin and exchange these values to produce the vector circuit walks we see in Figure 20.37.

We now have our twirling batons. A large movement in one direction in this biological space of internal energy is always accompanied by at least a small movement in another, and generally in the opposite direction. The areas in Figures 20.35 and 20.36 will add and subtract through their various changes in shape and direction, due to their required set walks, and as in Figure 20.37. We have confirmed those twirling batons. At least one dimension always acts contrary to the others to complete the circulation. The number of partitions and the quantity and density of mechanical chemical energy are consistently opposed. And what holds for the mass:number pairings will also hold for the mass:energy and number:energy ones. All three interact to cause mutual exchanges of values, forces, and potentials around the generation mean.

The Liouville constant, or number, L, represents the total phase volume for a complete ensemble. It therefore also governs the behaviour of all molecules. If one property increases, another diminishes to offset it so that the overall phase volume—and therefore the combined molecular force and energy total—is preserved. The net result is that any expansion in one direction, along any coordinate, is immediately matched by a balancing shrinkage on its conjugate axis for that dimension and for our required set walks and Ricci tensor divergences. The ensembles are left completely free to explore their various trajectories, but they will consistently be within the boundary set by the constant force and energy defined by this internal energy space. Potentials always increase when quantities decrease; and conversely; while the system shuffles its forces, its energies, and so its molecules across its various ensembles. The Liouville theorem states that no matter how the ensemble or system changes or develops, the sum of the two areas in those two halves of Figure 20.35 for each coupling of two dimensions is preserved.

Thanks to the Liouville theorem, our 0-, 1-, and 2-spheres can always act as Weyl tensors. They are boundaries. They establish the overall parameters for their interiors. They form the indiscernible points. Their interiors are the 1-, 2-, and 3-balls that they each contain, and which are then the Ricci tensor.

The Liouville exchanges produce the values that define the journeys both up the columns and along the rows that create the indiscernible points for the generation that we see in the fully biological 3 × 3 Owen tensor of Table 20:2:

 Constraint of constant propagation I: φ Constraint of constant size II: κ Constraint of constant equivalence III: χ Numbers of entities maintained I: n Tnumber:number τnumber:mass τnumber:energy Body material of organisms II: M τmass:number Tmass:mass τmass:energy Observed energies and behaviours III: P τenergy:number τenergy:mass Tenergy:energy

The values on the diagonals in the Owen tensor are the normal pressures. They are the times taken for the three required set walks, in and out, for each generation. Those promenades or required set walks create the numerical, mass, and chemical equilibria of BIDE, Kolmogorov, and Hardy-Weinberg. The off-diagonal or shear values are the Liouville excursions that produce those times and normal pressures for the internal energy.

Those diagonal values of Tnumber:number, Tmass:mass and Tenergy:energy, where rows and columns intersect, provide the backbone of all biological activity. That first Tnumber:number component has the two shear components τnumber:mass and τnumber:energy next to it on its row. They are the materials it works with. It presents them as quantities in real time to the outside world. It does so along its row, and through the linear planimeter as its 2-ball divergences. The two shear components τmass:number and τenergy:number beneath it, in its column, are the spanning of the polar planimeter. They set its sectors and its regions across the generation length as they establish the curls which are the rotations, rates and times at which the partitions and molecules in internal energy should be presented.

We see that the Tmass:mass component has τmass:number and τmass:energy on its row. It presents those to the outside world as its divergences, through the linear planimeter. It has τnumber:mass and τenergy:mass on its column, which establish its rates, rotations, and curls over time.

We observe more closely that τmass:number is under the simultaneous influence of:

• the polar planimeter in the numbers column which uses it to set times and rates for a generation; and
• the linear planimeter in the mechanical chemical energy row which uses it to establish masses in real time.

All the shear components have this double capability. The left-hand index is the row and the linear planimeter in real time, while the right-hand index is the column and the polar planimeter curving over a generation. The normal pressures, where rows and columns coincide, provide values for both planimeters simultaneously.

The various values from the baton twirls fill the six off-diagonal shear stresse, which all have this double capability. They all go poloidally, which is in time, up the helicoid axis and provide the internal energy. This is also for each position, in sequence, in the column of designated activities. These are again the intersections of the Weyl and Ricci tensors.

The 0-sphere for an indiscernible point is a Weyl tensor covering. It is all biological activity for the generation. That 0-sphere has its interior: its 1-ball Ricci tensor. This governs a distribution throughout the 0-sphere. That 1-ball interior, which is the three central values, is the sum of the in- and out- walks. It is the distribution of volume elements within the 0-sphere. It creates the indiscernible point through the ∫dN = ∫dM = ∫dP = 0 moving upwards through all columns. They are the pitch for the three components Tnumber:number, Tmass:mass, and Tenergy:energy that are the Owen tensor's backbone. They are the three twirling batons. As each entity measures itself at the beginning and end of each interval, those T's are the required set walks. They are the times it takes for its allowed set configurations to return to their distinct indiscernible points, and so to go through a complete circulation of the generations.

Figure 20.38

The four-dimensional biological space that surrounds the three-dimensional one holds the values from, and for, the in- and the out-walks and baton twirls that in their turn fill up the entire three-dimensional volume. The 4 × 4 Haeckel tensor that similarly surrounds the 3 × 3 Owen one holds the rates and times for each distinct moment in time, t. Each such distinct moment is a slice across the four-dimensional Haeckel tensor. It then produces the 3 × 3 Owen one of Table 20:2.

The sliced edge in Figure 20.38 is a tangent, at that time t, to the four-dimensional tensor … that is itself the 0-sphere that forms an entire generation. Being a slice and a tangent, the front facing edge tells us the rates and energy intensities at that moment. Its leading face holds the forward-pointing normal that tells us how rapidly it thrusts itself forwards into the three-dimensional space it creates at each moment. That normal establishes the values and rates for the various properties.

The diagonal that is the Ricci tensor and the interior 1-ball to the surrounding four-dimensional 0-sphere manifests itself in the surrounding physical world as a biological combination of (a) Tnumber:number which is an equilibrium age distribution population; (b) Tmass:mass which is a Lotka-Volterra or Kolmogorov equilibrium; and (c) Tenergy:energy which is a Hardy-Weinberg style equilibrium. Those are the twirling batons that are the three promenades or required set walks. They make spacelike contributions to the breadth of the steps in internal energy about their helices through the Ricci tensor; while also establishing timelike rises and overall rates and gradients through the Weyl tensor. The three together are again the normal pressures that create the overall length, T, for the generation. Their shear pressures are the observed biological phenomena that distribute themselves over those times.

The changes that pulsate through the Owen tensor move biological properties from point to point and moment to moment, and are responsible for the variations Darwin described:

It may metaphorically be said that natural selection is daily and hourly scrutinising, throughout the world, the slightest variations; rejecting those that are bad, preserving and adding up all that are good; silently and insensibly working, whenever and wherever opportunity offers, at the improvement of each organic being in relation to its organic and inorganic conditions of life. We see nothing of these slow changes in progress, until the hand of time has marked the lapse of ages, and then so imperfect is our view into long-past geological ages, that we see only that the forms of life are now different from what they formerly were.

In order that any great amount of modification should be effected in a species, a variety when once formed must again, perhaps after a long interval of time, vary or present individual differences of the same favourable nature as before; and these must be again preserved, and so onwards step by step (Darwin, 1869, p. 96-97).

The Owen tensor has three intersecting rows and columns. There is one for each biological dimension concordant with this physical space. The equilibrium age style distribution is the required set walk that encodes Tnumber:number. It uses the shears that accompany it in its row and column. It establishes both the constraint of constant propagation, which is vertically in its column; and the numbers of entities at any time, which is horizontally along its row. Its one requirement is that the number of partitions—or biological entities—at the beginning and the end of a suitable interval match so as to create the indiscernible point in number. The same pattern holds for each of Tmass:mass and Tenergy:energy with respect to their own rows and columns. The former establishes the generational oscillation in numbers and types of molecules to be partitioned, while the latter does the same for their configuration energy. Those are rules in this geometry.

Since the Weyl and Ricci tensors are four-dimensional, they state complete timeless states. They are indifferent to whatever arcane conditions the Newtonian-style firmament of creationism and intelligent design might care to impose. They are indifferent to that proposed regularity of absolute and linear clock time. The four-dimensional states are instead a set sequence of events which are free to flex and to stretch in their fourth dimension, which is measured along the curve all about Figure 20.38. An entire generation can therefore seemingly expand or contract at different rates when measured in absolute clock time, and as the apparent density of materials held at any time changes, through the Ricci tensor's volume elements, to satisfy the description for an indiscernible point that holds still in that indifferent four-dimensional equilibrium.

To requote Minkowski:

A point of space at a point of time, that is, a system of values, x, y, z, t, I will call a world-point. The multiplicity of all thinkable x, y, z, t systems of values we will christen the world.

Hermann Minkowski, 1908. Quoted in: Volkert, K., From Legendre to Minkowski–the History of Mathematical Space in the 19th century.

Instead of the x, y, z, and t of physical space, we have n, M, P, and t for our internal energy of biological space and its system of values. The system of values is measured both vertically through the columns for the generation, and horizontally along the rows for intervals of both clock and generation time. But whether measured vertically or horizontally, they are sensitive to directions and to magnitudes, and form a weighted average. That weighted average involves the columns. These are independent of the absolute clock time measured along the rows. Their weighted average is instead a statement of a time-independent sequence of events involving numbers, mass, and energy.

The constraint of constant size, κ, is centred on the Tmass:mass in its column. That is the result of its required set walk in mechanical chemical energy. That is the Helmholtz energy which is the reproductive potential, A. But as Figures 20.8 and 20.33 show, it is a balance of the varied forces, areas, and divergences created by the four shear components on its row and column. The absolute time for Tmass:mass is thus built up from the four sequences τmass:number and τmass:energy on its row, and τnumber:mass, and τenergy:mass in its column. The overall absolute time Tmass:mass is the result of navigating those four sets of configuration changes, which are the allowed sets and areas in real time. When this pattern is extended to all three dimensions, it gives the six interlinked configuration changes that are the six off-diagonal shear components in the Owen tensor.

The three values Tnumber:number, Tmass:mass, and Tenergy:energy upon the diagonal are the normal pressures. They are where a population measures its properties. They are the result of its various sequences after some interval. Those values also come from our three baton twirls which represent those promenades. They are its areas and its forces, its volumes and its energies. Each of the resulting six off-diagonal components moves each of the three dimensions through a set of configuration changes. The three normal pressures summarize those changes.

This is a four-dimensional geometry. The six off-diagonal shear pressures present a sequence of configurations. Those sequences are time-independent presentations of partitions, materials, and transformations. Any population wishing to be biological must configure them at some rate. Each will use its own set of components, distributions, and timings in biological internal energy. The weighted combination of those changes is the absolute time, T, needed to construct both that entity and the population. Each row-column intersection therefore brings all three sets of dimensions and their inverses together. One sets the sequences; the other the rates. The entire diagonal is therefore a summation and a boundary for all those off-diagonal and two-dimensional activities. It is the 1-ball to the 0-sphere.

As the three interlocking spheres in Figure 20.29 try to suggest, the tensor diagonal performs a double function. It is both (a) an exterior, and (b) an interior. It is both (a) a sphere, and (b) a ball. It is both (a) a boundary, and (b) a summation. It is both (a) a Weyl tensor, and (b) a Ricci tensor.

The diagonal to the 0-sphere is a 1-ball interior with respect to that 0-sphere … but … it is also a 1-sphere surround. It has its own 2-ball interior. The 1-sphere thus forms a boundary to all its 2-balls which are the six shears. So although that diagonal is a 1-ball interior relative to the 0-sphere; and although that same diagonal is a Ricci tensor to its 0-sphere; it is in its turn a boundary. It has its own interior. It is a Weyl tensor and a 1-sphere to the 2-balls it surrounds. That diagonal is therefore both an exterior and an interior. It is both a sphere and a ball. Those are the rules in this geometry.

The six off-diagonal areas and configuration components then do the same double act. They are both a 2-ball Ricci tensor interior to the diagonal; and a 2-sphere Weyl tensor exterior. They are the 2-sphere covering to the whole tensor of nine components. They thus provide the volume elements observed in this reality as biological entities and populations.

Every biological population uses its six, off-diagonal, two-component configuration interactions for its areas and divergences. They are its flux densities. They establish the 2-ball disk interiors to the 1-spheres bounding them that are their forces. They then act, as in Figures 20.35 and 20.36, to create the Liouville theorem behaviours. Those are presented to the external world as a designated series of biological events along the rows, and so as the 2-sphere boundary to the full set of nine tensor components. The observed ecological-biological events we see around us are then their direct 3-ball expression of populations of biological entities, complete with their masses and their energies.

Figure 20.39

The 0-sphere we see in Figure 20.39 is the set of three baton twirls. It is the full pitch, radius, and thickness moving about the helicoid. It is the complete circulation of biological entities moving poloidally, meridionally, and toroidally. It is the entire 3-ball or volume interior that creates the totality of the indiscernible point that is a biological generation.

We can deconstruct the four-dimensional 0-sphere represented in Figure 20.39 into its 1- and 2-spheres, and its 1-, 2-, and 3-balls. The 1-spheres are any bounding line on the surface, with the 2-spheres being the regions so bounded. The 1-ball is any length, the 2-ball is any disc, and the 3-ball any volume of mass and energy. That 3-ball is the complete set of biological events that are the indiscernible 0-sphere point that is the entire generation. Those are the rules in this geometry of biology and of internal energy.

Figure 20.40

Since this is a four-dimensional biological geometry, we view it in Figure 20.40 as the baton twirls produce the cylinders formed from the divergences which are the moving areas in time. We see them progressing helically in Figure 20.40.A. There is one such cylinder in each dimension. The shears provide the sequences that create the population numbers and the mass and energy fluxes. They are the areas and the divergences. They are also the lateral movements across the helicoid. They are its meridional aspect. They form the rotating discs or 2-balls. They also have 1-sphere surrounds. They circulate about the helicoid axis, creating the pitch. The two together pull the whole upwards for both the poloidal and toroidal aspects. The combination of the poloidal and the meridional is the toroidal corkscrewing through space and time.

The cylindrical areas rotate about their respective generational means. This has two effects. It firstly creates the volumes of energy that are the flux densities of divergences for the ball interiors. But it secondly creates the curls or circulations per unit area. We thus see, in Figure 20.40.B, each of the three biological dimensions producing its corresponding sphere exteriors and ball interiors. The areas create volumes as they rotate, poloidally, all about their means over time, and so from past to future. These are the 1-sphere and 2-ball, and the 2-sphere and 3-ball we again see in Figure 20.40.B. That is the entire three-dimensional phase energy volume from Figure 20.39 … and that is the Liouville theorem. These are all rules in this geometry.

The Liouville theorem, which is additive, insists that phase volumes hold constant. So if a Population 1 has the description P1(n1q1); and if a Population 2 has P2(n2q2); and if they both satisfy the theorem; then they must share the same Liouville phase volumes. They must also share the same boundaries for their forces. We can call the equality PL(nLqL). So we now have P1(n1q1) = P2(n2q2) = PL(nLqL). All possible populations and combinations, no matter what their differences in times and rates, share the same PL(nLqL). Their volumes are the same. We again have an additive constancy.

A mathematical aside

If the various flux densities in Figure 20.40 are equal so that PL(nLqL) always holds good, then the areas, volumes, and forces acting all about their boundaries are also equal, meaning Green's and Gauss's theorems are also holding good.

We now have R n • dm̅ = W nm̅dt from Gauss' theorem or the divergence theorem. But the populations must also satisfy Green’s theorem which links lines to areas. This is that both:

C m̅ dn + n dm̅ = R ((∂/n) - (∂n/))  dτ,

and that

C n dm̅ = -C m̅ dn = R   dτ. The same three hold between n and .

The Liouville ensemble is only a volume. It cannot operate without its driving forces. It cannot operate without its collection of fluxes, divergences, and curls. These shape it in space and time, and produce the entities, their morphologies, and their ecology.

If the various populations in a Liouville ensemble have an equality of behaviours, then they must also have an equality of forces. The Liouville phase volume, L, is therefore also a statement about the joint forces experienced by all members of the various populations and ensembles. This equality is guaranteed by the “Laplace operator”, ∇2.

The Laplace operator or laplacian is named after the French mathematician and mathematical physicist Pierre-Simon Laplace who produced the nebular hypothesis: the theory that our solar system emerged gradually from a vast incandescent gas rotating about its axis. As that gas cooled, it contracted and rotated ever more fiercely about itself. Successive rings broke off to condense, separately, as the various planets. Laplace had to describe his gas accurately to make the nebular hypothesis believable, which meant calculating his proposed gas's interior forces. He therefore had to develop a potential—his ∇2—that could state the intensities of forces and behaviours in every direction, and at every moment.

The laplacian states the equalities of forces and behaviours in the Liouville ensemble. If a substance is experiencing a force, then the laplacian states the net rate at which the object being worked on moves towards or away from some point. Since the members of a Liouville ensemble must stay together, then the laplacian over them all must be zero. That zero—i.e. ∇2 = 0—guarantees that their forces and behaviours are the same. There are then no arbitrary sources of materials or energy in the neighbourhoods or volumes they occupy. No matter how they interact, the volume totals for PL(nLqL) remain unchanged. So if a first member of the ensemble grows, then all others must grow in similar ways, leaving the ensemble relations the same. But there will soon be a complete ensemble in which they all do the opposite to preserve the mean. And if a second member either increases or decreases in energy density, then the others will do similarly, again leaving everything stationary overall. No population may push either itself or others out of L—which is their joint Liouville phase volume—by doing anything to change those shared values, for their forces seek a net equilibrium. When the laplacian is zero the net force or flux any two populations donate to or impose upon each other is zero. All such populations are equivalent at every moment t over T all about the circulation. They are then guaranteed to be members of the same Liouville ensemble for they will neither accelerate away from others, nor push others out with their independent forces or activities. Since a laplacian of zero guarantees that the equilibrium is being maintained, then no individual population can nudge the ensemble away from its overall Liouville values and means.

The 1-sphere we see rotating and moving poloidally and toroidally in Figure 20.40.A bounds a 2-ball flux or divergence pulsating about the generation mean. It respects the laplacian. It establishes the population's curl or circulation density: the amount of activity in time. The three dimensions come together to make the 2-sphere and 3-ball in 20.40.B. Every combination is simultaneously a covering and an interior. Each therefore makes its contribution to both the Weyl and the Ricci tensors, and so to the circulation that is the 0-sphere of Figure 20.39.

If we consider the dimensions in pairs, then:

1. Mass and number, M and n, come together as τmass:number for the divergence which is the mendelity, . The inverse is τnumber:mass which is the curl and the toroidal segment that states the rates of change and the proportionate contribution to the helicoid.
2. Energy and number, P and n, come together as τenergy:number for the divergence in nonmechanical energy which is the biopressure, . The inverse is τnumber:energy, which is again the curl and the toroidal rate of change and proportion of the generation.
3. Mass and energy, M and P, come together as τmass:energy to create a divergence or flux density that is the visible presence, V, and as the inverse, τenergy:mass, which is the curl or rate of change and the work rate, W.

The Liouville theorem handles whole ensembles. It does not specify the individual behaviours of either populations or entities. The various fluxes and behaviours it oversees must somehow be allocated matching rates and durations.

Figure 20.41

Where the Liouville theorem is additive and handles whole ensembles, the Helmholtz decomposition theorem is multiplicative and can handle the individual fluxes. It can guarantee the uniqueness of all fluxes. If the fluxes concerned are L, then the Helmholtz decomposition theorem insists that all the fluxes that contribute to any Liouville ensemble can be decompsed into two parts: (a) a divergence of L, and (b) a curl of ∇ x L. The divergence in the mass flux is ; that in the energy flux is ; and the curl in both is fixed via and/or MP. These are all fixed via τmass:number, τnumber:mass, τenergy:number, τnumber:energy, τmass:energy and τenergy:mass.

But as we see in Figure 20.41, while the Helmholtz decomposition theorem can guarantee the uniqueness of all fluxes, it cannot guarantee quantities:

• The divergence or flux density or per unit volume is the same at bottom and the top but the flux amount is different.
• The curl or circulation per unit area—i.e. the amount of crust we would get per area of pie—is again the same between at bottom and top, but the actual circulation or circumference has grown.

The Liouville and Helmholtz decomposition theorems can each individually guarantee uniqueness. But although the former can specify quantities, it cannot specify rates; and while the latter can specify rates, it cannot specify quantities. We can bring the two together by specifying the generation time, T, which we can do through Tnumber:number, Tmass:mass, and Tenergy:energy on the tensor diagonal. We then use the rates from the Helmholtz decomposition theorem to fix the quantities for the Liouville ensembles. The Helmholtz decomposition theorem then guarantees the uniqueness of species over the entire range of values that the Liouville theorem defines. All the ensemble members can then increase and decrease their properties, while maintaining their overall phase volume, L, which is at all times a flux total with both a divergence and a curl.

The 0-sphere and its diagonal establish the Ricci tensor which is the Owen tensor diagonal and the magnitudes over all three dimensions. This simultaneously establishes the Weyl tensor and so the rates, curls and divergences for the population magnitudes. That diagonal therefore holds the unique three values that define every species. Those three values brought together produce a “contraction” of the Ricci tensor known as the Ricci scalar, R.

The Ricci scalar is the weighted average formed from the 1-ball diagonal that is also the complete set of required set walks for any population. It is the single weighted value that states exactly how all its tensor components are behaving at any one time. It is unique to every tensor. If two populations have the same Ricci scalar at all times, then they abide by both the Liouville and the Helmholtz decomposition theorems. Since the Ricci scalar is also the Liouville constant, they always have the same material and nonmaterial quantities and intensities. They are the same population and the same species. By the theory of tensors, if two tensors share the same Ricci scalar, then they are the same tensor. The Liouville constant is therefore also the Ricci scalar, and it is the single number that guarantees that every species is unique.

Since the Liouville constant establishes the phase volume; and since it is identical to the Ricci scalar which establishes the identity of the relevant tensors, then we can call it the “evolutionary potential”, η. We conducted our Brassica rapa experiment to calculate its evolutionary potential. We measured it as η = 1.063 x 1013. By the mathematics of tensors; by the Liouville theorem which guarantees the molecular behaviour of the Heisenberg uncertainty principle; and by all the laws of science; then no other population can have that value without also being B. rapa.

We have now accomplished the following:

1. We have used the Liouville theorem to prove that reproduction is the return to an indiscernible point on a helicoid of internal energy.
2. We have used the Helmholtz decomposition theorem to prove that every species is defined with three, and only three properties: the divergence and curl which are the mendelity, m̅’, and the biopressure, p̅’, maintained over a generation, along with the generation time, T.
3. We linked both the above to the Ricci scalar to prove the uniqueness of all species through a single value, which is the evolutionary potential, η.

We have now accounted for the entire circulation of the generations. We have in particular demonstrated that there is a period when the mendelity or average individual mass over the population is declining so that the size of the entities is decreasing below the generation average, albeit always remaining positive absolutely, and so that (dm̅/dt ≤ 0) ∧ ( > 0). But we have also shown that, although the mendelity is declining so that each entity is thrusting itself into the surroundings with less force, that individual mechanical chemical energy decline is matched by a countervailing population-wide increase in numbers. And since the total hamiltonian must be preserved, then the Helmholtz energy, which is the reproductive potential, A, increases, per each individual, all through that decline in mendelity. In other words, they increase in their potentials. And because of that increase in potentials, the individual entities will shortly thrust themselves back out into the environment. This is all achieved because numbers are turning positive and increasing even while mendelity is declining. So, at those times of declining mendelity; and in spite of the decreasing entity size; both (i) the reproductive potential, A, and (ii) the number of partitions, n, are simultaneously increasing to give (dn/dt ≥ 0) ∧ (dA/dt > 0). This is an increase we seek in the number of biological entities maintained. We can render all this in words as:

## In the allowed set is at least one path such that mass is surrendered, and such that a further entity possessing the required set, and satisfying these four laws, results.

#### The Franklin cycle

Now we have our complete set of laws, maxims, and constraints, we can focus on the real world. We can set about proving that creationism and intelligent design are false because they do not respect the doctrine of A = B = C. The key is that we can only define a species by bringing together the Liouville and the Helmholtz decomposition theorems. The difference is that the former establishes quantities but not rates, while the latter establishes rates but not quantities.

We are using our helicoid to represent every population's journey through reproduction. This gives (i) a toroidal or circulating movement which is a combination of the poloidal or the vertical through time, and so from pole to pole; and (ii) the meridional or lateral, which is the energy and resources used to fuel that vertical journey. Those latter and lateral movements determine the radius and thickness of the torus that is then the entire helical and circulating journey from one indiscernible point to another. Every biological population and generation is thus composed of a set of twirling batons which form the helicoid surface that is that population and generation's set of interactions with the environment.

The 3 × 3 Owen and 4 × 4 Haeckel tensors encapsulate the key to our position, which is that everything is relative. But although Einstein might have refuted Newton's ideas on the surrounding firmament, he did so using Newton's ideas on force, distance, and therefore energy. Those twirling batons carry the forces, distances, and energies that create the three 1-ball interiors that are both (a) the Owen tensor diagonal and the values that establish both the Liouville ensemble and its constant; and (b) the two properties of divergence and curl that define all biological fluxes. Taken together they are the Ricci scalar that create the indiscernible point that in its turn contains all that population's time, forces, energies, and events.

We can direct our attention to the interactions all biological entities must have with the surrounding material and non-biological universe. They must take molecules in; transform them; return them to the environment; and then repeat with a new tranche of molecules.

Our first law of biology, which is the law of existence tells us that all biological entities and populations must convert energy into usable form so they can maintain themselves. Those conversion processes are the work interactions of those twirling batons. They are the 0-, 1- and 2-spheres that are the surface interactions with the surroundings. They produce the 1-, 2-, and 3-balls that are the body interactions the entities use to survive. The four-dimensional Weyl and Ricci tensors are where the two interact.

Every biological population is immersed in a universe in which it can maintain itself. Every individual entity can engage in whatever interactions will allow it to survive. Each simply receives and transmits the forces and energies it requires. Each population's helicoid of internal energy, in our biological space, therefore looks the same as any other. This is the biological version of the “principle of Copernicus”.

By the principle of Copernicus, as it is expressed in ordinary physical space, the universe as seen from just outside any one galaxy looks the same as that as seen from just outside any other. No matter how different each local galaxy's skies might look close to each of their individual suns, their night skies, as viewed from just outside them, are all the same. The same space with the same potentials permeates everywhere in the same and shared cosmos.

The uniformity in space with respect to viewing angle means that the surrounding cosmos is “isotropic”. And since the average density of galaxies and large scale matter is the same everywhere, then the universe is “homogeneous”. This prevailing isotropic and homogeneous nature means there is no favoured location or observer. No planet or galaxy can call itself the centre of the universe. This is the “cosmological principle”.

The biological version of the cosmological principle means that our batons twirl evenly with respect to every entity and species. Since each population and each generation survives, and so is orthonormal—i.e. uses its own self as its unit of measure—then every entity and every species sees exactly the same universe. Every species sees a universe around it in which it has everything it requires to maintain itself. That is the same for every population. Each is therefore surrounded by whatever 0-, 1-, and 2-spheres allow it to interact with the universe to create the 1-, 2-, and 3-balls that provide it with its force and energy.

Relativity nevertheless warns us that the absolute clock time, t, of the surrounding and nonbiological world does not flow evenly. If we call the past moment t-1, the present one t0, and the future one t1, then Einstein's theory gives us notice that we are far too ready to assume that time flows evenly past those moments. The duration we measure as the past, i.e. between t-1 and t0, does not have to be the same as the duration we will measure as the future, and so between t0 and t1.

Figure 20.42

Figure 20.42 shows our twirling batons from Figures 20.1 and 20.29, as well as the three-dimensional structures of memes and genes they create as they twirl, and that we first met in Figure 0.4 in ‘Before We Begin’. Those ellipsoids, made from the mass, energy, and number that create each population, show that our batons do not always twirl evenly. Biological space and its internal energy are not always isotropic and homogeneous. They are not spheres. Those ellipsoid shapes suggest the slight clumping of biological energy that is similar to that around our own galaxy, with its local concentrations of force, energy, and matter. Just like physical space and time clump to make the occasional galaxy, so also with the internal energy in our biological space. Those clumps are populations and species.

Our three 1-balls stretch across our 0-sphere from one indiscernible point to another. Reproduction, in our biological geometry, occurs over a measurable distance. That reproductive distance wends its way about our helicoid and our 0-sphere. It represents the work done on a specified quantity of biological internal energy to convert progeny into a set of progenitors that can then do the same.

Figure 20.43

Every population requires distinctive numbers of entities, and distinctive quantities of materials and energy. The work to create a 0-sphere, and so to reproduce a population, requires a definite number of entities. Every biological population, and generation, has its twirling batons. They form a surface that is their interactions with the environment. This in its turn creates the three 1-ball interiors that are (i) the Owen tensor diagonal; (ii) the values for the Liouville constant, and (iii) the values for the Ricci scalar. Each population, and each generation, will therefore have its own 0-sphere. They create its indiscernible point. That must then be distributed across its unit reproductive distance. Each generation is thus a specified distance travelled in biological space as the conjoined poloidal, meridional, and toroidal aspects of the circulation of the generations that we again see in Figure 20.43.

We have defined an ideal motion in our biological space—i.e. the simplest possible—as one in which the three batons twirl evenly. They maintain the same lengths at all times for a perfect sphere and a right helicoid. This keeps the rate, or ratio, between the absolute and the generational times—T and τ—constant. Biological events then create the circulations of the generations in an even and continuously repeating fashion from one generation to the next. This is the straight sides of the three curves for the baton twirls which are the sides of the right helicoid in Figure 20.43.A. It has the shape of the thread of a screw, winding and carrying biological internal energy forwards through time and space. As in 20.43.B, absolute clock time then passes evenly upwards along the arrow of time while the sequence of biological events passes evenly around the circle in 20.43.C. The former measures t as the clock time while the latter measures τ as the circulation of events that are the generation length. The two together are the journey from one indiscernible point to another, and the timeless repetition of states for our four-dimensional structure.

Einstein gives us further notice, with his theory of relativity, that we are far too ready to assume that generation distances, τ, also flow evenly. The measurements we made in our Brassica rapa experiment show that it is perfectly possible for two generation lengths to differ in the amounts of absolute clock time they each take to complete a generation. Ours varied between 28 and 44 days. If we now call the past point in a given generation τ-1, the present one τ0, and the future one τ1, then the generation distance τ-1–τ0 does not have to be the same as the generation distance τ0–τ1. So not only can clock times, t, vary externally as measured on clocks … generation distances, τ, can also vary internally and within biological and reproductive processes.

Figure 20.44

Since clock and generation times, t and τ, can both vary, then as in Figure 20.44, a helicoid can have a wide variety of shapes. Its exact shape depends on its pitch, and the proximity of its points and lines. If either or both of the absolute clock and relative generation times changes, then the helicoid will change its shape and orientation.

The right helicoid of 20.44.A has a regular shape with parallel sides. The conical ones of 20.44.B and C differ by having definite leans. They do not point to the same potential places in the firmament. They have aberrancies and variations which will be distributed throughout their populations and generations. But as in Figure 20.44.C, a population can maintain a right helicoid as an average summed over many generations. That is a statement of its Liouville ensemble. We call any population that can maintain such an average a ‘Franklin population’. We must carefully distinguish it from an Aristotelian population, which maintains a perfect right helicoid at all times. We will soon prove that every real population is a Franklin one, and that the Aristotelian variety is impossible.

The absolute clock time that the surrounding nonbiotic cosmos tries to impose must contest with the biotic sequences that define biological events. Biological entities and populations strive to go toroidally around the circulation as the biological time τ. They must, however, do so in a given clock time, T, which is poloidally. They must also be measured meridionally, and so through a given amount of resources and energy.

The time a population measures against a clock, to complete its generation, is itself (a) the sum, or the integral, ∫, of many individual infinitesimal moments, dt; and (b) is simultaneously the infinitesimal increment of the generation length, dτ. These two together account for (a) the differences in the vertical and the horizontal in the tensor, as well as (b) the differences between the local and the global versions of events. They are related via dt = Tdτ where T is some given generation length.

Our batons do not have to twirl evenly … and generally, do not. At the very least, solar flares and weather and climatic phenomena make one absolute clock moment different from the next. Absolute time may link the biotic and non-biotic worlds, but its effects are not evenly distributed throughout the biotic one. Those effects will then be recorded both globally and locally in the rows and the columns of the Owen and Haeckel tensors.

We need some way to record the overall speeds and accelerations our twirling batons are subjected to, so we can understand the forces imposed on biological populations. This involves (a) their velocities; (b) their accelerations; and (c) their jerks, which are any ongoing changes in those accelerations as they shift from one speed of twirling to another. The twirl of a 1-ball in time gives us the circulation length c, which is a 1-sphere and some part of a circulation. Those are the rules.

Every moment of clock time, t, that passes is also some distance, τ, about the circulation of the generations. We can bring them together and measure the overall circulation distance, c, if we use the method Newton and Leibniz recommended. It is the length all around our 1-sphere.

Figure 20.45

Given the inevitable interactions between the linear absolute clock time as imposed by the surrounding cosmos, and the relative circular biological time each population must create over successive generations, then our generation distances are unlikely to be ideal. They will instead have the irregular shapes we see in Figure 20.45.A. But we can measure them all with a combination of Newton's tangent straight line and Leibniz's osculating circle.

We first pick some random moment, t0, as the basis to make our measurements of populations and their surroundings. That random t0, acting as our basis, will allow us to observe, and to reconcile, the population's rate of transformation from the linear and absolute form of time to the circular and relative generational variety. We therefore need some way to use our basis to measure both (a) the linearity of absolute time; and (b) the circularity of the reproductive cycle. This is the combination of tangent and osculating circle.

We can measure both this linearity and circularity if we measure the population and its activities from a moment just before our basis of t0 to a moment just after. We in other words measure from t-1 to t1, and so right across our basis of t0. Absolute clock time tries to move straight ahead, from pole to pole, and so is tangential and linear. Biological time, however, seeks to move from indiscernible point to indiscernible point, and so tries to curve from τ-1 to τ1. The former is an infinitesimal increment in clock time and is dt. The latter is an infinitesimal increment in the circulation and is dτ. The two together produce the observable stretch of circulation or generation distance, which is the biological activity over the period, and about the helicoid. We denote that Δc. (That Δ means ‘a small amount of’, so that Δc then means a small segment of the circulation, c).

Since the circulation is being tugged both linearly by absolute time, and circularly by biological time, five minutes in any one generation need not be the same as five minutes in any other. A given set of biological events need not always take the same amount of absolute time from one generation to the next. Growth spurts and the like are always possible. We are likely to get different values every time we use a clock. This makes using one untrustworthy. We can nevertheless find the circulation's exact length if we follow the method Newton and Leibniz recommended.

Figure 20.45.B shows us focusing on a small segment of the circulation. We are determining the separate influences of the absolute and the biological kinds of time. We again designate our segment as Δc. It ranges around the present moment t0, and from a moment just before to one just after, which is from t-1 to t1 all around it, just as Newton and Leibniz suggested. The question is, how long is that amount of biological activity?

We can slice our circulation into a whole host of tiny consecutive triangles, each of length Δc. As in Figure 20.45.A, the entire circulation will then be the sum of all those Δc segments from the many consecutive triangles we draw.

We want both the Weyl and Ricci tensors. We are therefore interested in all the proportionate changes, and rates of change, between linear absolute time and relative generational time. We therefore want all those gradients. They transmit themselves all about the circulation and maintain the Liouville ensemble and Ricci scalar. That transmission of biological events all about the circulation, in those two times, is the source of all heredity. That is the Weyl tensor.

We slice the circulation into little triangles because as Newton and Leibniz pointed out, finding the length of each of those short Δc hops is trivial. We measure relative to our two axes, x and y. We then use Pythagoras' famous theorem. The triangle based on t-1, stretching to t0, has the breadth Δx-1 and the height Δy-1. The gradient or slope which is the rate at which the biological population brings the external world into itself and converts its Gibbs into its Helmholtz energies is the proportionate measure Δy-1⁄Δx-1.

But there is also the triangle based on t0 and stretching to t1. It has the breadth Δx0, the height Δy0, and the gradient or ratio Δy0⁄Δx0. The one measures from the past to the present, the other from the present to the future. Whatever x and y might represent, we now know by how much each of them changes at t-1, t0, and t1. We can express the gradient or rate of transform from nonbiological to biological as Δy ⁄Δx at each point. This is the rate at which the biological population is transforming itself and its resources from linear absolute clock time into circulating and relative biological time. This tells us how quickly external resources are being used to fund this biological activity.

Those batons do not twirl evenly. We already know that aberrancies exist. Every population has different degrees of linearity and aberrancy. Entities and populations suffer numerous and ongoing changes in direction, through a variety of directives they receive whether it be from DNA or through vagaries imposed by the environment.

Every journey through time from t-1 to t0 to t1, and simultaneously over a circulation distance from τ-1 to τ0 to τ1, has the following three attributes for its speed, direction, and degree of linearity and curvature:

1. Transform. Every population converts the nonbiological to the biological at some rate or velocity. Each is always engaging in two opposing streams of transformational exchanges:
1. The nonbiological and absolute linear must transform to the curving and biological, which are such metabolic processes as inhalation and assimilation.
2. The biological and curving must transform back to the absolute linear and nonbiological which are the converse processes of exhalation and excretion.
These must each happen at some velocity or rate. That net relative measure is the ongoing exchanges out of the nonbiological into the biological.
The ‘transform’ is the net absorption of raw materials in absolute time that draws the population closer to reproduction. It states how close to a straight line the generation is at that point through its engagement with the nonbiological as it achieves its mission. It is a rate or velocity.
Again thanks to the clearer notational convention Leibniz initiated, the two side lengths of the infinitesimally small triangles we draw to measure this rate are the two differentials dy and dx. They each tell us how quickly linear time is exchanging and transforming into the biological, which is the circulation. We can get the transform value from those two infinitesimal quantities very simply. We set one over the other to give dy/dx. That is the circulation's rate of transform between the nonbiological and the biological. It is more technically known as the ‘first derivative’.
2. Directive. Each population can only maintain a curving biological direction by interacting with the surroundings. But the second law of thermodynamics ensures that each is constantly pulled into a set of potentially deleterious exchanges. No population can maintain itself indefinitely. Two populations can easily have the same rate and transform, while one has a lower entropy, and so is showing greater success in maintaining its current states. We therefore need to know the circulation's behaviour both immediately before, and immediately after, every t0. There are only four possibilities:
1. The population can succeed in becoming more biological at any moment. Its circulation begins less than, or slower than, the linear at t-1. It uses the linear as fuel to accelerate itself to t0; briefly holds that tangent value; then continues to increase itself, afterwards, to t1 and beyond. The linear and the material thus allow the population to increase its overall biological state by allowing it to accelerate.
2. The population strives but fails to increase itself. It maintains its current biological state but the surroundings will not let it improve. The circulation begins less than the tangent at t-1; increases to touch the tangent and assumes its value at t0; but then immediately decreases away to stay the same side of that tangent at t1 and beyond; and so that the linear and its processes fuel it, but do not allow it to increase itself.
3. The population successfully resists the environment's incessant efforts to diminish it. It maintains its current biological estate. The circulation begins greater than the tangent at t-1; decreases towards the tangent at t0; touches it to assume its value; but then immediately increases away to stay the same side at t1 and beyond, so that it has used the linear to help maintain itself against the vagaries the same environment constantly imposes. The surroundings tried, but failed, to force the population to disassemble and to decelerate.
4. The population fails to avoid a decrease. It becomes less biological. The linear and the material overwhelm the population. Its overall biological state decreases. The circulation begins greater than the tangent at t-1; decreases and briefly holds the tangent value at t0; then crosses over and continues to decrease beyond t1 so that it has failed to use the linear and its processes to maintain itself at its previous levels.
Since these all involve ongoing attempts to change the transform, they are all accelerations. This property is technically called the ‘second derivative’. It has the symbol d2y/dx2. We call it the directive.
3. Aberrancy. As we already know, this tells us how close the circulation is to maintaining its circular—and so biological—values at any point. Aberrancy tells us what, where, and how firmly anything is dragging the population to and fro between the absolute non-biological clock time, and the circular biological generation time. It is the proportionate relationship between the two stretches of time t-1t0 and t0t1 and the two generation distances τ-1–τ0 and τ0–τ1. The greater is the curve's aberrancy, the more do the axis of aberrancy and the normal differ; and either the more extensively the environment is affecting the population, or the more successfully it is restoring itself across any interval. If there is no aberrancy then the forces acting on it are constant. The batons always twirl evenly. The population is consistently maintaining its biological nature, all without deviations and variations, such as when following a template. It is technically called the ‘third derivative’, and has the symbol d3y/dx3.

Just as do objects falling in the atmosphere, biological organisms and their populations must contend with variable forces stemming from the environment. Variable forces will always produce variable accelerations or directives … and therefore an aberrancy, which is a jerk. If, for example, an object falls in physical space and meets no air resistance, it accelerates completely smoothly because it feels a constant force. There is no jerk or aberrancy, and it feels no drag. If the aberrancy is zero, there is no extraneous force and we get the pure action of gravitational attraction. If, however, there is drag or other ancillary forces, then there is immediately aberrancy. We see a change in the gravitational behaviour caused by that extraneous air resistance. The third derivative cannot, under these circumstances, be null. There must be some proportionate change. In linear motion, such as with an object falling under gravity, any jerk imposed by the surroundings, such as air resistance, tends to oppose acceleration. The object in this case slows its descent. In uniform circular motion, the force tends to pull the object away from its designated path. Examining the aberrancies will always reveal any extraneous forces at work.

Creationism and intelligent design insist that all aberrancies, and all higher derivative forces, are either zero or insignificant. So all we have to do, as we did with Brassica rapa, is measure an aberrancy, and we have achieved our purpose for that jerk will be the force that drives evolution.

The transform, directive, and aberrancy are all rates. But the hamiltonians we need to establish species through the Liouville theorem are all quantities.

Each little triangle we create to measure the circulation in Figure 20.45.A, using the slicing method Newton and Leibniz recommend, hops over a small section. Each is therefore a shade inaccurate. If we want greater accuracy, then we must reduce the sizes of those triangles. The accuracy certainly increases, but so also do their numbers. Newton and Leibniz both realized, however, that we will eventually have an infinite number of infinitesimally small triangles. The total circulation length we are looking for is now the sum of all those infinitely many infinitesimally small distances, each of Δc. Thanks to Leibniz, whose notation and ideas were again a little more versatile than Newton's, any such sum of infinitesimals is again called an integral and given the symbol ∫. It also gives us the quantities we want.

We now have our circulation length as the integral ∫ dc. As does any such integral, this one simply says: “sum all the infinitely many infinitesimally small stretches each of length dc, all around c”. When we have summed them all, we have the circulation or generation length, c. This is also described by each of our three batons or 1-balls. Those three measures are Tnumber:number, Tmass:mass, and Tenergy:energy. If we take each of their integrals, we get the three values on the Owen tensor diagonal. We can then determine their weighted average. All of these are distances. We shall have the population's overall generation length, T. We can then ally that time and length to the above rates to produce quantities.

A mathematical aside

It is another rule of this geometry that all lengths and measurements follow the fundamental theorem on limits, which supports the entire edifice of the integral and differential calculus:

1. If a function u has a limit l and c is a number, then cu has the limit cl.
2. If u and v have the limits l and m, respectively, then u + v has the limit l + m.
3. If u and v have the limits l and m, respectively, then uv has the limit lm.
4. If u and v have the limits l and m respectively, and if m is not zero, then u/v has the limit l/m.
5. If u never decreases and there is a number A such that u is never greater than A, then u has a limit which is not greater than A.
6. If u never increases and there is a number B such that u is never less than B, then u has a limit which is not less than B (James and James, 1992).
Figure 20.46

Since physical space is again easier to imagine, we can think of biological populations as planets complete with mountain ranges. Each 1-ball and each baton twirls in its given direction. Each interacts with the others to create the two-dimensional surfaces and three-dimensional body volumes we see in Figure 20.46.

Each of those mountain ranges is a small section of the planetary surface … and a distinct portion of some circulation of the generations. Thus Populations I and II could be (a) two distinct populations living simultaneously; or else (b) the same population in two different generations; or else (c) the same population at distinct points in a single generation.

Quantities and rates influence each other. If we keep the rate the same then it will obviously take longer to walk about any larger mountain than it will about any smaller one. These variations in distances with respect to other distances build structures such as landscapes of different sizes and features. So coastal areas, where the height is low, do not produce many changes in height, i.e. in the z dimension. There is the occasional cliff, but if we travel just a few metres in either x or y, or even carry on upwards in z, those cliffs quickly come to an end. There is a quantitative association. Low altitudes tend to be associated with low rates of change in height. And, contrariwise, if we detect a high rate of change in height, we expect to be on a mountain. We will tend to expect the present ascent to continue. A mountain might have the occasional plateau, but if we keep going in x, y or z, the rate of change in height soon returns. We can travel quite far in either x or y, or even continue upwards in z, and the rate of change in height nearby is very similar. We therefore expect to see quite large changes in height in areas that are already quite high. We shall eventually show that the proposal Figure 20.46 makes, which is that Populations I and II are simply scaled up versions of each other, is impossible … and therefore that creationism and intelligent design are impossible.

If we take the ground as a basis or reference point and see a 1 kilogramme mass displaced 10 metres above it for 1 hour, which then falls back to the ground again, we know that some force acted to keep it displaced. We also know that more must be done to a 10 kilogramme mass to keep it displaced by 100 metres for 10 hours. There is a clear link—or association—between force, F, and displacement, d, to which we shall shortly return.

Figure 20.47

The large graph on the left marked ‘association’ in Figure 20.47 measures an entity's displacements from some reference position over time. It moves further away between t-1 and t0, and then reverses and begins to return between t0 and t1.

We can assess our entity or population's absement by measuring its displacement, d, over time, t. The association graph is therefore a record of the entity's total absement, or farness, over the period, relative to our chosen measurement location. We determine that total absement by again doing what Newton and Leibniz suggest and creating infinitely many infinitesimally small rectangles between t-1 and t1. Absement, therefore, brings together—i.e. associates—displacement and time. It is the ‘first integral’ of displacement, d, with respect to time, t, and is given by A = ∫d dt.

As we see in the two lower graphs in Figure 20.47, slicing the area underneath an association graph to find its area and total immediately creates a series of little triangles at the tops, where they do not quite fit. Those little triangles straight away give us the above values for the object's transform, directive, and aberrancy at each point. The transform or first derivative always tells us how fast something is changing at any given point, while the association or first integral tells us how much it has changed up until that same point. Absement and velocity are mutual inverses relative to displacement, d, both of them with respect to time, t. One tells us the total distance covered in the total time spent away, while the other tells us the rate at which further distances are being acquired. We now have information both on how long something has been away from any chosen reference location, and how rapidly it is transforming while it is away. We have both quantities and rates. One also indicates the past, the other the future.

It is a rule in this biological geometry that an association is the inverse of the transform. More technically, the first integral is the inverse of the first derivative and vice versa. It is also a rule, in this geometry, that if the one exists, then the other also does. Absements must therefore always be acquired at some rate.

If we return to the man and the woman with the shuttlecock in Figure 20.12; or to the soccer ball and the cushion falling to the ground in Figure 20.31; then some force, F, is in each case moving some object over whatever displacement, d, we are measuring. The objects concerned deform and change shape according to those forces, with various absements and velocities. Some force, F, pushes their molecules through some displacement, d, and so out of their original positions. All balls bounce because of the work done upon their materials, which again respond to the converse processes of absements and velocities. Those out-of-position molecules then seek to return.

Associating force with displacement, F and d, always produces the work done by and upon any objects or materials to produce the various absements and velocities held by those molecules. That work done is given by W = Fd … or … more correctly … work is the integral of force, F, with respect to displacement, d, as in W = ∫F dd. This is the first law of thermodynamics. We can record the effects in our tensor.

This formal definition of energy also means that the first derivative of any energy, with respect to displacement, recovers the force exerted over that displacement. Force and energy are also inverses. Their mutual and invertible relationship is also defined through displacement, d. This is the law of the conservation of energy, which is again the first law of thermodynamics.

According to the atomic theory, an object's behaviour always depends upon its molecular composition. Thus the shuttlecock's cork base is a highly elastic material. When the woman strikes it with her racquet she uses a force exerted over a distance to produce absements and velocities. The impact lasts only a few milliseconds, but it is enough to deliver about eight thousand pounds of force. The man, on the other hand, has almost no effect upon the cork base and so takes a lot longer to achieve the same purpose. He produces much less absement and considerably lower velocities on those same molecules.

Both populations moving across their circulations and balls, cushions and shuttlecocks behave in very different ways over time. A bouncing ball and a biological circulation are each sets of activities involving some material medium. That medium determines the speed at which photons and energy can travel across them. Since the energy involved cannot remain stationary, all biological entities must engage in those suites of metabolic and physiological processes that are characteristic of any population; of its circulation; and of its medium.

Molecules move at different rates. They therefore have different absements. Any rebound or return motion is driven by the acceleration each body receives, which depends upon the distances its molecules can move. Just as different biological populations exhibit different kinds of circulations and behaviours, so also do different balls bounce differently because of the different kinetic energies and momenta their molecules can each produce in response to forces impressed upon them. There are in each case differences between the external thrusting out effects and their accompanying internal configurations. It is the behaviour of materials across both space and time, and as energies flow across the rows and columns in our tensors. This is the difference between their Helmholtz and Gibbs energies.

Our indiscernible point, which states the energies and activities over a generation, is the work done on a population over a generation … which is a displacement, d, across that indiscernible point. It is the path integral of a force over a spatial trajectory which is the 1-ball. It is a part of the conservation of energy about that generation as progenitors produce progeny that can then do the same.

Both populations and elastic materials can restore their shapes after a deformation. Rubber balls have long chain polymer molecules that tangle and then untangle. Elastic materials lose very little molecular mechanical energy to random nonmechanical thermal motion. They again conserve energy. Their differences in behaviours between different materials, under force and energy, are structural.

A ball bouncing is a form of reproduction. A force applied in the z direction goes largely in x or y, instead of in z, as molecules are stretched, squeezed, and moved out of position. As the ball changes shape and its molecules transform, the chemical bonds increase their force of restitution. The molecules eventually reach their maximum deformation. That begins their return to their original positions. Their return to their original locations creates the bounce. These are its Helmholtz and Gibbs energies.

A ball bounces because the force it experiences, imposed from or by the surroundings, changes the displacements of its molecules. The molecules acquire a velocity, which is the rate at which they deform, over distances, under that applied force. It is a transform. It therefore states the material's responsiveness to those forces applied. It is a function of the material's overall relative tension, stiffness, and elasticity. This is the force, F, exerted per unit of the material's displacement, d. That responsiveness is the rate at which molecules twist, untwist, and change their positions under that force and energy. And … this first derivative of force with respect to displacement is also and immediately the second derivative of energy with respect to displacement, for that force was already the first derivative of some energy, with respect to displacement. We therefore have d2Edd2 = dFdd. The effect goes directly into the materials as newtons per metre.

The work done by a force working over a displacement and from one point to another equals the difference in energy the system builds up as it responds to that force. This is our 0-sphere.

Populations—just like bouncing balls—must reverse directions. First they have many small young; then they have fewer and larger adults; then they again have many small young. They must exploit and restore the materials of which they are composed. This is a bouncing in and out of the surroundings in time. It is a Helmholtz energy, A, and requires a conservation of energy across a circulation of the generations seen as a distance. This is again our 0-sphere.

When a ball bounces or a racquet strikes a shuttlecock, both forces and energies act in both distances and time. The first derivative of energy with respect to distance may be the force applied, but the first derivative of that same energy with respect to time is a watt. The time effect of one watt-second is a then unit of energy for a column in the tensor. One watt is the rate at which an object's velocity does work when seeking to move at a velocity of one metre per second, but which is consistently opposed by a force of one newton.

The bouncing soccer ball and the soft cushion differ in the way in which they distribute force and energy over time and over distance, and so between the rows and the columns, and between their Gibbs and Helmholtz energies. This is between their external mechanical, and their internal nonmechanical responses. The effect of a watt is absorbed by the materials. It is a rate of energy transfer or conversion between one moment and the next, and is a statement of power, not force. The watt elicits an internal configuration response and works upon the materials, but in time. If the materials can change their responses at any one moment, then they can affect how they respond in the next, which is their rates, transforms, and absements over time. If the materials absorb that energy of one watt by displacing their molecules and then returning, then there is a bounce; and if they do not then there is no bounce.

Figure 20.48

When our two workers left home together to go to their different offices, they initially had the same velocities. They walked side by side. They were parallel. They acquired absements at the same rate. But then similarly to what we see at location t0 in Figure 20.48, one of them accelerated. One chose remain in a given location. The other maintained the original velocity and continued onwards. And since one of them accelerated, their velocities became different. And since their velocities were now different, their rates of acquiring absement also began to differ. Absement, as the association or first integral of displacement, d, and time, t, can itself vary in time.

Figure 20.48 shows us taking up a selection of absement and measuring and investigating it over time. We can push the whole absement region out along the time axis, to see how it itself changes and flexes over time. We simply push the whole of absement, which is an area and a 2-sphere, outwards across a distance, which is then a 1-ball. A baton can also change the rate at which it twirls. We produce a 3-ball or volume. We can then determine its magnitudes and changes in magnitudes by using areas to infinitesimally slice that volume, which is to integrate it.

We are now finding out how something that itself changes with time changes over the period for which we study it. We can “conjoin” absement to time, which is to push it across the interval in Figure 20.48 from t-1 to t1 to see how it changes. Conjoining therefore lets us see how something that changes in time is itself changing in time.

As in the diagram, we can then use our slicing technique and project a series of infinitesimal squares or rectangles (in black) outwards and parallel to the axis to infinitesimally sum the conjoining throughout its volume. That gives us the total change in absement over the interval. Since the volume we create is determined by taking further slices along the time axis, that makes this conjoining the double integral, ∬, of displacement with respect to time. And since we are presently working with displacement, this is the absity, Ab, where Ab = A dt = ∬d dt.

A conjoining or second integral measures all changes in accelerations, which are directives. Therefore the inverse of conjoining, which is a double integral, ∬, is the directive, which is a change in the transform. It is a change in a transform. It is an acceleration of some kind, or a double derivative, and leads immediately to a change in conjoining or area. If one exists, then so also does the other. That is again a rule in this geometry of biology.

We can do the same again. We can find out how—and if—absity changes. If we now pick up our entire absity and push it out yet again along the time dimension, then the resulting ‘abseleration’ tells us about all changes in absity. We find out how it also changes or “distributes” itself in time. And if a double integral can represent a volume, then a triple integral represents a four-dimensional summation … or distribution.

If, for example, we are inflating a hot air balloon, we can easily draw a graph of all its different volumes at each point. Since volume is already three-dimensional, then recording the volume at each moment means we are working in four dimensions, one of which is time. We then know how that volume was changing as the balloon was being inflated. We can work out what three-dimensional volume changes took place at which moments, and as we varied the rate of inflation. This is the distribution of those values relative to time.

The atmosphere around us changes its density with its height. We can measure it at a host of different heights and locations, and record its density per unit volume at each. That gives us a rate at which its density changes across all those volumes and points. We can then compute the entire atmosphere's mass from those density readings by infinitesimally slicing it. We can work out the mass per each slice, as that density changes. We then add all those slices and determine the mass—which is now our variable or fourth dimension—as distributed through each slice. We slice and so integrate across a volume. But since that volume is already three-dimensional, then this is a triple integral, ∭.

Just as the transform is the converse of the association, and the directive is the converse of the conjoining, the aberrancy is the converse of the distribution. If ever we have an even distribution, then there is zero aberrancy relative to some dimension. And … if ever there is an aberrancy, then we know that some distribution is changing as surely as the mass density of the atmosphere at some point, or the rate at which we inflate a balloon. Those are also rules in this geometry of biology. The equilibrium between these various attributes is the key to Darwinian variations.

Figure 20.49

We can now take up the energy we procure by first associating force with distance, and integrate it for a second time, again with respect to distance. Energy is the association between force and distance. Since it is two-dimensional, then it is effectively an area. Those areas describe the energy landscape. The integrals and derivatives of force, F, with respect to displacement, d, tell us about materials and their behaviours. When we integrate for a second time we simply pick up a body of energy and transport it a over a certain displacement. But since energy is already a first integral of force with respect to distance, then we are conjoining force to distance. We get the single integral of energy and the double integral of force, both with respect to displacement: i.e. ∬F dd = ∫E dd.

As in Figure 20.49, when we integrate energy with respect to displacement, we push the energy out along an axis to create a volume. Under the law of the conservation of energy, which is the first law of thermodynamics, the transport of energy over a given distance produces something very similar to what meteorologists call an ‘advection’ of energy.

We now have Bernoulli's theorem. It is how aeroplanes stay up in the air. The wing's aerofoil induces the air to flow quickly, and so at lower pressure, over the top of the wing, while it moves more slowly, and so at higher and therefore supporting pressure, underneath. In the same way, a large bore pipe can transport a given volume of fluidic energy over a small distance; or else a small bore pipe can transport that same volume over a longer distance. The work done by the pressure difference between P1 and P2, per unit volume, in Figure 20.49, is equal to the gain in kinetic and in potential energy across that distance, and also per unit volume. The total energy delivered over the shorter length L1 is therefore equal to that delivered over the longer length L2, which is the equality of the advection. The area and the pressure, or driving force, contained in the materials and their structure at Location L1, at the beginning of the advection streamlines is higher, but the velocity lower, than at Location L2, at the end of the streamlines, where the area and the pressure are lower, but the velocity higher. The energy delivered at any time is the rate, or derivative, of this advection. By the same conservation of energy, we maintain an equivalence because if the energy intensity decreases at any point, then the distance over which it is transported increases, and conversely. As the area of delivery decreases, the velocity increases while the pressure decreases so that A1L1 = A2L2 … which is a transverse pattern familiar from the Ricci tensor. The various forms of energy interact so that 100 joules of energy carried for 10 kilometres is the same as 10 joules of energy carried for 100 kilometres. These two have complete equivalence, for they are both 1,000 joule kilometres. This is a longitudinal pattern familiar from the Weyl tensor.

The advection, as the conjoining of force and displacement, must have an inverse which is the second derivative of force with respect to distance, d2F/dd2. (It is also, of course, the third derivative of energy with respect to distance (and so the aberrancy with respect to energy)). This gives us a value for the stresses and the pressures that the materials endure under their deformation, and that cause their molecules to tangle, untangle, and deform, relatively, in those specific ways, and to both transport and to change those advections. It is measured in pascals.

We can now take up our advection and create a distribution by pushing it out, once again, with respect to displacement. This is the triple integral of force with respect to distance, ∭F dd; and the double integral of energy, again with respect to distance. This “biocapacitance” is the population's energy base or core. It is an extension of Bernoulli's theorem. Instead of streamlines interacting with areas over volumes to get an advection, we have stream-areas, interacting with volumes, and so giving four-dimensional energy dispensing hyperstructures.

We can measure this biocapacitance or distribution with the units of ‘joule metre metre’, which is better understood through the old Swedish unit of measure known as a kyndemil: the distance the average torch would last, which was approximately sixteen kilometres. Some torches could burn more intensely for shorter periods and distances, while others would burn less brightly for longer ones, all depending on the quantities and distributions of pitch and tar. But the manufacturer would have a set amount of pitch and tar, meaning the total light-distance in kilometres burned was fixed over all torches. The torch manufacturer can vary how many metres of pitch or tar are used to create each torch, as well as how many are created in each day, or over any time interval. This is then the light-metres that can be burned. Each particular torch can vary; and each user can make independent decisions on how to use up the particular stock of kyndemils in each torch in terms of brightness and distance. But each day and factory has a measure for kyndemils or light metre metres manufactured. In the same way, the same basic stock of biological materials and energy can dispense and distribute different amounts of itself at different times, and under different conditions, producing effectively different numbers of metres, with different possibilities for joule metres of carriage and advection in each.

The inverse of this biocapacitance or joule metre metre distribution is the aberrancy which is the third derivative of force with respect to distance, d3F/dd3 (or the fourth derivative of energy with respect to distance). These cause the variations in the above torches. While the pascal measures the forces that cause a body to change shape, that is only as a surface average. The pascal is a force per unit area, and is the force produced from that area as it is exposed to the surroundings. The pascal per metre pinpoints the different locations within the body, as each differentially feels, or generates, that externally applied force; and/or the pascal per metre states the pressure or the stress that each specific location can apply when it is active and exposed. If it has the same potential everywhere, then the pascals generated are the same at every point. The pascal per metre tells us the surface pressures that can be experienced at each location, as that location and its properties vary with the nature of the materials or its composition. It is the statement of the biocapacitance bases in time, and as the pinpoint energy distribution and energy construction per metre of its construction.

A force's behaviour, F, over time, t, when applied to a mass, m, is very different from its behaviour as the result of an energy applied over a displacement, d. All forces are defined by Newton's second law of motion, F = ma. If a force causes the displacement of mass, and so uses or produces energy, then it must also be active in time. A mass somewhere must be accelerating, also in time, or else an equivalent amount of work is being done.

All forces are known by their ability to displace masses and inertias in time, and are always aligned with acceleration, a, through F = ma. The woman with her racquet is only in contact with the shuttlecock for a short period of time. The man with his hand acts for much longer for he must first catch it before he can return it to the other side. This association between force and time is the first integral of force with respect to time, and is called impulse, J.

Impulse causes, and is equal to, any change in momentum, where momentum describes an object's inertial property. Force therefore states the rate of change in momentum. An impulse exerted across an interval of time causes a change in an object's velocity, v, which requires an acceleration. Impulse aligns mass with velocity which is the momentum, mv, measured in kilogramme metres per second, or newton seconds. It is the momentum maintained over that interval.

Impulse is the force multiplied by the time for which it is applied. It is J = Ft … or … more correctly … impulse is the integral of force, F, with respect to time, t, or J = ∫F dt. Associating force with time gives impulse as the summing, or integral, of force with respect to time. It is all the changes in the momentum, mv, of whatever receives that force. Thus the shuttlecock enjoys a smaller acceleration, with the man's actions, than it does with the woman's, for he can exert far less force than she. She has the much greater impulse and can impose greater forces on those materials.

If impulse is the association of force and time, then there must be a transform. If the force stays constant over any period of time, then the acceleration stays constant. But if the force changes then the mass undergoes a change in its acceleration, which is a jerk. We find the first derivative of any force, with respect to time, by finding the derivative of its acceleration, which is jerk, j. The transform or first derivative of force with respect to time is then mj or mass times jerk, and is known as “yank”.

When the man and the woman each strike the shuttlecock, they each give it different jerks, and therefore impose different yanks. Her racquet allows her to impose a much larger change in acceleration, which is a much bigger jerk. She therefore yanks that shuttlecock more severely. He has the lesser yank, and also exerts it over a longer period of time. Since he has the smaller impulse, he also has the smaller yank, and causes less of a change in momentum at each moment. Yank is measured as newtons per second.

We can clearly see the effects of these differences in impulse and momentum, as well as in yanks, when an insect hits the windshield of a moving car. By Newton's third law of motion, the insect's force upon the windshield is equal to the car's upon the insect. They are exerted in opposite directions. Since F = ma, the insect and the car will both accelerate, which is to change the rate at which they are each moving through space. The collision, however, is brief. The impact exists only for a short interval. The car and the insect will share properties from that common impact and will distribute them into their respective momenta according to their individual masses and velocities.

Since the car's mass is considerably greater than the insect's, its velocity is virtually unchanged by the force the insect impresses upon it. It will travel only a few micrometers in that time. Its internal structures are easily capable of absorbing the relatively feeble impulses, impacts, and forces imposed by the mosquito.

The car, however, is travelling those few micrometers directly forwards into the opposing insect … which undergoes a very different experience. Since the insect has a much smaller mass, it will put most of the force and energy it feels upon impact into an acceleration or an attempt to change in velocity … which is an attempt to move forwards through a far greater distance—but directly into the car—in that same time. In its case, that distance it tries to move, as it changes its velocity and its momentum, is directly into the car's incomparably greater mass. Since the force is the same but the insect's mass is so much less, the car subjects the insect to a much greater yank in that same time frame … which the insect's internal biological structures simply cannot sustain.

We can also conjoin force to time. This is the association, or first integral, of impulse with respect to time, mJ dt, and the double integral of force with respect to time as in mF dt. This measures the “portage” of the mass to which it is applied in kilogramme metres.

If two people are moving from the same home, one by 1 kilometre and the other by 10; and if they hire a special haulage company that charges the same amount to move 1 kilogramme of their possessions a distance of 10 kilometres as it does to move 10 kilogrammes of possessions a distance of 1 kilometre; then these two have the same portage of 10 kilogramme metres.

The inverse of portage is the directive of force and time. This is the second derivative of force with respect to time, d2Fdt2, and aligns mass with the time derivative of jerk which is called “jounce”; or also sometimes called the “snap”. The applied force is then known as “tug”. The woman striking the shuttlecock again uses her racquet to give a bigger jounce or snap to the shuttlecock, which is a larger change in its jerk, and so is a smaller tug, than the man gives. It is measured as newtons per second per second. A 10 kilogramme mass will need more tugs to accelerate and decelerate it to and from any chosen moving speeds than will a 1 kilogramme one, even though they might have the same portage in terms of mass-distance moved.

We can distribute force with respect to time, which is mF dt, and the triple integral of force with respect to time. It is the “impulsivity” measured as kilogramme metre seconds. A first way to understand it is through the dental surgeries or hairdressing salons often attached to dental or hygiene schools. They will sometimes offer the services of their most recent graduates at a reduced cost so that those graduates can gain experience; but will also make highly experienced practitioners available at a higher cost for those who prefer it. The times, costs, and numbers of experienced and inexperienced dentists and hairdressers are carefully priced, and allocated, to ensure that they all bring in the same revenue over each time period. The experienced ones therefore see less customers for shorter periods, while the less experienced ones see more customers for longer periods. This is similar to the man and the woman with the shuttlecock. She is like the high cost dentist or hairdresser who acts briefly but intensively, while he is like the low cost one who acts more frequently and over longer time periods. The end result is the same. The shuttlecock goes back to the opposite court. Time and portage adjust themselves appropriately.

A second way to understand impulsivity and the kilogramme metre second is through a variant of the poronkusema. Virtually the only thing that grows in the harsh northern environments of the steppes and the Arctic circle is lichen. There is little opportunity for agriculture. Almost the only thing that will eat the lichen is the reindeer. It thus provides everything, including its dung for fuel. If we now consider their dung, instead of their urine, then a herd of reindeer that stays relatively local and provides 100 kilogrammes of dung in 1 day, and that travels 1 kilometre to do so, is equivalent to a much more mobile one that travels 4 kilometres over ½ a day but only provides 50 kilogrammes; which is also equivalent to a much more constipated one that provides only 25 kilogrammes while travelling a ½ kilometre, and that takes 8 days to do so. These all have the same 100 value for kilogramme-metre-days of reindeer dung provision. Impulsivity therefore measures the different ways in which the same portage can be equivalently distributed in terms of the energies and masses moved over times.

The inverse of this impulsivity is the aberrancy of force and time and is the derivative of tug. It aligns mass with the time derivative of jounce, which is most often called “crackle”. The applied force is the “snatch” and is measured as newtons per second3. The woman again has a bigger snatch on that shuttlecock, and as ever for a shorter period of time. Her racquet helps her make the shuttlecock crackle, snap, jerk, and accelerate more rapidly than anything the man can do with his hands, and her forces, yanks, tugs, and snatches are all larger. And the 100, 50, and 25 kilogramme masses of dung will all need different snatches at the beginnings and the ends of their 1, 4, and ½ kilometre distances, over those 1, ½, and 8 day periods, all respectively, to produce them. Those variations in dung production are caused by the different snatches the reindeer exhibit to produce them. Those aberrancies are responsible for the differences in distributions.

Figure 20.50

We now have a complete set of seven values when force is applied to either distance or time. We have the force itself, and then its associations and transforms, its conjoinings and directives, and its distributions and aberrancies again in both distance and time. We know how it will respond, and at what rate.

Biological populations can now use their 4 × 4 tensors in conjunction with linear and polar planimeters, as in Figure 20.50, to measure themselves and each other. We can place each population around a common centre and measure them all, in both distance and time, with a single turn of our polar planimeter. If creationism and intelligent design are true, then every species must form the right helicoid we see in Figure 20.50.A. It is generated by turning a right-angle triangle about a central axis, and simultaneously drawing it vertically upwards along the axis. Every species will find its appropriate distance, determined by its genome. It will use masses and resources to form the breadth of an appropriate step, as in 20.50.B, as well as the thickness shown in 20.50.C. These determine its poloidal, meridional, and toroidal activity and intensity at each point. However … force acting in time is not the same as energy acting in time.

All scientific observations of the cosmos at large support the isotropic and the homogeneous proposals: that force, matter, and energy are everywhere the same, and everywhere have the same density. Creationism and intelligent design, however, breach the cosmological principle by advocating a special observer and location. Every species is deemed special. Every one has a privileged point of view. Each of the templates is intrinsic. On this alternative view, each has its right helicoid independently of all others. Each has its infinite and eternal properties. Each can maintain itself indefinitely in an equally eternal and unchanging firmament they all share, and whose features each finds especially fitted for itself. Each template is therefore independent both of the surroundings, and of all others.

A polar bear must determine whether or not reproduction with a grizzly is possible. Each must measure its biological space with sufficient accuracy, but each is obliged to use its own and relative units of measure. By the cosmological principle, every viable species surveys the surrounding biological universe, and every viable species sees everything it needs to maintain itself … which is exactly the same for all others.

Figure 20.51

The biological problem we have to resolve, with our twirling batons, is very similar to the one the hooprunner in Figure 20.51 faces. The twirling batons and the hoop that boy is running are each intrinsically circular. However, they must both move across a surface that is intrinsically linear, and that has a gravitational attraction that looks to make them each fall. They each therefore need a force to maintain them, and a conversion rate from the linear to the curved and back again.

As the batons twirl and the hoop rolls, they together create the Hooke biospheres we first met in Before We Begin. Those biospheres contain the mass, the resources, and the energy that every population needs to survive. As such, they all have transforms, directives and aberrancies, along with associations, conjoinings, and distributions. The Hooke biospheres are the biological populations maintaining themselves by running, or being run, through the surroundings. Their actions in the surroundings to maintain themselves are the strikes with the sticks that keep them running as hoops, and similar to the hooprunner.

We may now have a Hooke biosphere and a biological space, complete with twirling batons to energize them … but we cannot distinguish any one population from any other. They are all currently the same. None have any particularly memorable features to separate them from any other.

We again turn to Hooke who was certainly important in biology for his discovery of cells; his realization of the truth of fossils; and for his work with microscopes. But he was equally important in physics where he developed Hooke's law of elasticity, produced a balance spring for regulating watches, and created the pump that he and Boyle used to discover what became called Boyle's law. But perhaps his most important work was in gravity.

By the 1670s, Hooke was already espousing his great discovery: that gravity pervades all space. He was already proclaiming that the sun and the planets are attracted to each other with a force that grew as they got closer. Although he could only formulate his idea as an interesting mechanical problem, he had nonetheless produced a definite theory of gravity and planetary motion. He also saw the truth and argued for it as a universal force. He even considered that it might follow an inverse square law.

As Hooke did with his work in cell theory, he had stumbled, in physics, on a universal principle. He similarly saw, in biology, that the common factor to all biological entities, and the activities that sustained them, is that they are found in cells. He observed the character of thin slices of cork, which reminded him of the walls of a monk's cell in a monastery. Like his universal gravitation, Hooke's cells are a general principle common to all biological entities. Our Hooke biospheres are those cells taken in conjunction with the materials and the energy they need to sustain them.

Figure 20.52

Our twirling batons and our Hooke biospheres emulate gravity by projecting themselves and their properties across all biological space. They create the 1-, 2-, and 3-balls of Figure 20.52.A as the resources and energy that become, and contain, the biological entities and populations. Those then interact with each other and with their surroundings. They do so through the 0-, 1, and 2-spheres which are the surfaces of their Hooke biospheres running and being run through the surroundings.

As in Figure 20.52.B, those balls and spheres are uplifted, poloidally, in their various dimensions from one indiscernible point to another. As they hoop run upwards through time, they distribute forces and energies through biological space. Those complementary processes of spheres and balls are the interactions of the Weyl and the Ricci tensors.

Although Hooke had correctly realized the nature and pervasiveness of gravity, he could not do what Newton did in dynamics and mechanics, and that we urgently need in biology: provide a metric. Newton realized, unlike Hooke, that if he applied the Copernican cosmological principle to the entirety of space, then universal gravitation was unworkable. If he gave his universe an edge, then all stars and planets located there would only feel gravity on one side. They would be pulled away from that edge and would rush in towards each other … which is not what is observed. He also realized that if he gave his universe a centre, then stars and planets located there would only feel forces on one side. They would therefore pull those not at the centre in towards them. Those others would then either have to rush in to the centre, or else orbit to maintain their distances. Neither that rushing nor the orbiting is observed. He therefore constructed an infinite firmament with no edge and no centre. Creationism and intelligent design are impossible for exactly the same reasons that Newton's firmament is impossible.

Figure 20.53

We now know that although Newton's firmament neatly evaded his logical problems, it does not account for the observed data. The Russian physicist and mathematician Alexandr Alexandrovich Friedmann proved that Einstein's theory of general relativity made the Big Bang cosmology of Figure 20.53 inevitable. The universe is limited but expanding. It is also clumped into the regions we call galaxies. Biological space is similarly clumped into populations and species.

The imaginary spheres around Planets I and II can be arbitrarily redrawn to any size and will eventually encompass the universe. If the universe is isotropic at any one point, such as around Planet I; and if it is also homogeneous about Planet I; then the universe appears homogeneous and isotropic around Planet II. It must then be both homogenous and isotropic everywhere because the intersection region ABC must also be homogeneous. Although it is possible for the universe to be homogeneous but not isotropic, it is not possible for it to be isotropic without also being homogeneous. All galaxies and planets therefore observe the same phenomena. The same goes for all biological populations and species.

Since no point is privileged then an isotropic universe must have a fundamental cosmological time. It is applicable to all possible observers. They will all observe the same cosmological expansion which occurs at the same rate. There is one cosmological draw … and so there is also only one polar planimeter that governs all species and populations. They can all move from the beginning to the end of a generation, which is similar for all. They will all therefore vary only in their distances from the axis, which establishes the length of a generation. This is also to vary in quantities of resources, and in intensities of energies. Every possible generation can be expressed between its beginning and its end, and so from 0 to 1, relative to both itself and all others.

The Haeckel tensor we see in Table 20:3 recognizes these inevitabilities of biological times and measures:

 Biological time, τ ⇒ Power Clock time, T ⇓ ↑ over biochronometric distance ↓(generation length, τ) Forcefulness ← in chronological time (seconds, t) → 0 → TIME, τ ↓ TIME, T I Constraint of constant propagation, φ IIConstraint of constant size, κ IIIConstraint of constant equivalence, χ I Numbers of entities, N Tnumber:τnumber Tnumber:τmass Tnumber:τenergy II Mass of components, M Tmass:τnumber Tmass:τmass Tmass:τenergy III Joules of energy, P Tenergy:τnumber Tenergy:τmass Tenergy:τenergy

The Haeckel tensor reports all generation times up along the columns “biochronometrically”, which is a distance between 0 and 1. That is a rule in this geometry of biology.

If a population of either blue whales or mosquitos lives for two days, then that two day interval can be expressed absolutely along the rows, but must be expressed vertically as a proportion of the relevant generation length. Since that generation length is T = 4 days for the mosquito but T = 31 years for the blue whale, then that two days is a biochronometric distance of τ = 0.5 for the mosquito, but τ = 0.000177 for the blue whale. There is certainly some vast collection of mosquitos that has the same mass as a pod of 20 blue whales, but their values for energy, and their values for relative rates of change in T and τ for both mass and energy are going to be vastly different.

Figure 20.54

We now follow Leibniz. He used his “global” or “panoramic” view of force and energy to describe the conservation of energy and the many different paths a system can take. As in Figure 20.54, he used his global perspective to present his more comprehensive vision of his kinetic energy, which he called vis viva or “living force”.

Leibniz's vis viva produced a “quantity of motion” in terms of the potentials for effects moving bodies possess. For this reason he insisted on always working with mv2 rather than simply mv. He felt that unlike either force or velocity, vis viva was conserved. He was reluctant to give pride of place to force–and–velocity, through Newton's mv, because the way he looked at it, two bodies of the same mass and velocity moving in opposite directions, could appear to possess contrary quantities of progress, when in fact their striving for change, and their striving for motion, was the same. These two balls might bring each other to a standstill, but only because their strivings for motion were always the same. They donated that striving for motion to each other in equal measure, and it therefore stayed the same throughout. If this was not so, they would not have lost motion together. Therefore, their strivings for motion were always equal. They were always equally directed to all parts around them. That stayed the same throughout.

Leibniz used his ideas on vis viva, which we again now recognize as kinetic energy, to argue against Newton's contrary vision of momentum. In Leibniz's view Newton's momentum, measured as the simpler mv, could not, and did not, properly explain motion because it was destroyed as soon as direction changed. That was the advantage of his own preferred mv2. We now recognize that one of the differences between them is that although force and energy are linked over distances, forces acting in time are not the same as energies acting in time.

In Leibniz's view, since the mv2 he worked with removed all concern about direction, it immediately gave a more accurate view of events. It did not matter what direction a planet or other body moved in, it would conserve this very important property of quantity of motion. We now know he was more concerned with its kinetic energy, which is indeed conserved, for it is independent of either time spent or direction of travel:

Who can explain what is the essence of the attraction of gravity? No one now objects to following out the results consequent on this unknown element of attraction; notwithstanding that Leibnitz formerly accused Newton of introducing “occult qualities and miracles into philosophy” (Darwin, 1872, pp. 421–422).

Leibniz's kinetic energy reflects the work we must do to accelerate a body from rest, relative to ourselves, up to whatever its current velocity might be. The energy is entirely stored in motions and positions. Any energy expended is returned when the body decelerates back to rest or whatever initial velocity.

In the fashion of Leibniz, the two solid arrows in Figure 20.54 show many different populations and their biospheres picking their way through space and time and hoop running at time t0, which is the centre of the observation period between t-1 and t1. It is also the accompanying biochronometric segment of generation length ranging around τ0, and so between τ-1 and τ1. So where the time interval t-1t0t1 is a poloidal movement against a clock, the matching circulation length τ-1–τ0–τ1 covered is a toroidal distance about some specific generation length. The former creates the 1-, 2-, and 3-balls for the external Helmholtz energy, while the latter creates the 0-, 1-, and 2-spheres for the internal Gibbs energy. That entire circulation can be endlessly repeated. The population can hoop run its Hooke biosphere past this location endlessly.

The 0-sphere and its circulation, hoop running from indiscernible point to indiscernible point, is also a distance. The integral of that distance is the work we must do, in biological internal energy, to run a population from reproduction to some given moment. Since it is conserved, we can do this endless times. That energy can be stored. It is the ability to reproduce. The same amount will be returned when the body runs back around to the same indiscernible point. The energy is entirely stored in its position, behaviour, and structure relative to that circulation. This potential for storage in structure is the very definition of heredity, and is the potential and the energy for reproduction. That is the 0-sphere.

Although Leibniz's kinetic energy arises, as does all energy, from a force acting over a distance, the global view with which he derived it is far more connected with energy in time, t, than it is energy or force across a displacement, d. Energy can be transported large distances in time as heat, light, electromagnetic radiation and the like. It can therefore be transported around our spheres through its structure and its changes in structures. These are our 0-, 1-, and 2-spheres. These are our biospheres running through time and space. This is again the issue of heredity.

Even though energies can be stored and transported, they can only achieve their effects, upon arrival, by acting (A) within some material substances, and (B) over some displacement. The energies must become forces over those distances, whether this be internally and nonmechanically through some change in state as a reconfiguration of molecules, and so as a Gibbs energy; or externally and mechanically as a thrusting-out Helmholtz energy; or else through some combination of both, which is the juxtaposition of our spheres and balls.

When material substances absorb energy, the second law of thermodynamics insists that they must degrade at the hands of whatever force or event imposes that energy. There is a change in power, which is a change in energy in time in those material substances. The energy overall is conserved, but the forces or events acting in time to deliver the impulses are not conserved. However, by Newton's third law of action and reaction, they are equalized through momentum, which is mass times some rate of change. The loss in power and energy that is the increase in entropy must be identified in the surroundings, along with the inertia, and changes in inertia, that express it. The Hooke biosphere affects the surroundings so it can maintain its population.

Figure 20.54 also shows the Biot-Savart law and the Liouville theorem moving about the helicoid axis. So when we see, for example, a deer over any given stretch of time, we are observing a propagating current-element hoop running over some time period from t-1 to t1. It goes through some part of its life cycle which is also some distance from τ-1 to τ1. We immediately extrapolate a most likely life trajectory that summarizes the deer's general properties, including its forebears and descendants. This is Leibniz's global view. It produces the whole 0-sphere.

Leibniz's global method of extrapolating energies over periods of time therefore associates energy with time to give “action”, which is measured in joule seconds. We can determine this association or first integral of energy and time via Newton's third law of motion, the law of action and reaction. Every action is determined by integrating the energy over some time. We can even do this informally, such as with a deer and our implicit knowledge of its life cycle.

Action is the integral of energy with respect to time, ∫E dt, and predicts the most likely path a system will take by determining the way its energy changes over time, and from an initial to a final energy condition. In the case of our Hooke biospheres running through the surroundings, this is from indiscernible point to indiscernible point, which is its conservation. Action is twice the average kinetic energy held upon a path, multiplied by the time taken.

Since action is a summation of energy in time, it describes how a physical system changes. It almost always has one of three possible values. A system almost invariably heads towards a minimum; a maximum; or else it remains constant. The most common movement is towards a minimum, or else towards the shortest possible path. It is also conserved in time. Entropy—which is what something acquires so it can engage in its ongoing molecular movements—always, of course, heads for a maximum. Entropy affects action but is not conserved … excepting only in the absolute and ideal case that Clausius first used to define it. Carathéodory later used the state Clausius described to help define energy.

When many different observed biological motions are brought together over many generations, they result in the curve drawn in Figure 20.54, which is the surface for a 0-sphere and the action for a Liouville ensemble. It is our expectation. It is what is to be conserved across generations. It is a statement of the most likely path any population will take. We then use that observation to extrapolate our general expectation for any entity, which is the set of repeating values from minus to plus infinity, vertically upon the helicoid, and for all possible generations and their infinitely many indiscernible points which are all distances and circulation lengths. There is a constant action characteristic of each species.

Action allows us to use a “seen one deer, seen them all” method to determine the most likely behaviours, across time, for all entities of that general type. That is a set of proportionate changes, all with reference to the observed mean values. It is our Hooke biosphere for that deer population impacting the surroundings.

Since action is an association between energy and time, we know straightaway there must be a transform or first derivative, also with respect to time. Action and power are the two inverses. Power is some rate of energy use in time. All stocks of action require some rates of use. That use can be stated as a proportion, and so as the Δn/n’, Δ/m̅’ and Δ/p̅’ that carry us all around the generation. Those proportionate changes can also be stated absolutely, in some units, and so are the velocities and transforms. They are statements of power as watts, which is energy in time.

Leibniz's global or panoramic method is certainly useful, but it has its limitations. Actions and energies must show themselves in specific forces imposed upon masses over displacements. There must be a specific deer, and it must graze a specific amount.

A global event of the Leibniz kind is an entire panoply of local events flying by. Curved biological time is composed of a whole sequence of absolute linear and clock moments. Biological populations must therefore demonstrate some centre about which they turn to form a circulation, and which expresses the inevitable inertia of energy as a locus for force. We must see some specific deer going through some specific life cycle. Leibniz's global view does not always make this local set of events easy to zero in on.

Leibniz's approach, based on mv2 and the general ‘ability to move’, made little sense to Newton who preferred the observables of momentum, and of forces acting over times. He therefore championed an alternative way to understand action. In his view, the sensation we feel when we catch a ball is singularly responsible for its motion. It made little sense to him to say that “amount of motion” is caused by “ongoing ability to move”. If two balls bring each other to a stop, it is because of some real force, not because of some amorphous striving to move. That was, in his view, an intrinsically circular argument. No matter how fast a ball might move, its stopping power, which can be measured by mv, demonstrates the force it can exert. Newton felt that the only useful way to describe motion was through the rate of change of its quantity of motion. This is its acceleration, which also demands its direction. This is his F = ma.

Figure 20.55

Newton's method is admittedly far more effective in handling forces and the movements that forces create. His approach facilitates a “local” or “close-up” view of our biological space of internal energy. It can accommodate both the tangential and the circular, which are the two aspects of biological internal energy we see in Figure 20.55. One is the absolute, linear, nonbiological and chronological time we place on rows. The other is the relative, curving, biological, biochronometric and generational distance we place in columns. They come together to create the 0-, 1-, and 2-spheres, and their contained 1-, 2-, and 3-balls.

Newton's approach focuses not on time, but on changes in distances, and changes in momentum. Where Leibniz's approach showed us how energy is conserved in time—which Newton's approach fails to do—Newton's now shows us the forces and distances through which that energy both arises and is disposed. Newton's method shows us the instantaneous, and apparently linear, absolute correspondences in inertia and momentum; while Leibniz's shows us the circulating path the energy takes as the system responds to Newton's visions of tangential forces and inertia.

Although energy can be conserved in time, the forces that produce it cannot. Forces can cause the transports and changes of the objects to which they are applied, which is to move and to displace them in time … but the forces themselves cannot be conserved. The only way to transport forces is to first convert them into energy by moving them over some displacement. The resulting energy can then achieve its effects—including those of heredity—at a distance from where they are created. Forces do not have that transport capability, and are our 1-, 2-, and 3-balls.

Newton's approach records movements on a more direct moment-to-moment, and instant-by-instant, basis. It then determines action by calculating the average momentum upon a path, and then multiplying that by the path’s length. We arrive at the same association of energy and time, but we get there by examining momentum behaviour over a distance, rather than energy development over time. Action can be stated either as newton metre seconds, or as joule seconds.

Biological populations must somehow reconcile the absolute linear time Newton favoured with the curved biological variety Leibniz preferred. These are exactly the two fields we see in Figure 20.55 that make up the Hooke biosphere that runs through the surroundings.

Figure 20.55.A gives us the local perspective on the nonmechanical chemical aspect of biological energy. Figure 20.55.B gives us the equally local perspective on the very different mechanical chemical energy aspect of that same energy.

Thanks to Newton's local view, we can examine infinitesimal segments of each of these two fields as they move on their required set walks. They are both centred around time t0 and location τ0, stretching between t-1 and t1 and τ-1 and τ1 … which is the same stretch of time-distance we just considered in Leibniz's global view of our helicoid's surface.

The infinitesimal stretches we consider are always short enough to be linear. We can see, in this local view, the set of mechanical and nonmechanical chemical energy exchanges taking place in our Haeckel tensor. Energy and chemical components flux in at t-1 and τ-1, pass through t0 and τ0, and then leave at t1 and τ1 to establish the line and time intervals and segments of both energy and material molecules.

If we use L for the Liouville ensemble phase volume, P for the Wallace pressure or energy flux, and M for the mass flux, or Mendel pressure of chemical components, then these two fields are P-1L-1P1L1 and M0L0 of energy and mass, respectively, that the population places around itself in its biosphere. Those masses and energies must ultimately incorporate reproduction.

The Liouville theorem and the global view tell us that our two fields P-1L-1P1L1 and M0L0 must loop around our indiscernible points from the minus infinity pole to the plus infinity one. They must do so around the species mean values of n’, m̅’ and p̅’, and so all about the helicoid axis in internal energy.

The two fields P-1L-1P1L1 and M0L0 diverge before and after t0 and τ0, but combine as P-1L-1P1L1M0L0 over the infinitesimal biochronometric span t-1τ-1t0τ0t1τ1 to produce all biological events. Since our tensor rows and columns interact, biological entities must be sustained by these two different kinds of energy and force interactions.

The material components around any t0 and τ0 always form part of some allowed set. The resources in the field M0L0 always require some force to carry them over the circulation. The field P-1L-1P1L1 stretching across t0 and τ0 brings to M0L0 the energies it needs to carry the population across the biochronometric distance between τ-1 and τ1.

Since we always have a present moment t0τ0 where our two fields combine, then we always have a trail of biologically active molecules centred on our material biological field M0L0. They are bound up within the n entities at that time. This field M0L0 stretches back and forth between t-1τ-1 and t1τ1 either side of it. The one span is absolute and linear in time, the other relative and curved about the circulation. Those two outer bounds are the limits for the required set walks that surround and produce the allowed sets that sustain and propagate the current-element M0L0 centred on t0 and τ0.

Since every biological entity stretches between two infinitesimal limits either side of it, one in circulation distance and the other in time, then they are all ‘limit-points’ whose existence can only be observed in the material events ranging around t0 and τ0, and so between the time and distance points t-1 and t1 and τ-1 and τ1 either side. Those are the rules in this geometry of biology.

Since energy cannot be stationary, then neither of the two fields either (a) centred on M0L0, or (b) passing through it via its limit points P-1L-1 and P1L1 is stationary. Both fields move and oscillate continuously. But one is more concerned with forces and times, the other with distances and energy. They interpenetrate, but one is more devoted to forming spheres, the other to balls.

The energetic field P-1L-1P1L1 abides by all extant rules of absolute time. It is linear. It is not material. It is a carrier of energy. It rays into, and out of, discrete points. It is therefore (a) uniquely assignable to some entity at some point in time, and (b) non-repeating. It is the driving force from the stick running the hoop through the surroundings.

The field P-1L-1P1L1 is non-repeating in the sense that although it is always possible to recreate any given mechanical chemical energy interaction, such as we do in a later generation, the photons needed for any such interaction must follow the second law of thermodynamics. They must constantly and irreversibly dissipate. The energy used in that later moment or generation is not the same, even if some of the materials used are. There is always a new and different strike of the stick to keep the hoop running. The indiscernible interactions that create an indiscernible point must therefore be recreated using a completely different tranche of energy on each occasion.

The energetic field P-1L-1P1L1 does not in itself follow the doctrine of indiscernibles. But it nevertheless succeeds in forming the 0-sphere by ensuring that it is constantly present everywhere. We can always strike the hoop with the stick and keep it running. P-1L-1P1L1 enters and leaves biological populations through the apertures. It brings to M0L0 the energy M0L0 needs at every time t0 and location τ0. It forms the limits for the M0L0 circulation. It penetrates M0L0 all along its length, so bringing to it the force that that latter material inertia needs. P-1L-1P1L1 thereby provides the energy those material components need for the columns, so they can propagate the three constraints of constant propagation, size, and equivalence. P-1L-1P1L1's net activities over both distance and time in all directions —T and τ—sum to zero to create and conserve both a circulation and a generation.

The alternative curving and material field M0L0 is constantly penetrated by the energetic and linear field P-1L-1P1L1 so that the resources and chemical components that it gathers from the surroundings can make the hoop roll. It follows the doctrine of indiscernibles. Its line segment of molecules and chemical components form a hoop and a 1-sphere that is circulating and repeating. It is always busy propagating itself over distance. It enters, forms, and then leaves biological entities and populations using the orifices. It is a structure. It creates the circulation of the generations.

The field M0L0 has the potential to circulate endlessly. It is again the hoop that is run. It can realize indiscernible points and all distances up and down the helicoid. Since its molecules are constantly being refurbished; and since it loops constantly around indiscernible points; then any entity, at any point, will eventually be replaced by one above it in the helicoid that is essentially indistinguishable. That replacement entity will be made from a completely fresh batch of chemical components. But it is indiscernible. It is indiscernibly linked to those both before and after. That is the 0-sphere running through the surroundings and about the helicoid.

No matter how infinitesimally short the length of time, the field M0L0 always has dual sets of limit points τ-1 and τ1 and t-1 and t1 either side of it. It is a ciruclar hoop running. Since it is always busy propagating, then it always has another and nearby limit point just beyond any current one. That one just beyond is part of the overall circulation that is the Liouville collection of indiscernible points. They define its length and properties. However, it can only circulate if it constantly receives a force or torque derived from its supporting P-1L-1P1L1, and which is always linear. It is a hoop. But it needs to be run.

Given this constant arrival and departure of surrounding limit points, there is no absolute origin for any portion of biological mechanical chemical energy flux in any M0L0. As in Figure 20.55.B, any components that are removed, over any interval, will always leave some behind them that continue. The hoop is always intact. That successor set of components can be followed both forwards and backwards beyond any proposed t-1 to t1 or τ-1 to τ1 limit points either side. There is always a hoop, even if its material and chemical components are never the same.

And in the same way, any components that arrive, over any interval, are always added to some prior set which can also always be followed backwards or forwards beyond any limit points. There are always, therefore, components that both pre- and post-exist any given of stretch of distance or time-interval t-1τ-1t1τ1 around any t0τ0.

Since not all material components can ever be removed from any M0L0, there will always be some continuing on beyond any t1 and τ1. There is again always a hoop running. And when those continuously maintained material biological substances are followed, we will always loop far enough to find some apparently different entity on some different helicoid step, using some different tranche of energy, but that is nevertheless indiscernible through occupying some continuously connected and indiscernible point over that distance. That prior (or following) entity and indiscernible point is then the apparent source (or destination) for the components under study … but with another just like itself, complete with its own limit points, lying just beyond it and so on and so forth. And since the distances we measure for, and about, this flux are in fact also intervals in time—measured both absolutely in clock time as t and biochronometrically as τ—then there is always both (i) a prior (or following) moment, and (ii) a prior (or following) entity serving as the source (or destination) for any selection of biological components. So although it always seems possible to assign the material resources and components within a population to some specific entity or entities, those resources do not, in actuality, originate, or terminate, with any of them; and nor at any moment. By the fourth law of biology, which is the law of reproduction, they were all in their turn reproduced at some earlier moment, or can continue to some later one for these are also distances. The flux density of mechanical chemical energy—which we know we can measure as the mendelity of —therefore at some times converges. This is to tend towards zero (as in → 0) because more flux appears to depart in each moment than arrives. This is the stage in the cycle in which overall entity size is declining. But while individual entities may disappear into zero, which is to converge and condense into discrete non-biological points in the surroundings and dissipate; the population overall may not do so … unless it is becoming extinct. The hoop must keep running no matter what happens to individual components.

The population wide material biological components that maintain the continuous stretches that make up any M0L0 never simply vanish. They have no identifiable sink into which they simply vanish. They always continue. Biological materials always emerge from other biological materials. That is the field M0L0.

Biological entities may also never do the opposite and appear as if from some distinct and identifiable source. There was always some cellular entity in existence before each moment. Anything else would be to appear from zero and from some discrete but non-biological point directly out of the surroundings. That would be spontaneous generation which Louis Pasteur refuted in his famous experiment.

Since spontaneous generation never occurs, then biological entities always emerge from prior biological components. They are not fresh-formed. So by the first maxim of ecology, which is the maxim of dissipation; and by the fourth law of biology, which is the law of reproduction; then while material biological flux may converge, that convergence never reaches zero. It eventually reverses; becomes a divergence; and then appears to expand out of some point or entity that is then the apparent source for any current growth, and so the current divergence.

The two fields P-1L-1P1L1 and M0L0 outlined in Figure 20.55 above—one circulating, the other linear—interpenetrate. They work together to create the entire circulation of the generations that we see in Figures 20.39 and 20.54, and which is the universe of biological events. One is in time, the other is in biochronometric distance.

Each entity in a population is always at the centre of some line-segment of distance, and is always either being reproduced; growing; or else is a part of some broader set that although it might be dissipating, is also—or soon will be—reproducing, so that any current limit points are themselves always at the centres of their own surrounding limit points. That is the character of this helicoid. These are statements of structure, and those are the rules in this geometry.

Every population must see the same thing when it looks at itself; at others; and at the surroundings. Since each is entitled to use either itself or another as a basis, then each must see both the ability to survive and the ability to reproduce the next generation. So every generation of Lepus americanus, for example, must find itself at a similar point within the ten-year cycle it creates with both woody browse and Lynx canadensis; and the same must hold for those other two relative to L. americanus. If such properties do not hold then predators in one generation would not enjoy the same prey their predecessors did; and succeeding grazers might have nothing to browse. All populations must share the same surroundings, which must act the same to them all. They must all measure the same polar planimeter. They must all be able to measure from the beginning to the end of a generation. This is from 0 to 1. They must all therefore use the same ruler.

The global and the local views of Leibniz and Newton are simply different views of the same sets of phenomena. Leibniz's global view compares developments across absolute clock time as energy changes from P-1L-1 to P1L1. It measures them as a power and as a Gibbs energy for interior transformations up the tensor columns for a generation, τ. These are the whole tensor, the diagonal, and the six off-diagonal components. It involves the Weyl tensor and the 0-, 1-, and 2-spheres.

Newton's local distance–momentum view instead involves the circulating field M0L0 and all observed changes between τ-1 and τ1. It is the drive and forcefulness of the Helmholtz energy along the rows and in time. These are the diagonal, the six off-diagonal components, and the whole tensor. It involves the Ricci tensor and the 1-, 2-, and 3-balls.

These various aspects of spheres and balls, global and local views, Weyl and Ricci tensors, absolute and generational times, and mechanical and nonmechanical forms of biological energy cannot really be separated. But although the two fields P-1L-1P1L1 and M0L0 interpenetrate, we shall from now on refer to the construction of spheres and distances as ‘metabolism’, for it is the spheres that establish the spaces within them, so mediating the interactions between their interiors and the surroundings. We shall then refer to the maintenance of balls as ‘physiology’ because they are the energetic interactions that transport the forces that then work over distances to maintain both the material components and energies contained within any sphere.

Metabolism, for our herein purposes, means a path of constant pressure mechanical energy interactions in which molecules pass into; or out of; or else cross over; whatever boundary is drawn about a biological entity or entities to produce some sphere. Metabolism passes exclusively through our orifices and is whatever ongoing process or processes move material-chemical resources into and/or out of the surroundings over some distance for some M0L0. Metabolism is the construction and maintenance of a hoop.

Physiology, by contrast, is any activity that changes a biological entity's, or population's, quantities of energy, but without any accompanying changes in its mass of chemical components retained. It is, therefore, a constant volume interaction and covers all energy transports using an internal gradient and potential. It passes exclusively through our apertures and is any nonmechanical path that allows an organism to maintain a 1-, 2-. or 3-ball, and forming the provisioning and sustaining energy field over some time interval and for everything between the limits P-1L-1 to P1L1. Physiology is the running of the hoop.

A mathematical aside

Metabolism is, more formally, the rate of change of the mass flux or mechanical chemical energy, and is dM/dt. And since the stock of chemical resources maintained in any stretch M0L0, and so over any interval t-1 to t1, is a mendel of U kilogrammes, then the metabolism is d2U/dt2 = dM/dt kilogrammes per second per second.

Physiology is the rate of change of the Wallace pressure, or nonmechanical chemical energy, so giving dP/dt = d2H/dt2 watts per second or joules per second per second where H is the total quantity of energy, in joules, held over the entire interval, so being the population, or entity's, enthalpy or energy content: i.e. H is its ‘biological enthalpy’ or ‘biothalpy’.

Figure 20.56
• It is a global truth—as Galileo noted, and as is now immortally enshrined in Newton's first law of motion—that every object continues in a right line, and so on the tangent plane we see in Figure 20.56 … unless disturbed from that right line by some force. The hoop keeps running in the direction it is going on that tangent plane. That force must then act over both distance and time. It also acts in an isotropic and homogeneous universe.
• It is an equally global truth—as immortally enshrined in Newton's law of universal gravitation—that as again in Figure 20.56, every rolling hoop and ball will be pulled by gravity so that it accelerates out of a right line and hugs the earth's surface. It will follow that global curvature and circulate about the earth.

The above two tendencies combine. Most architectural, geographic, and surveying projects do not need to account for the earth's curvature. The earth's surface can for all practical purposes be looked on as flat. However, an aeroplane or sea journey that does not take that curvature into account will be inaccurate.

When we bring the above two tendencies of straight line and circular motion together, we can draw an entire tangent plane to the earth's surface at any point. That tangent plane is then the plane along which all those rolling balls and hoops will move, and no matter from what direction they approach the normal. It will then always look, locally, as if the hoop is ignoring the earth's intrinsic curvature and making Galileo's doctrine of straight-line motion true, locally, everywhere. It will always look as if the hoop is going directly outwards along that tangent plane. However, that will still not be true globally anywhere. If the hoop keeps running, it will go all around. The earth's curvature may not be detectable locally and at the scale of any rolling hoop, but it still exists globally.

Now we have our two fields P-1L-1P1L1 and M0L0, our biological space of internal energy has what is effectively a set of straight lines for the former, and circles for the latter. They are the absolute and linear on the one hand, and the relative and the curved on the other. Seen in one dimension, they are the straight line and the curve, and our 1-ball and 1-sphere. Seen in two dimensions, they are the square and the circle and our 2-ball and 2-sphere. And seen in three dimensions, they are the cube and the sphere and the 3-ball and 0-sphere. They meet in the helicoid's indiscernible points. But for every movement in linear time, dt, there is some movement in curving biological time, dτ, about some circulation of length T. These are the possibilities of our twirling batons.

Figure 20.57

Gauss was the first to properly reconcile (A) the straight, the rectilinear, the cuboid and the absolute, with (B) the curving, the circular, the spherical, and the relative. King George III of what was then the United Kingdom of Great Britain, Ireland and Hanover commissioned Gauss to survey the royal lands in Hanover, then part of his Danish possessions. Given that the earth is constantly curving, Gauss wondered about the accuracy of his measurements.

The upper part of Figure 20.57 shows the reverse side of a ten Deutsche mark banknote issued in Gauss' honour and depicting the “heliotrope” he invented to overcome the measurement problems he saw in fulfilling King George's request and reconciling the linear and the curved. The lower part is an enlargement of the bottom right of the banknote and shows the map Gauss provided.

Gauss solved his problem by covering the land with a large series of triangular grids. He then used the mirrors in his heliotrope to reflect light from one point to another. He could now measure far greater distances more accurately than had previously been possible. Based exclusively on the area measurements he was making in only two dimensions with the attestably linear properties of light, he could determine the curving movements about the earth, into and out of the third dimension. Granted that the earth has a radius of approximately 3,963 miles, 6,378 kilometres, then if we could run our hoop 1 mile, 1.61 kilometres, directly outwards upon the tangent plane in Figure 20.56, our final position would be 7.98 inches, 20.31 centimetres, above the earth's surface. The earth always seems flat everywhere, locally, but its curvature is always approximately 8 inches per mile, or 20 centimetres per kilometre.

It is very easy to picture hoop runners, but they have to make sense in our biological space. The hoop runner certainly has the problem of keeping the hoop upright and in motion, but what are the biological equivalents of (a) upright, and (b) constant motion? Our problem, now, is to reconcile the linear and the curved in biology, but in meaningful ways. The Liouville ensemble and its energies work additively, while the Helmholtz decomposition theorem and its fluxes work multiplicatively. The energies needed move linearly with the field P-1L-1P1L1, while the material components move circularly with the field M0L0.

Every biological entity must measure at least one other entity—its predecessor—as being sufficiently close to it in its neighbourhood. That was the one that reproduced it at some prior time t-1, and point τ-1 in the same circulation. And if the population wishes to maintain itself, some must reproduce. At least one progenitor must measure at least one progeny one at some following time t1 and location τ1. Those two sets of points either side of t0τ0 must lie sufficiently close to its present location to make that space and that neighbourhood linear and flat—i.e. sufficiently non-evolving—relative to the population concerned. The local neighbourhood t-1τ-1t0τ0t1τ1 must be flat locally. It must be locally flat even though the population must be rotating globally to complete the circulation. This must hold at every point over both the absolute and relative times and distances involved.

One of our fields is linear, the other circulating. They must be reconciled at every moment. Every biological population therefore needs a ruler so it can establish the straight line intervals of absolute clock time it needs. It must also establish its relative sizes so it can measure itself and all others. Every biological population also needs a protractor to ensure that it rotates sufficiently about itself at each point to maintain its population and its generation.

We are now looking to measure the curvature that creates the circulation of relative biological time about absolute linear clock time. This is the energy that the field P-1L-1P1L1 provides to M0L0 so the latter can circulate and allow the entities concerned to reproduce.

We have no other reference but ordinary physical space. It may seem very obvious that a straight line in that ordinary physical is importantly different from a circle. It may also seem very obvious that gravity always pulls straight downwards. We know that plumb lines exist. Those are then a definition of straight in ordinary physical space.

There is also never any doubt when one mass in ordinary space is greater than another. We can tell this from its weight. These things may all seem very ordinary … but our biological space has nothing similar.

Biological organisms cannot reproduce, and we cannot discuss evolution, until we have some concept of what is straight and ordinary in our biological space of internal energy. We need some concept of when entities do or do not curve away or towards some natural or unnatural path, or some expected and unexpected neighbourhoods. Our biological space has no way, at present, to establish a preferred direction for any set of activities; and nor can it indicate what a straight line might be, nor even when one thing is greater or smaller than another. We must establish a biological geodesic: a shortest straight line for the biological equivalent of gravity.

Gravity and evolution are similar in that neither is very strong, locally. It is not possible to see evolution at work in any local moment. But Darwin argues that even though we cannot observe the process overtly, the species and the populations we see around us are evolution's handiwork. They are conglomerations of biological energy in distinct regions and neighbouhoods in biological space.

Although gravitational attraction can create large-scale galactic and cosmic structures, it is so weak that Newton’s universal constant of gravitation, G, is notoriously difficult to measure. It must invariably be measured locally, where it is barely observable. Individual objects, such as two balls lying on a table, do not exhibit much in the way of their mutual attraction. Other forces, such as friction, generally have priority.

Figure 20.58

The first to give an accurate value for Newton's gravitational constant, based upon demonstrating its local interaction, was Henry Cavendish. In 1797–98 he conducted a justly famous experiment which he originally used to measure the earth's mass and density. He felt that an improved constant would improve astronomical calculations.

Cavendish used a torsion balance originally conceived by the geologist and seismologist John Michell along with some lead balls that he arranged as in Figure 20.58. He used the curved movement their mutual attraction induced to measure the value he wanted.

Cavendish attached his two smaller, one inch, lead balls to a six-foot rod. He attached the rod to a fine wire. He then brought the smaller balls close to two larger fixed 350 lb lead spheres so that their gravitational attractions would induce the wire to twist. He calculated the angle with the mirror attached to the wire. The inertia of the two small balls would cause them to go slightly beyond the balancing point and oscillate slightly, which he could also measure in the mirror. Since he knew the wire's spring coefficient; the masses; and the distances through which the small balls moved; he could calculate the forces involved, and so the earth's mass and density which was responsible for all their weights. In 1873 other scientists repeated his experiment and used it to calculate G to within 1% of today's accepted value.

Cavendish demonstrated that although the attractive force that one ball exerts upon another is never zero, given the myriad other objects each one has to interact with, it is not generally worth computing. The same goes with evolution and entire biological populations. Their interactions with the surroundings are generally of primary importance. The potentially evolutionary interactions which are a consequence of their movements relative to each other are generally not worth discussing. But that does not mean there is no such effect. This is again the error that creationism and intelligent design make.

Figure 20.59

We can in fact show that if creationism and intelligent design are true, then all generations and biological populations must follow the patterns of behaviour we see in Figure 20.59.A. It is a movement, a circulating trajectory, across the fields and contours of the surrounding space that is technically called a ‘4-point contact’.

Gravitational theory tells us that all space is created by the gravitational field around some mass. All movements in space must then be towards or away from some object. Every movement must cross the field lines of whatever mass has created that field.

A ‘1-point contact’, seen in Figure 20.59.B, is a simple crossing of a curve and/or a straight line. This could be one object falling in another's field, or else one moving across the lines of force created by another. Thus a ball falling in space towards the earth would make a series of 1-point contacts with the countour lines at every height. Or two moving bodies can temporarily collide through having separate paths, and separate rates of change. When they collide, they hold a common magnitude, relative to the body creating that space, for an infinitesimal moment. They then go their separate ways. They exchange a magnitude via their rates of change. But since they do not have a common tangent they do not have a common rate of change or transform.

In a ‘2-point contact’, the two curves are tangential. If we throw a ball up, it reaches a maximum height; it is tangential to some field line at that point of greatest height; and it then begins to move away from that highest tantential point. The general principle is that one of the contact lines is more linear than the other. That one has the smaller rates of change both before and after contact. It therefore goes through a greater range of values. After their infinitesimal moment of contact, where they share both a magnitude and a rate of change, the two head off to their separate maxima and/or minima, exhibiting their different behaviours.

If two populations make a ‘3-point contact’, then they exchange rates and values. One approaches the point of contact being slightly more linear, while the other is slightly more curved. The more curved one forms an osculating curve at that tangential point of contact. They interchange so that the initially more linear one becomes more curved as it leaves, and vice versa.

In a 4-point contact, the two curves are more than merely tangential. This is like tossing two balls simultaneously up side-by-side, and then catching them again simultaneously, but also so that one goes up much higher than the other. If they rise and fall in the same time frame; but if one has to travel further than the other; then one must move faster than the other. One curve therefore sits inside the other. They both reach their maxima at the tangent so that their derivatives are equal and they share a common vertex and maximum.

If creationism and intelligent design are true then biological entities and their populations and generations can only be free from numbers, and follow a template, if they have constant 4-point contacts, relative to the environment. If two orbits about any given point are each to be endless and infinite, then they must be 4-point contacts relative to each other, so they are always parallel. The straight lines tangential to the two curves, as well as those radiating from their centres, are then their templates. Their instructions and energies arrive from infinity and impose their behaviours. They must also be without curl.

A straight line in biology must show some semblance of the behaviours we associate with a ruler in ordinary physical space: the archetypal straight line measuring device. It is also associated with gravity, because if we hold a ruler gently at one edge, its mass induces it to fall straight down. It acts like a plumb line. That plumb line concept indeed shows that straight and gravitational attraction are intricately and inseparably related.

Figure 20.60

Figure 20.60 shows the relationship between a plane and the helicoid of internal energy we are using to construct our biological space. They are both the kinds of ‘ruled surfaces’ that allow for exactly the measurements and movements we need to create the gradients, properties, and Weyl and Ricci tensors that show that creationism and intelligent design are simply impossible, and that also positively prove evolution.

Figure 20.60 shows one twirling baton in action for one dimension. All three work the same way. They can create the 1-, 2-, 3-, and 4-point contacts.

The helicoid and the plane can both always be described, locally, as the set of points swept out by a moving straight line. The flat plane in Figure 20.60.A grows from a baton held straight and then pushed ahead parallel, i.e. orthogonally, to its length. When we do this with all three batons, we create the x, y, and z or length, breadth and height dimensions of this ordinary physical space. These are straight forward to create and describe.

The helicoid in 20.60.B grows from a baton simultaneously rotating about a vertical and orthogonal axis, while uplifting vertically about that same axis.

‘Straight’ in these contexts means that the rate of change with respect to some other given dimension is zero. It is possible to move in x without moving in y and so forth. Such surfaces are ruled, and are also sometimes called ‘scrolls’. Since our biological universe can provide movements that are zero in all dimensions with respect to all others, then it is a scroll. Just as we can move in x separately, y separately, and z separately, we can increase and decrease numbers separately; mass separately; and energy density separately. This creates the sets of points that are the 1-, 2-, and 3-balls that reflect such spaces.

The plane and the helicoid are both scrolls. Every point lies on some straight line that is completely embedded in that surface. This gives each of them an impression of sameness and ‘flatness’. A helicoid therefore maintains the Copernicus principle. It will accommodate an isotropic and homogeneous biological universe. The twirling batons we first met in Figure 20.1 are the ones twirling here, and create exactly this space.

The embedded line that gives the scroll its flatness property is called a ‘ruling’. Any tangent taken from a smooth and ruled curve on such a surface is also ruled: i.e. embedded in the surface. So if an association, conjoining, or distribution is embedded, then so is the accompanying transform, directive, and aberrancy and vice versa. They all hold. If we have a rate of change that is zero, then any of its integrals and derivatives will also be zero, both relatively and absolutely. This is true for all populations both with respect to themselves and their successive generations; as well as with respect to each other as they measure each other.

We have represented a biological journey through the generations with a helicoid, which we can create by taking a ruling on a scroll, and drawing it smoothly upwards through space so our twirling batons can create the coverings for the balls, which are the 0-, 1-, and 2-spheres. This has the same effect as the process in Figure 20.60.B, where we add a twist to a section of a plane. We can now rotate the plane equally smoothly about its central axis, while still uplifting the scroll and its ruling to create a helicoid, along with the Weyl and Ricci tensors. This combines the straight line and the circle. If we explain the plane we explain the helicoid and conversely.

There is a big difference between the horizontal and the vertical measurements that we can make on our biological scroll. All measurements upwards, and so at 90° to the horizontal, are poloidal. They go from pole to pole and are pure measurements in time. They are points that lie directly above and beneath each other. They describe states that endure in time by being timeless and unchanging. They allow us to state how long certain features have been maintained, all in terms compatible with the Weyl tensor and so with heredity. If a gene lasts indefinitely in time, it will move vertically and indefinitely upwards for all its indiscernible points. This is what creationism and intelligent design propose. All biological populations are unchanging. They all have features that persist endlessly in time. This creates a right helicoid.

All horizontal measurements on our biological scrolls, i.e. at 90° to the vertical, are pure measurements in space. We call these meridional. They are entirely structural. They allow us to describe properties, structures, attributes, shapes, and distances that do not involve changes in time. So this creature has four legs; that tree has rootlets; and the ones before and after them are the same. They are compatible with the Ricci tensor's volume elements. They are the morphological and ecological attributes.

Biological movements are neither purely temporal, nor purely structural. They involve elements of both. They are toroidal. They can pass between populations and entities over time. They eventually produce a circulation of the generations, which is a set of changes in states and structures over time. Toroidal movements are therefore always inclined at some angle across the scroll, and are partly poloidal and partly meridional. The same features keep repeating over time. They are hereditary.

We can see the effects of twisting the plane into a helicoid in Figure 20.60.C. It looks, there, as if the helicoid has a slope. Our batons are being drawn upwards at some rate through time. That would appear to be a structural change in time and so at some rate, which is an entire biological velocity field.

The distance x1 to x2 looks straight upon the plane it originates from … but it looks curved out of position on the helicoid of internal energy. It looks as if it has been carried round to x3. It now looks as if we have three non-linear points x1, x2, and x3, with all the implications this has for transports of states over time, which are events governed by our Weyl and Ricci tensors and our balls and spheres which all interpenetrate and affect each other.

Although the flat plane and the helicoid are very similar, they have an important difference. We can set a ruler and a protractor down anywhere on Figure 20.60.A and draw a first line; rotate our ruler through any angle; draw another line through that same point; rotate again; and repeat as often as we want. All those lines pass directly through that point. The plane is the only surface that can completely contain (at least) three distinct lines through each of its points.

Our helicoid initially appears very different. If we use a ruler and protractor in a similar way, it initially looks as if only two lines will pass through any one point:

1. There is the ruling that stretches outwards from the axis and so that generates each point structurally and spatially. This is the meridional line of latitude that links all locations in space at that same given time.
2. There is also the ruling parallel to the axis as the generating line is drawn upwards. This is the poloidal and vertical line that links all locations that seemingly share the same distance or penetration into the circulation.

Indiscernible points on the helicoid share the same latitudes and longitudes. The longitudes are separated only by their pitches. They appear when we apply the twist.

The helicoid's poloidal lines extend up and down through apparently empty space. They are seemingly off the helicoid. They pass through the kingdom of absolute linear clock time. They pierce up and down, through time, from one indiscernible point to the next. The distance between them, which is the pitch, tells us how long it is going to be before we return to the same state. Those states can be indiscernibly substituted for each other, leaving everything else the same. This is the helicoid equivalent of the protractor and the ruler for we can always get to the point next door using the pitch. The pitch substitutes itself for the angle through which we rotate the protractor. The outwards and inwards journey from S-1 to S0 and then back to S1 in Figure 20.60.B is an out- and inwalk for a generation. The horizontal component is number, mass, and energy. The vertical component is time. The two together are the biological journey.

When we apply the twist to create our helicoid, S-1 and S1 end up with the same values. They are directly above and beneath each other at the periphery, with S0 being close to the axis. The journeys on the two halves of 20.60.B are then biological converses. They are the reversible journeys in mass, number, and energy that all biological populations must undertake. All the points above and beneath each other are now timeless states. They belong to the helicoid across the space, which is the pitch. All such points can be repeated through being indiscernible. The biological equivalent of gravity is then the force to repeat those points.

Since a helicoid is made from a plane, it is as extensive as a plane. It has no boundary and is a “complete surface”. Anything we prove on the plane we prove on the helicoid which defines the repetition of states. We can now use our two complete surfaces—the helicoid and the plane—to describe biological populations, and to show that one free from Darwinian fitness, competition, and evolution is simply impossible.

Since we can untwist an entire circulation to create a rectangular area, we can represent a population's genome with a single strip's horizontal width and relative position. No two species can now have the same width and the same meridian without having the same means and other values, and so lying directly on top of each other at all times. To be above and beneath each other in all ways, and at all times, is to have the same timeless states and structures. It is to have the same energies, forces, and distances. It is therefore to undertake identical processes which is to have the same 0-spheres and indiscernible points. As in Figure 20.60.A, the rectangular strip for a given species stretches up and down indefinitely, and is independent of any other. And if it is twisted to form a helicoid, it produces the constant and infinite repetition of its indiscernible points. Once again, if two such strips or species coincide in every way, then they are the same.

If one helicoid now leans relative to another, then since they are both infinitely long, they will eventually run into each other, which is to evolve. And since each dimension can be freely exchanged for any other, then all distinct species must have all their lines parallel, and they must each be enclosed within distinct rectangular strips similar to those in Figure 20.60.A, which are the right helicoid of Figure 20.52. If we can prove the right helicoid is impossible, then we immediately prove that creationism and intelligent design are impossible.

If two species now have the same sets of generation values and the same width, which is the same range for a helicoid step from minimum to maximum across the generation, then they have the same associations, conjoinings, distributions, transforms, directives and aberrancies at every point, both absolutely and relatively. They are the same genomes constructed from the same chemical components, which are being absorbed and emitted at the same rates. By all the laws of chemistry and physics, they are then the same. Therefore, if creationism and intelligent design are true, then all locations and distances on those strips must remain fixed for each species, and must always have the same distances and angles.

Figure 20.61

Biological entities are always being surrendered to the environment at some given pace, with more always being born. If biological matter is never stationary; and if it is always looking to complete a circulation; then it is very like the earth's surface in Figure 20.61.A which also goes around from point to point.

If we look at the earth carefully, from this vantage point, we can see that, like a biological circulation, no part of its surface is flat. Although it appears flat, linear, and normal, locally, it all curved everywhere, globally. That curve imposes a directive and an acceleration all around it for all objects upon it. They can move nowhere without changing their relative magnitudes in space.

We need a general way to determine the surface and regional accelerations about any proposed surface, and simply through points in neighbourhoods holding specific relationships to each other. It is the rate at which the linear, material, and nonbiological is converted into the curving, biological, and hereditary. That curvature is biology's velocity field. It is the cause of its circulation of the generations.

Gauss realized that the space around us is deceptive. It deludes us into thinking we are measuring straight, locally, when the space's global nature is inducing us to move—as the ships and the hoop runner in Figure 20.61.A are—in a curve:

1. The hoop runner running along the helicoid in Figure 20.61.B believes that he is on a straight line and a tangent. We have to identify that straight line behaviour so we can measure the transform from linear and absolute to curved and relative.
2. The curve the hoop runner is in fact running on imposes a directive or acceleration. The spirit level in Figure 20.61.A is the straight line tangent or baton that measures the directive by heading off into outer space upon either side. It allows us to treat the the local space as if it is smooth and even everywhere.

Gauss pointed out that we can soon determine the amount of curvature, or rotation, being imposed anywhere on earth, or on the hoop runner on the helicoid in Figure 20.61.B, by imagining ourselves, from our external vantage point, drawing a circle right underneath him. The circle's radius is the boy's current rotation at that point. We will then also have a normal we can use to measure how tall he is as he protrudes up from the earth. We will also have the tangent and an entire tangent plane that tells us his exact speed and direction. We will in other words have our set of tangent vectors to describe all associations, conjoinings, and distributions, as well as transforms, directives, and aberrancies.

Gauss realized, however, that the above method could only find a line’s “extrinsic curvature”. It is extrinsic because we measure from outside it. We must first step up a dimension, to this three-dimensional world we inhabit, before we can use it.

The extrinsic method is unworkable when assessing whether or not our own three dimensional world is curved. We cannot step up to any fourth dimension to assess the curvatures being imposed all around us. Gauss nevertheless offered a solution. He introduced the idea of “intrinsic curvatures”.

A circle's curved nature seems obvious looking at it externally, but it would not necessarily be noticeable if we examined it longitudinally: i.e. by placing ourselves along the circle itself. All we will ever know is that there is another point in front of us, always at exactly the same distance away. We never know any different, and would never perceive that we had gone all the way round again to the beginning. Gauss thus declared that it has “zero intrinsic curvature”.

Figure 20.62

Gauss next dealt with the problem of measuring intrinsically in two and three dimensions. He pointed out that as in Figure 20.62, no matter what surface we are on, we can always draw a large circle all about ourselves. We can then measure its radius, r, its circumference, c, and its area, A. This allows us to calculate what the circumference and area should be by using the standard formulae c = 2πr, and A = πr2. We then measure our circles; calculate those values; and compare the one set to the other. If they are the same, as in Figure 20.62.A, then the surface underneath us is flat. It has no curvature. But if, as in 20.62.B, the surface curves gently away so that the circle is smaller; or else if, as in 20.62.C, it curves upwards so the circle is larger; then we can calculate the exact amount of curvature from the differences in the measured and calculated values. Gauss called this its “intrinsic curvature” and proved the theorem that: ‘An inhabitant of a 2-dimensional surface can measure the curvature of their universe by using only a ruler and a protractor. This curvature is intrinsic to the surface’. So no matter how flat the space around us might look, we can now find out if it is curved. We measured Brassica rapa's space as curved. It is therefore not a population of the right helicoid.

It is perfectly possible for Figures 20.62.B and C to combine. The surface then curves first one way, and then the other. The measurements we make could now end up the same as the values we calculate, while the surface is still curved everywhere. Therefore: two surfaces can easily have the same calculated curvature, yet be completely different when measured everywhere.

Gauss’ pupil Riemann solved the problem of how to handle surfaces that were different, but that had the same values for their curvature. We draw tiny little squares and other parallelograms all about the surface. We then measure them in both an inwalk and an outwalk. We next calculate the values in each direction; and then compare calculations to measures. This gives the “Riemann curvature tensor”, which is an intrinsic measure.

We can now state the problem with creationism and intelligent design. They are correct in seizing on the local truth: every one-dimensional line and curve is without intrinsic curvature. Time, whether biological or nonbiological, always seems just the same. Creationism and intelligent design are also correct in insisting that, looked at locally, a curved line is always exactly like a straight and flat one. Any apparent curve is an irrelevance. Every stretch of time is also congruent with an ideal template, with no attributes. No matter how any such line looks to us externally or extrinsically, it would only “have” a curvature if we could observe some intrinsic difference in the distances between points right there along the line itself, and using Riemann's methods.

But this is then where creationism and intelligent design are in error. They deny the existence of intrinsic curvature. They therefore deny that the circulation for any given species could ever vary. So all we have to do now is prove that those circulations can indeed vary, and that the space that creationism and intelligent design describe is impossible.

If we notice some difference along a line, intrinsically, we have immediately moved to (at least) two dimensions because we are now measuring a difference in, for example, intensity along the line itself, which is automatically a second component. We now have two dimensions and are speaking of divergences, deviations, linearities, aberrancies, areas, displacements, and vectors. We would immediately have (at least the) first, second, and third derivatives which are measures for its linearity and transform, and its curvature, directive, and aberrancy. We can now take measurements and calculate curvatures. Those would all be timeless and intrinsic properties of that space.

Figure 20.63

Using Riemann's methods, we can measure the curvatures in—and so the accelerations that are imposed by—the three surfaces in Figure 20.63. All the little squares we draw on the flat plane in Figure 20.63.A produce exactly the same values everywhere, no matter which direction we care to measure. It is therefore flat. It has a “mean curvature” or “mean acceleration” of zero because its curvature is exactly zero everywhere. It is a “minimal surface”: one that seeks to minimize the area all around it in the sense that it can support perfect circles, which are shapes that bound the maximum possible area within the shortest possible lines.

The surface in Figure 20.63.B has different curvatures all around each point. Since the curvatures all around each point are different, then its mean curvature is not zero anywhere, and it is not a minimal surface.

Figure 20.64

The helicoids in Figure 20.63.C and Figure 20.64 are a special case. As with all helicoids, they are each generated by rotating a straight line about a central axis, and simultaneously uplifting it up that same axis.

Figure 20.64 clarifies the two important helicoid-generating processes. Figure 20.64.A looks at it head on and shows the inner and outer circles that produce the helicoid steps in 20.64.B. Since they are circles, they both have exactly the same aberrancy and curvature. They neither approach nor leave each other at any point, and will never intersect. They are parallel. The doughnut shape they produce states maximum and minimum values on a helicoid, along with implied rates of change in and for internal energy. It could support a species. These are the meridional movements. When we twist it we get a right helicoid. This right helicoid of internal energy is what we have to show is biologically impossible.

When we pull or twist the flattened two-dimensional spiral or helix in Figure 20.64.A out of the plane and draw it upwards along the vertical axis, it becomes the three-dimensional helicoid in 20.64.B. This is its amount of “torsion”. That torsion produces the pitch, which is the rate at which it moves, in time, from pole to pole. And since the helicoid is generated by both a straight line and a circle, then it retains those two essential properties all throughout itself. We have produced a homogeneous and isotropic space that can again support a species:

1. For every point the torsion produces by moving a specific distance upwards, there is an equivalent one at exactly the same distance downwards.
2. For every distance the rotation produces forwards around the axis, there is an equivalent one at exactly the same angle backwards.

A mathematical aside

If the helicoid has the formula x = ρsinθ, y = ρcosθ, and z = kθ, then the arc length, which is the elapsed generation length, τ, is (ρ2 + k2)½ × θ; its curvature, c, is ρ/(ρ2 + k2); and its torsion, t, is k/(ρ2 + k2). By Lancret's theorem, the ratio c : t is constant for all helical curves, so that c/t = ρ/k.

Granted that all points on a helicoid have equivalent or indiscernible points both forwards and backwards, as well as upwards and downwards, then they all sum together, using Riemann's method, to produce a zero mean curvature for every point. So while the helicoid may appear to curve gently everywhere in all directions, if we use Riemann's method and measure in all directions, the sum of all their curvatures, in all directions, is zero, making the helicoid exactly like the flat plane. This is why two squirrels chasing each other around a tree, for example, will describe a helicoid. They are each striving to move as fast and as straight as possible; and they are each striving to cover as great a distance as possible in the shortest possible time. Their spirals are the straightest lines for that surface, and the helicoid minimizes lines and areas everywhere. Every point has the same mean curvature of zero. And since this holds for all points, then the helicoid has a zero mean curvature everywhere. It is therefore a minimal surface with zero mean curvature. It and the plane are the only two minimal surfaces through both having zero mean curvatures. Whatever we prove on the plane, we prove for the circulation of the generations that form the helicoid. We can in particular prove that biological populations will always try to follow the geodesic or straight line path for a helicoid.

Figure 20.65

The helicoid as a minimal surface means that, as in Figure 20.65, we can freely apply to it the three different coordinate systems we first met in Figure 20.18: the rectangular, the cylindrical and the spherical. We see those three reproduced in 20.65.A, B and C respectively.

Our surface of zero mean curvature declares how our biological space of internal energy must behave if creationism and intelligent design are true. It helps establish what we must measure.

Figure 20.65.D shows a coordinate system being applied to the entire helicoid, while 20.65.E shows a section ready to measure the divergences and the curls of the allowed sets that result from the required set walks. It is a statement of the internal energy in our space. It means that we can measure everything we need, in both entities and populations, without fear of distortion. It does not matter what our batons get up to. No matter what linearities, curvatures, or aberrancies are involved, the helicoid of internal energy allows us to measure any population, with all measurements being guaranteed accurate.

A “manifold”, as defined in Riemannian geometry, is a geometric object that allows us to take any local and infinitesimal subset of space which then behaves exactly like this familiar space here all around us. Our helicoid of internal energy is therefore a manifold.

A “Riemannian manifold” contains both a “metric field” and the “metric tensor”, which allows us to take the kinds of measurements we need for any surface with any number of dimensions. They allow us to determine shapes, distances, and curvatures accurately. The metric field allows us to take measurements for our circles and our squares, and so that we can gradually build up the true picture of the underlying Riemannian surface. It allows for measuring distances, finding areas and volumes, taking tangents and normals, and even integrating and differentiating. The metric tensor then uses Riemann's method to produce a set of numbers that describe every point, and no matter how many “folds” or “dimensions” it may have. We can select axes and coordinates so that any number we allocate to the surface by measuring in a first direction, is equal to the number we allocate by measuring in the opposite direction. It means that if we have a basis and then measure 2 units in the first direction, then the basis in the other direction will measure ½. So for any general measurement, g, we make as an α and a β, which we can denote as gαβ, we will have a reverse measurement gβα so that gαβ = gβα, and no matter what the basis, nor how much one or the other changes.

Using Riemann's methods as applied to manifolds, we can take any surface and draw an entire tangent plane so we can determine the amount and rate at which that surface is curving, in all possible directions. As in Figure 20.56, it is even possible to draw entire two-dimensional tangent planes to the three-dimensional Riemannian surface. The tangent plane allows an object to approach the normal from any direction. The normal emerges at 90° to allow us to create areas or divergences. It also allows for movements up and down, in that third dimension, to create volumes. We can now take tangent vectors and create entire “tangent vector spaces” at every point. It is these Riemannian methods that allow us to measure the earth's curvature as 8 inches per mile, or 20 centimetres per kilometre. When applied to our biological and internal energy space, it will allow us to state what behaviours we can expect in each and every dimension. We can also measure the biological dependency on numbers.

We are not limited to three dimensions. The metric tensor that defines a manifold and/or surface, by using tangent vectors as its inputs, can be taken to a four-dimensional manifold. We then have three-dimensional tangent vector spaces that are exactly like this world we inhabit, along with all its rates of change. Our helicoid of internal energy has its three-dimensional and tangential vector spaces which are again like this one. A helicoid requires only that we constantly displace a surface around some fixed axis, while also displacing it parallel to that same axis with a velocity that is proportional to its angular rotation about that axis. We now define our geometry for biology as a manifold whose tangent vectors are the space we observe around us, complete with Weyl and Ricci tensors and indiscernible points for the circulations of the generations.

A plane has a mean curvature of zero because the sum of all possible curvatures around all its points is always zero. The helicoid, however, only has a mean of zero because its sum is zero in all combined directions … but without necessarily being zero in each distinct direction. The helicoid has its zero mean curvature because it moves from indiscernible point to indiscernible point within a given time period. The helicoid's spiral always seems evident extrinsically. But because it constantly folds about itself, it and the plane are identical intrinsically. They are indistinguishable. They both always contain infinitely many identical points whose mean curvature is zero near to every other one.

Figure 20.66

As we see in Figure 20.66, we have now added a fourth way of measuring the biological activity in our configuration space to the three we had—the rectangular, the cylindrical, and the spherical—back in Figure 20.18. Our latest helicoidal system adds both the pitch and rotation measures to state how far around and how far up each point is from another, as its indiscernible point. We can measure any chasing squirrels or twirling batons. These four are all ways of measuring the same space and object: our biological circulation of the generations.

The three dimensions in our configuration space are number, n, mass or mendelity, , and visible presence, V; although it will sometimes be preferable to refer to this last through the biopressure, . They behave exactly like the x, y, and z dimensions in ordinary space. Whether we think of our biological population globally, with Leibniz, as moving through its Liouville phase volume, or locally, with Newton, as creating the intersecting fields of M0L0 and P-1L-1P1L1, our biological potential μ is the sum of all the biological forces acting on a population. Just as we have a gravitational experience, F, at every point, so do we have a biological one, μ. It is the sum of the infinitesimal increments in the three constraints of constant propagation, dφ, constant size, dκ; and constant equivalence, dχ, so that μ = dφ + dκ + dχ. It is therefore the sum of all the forces transforming the population at any time. A population is always at some point in our three-dimensional biological space of n, , V. Since μ is our biological potential; and since the three coordinates for our space are n, , and V (or, equivalently, n, , and ), then we can always describe a population's complete state with the triplet of values μ(n, , V), which is always some point in our internal energy space. If we now use L to denote the Liouville phase volume for any species, then all that species' locations for all possible generations in our isotropic and homogeneous space, with its timeless states and indiscernible points, must satisfy the relation μ(n, , V) = L. We now have a set of paths for that population, and a set of gradients, values, and behaviours for its attendant Weyl and Ricci tensors.

The statement μ(n, , V) is the declaration that the population's behaviours and transformations can always be summarized as the net change in its numbers, n, in its mass flux or mechanical chemical energy, , and in its energy density or visible presence, V. This V is the sum of all those changes in its Wallace pressure or energy flux or nonmechanical chemical energy, P, as are not caused by changes in either numbers in the population, or their ongoing changes in the numbers or types of chemical components retained. Those are therefore the three forces in numbers, mass, and energy our three batons create as they twirl about to create this biological space.

Figure 20.67

As in Figure 20.67 (as well as Figures 20.34 and 20.36), all the points a population occupies in its circulation are linked by lines that represent its forces in metabolism and physiology all about that circulation. Those lines are the equivalent of tossing a ball up and down or a planet orbiging about a sun. The entire set of rates of change, all across a given generation, is now the engine that carries that population all around its circulation of the generations. It is the behaviour that allows its members to be indiscernible from each other over time, and thus to be members of the same species.

If creationism and intelligent design are true, then it must be possible to describe a species' metabolism and physiology independently of anything happening to any specific members or numbers: i.e. independently of n. We must construct some alternative to μ(n, , V) = L to depict a population free from numbers.

We are looking for a biological straight line. Whichever coordinate system we or a given population use to measure in our space, a population is always moving between two points in our biological space that are infinitesimally close together. We call the initial point i, and the final one f. Those points are subject to the three forces of numbers, mass, and energy that pervade our biological space.

We are now applying Riemann's measuring-forwards-and-backwards strategy. The points i and f are in each other's neighbourhoods, where the neighbourhoods are a Riemannian manifold which is always flat and local. Whichever of the four coordinate systems we use to describe our population at any two nearby points, they will each have a value for n, , and V. We can therefore describe the “configuration distance”, c, between our initial and final points, i and f. We can write this as c(i, f). It is therefore a distance between two configurations or states of internal energy in biological space involving numbers, mass, energy, and time. We now want to make that distance as small and short as possible, so it meets the definition of ‘straight’.

As the population leaves its initial state i and transforms into its final state f, its configuration speed depends on how rapidly it approaches f and leaves i. Since we are using tensors, we can use either location as a basis. We therefore have two different ways of describing the same event. Either (a) we can use the initial location, i, as our basis and describe the configuration change as occurring from i and towards f, which is ci; or else (b) we can use f as a basis, and instead describe the situation as the approach towards i, and from f, which is cf. When we measure across a complete generation, which is from indiscernible point to indiscernible point, those distances will be the same even if the numbers look different because of the difference in bases. They are the same. Only the descriptions are different.

We now apply the Riemannian reversible measurement principle. We can express our configuration distance c(i, f) in terms of the configuration changes it expresses, but based in either ci or cf as is our choice. If there is a change from 100 to 200 then it is a proportionate change of either +100% or -50%, depending which way we choose to express it. Since we want to know the rate of change at every point, this gives us a choice of the two partial derivatives ci = ∂c/∂i and cf = ∂c/∂f. These are measurable values that describe the change from each point and viewpoint, and so in each direction.

The arrow of time that drives our energy field P-1L-1P1L1, and so which sets the direction in which molecules and energy dissipate, is the direction in accord with absolute clock time. It has a favoured direction. Two events may be either two seconds or two years apart whichever direction we measure them, but there is no doubt which one is in the past relative to which.

Since our configuration space clearly has a favoured direction, it is a vector space. If we go in one direction, using ci, we will be going forwards through the generation and we will see a biologically expected set of behaviours and configuration changes. But if we go in the reverse cf direction, and measure towards i, then we are going backwards, and we will see the reverse.

Although every biological entity comes into being, and every one of them dissipates, there is a vital distinction between our two directions. If we pick the ci one to move in, which is from any i to any f; and if we keep keep going; then we will eventually observe every entity—without exception—dissipate into the surroundings. We may see others appear, but we see them all dissipate. Maxim 1 of ecology, the maxim of dissipation, expresses this as ∫ dm < 0.

There is another important aspect of this ci direction and movement from initial to final. There may be a certainty that we will see every entity dissipate …but even though every entity is born, there is no certainty we will see a reproductive event when we follow each one. Not all entities reproduce.

If we now go in the opposite cf direction—which is from the final state f and towards the initial one i—then although we might be going backwards through absolute time; and although we might be going backwards along generation distances; reproductive events are now certain. Every single biological entity comes into existence and is reproduced at some point, even though they do not all themselves reproduce. We have now stated that proposition rigorously and measurably.

Since the two configuration directions ci and cf, are completely different, then all distances and required set walks are again vectors. One direction will see energy conserved as it is transformed from one condition to another over time, while the other will see the forces that act over those distances to produce those changes in condition. Either way, we must establish a basis of measure. This is a set of unit vectors. We also want a flawless system of measure to describe them.

We procure the basis we need, for our unit vectors and our tensors, in the standard way. We measure in each direction. We then find the mean value. That is our basis. Since it is the mean value, then ci and cf will automatically give the same values relative to each other. They have different numbers and rates of change in their separate directions, but in terms of each other. They will differ only in that we can always tell which one precedes which about the circulation. Their descriptions are different, but the states they describe are the same. If one is one-half measured one way, the other is double measured the other. While the numbers attached to them might be different, the distance they describe is the same.

If we now take a large number of such measurements, and gather up a large collection of mean values for ci and cf, we will gradually build up the tensor that describes the entire generation. We will describe the same situation both forwards and backwards, all about the circulation. We will eventually get our basis of measure. That allows everything to be the same. Those are the three weighted values for numbers; for mass; and for energy density. They belong to the entire circulation.

We now have our basis vectors and units of measure for this space, relative to that population and its circulation. The population can also measure any other, just as all others can measure it. We now have the values that create the twirling batons and straight lines for our populations and their interactions. These are what we can use to create our biological space of internal energy.

Now we have those units of measure, we can restate the claims creationism and intelligent design are making. We can restate them as the assertion that they are only interested in those behaviours and relationships that are independent of numbers, and so independent of any and all changes that might be applied to which they regard as always being the same. If such proposals are valid, then we should be able to identify some specific subset, σ, of all general Liouville behaviours that pinpoint those number-independent interactions. We have indicated that subset with the line of arrows of constant height about the circulation in Figure 20.67.

We already know that if we want to isolate the behaviours that are independent of numbers, then we must apply some extra constraint, c, to the entire Liouville phase volume so that it does not produce the full range of values in μ(n, , V) = L but only instead the subset σ(n, , V) = c where the set of values in n is fixed. We will then have the complete set of interactions in mass and energy that preserve that species when its numbers are constant and/or all its changes are irrelevant. We will also be able to take measurements to prove the case either way.

A mathematical aside

We can now of course apply the Lagrange multiplier technique to solve for μ(n, , V) = σ(n, , V) = c. Since the proposed circulation in biological potential must satisfy that condition, we need to maximize μ(n, , V) subject to σ(n, , V) = c, which is a specific set of normals and tangents. If we apply the technique and use δ to represent the Lagrange multiplier, then we must solve the further function:

Γ(n, M, V, δ) = μ(n, , V) + δ(σ(n, , V) - c).

Since we already know n—it is fixed—we only have to determine those points where the partial derivatives in Γ are zero. We will immediately have the complete set of interactions between metabolism and physiology for that population, and its ensembles, that are independent of number. We will thus define the circulation of the generation in terms of those values for mendelity and biopressure that allow the entities in all ensembles to exhibit those behaviours that are truly independent of number. We can then select any population, such as Brassica rapa, and measure it to see if any of its entities consistently maintain these predicted values.

There is another way of describing this. It puts the realization that numbers can be fixed to good use. If creationism and intelligent design are true; if numbers and quantities have no influence on any biological population's essential properties; then we can remove the entire row and column in the Owen tensor that allocate changes in numbers. We can remove both that row and that column from consideration. Those are the no-go areas in Figure 17.1.

We now have a definition. That proposed set of transformations, i.e. without the influence of numbers, is the essential development, λ, we want. That essential development is how populations must behave if they are truly free from numbers. We can find it at any time using derivatives such as dndt, and partial derivatives such as ∂n⁄∂t, along with their higher order versions d2ndt2 or ∂2n⁄∂t2 and the like.

Our fourth maxim of ecology, the maxim of apportionment, states that a biological population will increase from an initial to a final value for three reasons: (a) increases in mass; (b) decreases in competition; and (c) the essential development, λ. This is the same set of mass and energy transformations that allows a group of biological entities to maintain themselves as members of a Liouville ensemble, and so to be indiscernible over time by moving constantly over a specific set of indiscernible points.

We can now say that the straightest and the most direct journey to the next indiscernible point is the essential development, λ. It has no need for—and avoids—all variations. It is the 2 × 2 essential development tensor and set of values we see in Table 20:4:

 σ(n, M, V) = c Constraint of constant size, κ Constraint of constant equivalence, χ Metabolism Tmass:mass τmass:energy Physiology τenergy:mass Tenergy:energy

Biological creatures may live in our three dimensional biological space of n, , and V, but they also live in ordinary physical space with its standard x, y, and z dimensions. That is where they thrust out their forms and shapes, and where they engage in their behaviours. It is where we measure them.

Each population must have its own domain for its essential development, λ, on our helicoid of internal energy. Each must have its own range to determine those physical characteristics and behaviours it must express in ordinary space and time. Those values declare its localized configurations, transformations, and rates of change.

Each population now navigates its way between the potentials and the possibilities made available to it, by its essential development, whilst all the time interacting with the environment. Each interacts in both biochronometric distance and absolute time, and so with both force and energy. Each population thereby converts the potential energy of its position, relative to its generation mean, into the kinetic energy of moving about the circulation within those surroundings. The forces and the energies exchanged over both distances and times are the essential development contained in the helicoid steps, and so in the biological space.

We are nearly in business and can soon run our experiment. This essential development, λ, that is the configuration mass-energy of the helicoid, is:

1. the biological activity that excludes the no-go areas in Figure 17.1;
2. the circular or regularly shaped curves in Figures 20.3 and 20.35, and so the weighted values for , and that the polar planimeter uses to calculate its areas;
3. the flat plane of four quadrants in Figure 20.25;
4. the surface of one of the three intersecting spheres again in Figure 20.25;
5. this plateau of constant height in Figure 20.67.

Table 20:4 above only has four values. They are the interactions of two forces: the mechanical and the nonmechanical. All we have to do to prove creationism and intelligent design either way is measure those two biological sequences and sets of divergences, τmass:energy and τenergy:mass, and the two times or cycles over which they change all about a circulation or generation, which is Tmass:mass and Tenergy:energy over a generation. The former two state the events that must occur; while the latter two state the times they must take. That overall rate of biological activities now uniquely defines the population.

Table 20:4 tells us that we now only need to isolate the four components that define the observable values for mass, energy, metabolism, and physiology in any population and we will have the T, , and that uniquely define that species. These are τmass:energy, τenergy:mass, Tmass:mass and Tenergy:energy, all of which are events and sequences that are most easy to measure.

Table 20:4 also tells us that since we are looking to make numbers irrelevant so we can validate creationism and intelligent design, then our initial and final points i and f, and our initial and final configurations, ci and cf, must always occupy certain very definite neighbourhoods in our space. Just as Carathéodory defined entropy and the second law by saying “in every neighborhood of any state S in an adiabatically isolated system there exist other states that are inaccessible from S”, so also can we say that while every population is most likely surrounded by many possible states, if any of them truly want to be indifferent to numbers, then they must regard some amongst them as unavailable and inaccessible. Those are the ones that involve changes in numbers.

We must now put together a method for separating all transformations involving numbers from all those not involving numbers. We call this method the “sieve of Aristotle”.

The sieve of Aristotle is named after the “sieve of Eratosthenes”, an ancient process or algorithm for finding all the prime numbers up to and including a specified integer. To find, for example, all the prime numbers under 30, we first write all the numbers from 1 to 30 in a list. We start with 2, which we know is a prime number. We leave 2 standing at the head of the list, but strike out all the even numbers because they are all multiples of 2, and so cannot be primes. This 2-cut removes 4, 6, 8, etc as candidates. We are left with only the odd numbers between 3 and 30. The first number left standing after any such cut, in this case 3, must be a prime, because otherwise it would have been struck out in the previous step. But no multiples of 3 located after it can be primes. We therefore strike out 9, 15, 21 and so forth. The next number left standing after this 3-cut, 5, must again be a prime, because it is the first after a cut. It is the first that is not a multiple of either 2 or 3, our two previous cuts. We now cut 25, which is 5's only remaining multiple. The next number left standing is 7. Since all its multiples (14, 21, 28) have already been removed, we move on. The only other numbers left still standing are 11, 13, 17, 19, 23, and 29. None have multiples in this set, and none are multiples of any others, so we now have all the prime numbers. We have successfully sieved out all the elements we do not want.

If a population is going to be free from variations in numbers, then the configuration distance c(i, f) must always follow certain rules. It must lie within the bounds set by the sieve of Aristotle. We shall call the range set by those allowable bounds the “concession of Aristotle”. The precise descriptions for the sieve and the concession will differ according to which coordinate system we want to use. But it will always possible to determine a change and a transformation that keeps one dimension, in our case numbers, constant. So just like we can use Gauss' and Riemann's system to calculate the earth's curvature, we can calculate what any change or transformation free from numbers should be, and all within suitable ranges for a sieve and a concession. We simply leave constant. We then measure back and forth to determine those ranges of values in the other two dimensions, which are their areas and parallelograms. We can then measure some real population, such as Brassica rapa, to see if it matches the predictions for the sieve and the concession. If creationism and intelligent design are true, then our real world measurements will fall within the concession applied by our sieve. If those doctrines are false, they will not. If creationism and intelligent design are true, then the following must all hold:

1. The plane.
All our measurements must remain on a flat plane. We must stay strictly in two dimensions and must not go vertically into a third. We must therefore have c(i, f) =u + v+ , where can always be taken as either zero or unity for it never changes.
2. The cylinder.
We must have a right cylinder that keeps its radius constant, its angle constant, or its height constant. We can thus say that we can go up and down on the cylinder, and all the way around it, but not into and out of it. So there can be only the angle of turn needed to produce the points on a constant circle, combined with the height for a line of ascent on a single right cylinder. We must therefore have c(i, f) = a(cos v + sin v) + u, where is unity but u is always zero, or vice versa, or both have the same such value.
3. The sphere.
Either the radius or one of the two angles must remain constant. We must therefore have c(i, f) = a sin i sin v + a cos i sin v + a cos v … and where is again either zero or unity.
4. The helicoid.
The helicoid must have a constant pitch, and must also always fall upon a right cylinder. This is given by c(i, f) = au (cos v+sin v) + v, where is once again set to zero or unity.
Figure 20.68

No matter which coordinate system we choose, the above claims, sieves, and concessions are all trivial to measure in any experiment and to see if they are so. It is also easy enough to determine values for the Weyl and Ricci tensors. Figure 20.68 shows us establishing the flat plane and parallelogram that Riemann suggests we can always draw to take accurate local measurements in our biological space, no matter what its ingredients or coordinate system, so we can apply our sieve.

We still need our straight line, however. We need to guarantee that we can move in any one dimension, independently of all the others. We are discussing our biological space—our biological medium of internal energy—itself. We are discussing the components that all biological entities are made of, which is the space and the surface that makes up our biological medium of internal energy.

We can easily determine the entire surface in the fashion Riemann suggests. We draw infinitely many infinitesimally small parallelograms upon it and measure forwards and backwards around them all. We measure each one at time t0 along our n, , and V axes using both our linear and our polar planimeters, or any other instrument we choose. The mass flux at any one of them is M0(t0).

Our mass flux must also exist at the two nearby points i and f. Since one is an earlier point, and the other a later one, we can call them M-1(t-1) and M1(t1).

We could easily have arrived at t0 from either t-1 or t1 with no effective difference, seen locally. And since we can measure directly towards those two points, then everything we measure at t0 has a component relative to both those directions. That is to say, we can measure a set of component changes ct-1 and ct1 so that any infinitesimal variations in distances disappear, locally. We can help calculate what values will be consistent with any sieve and concession we impose.

Gauss' experiences in Figure 20.57 surveying for King George III, and that led him to invent the heliotope, now make themselves relevant. Gauss saw that it was simply not possible to measure his areas—which are in our case our divergences and movements about the helicoid's circle—without also incorporating movements in the third dimension, which are in our case the torsion and pitch as the curls. The movements in our space thus distribute themselves between absolute clock time and relative biochronometric distance to complete a generation.

Every nearby location has its set of neighbourhood points, just beyond it, that carry the motion all around the surface in both distance and time … but in their different ways. As Figure 20.68 indicates, the point M1(t1) does not have exactly the same coordinate axes or indiscernible points as M0(t0) when considered globally, although the two are indistinguishable and flat locally.

If we consider the earth, then two neighbouring semi-detached houses might make a snug fit, seen locally. But that does not mean there is no global curvature across them. It exists all across them. In the same way, two people starting in the same place and taking a 5,000 mile, 8,000 kilometre, journey to two locations only 5 miles, 8 kilometres, apart may well take the same plane, boat, and train journeys almost all of the way, and so follow almost the same curvatures. Their paths, however, will eventually diverge. Their curvatures about the earth will then have slightly different values. Their final locations will give them a curvature difference of 40 inches, 102 centimetres. This may be a vanishingly small proportion of the original journey … but it is not nothing. If we undertake this journey enough times, the difference will accumulate. Eventually, the size of the difference will be equal to the length of the original journey. Darwin claims that as evolution.

We measured similar differences in our Brassica rapa experiment. The journey around a circulation is the equivalent of a journey around a planet. If one set of B. rapa plants remains continuously at its longest 44 day generation length for 100 years, while another set remains equally continuously at the shortest 28 day generation length, then the former will complete only 830 generations, while the latter will complete 1,304. If they kept this up for 1,000 years, that would be a difference of nearly 5,000 generations That would certainly seem enough to provide scope for the development of more than a few heritable traits.

Every location is always flat locally. Generation distances are also differences in time. We can always take tangents and normals using any of the four methods. However … we also know that any conclusions we draw working entirely locally, all around the surface will fail globally, for the surface is curved. There is no easy way to know, locally, how long any given generation length is. This is all part of the sieve and concession of Aristotle, and our attempt to pin down relevant traits and behaviours.

There is a clear difference between:

• not taking the earth's curvature into account because it is not necessary for some particular task; and
• not taking the earth's curvature into account because of a convinced belief that the earth really is flat.

This latter is the error creationism and intelligent design make. It is also the basis for applying the sieve and the concession of Aristotle.

We are getting closer to the straight line, the circle, and the various forms of biological contact we need. We can define a tangent plane for the number dimension.

Creationism and intelligent design are so convincing because we can—and we do—get an acceptably accurate idea for any biological population by ignoring all its ongoing losses in numbers. We can describe both a blue whale and a mosquito by attending only to its principal morphological features, without taking numbers into account. That interrelation of mass and eneergy is the essential development, λ. But this is very different from insisting, as creationism and intelligent design do, that since ignoring numbers gives us an acceptably accurate and local idea for any species, then that must also be the complete and global description. This is again the error that creationism and intelligent design make.

Figure 20.69

An Aristotelian population incorporates no curvatures in the number dimension. Its supporters insist that its flatness in numbers is the cosmic reality. We show our two biological fields P-1L-1P1L1 and M0L0 in Figure 20.69. They come in from infinity. They have their different scopes, properties, and powers. Each of P-1L-1P1L1 and M0L0 can fulfill whatever criteria are needed for a population free from Darwinian competition, and so that we can apply the sieve of Aristotle. As ever, P-1L-1P1L1 forms the limits to, and energizes, M0L0 across t0, and so over the entire period from t-1 to t1. They are the mechanical and nonmechanical aspects of internal energy that sustain any population. They must, between them, interact so they account for the mix of linear and circular forces biological populations experience as they express the prevailing biological velocity field while striving to complete their circulations.

We can now contrast the global and the local consequences of this creationist and intelligent design template. One of its consequences is that the sources for both P-1L-1P1L1 and M0L0 must be infinitely far away, globally. They are infinitely far away relative to all local behaviours in the sense that while they can affect and drive local behaviours, local behaviours have no reciprocal influence upon them. In that sense the sun is also infinitely far away from us because while the sun's heat reaches and affects us very greatly, the heat we radiate back has no effect on the sun. Looked at locally, it is infinite.

Both of the fields P-1L-1P1L1 and M0L0 that creationism and intelligent design propose are indifferent, locally, to all forces. Objects respond locally to those infinitely sourced instructions, which are therefore received as parallel at all times. Both P-1L-1P1L1 and M0L0 always move straight ahead upon their selected and designated paths no matter what happens locally. The field P-1L-1P1L1 of Figure 20.69.A provides the sum of the Gibbs and Helmholtz energies, with the field M0L0 of Figure 20.69.B siphoning off the Helmholtz fraction it requires so it can exchange all needed chemical components back and forth with the surroundings and construct the entities. It uses the rest to configure itself. This is Newton's firmament.

Since all locations infinitesimally close to any given location behave the same way, all the proposed biological line segments that the two fields fields P-1L-1P1L1 and M0L0 can create are flat everywhere. They act as if they emanate from the firmament. The forces acting across them are all parallel everywhere. The local view, therefore, is flat and unchanging everywhere. All previous, and all succeeding, generations, ad infinitum, will behave in exactly the same way, and independently of any and all local changes or variations. That is what creationism and intelligent design assert for biology. It is what we must measure if they are true.

If P-1L-1P1L1 and M0L0 are going to affect biological entities, then they must both act over both distances and times. They must also carry the population around an entire circulation. While the field P-1L-1P1L1 provides all heat, light, and chemical and other forms of energy both potential and actual, the field M0L0 uses its specifically mechanical form of chemical energy to interact with those same surroundings, thereby drawing in the needed resources.

Figure 20.70

According to creationism and intelligent design, the energy for M0L0 comes in from the infinitely far away firmament. It behaves like the ancient view of gravity we see in Figure 20.70.A. Its lines of force come from an infinitely far away template, and therefore seem parallel everywhere when received locally, driving all populations and entities to interact in their specified ways with the surroundings.

As long as we keep to this local perspective, based on a particular vision of the firmament, then we fail to appreciate what Newton taught. Gravity, seen locally, appears exactly the same everywhere locally. It is always flat and parallel, locally. The smaller the area we consider, locally, the flatter it seems to be. When the area is small enough, the earthly curvature disappears altogether. Not even standing on a ship and looking out to sea punctures this illusion. It appears so flat that no number of hills and valleys is sufficient to convince a ‘Flat Earth Society’ advocate that, throughout them all, the earth's surface really is gently curving all about its centre. Creationism and intelligent design make the same claims in biology.

When we think of mechanical chemical energy and of its interactions with the surroundings in this way, we fail to appreciate the true significance of gravity always pulling directly inwards to the centre of any mass in the way that Newton taught and proved. We instead believe what the Ancients believed, and that the creationist and intelligent design advocates continue to affirm, to this very day, in their approach to biology: that the earth—and its biology—is a vast and level flat plate, with everything always falling ‘straight down’ as if governed from the firmament, and so by some infinitely far away and abstract template. They insist that we observe no change anywhere as we, for example, move from where the car is busy falling to where the ball is busy falling, or vice versa. We can go all around the earth—which is the circulation—making such observations and end up right back where we started, always seeing this same thing. We are then ready to go all around again, believing the firmament to be flat, unchanged, and unchanging. This is the creationist and intelligent design vision of the circulation, which can be infinitely and invariantly repeated.

Another feature of gravity, when viewed locally, and with the traditional, infinitely far away, firmament-based but local view perspective that creationism and intelligent design still support, is that all objects dropped, simultaneously, from any given height will all accelerate at the same rate towards an apparently immobile earth. A falling body's ultimate speed depends only on the height from which it is dropped. We see the dropped object move, but we do not see the earth move. In the same way, or so creationism and intelligent design assert, we see biological objects transform according to their templates, but we do not see the templates transformed as a result. Species are therefore invariant. That is the false claim those doctrines make. We will soon take measurements to show that falsity

Since a larger falling object has a greater mass than a smaller one, then just like a car that casually swats an insect aside, the larger object will of course have a consistently greater store of momentum and kinetic energy all the time it falls. A larger object will also wreak more havoc, by exerting more force, and so by doing more work, when both it and the smaller one hit the ground. This is the reality that creationism and intelligent design ignore.

A larger object might have more impact and do more work when it lands, but it will still only hit the ground at exactly the same moment as any smaller one released at the same time. They have the same acceleration. On this point, creationism and intelligent design are correct. The smaller object simply starts with less gravitational potential and momentum; will land with less impact; will have less kinetic energy; and so will have less ability to do external thrusting work. Creationism and intelligent design ignore that fact that since it has much less Helmholtz energy, it will impose and undertake a smaller number of externally oriented transformations. When viewed locally, therefore, any two objects will appear to fall side by side, irrespective of their masses, and will be parallel every step of the way. They will show no differences until they hit the ground. They will form the rectangular areas shown in Figure 20.70.A. That is the description to which creationism and intelligent design hold.

The biological equivalent, in our space, to two objects falling side by side through gravity is two populations circulating precisely parallel at different distances on a helicoid, with the same turn of a polar planimeter. If two populations start off at the same time, one with a 50 day cycle and the other with a 100 day one; and if the first is half-way through its cycle match when it is 25 days in, and the other is half-way through when it is 50 days in; then these two matc each other. They are each covering equivalent amounts of their circulations in equivalent amounts of time so that dt = Tdτ. We have 25 = 50 × ½ and 50 = 100 × ½.

Creationism and intelligent design then insist that even though larger biological entities, further out by the periphery, have more impact on the surroundings than do smaller ones closer to the axis, those differences in impact are irrelevant to the entities themselves. Effects on the surroundings are irrelevant. Larger and smaller, and many and few, make no difference to the template. All entities and populations do what the firmament and the template dictate. All entities go through their circulations in the same ways. They all interact with the surroundings in conformity with their templates and the firmament. On this approach, it makes no difference to whales or to their template that they have to do so much more work, and take so much longer, to get half-way through their cycles when compared to mosquitos.

This claim from creationism and intelligent design is very like saying that a car crashing into the ground and an insect crashing into the ground have the same effect. And … this is how we can disprove them. We can distinguish rows and the columns in our tensor, and measure the different effects.

When we change perspectives and view gravitational attraction globally, as in Figure 20.70.B, then we begin to appreciate that the local but firmament-based view is quite mistaken. Gravity is not a parallel force emanating from infinity. Its force and behaviour depend on the size of the mass originating it. The same holds in biology.

The global view of gravity reveals what Newton taught: that it is always directed towards an object's centre. Furthermore, the earth does not remain static as objects fall towards it. It instead moves towards whatever it interacts with, even if only microscopically. Every object moves relative to every other.

Biological species behave more like the global view of gravity, which is based on Newton's law of universal gravitation and its doctrine of the mutual attraction of all bodies. Gravity then consists of two separate events or components, both arising locally. They do not arise from any firmament. Each body acts locally and contributes one component to the total global force which is F = F1 + F2. Each local F1 and F2 contribution depends upon (i) each body's own mass, m1 or m2; and (ii) the distance between them, r.

Newton expressed gravity's mutual and global reciprocity—which is also the basis of Darwin's theory—by saying:

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

Sir Isaac Newton, The mathematical principles of natural philosophy, 1689. Translated into English by Andrew Motte, First American Edition, New York, 1846.

Darwin preaeches what Newton teaches, which is that:

• The ball in Figure 20.70.B pulls the earth towards it with the component, F1.
• The earth, meanwhile, contributes its own component F2 towards that same force F between it and the ball.

All entities reciprocate with their populations, and all populations reciprocate with each other. According to Newton, the two locally-based components, one donated by each body, come together to give F = F1 + F2. But since the earth's mass, m2, is vastly greater than the ball's, then just as an insect colliding with a car finds that the car dominates it utterly, so also does the earth's gravitational contribution utterly dominate all small objects located on it. The earth's component is so vast that a falling ball finds it near impossible to persuade the earth to move. The earth thus remains effectively stationary. The earth's far superior F2 component instead compels the ball to move in towards it while it does nearly nothing.

The global view states, however, that the ball does not do nothing. The Weyl and Ricci tensors work so that the total gravitational force is the coming together of F1 and F2. Its magnitude is F =F1 + F2. The ball's F1 component might be minuscule, but it still exists. However, the earthly F2 component is the only one the local but firmament-based creationist and intelligent design view recognizes. We shall soon isolate that equivalent error in biology, and show how to measure the similar effect each distinct entity has on its supposed invariant template.

Figure 20.70.C tells us another of gravity's important global consequences that the local but firmament-based view ignores. Since gravitational potential varies with height—which is distance away from the attracting object—then the tops and the bottoms of all objects are at different distances from the attracting object's centre of force. They immediately feel different effects, and different gravitational forces. This difference always exists, but is generally imperceptible because most objects in daily life are rigid. They are also small relative to the earth's size; and the earth's gravitational field is in any case weak. Nevertheless, the force at the bottom of all objects is always greater than it is at their tops. Gravity therefore distorts them all by stretching them all out longitudinally. The bottoms therefore accelerate downwards more rapidly than do the tops. The effects only increase as objects fall. A black hole is quite powerful enough to tear any object near to it apart. This is a change in the volume of a given flux amount, which is a divergence. We can measure it, which is what we did in our Brassica rapa experiment.

There is a similar lateral effect. Gravitational attraction makes objects lose in girth, latitudinally. This is because, as in Figure 20.70.C, the attracting mass always pulls the outsides of the attracted one closer in, towards the central line of action, as that attracted one approaches. The effect is again imperceptible in small rigid objects placed in the earth's weak gravitational field. The net consequence, however, is that two objects falling side-by-side will create the toroidal area between them shown in Figure 20.70.C; with each one also individually feeling the shearing tidal forces that gravity imposes. We may choose to ignore this locally, but it is nevertheless real and measurable. It is again a divergence.

Gravity always acts from the centre of one object to another. It thus imposes a divergence. The right-, left-, front-, and back- sides of any object, at any one height, will all tend to narrow at exactly the same rates and paces. Through its Ricci tensor, it causes them to change shapes and volumes. But since all those edges share the same centre of action, then there is no net force making the object spin, turn or roll about itself. The force is directly in and out of the centre. Every part remains in balance as it either diverges while being lifted upwards through the field, or converges while falling downwards through it. Although gravity can pull objects directly towards a centre, it has no ability to rotate objects. It has the power of divergence, but has no power to curl. This is an important distinction.

The law of universal gravitation is not the only force that works on orbiting bodies. We still have our hoop running and our twirling batons.

Gravity pulls straight in to the centre. It can only impose a linear acceleration and a linear momentum. It cannot, of itself, make things begin to orbit or rotate. It can only pull them in. Each of its infinitesimal increments can act tangentially and contribute to linear and absolute clock time. Left to its own devices, however, a linear force cannot circulate. But our batons keep twirling. Planets orbit and rotate.

All bodies have an angular momentum, either by spinning about themselves, or by orbiting. That is the force that pulls all populations around the circulation. It is also the force that twists the flat plane into the helicoid.

Figure 20.71

Figure 20.71.A depicts an object responding to a torque of the kind we clearly need if we are to account for biology. It creates our baton twirls, our helicoid, and population circulations. These forces are quite unlike gravity. They make objects both orbit and spin in our biological space. They provide a circulation density. This is an intensity of rotation at and about each point.

As with the two down arrows in Figure 20.71.A, the fallings or orbitings that any object undertakes are due entirely to gravity. But if that same object spins and/or rotates about itself while falling, then some alternative force—a torque—is at work. Without a specific torque or force upon one side no object will rotate. It will simply fall. All biological materials, however, travel around the circulation between t-1τ-1 and t1τ1, and so are being rotated. There must, therefore, be some other force. The hoop runner has to keep the hoop rolling, and also has to prevent it from falling.

The earth in 20.71.B does more than just orbit. It also rotates about itself. Its processions give us the years, but its rotations, whatever their cause, give us the days. The two together give us the variations we experience as the seasons and the days. That is a non-gravitational form of energy density and intensity. It is the cause of biology.

As the earth does in Figure 20.71.B, any object subjected to rotational forces will constantly present first one face, and then another, to the sun. It will circulate constantly about itself, as a distinct object, and have a measurable curl relative to the sun.

There is, however, a vital difference between objects that curl and rotate because:

1. external forces cause it to do so;
2. it has forces emanating from within itself and so does so under its own steam; and
3. the field or space in which it is embedded creates a path that torques the body, with the body otherwise being stationary.
Figure 20.72

We see the last of the above possibilities in Figure 20.72. As a population proceeds about the circulation, it transformts through the configurations c-1, c0, and c1 which are its past, its present, and its future states. Since it holds each of those states at t-1τ-1, t0τ0, and t1τ1, we can also describe it using its biochronometric displacement, d, from the beginning of its generation. It is also sensitive to direction.

We already know that a helicoid only has a mean curvature of zero because it sums its indiscernible points forwards and backwards, upwards and downwards, and left and right. Since the population goes around a circulation, then its torques must also sum to zero to create the indiscernible point.

The indiscernible points t-1 and t1 at the beginning and end of a generation are the biochronometric distance we can measure between d = 0 and d = 1. As in Figure 20.72, this gives us the two population configurations c-1 and c1. Since the configuration change c-1 to c1 is now a distance; and since it takes a measurable amount of time; then that change in configuration is also a velocity. It must incorporate the curl or circulation density at each point.

As again in Figure 20.72, we can now distribute both the amounts and the rates in numbers, mass, and energy across that entire period and determine all values. Those beginning and ending points are indiscernible. And since c-1 and c1 are the same, then energy is being conserved across that distance … which is immediately the first law of thermodynamics. And since the differential of energy with respect to distance is always force, then the force at every point about the helicoid, stated relative to our basis, must be driving the population so that it moves to conserve all its values between the beginning and the end of a generation. That conservation of energy must include the curl: the circulation density.

Figure 20.73

We wish to set some A equal to some B, and then some B equal to some C so we can determine if the A equals the C. As in Figure 20.73, we can easily measure an A for every population, all about its circulation of the generations. We use the population itself as its own basis to measure its own circulation between t-1τ-1 and t1τ1. It will give us its set of normal pressures.

We place a set of axes that we can call i, j, and k (the most general and versatile way to represent them) within the population so that it can measure its own properties with itself as its basis all about the circulation.

The i, j, and k axes are based on its own population means measured relative to itself. There is one each for numbers, mass, and energy over a single generation. Since we are working with tensors, any of our three dimensions can go on any axis. We can then use the means to state all population values as displacements and vectors relative to that measured generation mean. Those will be values entirely within the population as it compares itself to itself between t-1 and t1. If it measures the same values at the beginning and end of the interval it can conclude that those properties have not changed over its generation, relative to its own self. It then knows it has stayed the same.

If we suspect that some specific snake has a template, then we need only measure many of them over numerous generations and we can gradually determine what is common to them all, which is this Liouville ensemble. We can then use the Liouville ensemble for our B. This gives us a set of I, J, and K axes and unit vectors to measure against based on its multitudes of generations and entities and their coordinated means and values. That is their template. We can then see how closely any one generation, such as the i, j, and k above, matches this I, J, and K set. We now have a B that we can compare to any A for any population.

A mathematical aside

Our displacement (or position vector), d, at any time t0 is given by:

d(t0) = rcos(γt0) I + rsin(γt0) J + ((ργt0)/2π) K

where r is the helicoid radius at that point, γ is the rate of rotation, and ρ is the pitch. The circulation length is τ = (ρ2 + 4πr2)½.

The population's speed about the circulation is S = γτ/2π, while its tangential velocity—which is its speed in linear time—at any point is V = Si.

The population's movement on its three independent axes i, j, and k is fully defined by V (and therefore by S) in combination with its own rotational velocity, ω, relative to its three axes, which therefore has the three components ω = ω1i + ω2j + ω3k.

The meridian always lies at the angle tanθ = 2πr/ρ.

If s is the arc length about any point of interest, then the unit tangent vector for the space is T = (dd/ds) / [dd/ds]; the unit normal vector is N = (dT/ds) / [dT/ds]; and the unit binormal vector is B = T × N.

The curvature at any point—which helps give us both the aberrancy and the speed in biological time—is 4π2r2 (which is also [dT/ds]); while the torsion is 2πρ/τ2 (which is also [dB/ds]).

Unfortunately, two sets of values are not enough to create a standard. If measurements made against the axes i, j, and k differ from those made against I, J, and K, we have no immediate way of deciding which one is correct.

Our helicoid of internal energy embedded in space, that we see in Figure 20.72, can be our arbiter. It is a scroll, complete with rulings. It has its complete set of tangents. Those rulings and scrolls tell us what is flat in the surrounding space. We can easily use it to determine the values for that flat and homogeneous isotropic space anywhere and everywhere. We can therefore calculate three mutually orthogonal flat axes. They are traditionally called the tangent, T, the normal, N, and the binormal, B. These are the embedded vectors that belong directly to the helicoid we see in Figure 20.70. They are direct properties of this space and its path. Those three values are independent of both the population and the species. They belong to the space. Any movement away from those flat rule and scroll properties in our embedding is an acceleration … and … by Newton's second law, where there are accelerations, there must be forces. The values based on the T, N, and B set of axes, if we can properly define it, can then be our C.

Figure 20.74

As in Figure 20.74, we want to examine any population and then gradually sieve out the ones that follow the creationist and intelligent design proposal from the ones that follow Darwin. This is exactly what derivatives and partial derivatives are for.

Our overall Liouville population is μ(nV). The subset we have proposed as free from all influence in numbers is σ(nV) = c. Darwin suggests that when an entity is lost, the survivors will increase their individual biological potentials to make up the deficiency. If this is so then since the third and fourth maxims tells us that the numbers lost from μ(nV) are given by the partial derivative ∂n/∂t, then it can be measured. That is our sieve. That is what we measured in our Brassica rapa experiment. We applied it as our sieve to find the differences between populations. We can use it with our collection of 1-, 2-, 3-, and 4-point contacts, in conjunction with our associations and transforms, directives and conjoinings, aberrancies and distributions:

• If the Darwinian competition model holds good, then after a given and determinable period of time there will be zero plants following the constraint values σ(nV) = c, with all following μ(nV). We will have sieved them all, and there will be none following the creationist proposal of the Aristotelian template.
• But if the proposal of the Aristotelian template holds good, then the surviving entities will instead display the constraint values, σ(nMV) = c. They will gradually sieve themselves out away from the values Darwin suggests. There will eventually be none left following Darwin's proposed μ(nV).

Some of the force from the hoop runner's stick is intended to stop it from keeling over, while some is directed towards maintaining its forward motion. Helmholtz took an important step forwards in separating these in 1858 when he proved a precursor of what later became known as the ‘Helmholtz circulation theorem’. He proved that once a material line element or “vortex filament”, as he called it, in a fluid had become ‘aligned’, so that its forces and energies followed certain patterns and behaviours, then it would always remain aligned and parallel in that sense.

Figure 20.75

Populations A and B in Figure 20.75.A, show the parallel alignments Helmholtz proved. They are the biological equivalents of two objects going around something at the same orbital speed. Since the two circulations are equidistant, the populations share the same centres and means. They form a 4-point contact. But since they constantly hold different positions, they have different velocities, accelerations, and torques. They are on the outside and inside, respectively, of the helicoid. Every point on the two circles has some displacement, d, about the circulation. Population B, with mass mB, is an “anchor population” meaning its values are consistently closer to the mean.

The Helmholtz circulation theorem is about matching torques. It is the declaration that if some first parcel of mass, mA, in a fluid, such as an air mass, has twice the mass of some second parcel, mB, at some point, and measured relative to some basis; then mA will always have a similar and proportionate relationship to mB. Once a vortex filament has twice the mass, then it will remain twice the mass at that point, unless some force acts to change its state. The two circles in Figure 20.75.A represent this by being always equidistant and parallel in these 4-point contacts. Their torques are constant. They share the same mean and axis. But since one has to swing back and forth across greater distances, they exhibit variations around their shared mean.

Figure 20.75.B shows that Helmholtz's original circulation theorem was a little incomplete. By the law of universal gravitation, the moon pulls on the earth. It also imposes a torque. Points further away enjoy a differently sized torque from those closer.

The gradient force the moon imposes on the earth causes our tides. But since the moon is relatively small and weak, those gradients are weak and relatively flat. They are not powerful enough to affect the rocks and continental landmasses. All the moon can do is create the earth's tides. They do, however, have a frictional effect. They scour the earth's ocean basins and slow the earth down in a constant and measurable secular acceleration. That tidal friction is a measurable aberrancy.

The earth exerts similar and reciprocal effects upon the moon. But since the earth is much larger than the moon, it has a correspondingly more powerful torque effect.

The moon is a rigid body. We never see its dark side. However … if the moon had zero torque, and so zero rotation, relative to the earth, then we would see its dark side. Since we do not, then the moon must have a torque causing it to rotate about itself at every point, and that exactly matches its orbit. The earth must therefore (A) pull the moon into an orbit about itself that lasts a month; while (B) simultaneously imposing upon it a torque whose net angular momentum and intensity make the moon rotate at a precisely matching speed, so we never see its dark side.

The moon once rotated about itself very much more quickly. Because of that rotation and intensity, it showed its various faces to the earth. But the earth's very much stronger gravitational field exerted a drag force and an aberrancy upon the moon which slowed it down to its present rotation where its orbital and rotational speeds match so that it only shows us its one side. It now has a constant rotational intensity all about its circulation, which is an even circulation density or curl.

Gaseous bodies behave in the same way as rigid ones. In 1867, Kelvin developed Helmholtz's original circulation theorem and extended it to cover all torques, curls, and circulations. He extended it to non-rigid bodies, such as atmospheres and gaseous planets. He proved, as in Figure 20.75.B, that due to the conservation of angular momentum, when gaseous or fluidic bodies orbit about some larger object, they behave in exactly the same way as rigid bodies. They will amend their circulation densities so that even though each independent material element or parcel could rotate independently about itself, they instead combine and act jointly as they orbit about whatever is attracting them. They act as if they are rigid bodies and constantly keep the same face towards that axis or centre. The body will be permanently in darkness on the side facing away from the sun, with the other side being permanently in the light. Since its rotational and orbital speeds match, it is without curl relative to its central axis.

If a biological population is to complete a circulation, it must change its behaviour at every point so it can indeed go all the way about. Relative biological time must curl the absolute linear variety about itself. But however a biological curl expresses itself within our biological geometry of internal energy, it will be based upon differences in displacements across the various dimensions, and as measured from some central point or axis.

Figure 20.76

The surfaces in Figure 20.76 use gravity and their shapes to suggest the effects of torques and rotational behaviours. Those can impose a variety of torquing forces on anything travelling upon them, such as in 20.76.A when roads and railway tracks have “superelevations” (or “cants”, “cambers”or “bankings”) built into them. These are designed to counter the centrifugal and other forces vehicles can experience due to the changing landscape. The forces induced would push vehicles off tracks or roads and/or overturn them, but the superelevation's tilt moves the body in the third dimension and so allows gravity to exert the force needed to keep the vehicle on its path. The built-in angle allows the car to remain glued to the tracks when moving at the design speed. If a biological template is to move a population around a circulation, then it must exert similar forces on biological materials. They must also be measurable.

The upper and right-hand parts of Figure 20.76.B shows another kind of torque. That one does not move into the third dimension. It is, therefore, a two-dimensional curl. The “slewing curve” or “slewing function” which shifts a section of track or roadway sideways from one line of progress to another is an example. The slew first increases progressively within the slewing zone, and then decreases, so the slewed and unslewed portions are joined with a smooth reverse curve. The body being slewed moves relatively imperceptibly, locally, from a first line of progress to another. The end result is that the space has imposed a curl or differential rate of rotation about the object in one direction on one side; followed by a curl and matching differential rate of rotation but in the opposite direction upon the other.

The “transition curve” or “transition function” on the right-hand side of Figure 20.76.B has a broadly similar effect to the slewing one. It links two curves of two different radii, such as when a road or railway track picks its way across bumpy and/or circuitous terrain. Such transition curves often join a straight line to a sharp corner, or else curves of different sharpnesses or directions to each other. They change the line from a first or initial radius to a second one all across the transition zone much more gradually. They can be as long or as short as desired. They damp down or remove all jerks, sudden changes, and centripetal forces at each local point. Both the original curves and the transition are again variations in the surroundings that institute a curl.

If the vehicle or object upon any path is enclosed; has good suspension; the path or track is well built; and the driver moves the body upon the path at the correct speed; then the road or track design can create the illusion that the vehicle is always moving straight ahead, locally, and that there are no lateral forces either locally or globally. This almost obviates the need for the driver to steer. But although the car, its driver, and its passengers may appear, locally, to keep moving straight ahead, forces are being imposed by the surroundings. Those forces can be measured, both locally and globally. And since they depend on displacement, they can also be different at different points across the body or vehicle … which is for it to have a curl—or set of variations—imposed upon it by those surroundings. We then have a local straight ahead movement, but a global curvature.

Slews, superelevations and transitions curves are variations in energy and intensity imposed exclusively by the surroundings, and in spite of all contrary local perceptions. Riders with only immediately local views can gain the impression, when slews, superelevations and transition curves are carefully combined, that they are always moving straight ahead, locally, with all changes in motion and direction—which are curls and accelerations—being effected not by the object, but globally in the surroundings. If a population is to be free from changes in numbers, then our helicoid of internal energy must incorporate such superelevations and other structures to keep populations on track all about the circulation, and in spite of any changes. It must incorporate such superelevations and other structures to keep populations on track all about the circulation, and in spite of any changes. They must be measurable. And if creationism and intelligent design are true, they cannot be three-dimensional.

If creationism and intelligent design are true then they are collections of slews and the like that emanate from an infinitely far away template so they can direct biological entities and populations from one point in the circulation to another. They must convert the linear and absolute into the relative, the biological, and the curved. They must do so without ever moving into the third dimension of number.

A helicoid results from applying a torsion to a circle to uplift it along the third and temporal axis. The simultaneous rotation and torsion produces the rates in time for the required set walks from minimum to maximum values for each of the dimensions. These are all measurable as derivatives and partial derivatives. Any objects ascending out at the edges must cover greater meridional and spacelike distances than any anchor objects closer to the central axis, which will instead cover greater poloidal and timelike distances. Anchor populations will always have larger timelike gradients but smaller spacelike ones. These are also what we sought to measure in our Brassica rapa experiment. We sought for aberrancies.

Figure 20.77

Our biological force must move through our space, like gravity, and draw populations around the circulations of the generations. Figure 20.77 shows the 4-point contacts that a population of entities must undertake if it is to be free from changes in number. They produce the Tnumber:number, Tmass:mass and Tenergy:energy values that come together to create the overall generation length, T. However, if creationism and intelligent design are true, then Tnumber:number, τnumber:mass, τmass:number, τnumber:energy, and τenergy:number will have no effect. There are no superelevations, slews and the like involving that dimension. That is the definition of constant speeds and parallel in that dimension, and therefore of 4-point contacts.

By the Helmholtz and Kelvin circulation theorems it is only possible to maintain the Liouville ensemble that creationism and intelligent design need if the various properties distribute themselves evenly either side of the various generation means. Therefore, if some property is far from the mean on one side for one generation, it must have an equivalent value on the other side in that same generation. It must have both a greater range and a greater rate of change so it can maintain its alignments. If not so, then the Liouville phase volumes and values cannot be maintained.

Generation I in Figure 20.77 is an anchor generation. Its mass and energy values are always closer to the mean. It must have the smaller ranges and rates. But Generations I and II have the same mean, they are aligned in the sense meant by Helmholtz and Kelvin.

Generations I and II are aligned whenever they each hold a common generational average value. So if ever Generation II has, say, twice or three times Generation I's mass at some point, then it must have the inverse of one-half or one-third at some other. And if Generation II has two or three units more than Generation I at some point, in some property, then it must have two or three less at some other or they do not remain aligned.

The two generations must show the same pattern for their energy. They must coordinate their torques in these ways so that all their relative differences, again relative to the generation mean, maintain both their defining mean and their Liouville constant. They will then be constantly aligned. If the various populations do not coordinate their rates and quantities across their generations so they remain aligned and parallel, then some force is carrying them off their common helicoid of internal energy, and out of the ensemble. There is an aberrancy.

Kelvin proved that aligned generations will be irrotational relative to each other. They always show each other their dark sides. This is a complete range of behaviours throughout the bodies. They will exhibit the same rotational and circulating densities at each point about their respective generations. They will never look different, at any point, to each other. They never in other words vary with respect to each other.

By the law of universal gravitation, a bouncing ball has an atomic behaviour whose field of interactions is similar in scope to its gravitational ones. Both of these are both local and universal. When a bouncing ball is one metre away from its contact with the earth, but is still falling under their joint gravitational attraction, then a molecular and electromagnetic interaction between them is already in place—albeit amazingly weak. Their two electromagnetic influences are already interacting. They are already interacting electromagnetically before impact; and they will continue to interact electromagnetically after impact. It is simply that, while away from each other and still falling, other forces, such as thermal, friction, wind, and the gravitational are far more significant. But the atomic and molecular interactions are still not zero. They can still for example exchange heat energies through a temperature gradient.

When a ball bounces on the ground in ordinary physical space, the only thing we see, externally, is the mechanical reversal of its direction. This is a manifestation of its Helmholtz mechanical thrusting out energies, and of its Ricci tensor. But the entire interaction is also a product of the ball and the earth's atomic and molecular interactions as they transform.

The ball's bounce is an interaction between the molecular configurations and interactions of both the ball and the ground. But no bounce can happen without the more interior—less visible—Weyl tensor elasticity and transformation properties that affect entropy and the Gibbs energies. The same holds for a population that uses the surroundings to “bounce” itself from generation to generation, and so from indiscernible point to indiscernible point.

Both a ball and a population's engagement with the surroundings are intrinsically molecular. When two atoms are more than approximately one hundred atomic radii apart then their mutual interactions can be taken as being so close to zero that they might as well be zero. However, the electromagnetic fields that make up the ball and the ground—and an entity and its population—do not suddenly drop to zero. They each extend well beyond the radii of their various atoms. They each extend out to infinity. It is simply that beyond a critical threshold of a few atomic radii, other forces become far stronger.

Atomic fields are admittedly notoriously weak … but they are never zero. This assumption of zero is the simple error of principle that creationism and intelligent design insist on making within biology.

Just as we can safely, and locally, ignore a bouncing ball's electromagnetic interaction with the earth, when it is one metre away; so also can we safely ignore that same ball's independent gravitational influence upon either the moon or Sirius. It instead affects external objects through the joint centre it shares with all other masses constituting the earth. Indeed, so weak is gravitational attraction, globally, per unit of mass that we can place two balls side by side here on earth, locally, and safely ignore their mutual gravitational attraction. The earth's much larger mass is far too powerful for their distinct effects on each other to be observable. The earth's friction in all local neighbourhoods is so great that those two balls will not slide closer together because of their joint gravitational attraction. But that does not mean that it is zero.

Since gravitational attraction is so weak, two distant stars or other celestial objects do not have to concern themselves with any balls or other such objects placed upon the earth's surface. They need only consider the earth's entire population of molecules. They can consider the earth and all its entities and objects as one discrete whole, and as one mass. They can treat the earth according to its centre, and irrespective of all else. Nevertheless, the earth's net gravitational influence changes with each ball and entity placed on it or removed from it. If we keep removing a ball, then we make it easier for the sun to reel the earth in. And if we keep adding a ball, we will eventually get a black hole.

A spacecraft the size of Rosetta does not have independent interactions with the sun or any other celestrial body when firmly placed upon the earth. However, once detached from the earth its interaction with the sun and all other bodies is sufficiently powerful for it to rendezvous with a comet. Our biological space must have a similar interaction between the small and the large, and the local and the global.

We may on the whole treat gravitational attraction differently, but it is ultimately an attractive force transmitted between any two atoms in space. In the same way, evolution and heredity are generally large scale forces between large regions in biological space, but they are also and ultimately forces at work between atoms and specific point locations. The assumption of local zero is again the simple error of principle creationism and intelligent design make within biology.

Although gravity in physical space cannot directly cause curls, torques, and rotations, it can nevertheless overlay its responses. We can now express the proposal of the Aristotelian template—which is creationism and intelligent design—in our tensor language. The proposal insists that its infinitely far away template, and only its infinitely far away template, elicits all species responses. There are no extenuating circumstances. There is no need, on this basis, for either a Weyl or a Ricci tensor. We can therefore identify creationism and intelligent design as the proposal that those two tensors are zero.

The Weyl tensor is timelike in our space. It governs hereditary behaviour: the pitch, lengths, and timings of the flows and handoffs from biological event to biological event across biological space and time. It forms the 0-, 1-, and 2-spheres. It is the component of biological behaviour that describes the transformations of inertial biological frames as entities feel, and respond to, the effects it imposes. It changes the shape and intensity of the forces and energies that the material objects that the Ricci tensor in its turn assembles will all feel once they have been placed in the space that the Weyl tensor in its turn defines and creates.

The Ricci tensor has the more particle-like and spacelike manifestation across populations in our space. It forms the 1-, 2-, and 3-balls and governs the locations, sizes, and movements of both entities and populations and the size and density responses of all those volume elements.

Unfortunately for the creationist and intelligent design proposal, matter always attracts. Gravitational force is transmitted through every location, whether matter is present in that location or not. The Weyl tensor, that transmits it, handles gradients and the transmission of gradients. It therefore represents that part of the spacetime curvature that can propagate from one location to another through, and then beyond, any given parcel of matter. It can do so whether matter exists within that designated volume of space or not. This is the transmission of heredity.

The problem with the creationist and intelligent design's proposal that the biological expression of the Weyl tensor be zero is that Weyl showed, when he proved it, that the absence of matter in some given location does not immediately bring gravitational intensity in that same location down to zero. In the same way, just because a specific trait is not being expressed in a specific entity or location does not mean it is not being transmitted through that entity or location.

Since the Weyl tensor carries information about gradients, then if it could ever be zero in any one location, there would immediately be zero gravitational attraction everywhere. There would be no gradients, and no propagation of gradients, anywhere. There would simply be no bodies anywhere at all exercising forces on each other. If there were no attraction then there would be no objects anywhere throughout all possible spaces. There would be no space anywhere for there would be no matter anywhere. This proposal of a flat or zero Weyl tensor means no gravity is being transmitted anywhere because there is nothing, and there are no bodies, anywhere. To similarly deny the propagation of fitness and evolution through reproduction is also to suggest that there are no biological beings anywhere.

The Weyl tensor shows how matter can curve space. Its propagating gradients define matter's possibilities and potentials. If we for example remove the sun from the Milcy Way; double the sun's mass; and then return it to the same location; the gradients it feels across it when we return it were already embedded in that location by all of space. Those gradients across it immediately tell the sun how to conduct itself in that location. Therefore the Weyl tensor can never become flat or zero in the way suggested by creationism and intelligent design. Gravity—which is simply another term for universal mutual material attraction—will always propagate across free space; and it will always impose a curvature. Its strength and curvature therefore depend entirely upon the sizes of all masses everywhere within space, from the smallest of molecules to the largest of galaxy clusters. It can again never be zero anywhere if there is matter anywhere. A propagating gradient always exists.

The biological equivalent to the impossibility of the Weyl tensor being zero is that no biological entity can be without heredity. All organisms pass and receive heredity through and from all others. Most of the time, however, biological entities are found in populations and in regions that mean this global truth can be easily ignored because of those more obvious effects. Most of the time, the hereditary influences or gradients of one organism directly upon another are zero, and can be taken as zero. We can consider all such organisms locally, and as if they are without mutual hereditary effects. But that does not mean the effects between any two are always in fact zero. All DNA is hereditary and all DNA either influences or is influenced by all others.

This becomes a discussion of how a vacuum is defined. The Weyl tensor and its effects are implicated in the cosmological constant that Einstein introduced when he first presented his general theory. He used his cosmological constant to provide an energy density to counteract the constant contraction gravity implied, and so he could recreate Newton's firmament. When Friedmann and Edwin Hubble proved, theoretically and experimentally, that the universe is expanding, the cosmological constant was removed from the equations. But since contemporary observations imply that the cosmological expansion is accelerating, rather than decelerating, then many cosmologists have reintroduced the cosmological constant. Its effects are sometimes referred to as “dark energy”.

The cosmological constant describes a vacuum's mass and energy density in grams per cubic centimetre, and electron volts of energy. It is the fundamental desire to diverge that universal gravitational behaviour impresses upon any space. It is a scaling factor that specifies the relative distance of galaxies as a function of time, and such that if two galaxies are twice as far away at any succeeding time as they were at some prior time, then their mass-energy densities reflect that increased distance.

The Weyl tensor governs the curvature of space. In the same way, biological entities affect each other hereditarily through a combination of their masses and their configurations. They do so as a function of their hereditary distances from each other. Some are so close that, as with the a ball and the earth, or the moon and the earth, or the sun and the moon, we can view one as the sole and proximate cause of the other's properties and behaviour, for that is how it appears locally. That does not belie either heredity or gravity's powers, or their realities, when viewed globally.

The similar idea that the Ricci tensor can be zero needs handling with equal care. The Ricci tensor can appear to be zero, or vanishing, in certain specific locations because it is entirely possible that there is an insufficient density of matter, or mass-energy, in those locations for those specific locations to act as if they contain masses that could have distinct gravitational sourcing effects. If matter has not converged in some location then it is not a source point for gravitational attraction. That location can still, however, respond to the Weyl tensor and pass on a gradient.

Just as the Weyl tensor states the general properties of space and its combined effects, the Ricci tensor tells us, more accurately, the direction we should move in if we want to see space get denser, or less dense, around given points. It tells us where we should move if we want to find a source of gravitational effects by telling us the local divergences. Just as the European Space Agency could search for the specific influences from the comet 67P/Churyumov–Gerasimenko and then plan a course to make contact, so also does the Ricci tensor tell us the specific properties we must search for if we want a specific and material exchange of heredity.

The directions of divergences, which can be observed locally, is exactly what we see, biologically, with the initial and final configurations ci and cf. These establish behaviours in neighbourhoods both between that population and the surroundings, and between the entities themselves. If we want to see constant reproductions we can go in one direction, and if we want to see constant dissipations of entities we can go in the other. They are vectors that indicate directions in space for the specified biological events that the Weyl tensor transports. The Ricci tensor thus achieves a similar effect for biological matter.

The biological Ricci tensor can act as if it is tending to zero, or even be zero, in some location. But since it is also a rate of change, then its tending to zero simply means that, on a population-wide basis, it is most likely about to reverse in some specified rate of change. The Ricci tensor again states a divergence or flux density of biological materials. It cannot be zero, or tend to zero, absolutely everywhere in space, because gravitational attraction must emanate from some location or other. The Ricci tensor is the microscope that tells us where those density holding locations are. If it is tending to zero, so that matter is diverging, then there must be other locations in which it is converging. Those convergent locations then have a significant effect on their surroundings, be it gravitationally or in heredity, through the density of the volume elements they define. They contain matter and are a source. Like the Weyl tensor, therefore, the Ricci tensor cannot be zero everywhere.

Creationism and intelligent design perpetuate the doctrine of the Newtonian firmament within biology by suggesting that completely flat and empty biological spaces are possible. This is the error.

Even when real physical space appears empty, the Weyl tensor transmits effects the curvatures and effects of all space. There is also always a Ricci tensor concentration somewhere, and that is ultimately responsible for sourcing and receiving those transmissions. In the same way, all biological entities transmit hereditary effects through reproduction, firstly by being reproduced, and then by reproducing.

We can also run experiments to demonstrate that creationism and intelligent design are false, and that the biological versions of the Weyl and Ricci tensors that work in internal energy cannot be zero anywhere but must always have some value everywhere. We ran our Brassica rapa experiment to show that there were indeed no zeros anywhere.

Those complementary processes of circles and lines, spheres and balls, and Weyl and Ricci tensors can only take place across both (a) distances, and (b) times. This immediately gives us things that are extraordinarily easy to measure so we can assess the validity and viability of creationism and intelligent design, which define biological populations by focusing entirely on rates. They refuse to specify quantities. Beyond insisting that all biological entities must follow some form of template, they do not make testable predictions.

By their very nature, creationism and intelligent design assume a smooth and even universe everywhere in internal energy so that their templates are followed. No matter how many entities are lost or gained by any population, and no matter how many variations in circumstances there might be, they insist that their templates rule supreme. That continuity is the definition of smoothness and evenness.

Whether or not creationism and intelligent design are valid, the Weyl and the Ricci tensors are responsible for distributing the full range of energies and effects about the circulation. Since the Ricci tensor is the more spatial and structural, we need to know the effects its volume elements can have:

1. in the three x, y and z dimensions of ordinary physical space; and
2. in the three n, , and V dimensions of our internal energy biological space.
Figure 20.78

We again concentrate on internal energy. We can determine the effects the Ricci tensor imposes materially, and in physical space, by seeing what our three twirling batons can achieve in a real case. We can for example consider the Cannabis sativa plant shown in Figure 20.78.

Since the Ricci tensor gives all biological populations their material substance and presence, it gives the molecules within a population's internal energy their required kinetic and configuration energies all across the three dimensions of x, y, and z, or length breadth and height.

We now let the Ricci tensor's volume elements manipulate Cannabis sativa's internal energy so it grows. But as in the middle of the diagram, we constrain it so it only grows laterally in x. It does not get taller or broader, so it does not change in y or z.

The increase in Cannabis sativa's x-distance requires both mass and energy. It is an expression of the mechanical chemical energy aspect of internal energy. The plant is therefore expressing an inertia in x. Its force in x is its distribution of a mechanical mass and energy, as a mass flux, across that increased x-distance. That energy and inertia will of course be influenced by any underlying curvature of space, also in x.

As on the far right of Figure 20.78, we could instead constrain Cannabis sativa's internal energy so its Ricci tensor's volume elements only permit it to grow vertically, in the z-dimension, leaving both x and y unchanged. This again requires a change in the mechanical chemical energy aspect of internal energy, this time as a distribution across z. The plant is now expressing an inertia in z, which is a force in z, again influenced by any underlying spacetime properties.

The distribution of these various forces and energies in x and/or y and/or z across time requires the interaction of both the Weyl and the Ricci tensors. Those then state the impulse and the power that the internal energy deploys across the various physical dimensions, and all according to the inertia, and rate of change in inertia, displayed per each.

We first consider physical space. If Cannabis sativa's total mass-energy increases as it grows, then it must draw resources in from the environment. This is a use of the Helmholtz portion of its internal energy. But if its hamiltonian—which is its total energy—remains constant, then as its mendelity increases its biopressure must decrease. It must reconfigure itself and divert energy towards growth. We can look for the sources and resources it uses for all its transformations, whether internal or external. They will express themselves across the three dimensions and in time, and as both a force and a power.

We now turn to Cannabis sativa's behaviour in our biological space. The population cannot do anything in the physical space of x, y, and z over any time interval t without also doing something in our biological space of n, and V over that same interval. Its internal energy changes. This is again the Weyl and Ricci tensors.

Just as Cannabis sativa can use its expressions of force and energy across distances and times per each dimension in three-dimensional physical space so it can grow and change over time; so also does it use those same forces and energies to grow and change in our biological space of internal energy. We can measure its numbers, masses, and energies in both absolute clock time and biochronometric distance. Each change in each biological dimension will then state C. sativa's energy, force, and inertia for that dimension over both distance and time.

Every change in number, or mass, or energy density requires energy and force in distance and time, which is internal energy. Biological expressiveness is ultimately a force in the molecules gathered in the x, y, and z dimensions in physical space … but those same molecules are also partitioned out in our biological space of internal energy, and so between whatever entities compose that population, which is the n, , and V biological dimensions.

Figure 20.79

Figure 20.79 shows some of our experimental results as a baton twirls in our biological space of internal energy. It depicts Brassica rapa's behaviour in numbers, n, over a generation. Entities are first lost to the environment. Molecules and energies therefore leave the system and as shown in the right-, up-, and left-pointing arrows. The population ultimately reaches the minimum for the cycle.

We can use the generation mean as a basis or reference. The ongoing losses can be expressed as displacements, d, relative to that mean. And since energy is being lost to the environment, then the environment is exercising the force, at each displacement, that causes those losses. It also does so over some time interval. That number loss therefore has both a velocity and an acceleration.

As with the down-pointing arrow in Figure 20.79, Brassica rapa eventually responds to the imposed number losses by reproducing. Numbers then increase above the mean, reaching the maximum displacement for the cycle. This requires an intake of mechanical chemical energy, which is again a force at, and over, a displacement. That displacement is a biochronometric distance about the circulation.

Since population numbers oscillate up and down, they vary around that mean. They vary between the generation minimum and maximum. They always have both a size, which is the number; and a displacement, which is the current displacement, either positive or negative, from the mean.

Brassica rapa see-saws continuously in an expression of Archimedes' law of the lever. The distance from the mean is again the displacement; while the size at that point is its expression of mass-energy. The sizes and displacements are therefore a potential. Creationism and intelligent design deny that this potential can have any influence.

If we want to either prove or disprove either of Darwinian evolution or creationism and intelligent design, then we must have an accurate measure for these proposed displacements and potentials. Since that potential is stated relative to a specific reference, then it is a force. We now call that force due to that displacement a “portage”. We met it earlier as the conjoining of force and time, mF dt, and measured in kilogramme metres.

Just as the strength of gravitational attraction varies with height, so also does the magnitude of portage vary with displacement from the mean. A given object's height in a gravitational field, h, is a displacement in a specific direction in the intensity of whatever field is surrounding that mass. Our portage is also a displacement in a specific direction in the intensity of a biological field. It is the equivalent of mass times height, mh. The resulting ‘kilogramme metre’, mh, in a gravitational field is the gravitational potential whose exact expression, as a weight and so as a force, depends upon the gravitational field, g, in which that mass is inserted, thus producing mgh.

Jupiter and the earth have very different gravitational fields. A one kilogramme mass held one metre above each produces a one kilogramme metre portage that nevertheless results in a different weight, and therefore force and energy, above each planet. Every planet can also produce a see-saw whose weights counterbalance, although the gravitational field concerned determines the sizes of the forces that achieve this. The final results of any portage always awaits the specific field, g, in which the mass is inserted. The full gravitational potential is always mgh. The speed of movement or transformation depends entirely upon the field, and the acceleration or curvature it provides. The portage is therefore a coherent statement of the position within a field, and the subsequent effect in force and energy.

Biological portage is the equivalent of the kilogramme metre. It is the size times the displacement from the mean. The forces, energies, and behaviours ultimately depend upon the intensities of the different biological fields … which are the various populations. Thus a given distance from the mean has different effects within different populations.

Every population does a required set walk, which is the twirling of a baton and a sequence of portages through internal energy in each dimension. The sizes and displacements in each dimension express themselves through the forces that those distances create in that population, which is then an energy resulting from the portage. That interaction is an integral of that force with respect to that displacement. And since it is an integral of force and distance, F and d, then it is an energy.

The portages over any circulation increase, decrease, and return to the same value, which is the same indiscernible point. That sequence of portages is an expression of the total absement: i.e. of the integral of that size and that displacement about the generation, and all about the population's mean value. Energy has been conserved over that distance, and as a result of the distance travelled between one indiscernible point and another. In the case of numbers, this set of portages over the generation defines the constraint of constant propagation, φ.

There is a similar absement, plus its conservation of energy, in each of the other two biological dimensions of mass and energy density, and V. They are the total population mass and energy fluxes and required set walks. They are the sequences of portages between their minimum and maximum values. They are, respectively, the constraints of constant size and equivalence, κ and χ. Those three sets of portages are the statements that are our twirling batons of internal energy.

We now call the required set walk that produces the total energy used for the absement in each dimension—which is the sequences of portages that are the first integrals or association of force and displacement—the “promenade”. A promenade is therefore an expression of the increases and decreases in the Ricci tensor's volume elements; with the time this takes immediately being an expression of the Weyl tensor. And given that a portage operates at a displacement which is a distance, then every promenade—which is a complete baton twirl—has its attendant velocities and accelerations, its absities and its abselerations. The helicoid and the circulation of the generations, which is the biological field, is the product of the three intersecting twirling batons. Our biological internal energy is that set of promenades.

Mendelity and biopressure, which are shared within each population, increase and decrease over their respective promenades. So also do population numbers. Values for mendelity and biopressure therefore depend upon every entity in every population. No entity is zero with respect to those averages; their displacements; their portages; and their promenades. All entities contribute, directly, to their indiscernible points and 0-spheres, and so play a part in the conservation of energy. All influences are therefore transmitted. This is heredity. We must now find a way to measure all such effects.

Just as all material bodies contribute to the gravitational attraction exerted by any planet, so do all entities in any population contribute to every biological measure and dimension, including portages and promenades. In almost all practical cases, however, the specific influence any particular biological entity has upon any other can be ignored. Nevertheless … the Cavendish experiment calculates Newton's universal gravitational constant, G, from two small balls, whose effects on each other are generally beyond the range of detection. Darwin's theory of competition declares that biological entities have similar effects upon each other. By the same token, therefore, portage and promenade are not zero for any biological entity. All entities affect each other through heredity and the surroundings. But most of the time, it can be ignored.

Creationism and intelligent design now claim that while portages and promenades in the two dimensions of mass and energy define a species, those in the third dimension of numbers have zero effect. This is becoming increasingly easy to measure, and so to show that it is an impossible claim. All we now have to show is that a stationary or non-twirling baton for numbers is impossible. This means no curl in numbers, so that the partial derivative ∂n⁄∂t = 0. That is something we can measure.

Every population needs energy to navigate a generation. That energy must be stored as a potential in its DNA. It is a combination of the Gibbs and Helmholtz energies which, thanks to Black, we already know how to measure. We call the sum of those energies, which is a hamiltonian, a ‘generon’, G, to make it clear that we are concerned with a circulation and a generation—as a biochronometric distance. A generon is the total energy in an indiscernible point.

As in Figure 20.46, the energy in a generon is a structure and a landscape. It is formed—as is all energy—from an association or first integral between force, F, and displacement, d. It is again independent of time.

Figure 20.80

Figure 20.80 shows the clear distinctions we must make in the different kinds of baton twirls and movements of internal energy over a generation, and so that we can unravel evolution. Location t0τ0 shows the population (a) being carried poloidally upwards in time; (b) responding meridionally with a divergence and a curl in both its mechanical and nonmechanical energies; and (c) moving toroidally about the helicoid.

We must carefully clarify the various aspects of the helicoid's distances and movements as enshrined in a generon, G. That energy distributes itself through the 0-sphere of an indiscernible point, and so across the generation, between t-1 and t1, and between τ-1 and τ1. It moves courtesy of the two interpenetrating fields P-1L-1P1L1 and M0L0.

The meridional or spacelike movements at any point and time t0–τ0 have three aspects:

1. The biological force or torque, F, which is the differential of energy at that point, and that mobilizes the population.
2. The curl that carries the population about the meridian.
3. The divergence that moves the population either closer to or further away from the main axis and meridian.

Biological events are therefore subject to the unique torque that keeps them on the helicoid of biological internal energy. They are governed by our four laws of biology, our four maxims of ecology, and our three constraints (of constant propagation, size, and equivalence).

The force molecules experience creates the curl that swings them around given mean values at a given rate to create biological entities via P-1L-1P1L1 and M0L0. Their sum is the total stock of available generational energy, G. It is the the potential the population holds in DNA, summed in the Gibbs and Helmholtz energies.

The generation begins at τ-1. The population has an initial energy stock of G joules. As the generation gets under way, we take the minuscule amount -ΔG out of the populations's stock. We distribute it amongst the n entities so they can follow their DNA programming and go about the circulation towards τ1. They then use that energy for their metabolism, their physiology, and all their ecological interactions. These carry them across a biochronometric distance at τ0, and so gradually about their circulation and closer to τ1.

Subtracting a proportion of energy, and then distributing it amongst the entities, so they can travel their selected distances, is to differentiate that energy with respect to that circulation distance or portion of the generation. It is necessarily distributed (a) poloidally, (b) toroidally, and (c) meridionally. Since energy is the integral of force and distance, then this toroidal differential of energy, i.e. with respect to generation distance, means we will observe the entities receive and use that energy as a force, which is in this case the torque in internal energy that moves them about the circulation.

The torque arising from the use of energy will carry the entities both poloidally, which is in time, and physically or meridionally through ordinary physical space. But they will also associate toroidally, and so about the helicoid circulation within our biological space of internal energy. It is therefore some part of a generation.

We can see that the above understanding is entirely correct because if we use Riemann's strategy; turn ourselves around; go back in time; count the entities; restore them back to their previous states; carefully remove from them the forces they are each exerting; drag all that jointly expended force back to τ-1; and then place it all back in the stock we originally removed it from; then everything goes back to exactly the way it was. To multiply and to add in this way is to integrate poloidally, toroidally, and meridionally: i.e. timelike, spacelike, and biologically. When we do this with respect to the same biochronometric distance, we get all of our original energy, G, back. It has been conserved.

If we now repeat this exercise of removing a stock of energy and distributing it amongst the entities all around a circulation, then they will sometimes in the net receive energy and diminish the stock, and sometimes in the net respond by returning some from the environment back to us and increasing the stock. They will sometimes lose and sometimes gain, but over the generation as a whole the amount they add equals the amount they remove so that by the time we get to τ1 everything is the same. The population is then in the same state, and energy has been conserved. We now have our 0-, 1-, and 2-spheres, and 1-, 2-, and 3-balls along with Haeckel and Owen tensors and the rest. This is our indiscernible point.

Energy

The association of force and displacement, F and biochronometric displacement, d, produces energy. Populations disperse, tangle, and untangle their molecules in different ways. Some reconfigure, very readily, under force and energy. Others show much more resistance. These affect the Helmholtz and Gibbs energies, the Weyl and Ricci tensors, and the Liouville constant and the Ricci scalar. An association between force, F, and biochronometric displacement, d, produces the generon or energy needed for a generation. This energy is therefore the first integral of force with respect to the distance that is the circulation of the generations, and that conserves energy.

Biological force/Torque
The transform and first derivative of energy with respect to biochronometric displacement must also exist, and is the force and torque that carries the population around the circulation. And since energy is never stationary, then the torque is the population energy content at any point. We simply state that energy content all about the circulation, in terms of that circulation length, and we have the torque at each point. That is the first derivative of that energy at that point. It is also what we set out to measure in our Brassica rapa experiment. If we refer to our standard population, then we can define the standard size of this torque as the energy that allows that given population of 1,000 entities to maintain both its mass and its energy for the one second that is also 1,000th of its circulation.

Competition
The transform of force with respect to displacement, which is also the directive of the energy, also exists. Force's rate of change with respect to biochronometric distance, all about the generation, is a statement of the population's materials and molecules, and so of its genomic and genetic composition. This is the population's molecular responsiveness to inputs of energy, to its transformations, and to its numbers. As the first derivative of force with respect to biochronometric distance, it is the readiness and the rate at which the population's molecules will deform, transform, and then return to their original positions all about the circulation. This is the tensile nature of the Hooke biosphere … the rate at which it will flex, reshape, and transform as the population goes about its circulation. Competition therefore measures this rate and energy transform, again given population numbers; given that circulation length and given the surroundings. And insofar as it states an entity and a population's responsiveness to energy and to force, then we define this first derivative of the biological torque as “competition”. It is the rate at which the Hooke biosphere changes with circulation distance. We now have a clear definition for that vitally important concept in Darwinian exegesis.

We conjoin force and biochronometric displacement for Bernoulli's theorem of Figure 20.49. There is an equivalence, through the Weyl tensor, between energy transported and generation length. If one increases, another diminishes, while maintaining equivalence in areas, times, and intensities of delivery.

A population can use its Weyl tensor to increase its toroidal distance and generation length at any time. Brassica rapa varied between 28 and 44 days. The population then carries its energy stock over greater and larger distances. There will be countervailing changes. Just as air flows adjust their pressures and speeds either side of an aeroplane wing, the work done and the total energy delivered over some shorter generation length is equal to that delivered over some longer one. This is the longitudinal pattern familiar from the Weyl tensor. The equivalence is achieved through through the Ricci tensor's changes in the numbers, masses, and energies of the entities supported in that population. A population of long generation length but with a small number of larger entities is energetically equivalent to one of shorter generation length, but with an increased number of smaller entities as the Ricci tensor distributes that energy, again so that the total energy and circulation distance is constant.

Fitness
The inverse of the conjoining in force and displacement is the directive and second derivative of force with respect to biochronometric distance (and the third derivative of energy with respect to displacement, making it the aberrancy, in displacement, with respect to energy). It is the toroidal derivative of competition. It is the rate at which competition changes all about the circulation, and as its entities interact poloidally and meridionally, with the surroundings to take on and give off energy. This is the set of forces and events responsible for the specific molecular accelerations as they tangle, untangle, and deform in their specific ways within the entities and populations, again all about the circulation. It is the differential rates at which the Hooke biosphere accelerates in its different directions, and at each point about the circulation. This second derivative of the biological torque is “fitness”.

Heredity
We can then create the distribution, which is force's third toroidal integral. This is the triple integral of force with respect to biochronometric distance. (It is additionally the first toroidal integral of the advection, or its association with respect to biochronometric distance). We have now defined heredity. It is energy's double toroidal integral: the double integral of a population's energy stock with respect to its generation distance.

Heredity allows a species, as a collection of populations and generations, to allocate and to reallocate its energies, its compositions, and the resources it makes available poloidally, toroidally, and meridionally. This heredity measures a population's ability to vary both its generation length, and the energy it disposes over that selected generation length … but all without any substantive change in its structure or capabilities, for all these states are equivalent under force and energy. The masses and the energy densities can then also vary. This is what we proved in our Brassica rapa experiment.

Every population deploys its heredity in one or another generation length, always measured between 0 and 1. Since every generation and population can measure every other, then each is always at some point about its own central axis. Every species has its values for n, , and which it uses to state both its own size and displacement at each point about that generation, as well as those of others. Each uses its own values as its basis.

A mathematical aside

All population and generation shapes, sizes, positions, and behaviours can always be stated, relative to each other, with the triplet of values x = r cosθ, y = r sinθ, z = cθ. As we have learned: that angle θ is simply a relationship between our various values.

Flounce
The aberrancy of the force that both drives, and results from, heredity, with respect to biochronometric distance, is the triple derivative of that force or torque with respect to distance. It measures the rate at which the torque deviates and then corrects itself as a population tries to maintain itself while measuring both its own self and all others. This aberrancy states the rapidity with which—and the amount by which—an entity and its population respond to changes in circumstances, always relative both to its own self and to others.

We term the rapidity with which a population changes its energy at any point, but with respect to its circulation or generation distance, its “flounce”. It is the triple derivative of force with respect to generation distance. It therefore measures the aberrancy with respect to force or torque.

If creationism and intelligent design are true, then all flounces, fitness and competition must maintain the same heredities, advections, and torques, respectively, which is the same generon (i.e. quantities of energy per distance)—and so the same pitch and helicoid radius—from generation to generation, and across each species when measured from any other. Any one species can also use its own values to define all others relative to itself, for they are all similar in emanating from the same firmament and so have the same integrals and derivatives at all times.

Figure 20.81

If a population is to follow a template and be free from all influence of changes in numbers, then as in Figure 20.81.A its mechanical chemical energy must align with the infinitely far away firmament. Mechanical chemical energy must always be locally parallel. The stock of chemical components must increase and decrease without changing the ways in which they are configured. They must be straight-line and parallel relative to the generational centre so they emulate gravity seen flat, locally. All increases and decreases must happen without curl.

If creationism and intelligent design are again true, then mechanical chemical energy must act exactly like gravity. It must be a fully conservative vector field. It must define, for each species and each population, a constant generational mean. There must be a centre of attraction to and from which it diverges in straight-line and parallel motion. Biological masses must then increase and decrease with no changes relative to their defining means. That alignment means they must have very specific masses, and rates of change in masses, at very specific points all about the generation.

Mechanical chemical energy may have divergences, but must create invariant alignments with its population means. No population may move faster or slower about the biological inertial frame of reference and its circulation than the firmament dictates. As again in 20.81.A it must be both non-solenoidal and irrotational. Or alternatively … the chemical components in each and every population must at each point follow the standard laws of science at that point. The population will be drawing more components in, or releasing them, at each moment, but at that same moment, the population appears simply to be preserving itself indefinitely in that same state.

The entities that anchor a given real population and its species are those whose flounces maintain the most circular of its values, all about the generation. They cause the least aberrancies and also have the greatest distributions. Those with greater deviations in their flounces—i.e. that have the greatest aberrancies—are the least likely to be heeding the firmament and to be following a template. They are, in fact, the most likely to be leading any ongoing evolution in species. We can easily measure this value for it is simply the triple derivative with respect to generation length. If creationism and intelligent design are true, then this should always be zero. It was not zero for Brassica rapa.

Each promenade and complete baton twirl in the mechanical chemical aspect of internal energy produces a 1-ball which is a linear range of values. It is the required set walk. But each can only produce its 1-sphere with its 2-ball of its allowed sets and divergences by working in conjunction with at least one other dimension. Those are areas and volumes.

Creationism and intelligent design insist that number is not a viable force. It is not a relevant dimension. Therefore, mass and energy must somehow coordinate biological internal energy without use of number. There must be no areas or volumes incorporating numbers, which is a flat plane in this geometry. Mass and energy must therefore follow prescribed behaviours. That is the very definition of a template.

Mechanical chemical energy does not, and cannot, act alone. If a population or generation must take on a mass of specified chemical components, then it is also obliged to take on specified quantities of nonmechanical chemical energy so that it can maintain its chemical bonds, activities, and alignments and equivalences in energy. If it does not, then the population is changing its physiology, which is to evolve.

Nonmechanical chemical energy must now join the mechanical in maintaining appropriate poloidal, toroidal, and meridional values. If a population must have very specific masses at very specific points all about the circulation, then it must also have very specific biopressures. If populations are to be free from all influence of changes in numbers, then no two generations may accelerate or decelerate away from Helmholtz-Kelvin alignment. Nonmechanical chemical energy must define the rates and quantities in energy that establish the species genome. If it does not, then τmass:energy and τenergy:mass will have altered, which in their turn alter Tmass:mass and Tenergy:energy … which is again to evolve and to change generation times.

If mechanical chemical energy must be fully conservative and be nonsolenoidal and irrotational, then nonmechanical chemical energy must match it by also being fully conservative. As in Figure 20.81.B, it must do exactly as its template declares. There is a specific pattern of behaviour, and it must exhibit direct lines of force between itself and its generational mean. It must carry all its entities about the circulation. It must never toss and spin them, relatively, by changing any of their values away from its designated means. Nonmechanical energy must be perfectly cylindrical at all times and be ‘rotational’ and ‘solenoidal’ relative to its constant generational means. If creationism and intelligent design are true then all populations must be (a) non-solenoidal and irrotational in mechanical chemical energy and the mass flux; and (b) solenoidal and rotational in nonmechanical chemical energy and the Wallace pressure. Or alternatively … the energy in each and every population must bind those specific components in an intrinsically biological fashion, but always while following the standard laws of science at that point. That energy must be linear and remove at least some photons while replacing others, but must do so without affecting the number of components held at any one time so that the population can then move about a circulation.

We now have the two conditions creationism and intelligent design require. We can therefore build a biological inertial frame of reference that allows biological entities to take the same kinds of journeys, relative to each other, and in our biological space, that Rosetta and other celestial objects take relative to each other in physical space. If biological entities wish to be independent of their numbers yet remain in the same frame—which is to remain members of the same species—then they must maintain constant motions and orientations with respect to each other. They must not accelerate relative to each other. They must maintain the same required set walks, allowed set divergences, and curls in our biological space, and the same transforms and associations, directives and conjoinings, and aberrancies and distributions at all times … including in reproduction. Only if they can do all this will they be stationary relative to each other, and so be members of the same species. These values are also very easy to measure and test in any real case, and such as we did with Brassica rapa.

If the entities in a given group are to remain members of the same species, then their mechanical and nonmechanical chemical energies must each go all around their independent circulations as measured by their distinct population means and axes i, j, and k. They must also end up in the same place in biological space, relative to some definite and shared species axis I, J, and K. Those species axes are their means n’, m̅’, and p̅’ which provide the needed unit vector values , , and . Those two sets of vector unit normals—i.e. i, j, and k and I, J, and K—must always coincide. If they do not, then we have evolution, for we have an A that is not equal to a B, and things are not equal to each other.

Figure 20.82

Populations and species can form landscape-like energetic structures in our biological space. Populations are the equivalent of regions and galaxies. If we want to positively prove Darwin's contention that biological entities are dependent on numbers, n, and therefore that they evolve, then we must isolate those number effects. The essence of Darwin's contention is then that populations will have spacelike or meridional effects on each other. Competition means they will attract and repel each other in the way Newton proposed in Figure 20.82, as a test for gravity.

Newton suggested that his law of universal gravitation could be tested with the ‘attraction of mountains’. He argued that when a plumb bob is held above level ground, it falls vertically downwards because only the earth directly underneath it will attract it. It heads straight towards the earth's centre. But he also said that when it is carried close to a mountain, the bob will slowly deviate from the vertical entirely because the mountain's mass also attracts it. The mountain's centre competes with the earth's.

In 1772 the then Astronomer Royal Nevil Maskelyne suggested an experiment to observe the attraction of mountains; to check Newton's law; and to deduce the earth's mass. The Royal Society commissioned Charles Mason—who later helped establish the Mason–Dixon line—to select a suitable mountain in the Scottish Highlands. Mason chose Schiehallion, just north of Perth, because of its symmetric shape and relative isolation. Maskelyne then spent four months in the summer of 1774 taking the necessary measurements.

After Maskelyne had submitted his measurements, which validated Newton's theory, the Royal Society hired the Westmoreland mathematician Charles Hutton to do the necessary calculations and to determine the earth's mass and density. He developed the concept of contour lines and calculated the earth's density as 4.5 grams per cubic metre. This is within 20% of today's accepted value.

If Darwin's comparable theory is true, then populations will change their behaviours with changing population sizes, which are the equivalent of changing mountain landscapes. Every distinct species will be a distinct mountain of energy in our biological internal energy landscape. Each thereby affects all others with its attraction similar to mountains.

But if creationism and intelligent design are true, then populations will ignore all such variations in other populations. Their own landscapes will be unchanging. Each will always, and instead, respond directly vertical and parallel, like an undeviating plumb bob. Each will always act directly to the centre, and straight and parallel to the firmament.

If Darwin's theories of competition and fitness are true, then we should be able to measure a similar value for the universal attractiveness and/or repulsion between biological entities and populations of different sizes and masses. This means verifying that the ΩT’ term in the various equations from the refutation give the predicted results. The Ω states the effects of changing population sizes, while the T’ states the effects of differences in generation lengths. If we can isolate these, then that Ω will be the biological equivalent of Newton's gravitational constant. Its changes and rates will state competition and fitness.

Figure 20.83

Figure 20.83.A shows aspects of circulations in Minkowski's four-dimensional spacetime. Each property is a ‘particle’ that has a ‘worldline’ that is a trail of its ‘events’: a record of its history. That history is its baton twirls or promenades.

According to Minkowski's four-dimensional spacetime geometry, a worldline, such as our promenades about a generation, is a “timelike curve”. It is a series of linked events. It states both the x, y, z of the space an object occupies, and the time, t, it is there. Its cross-section gives us a normal, which states the prevailing properties; while its tangent indicates the velocity and the acceleration it enjoys in whatever space is represented by the coordinates. This curve gives us all the coordinates our hoop runner will occupy, and the times he will be there.

Each proposed worldline in Figure 20.83.A is a four-dimensional construct that is not directly visible in real time. As a promenade and a worldline, it simply summarizes the spacetime behaviour of the object it represents. Only the individual events—the portages along that promenade—exist. Each distinct portage can be determined from the promenade, or worldline, according to the field or object concerned by reading the values for that instant. The helicoid and its poloidal, toroidal, and meridional movements indicate whatever forces a population is enjoying in this three-space as it prepares to move to its next state. Those are the values n, , V (or ), and t. A promenade therefore tells us an object's location, velocity, and acceleration within a circulation.

The four-dimensional line in Figure 20.83.B tells us the relative distance or difference between two objects at specified moments in time. If one is a buck and the other a doe, then the distance between them is a line that represents their differences. It could be their masses. Those differences in promenades sweep out a ‘worldsheet’. Their relative changes flex and transform the worldsheet according to whatever laws or events govern their relative changes. If their two values are ever aligned, which is to have the same value, then the line of course becomes a point for that event or moment in spacetime. But the worldsheet is a catalogue of all their relative differences over time, which is their relative absements and presements. In the case of a buck and doe, we are graphing and promenading their relative sexual dimorphisms. We can now state the values, forces, and energies of the one relative to the other, and in whatever units we or they choose.

These principles can be extended to several dimensions to produce a multidimensional ‘brane’ (or mem-brane). As in Figure 20.83.C, our three dimensions create a ‘worldvolume’ that states all totals and rates of change over the period. When that interval is a generation we have the mass, numbers, and energy that define both of the Liouville and Helmholtz decomposition theorems.

A worldline’s slope or tangent is a transform. It indicates a property's velocity, or rate of change, at that point. If the worldline is straight and parallel to the time axis, then the object is not changing in time in that property. The only contributor to absement is time. There is no transform or first derivative, and no velocity or rate of change. This is the claim creationism and intelligent design make regarding number. The proposal insists that the number promenade should be straight and parallel to the axis. It should show no change in values. We can easily measure this to see if it is true.

If, however, a promenade or worldline has a slope, then it has a transform which is a first derivative and a velocity or rate. If any slope is changing, there is an acceleration and a directive. Since the absement is a poloidal association, i.e. a first integral with respect to time, we can soon determine it. We also have the acceleration, jerk, absity and so forth for that dimension. We can then determine the forces that cause them to change … except that … those displacements, absements, rates and so forth are all values stated with respect to time, and not with respect to distance. They do not, therefore, tell us about the conservation of energy, and so they say nothing about heredity.

Action

Forces and energies have different behaviours in distance and time. Energy has an effect in time as action.

Although energy can never remain stationary, it nevertheless makes complete sense to speak about the behaviour of a discrete body of energy, of say some 100 joules, moving over a definite period of time. The system transforms from one condition to another as energy is necessarily conserved.

Figure 20.84

A population can express a variety of actions over time as its generation length changes. Figure 20.84 shows a whole host of different available trajectories over which we can fire a cannonball. We can easily videotape the results using a camera that has a fixed number of frames per second. Flights with longer trajectories will therefore use more frames.

Our cannonball takes different amounts of time—and therefore also a different number of frames—to travel its different trajectories. It has different magnitudes, and different rates of change, in its potential and kinetic energies at each point.

The cannonball's potential energy first increases as it climbs higher, and then reverses and decreases as it falls towards its target. But there is almost always some point—such as the beginning and/or end of the flight, or the highest point—where one or the other of the kinetic and potential energies is zero. By Newton's first law of motion, there must then be some point at which, no matter what is happening to its kinetic energy, its rate of change is zero.

The highest point on any trajectory is marked by (a) zero for the height component of the cannonball's kinetic energy, and (b) zero for the rate of change in potential energy. These are both zero because the cannonball is about to reverse, under gravity, and begin falling down again. This is true for all possible trajectories. At each highest point the forward motion continues unabated while the vertical motion ceases so it can reverse.

The cannonball trajectories in physical space make it clear that some go high while some go low, and that they take different amounts of time. This means different velocities, accelerations, and the like. There are certain things held in common, but it must also mean different quantities of energy and action. These are variations. In spite of all those variations, there is still something intrinsically the same in the various potentials and possibilities, and in the way they develop. The cannonball's mass is of course one important contributor. They are the same journey undertaken by the same cannonball between the same two endpoints. Only the energy inputs and angles of fire vary. There may be infinitely many possibilities, but there are clear maxima, minima, and rates of change.

If we are now given a selected number of frames and are then told how many frames there are on that path, we can eventually reconstruct that entire trajectory, from the given frames. Action is thus the first integral of energy and time, ∫E dt.

We can recreate the trajectory because its potential energy depends upon its height, as mgh, while its kinetic energy depends upon its speed and direction, and is mv2/2. Low and flat trajectories and high and looping ones will have different proportions of each of those two energies. The two coordinate on every trajectory, but they will also change at different rates according to the demands of each. Those differences will show in the sequences of video frames. This is effectively to integrate the whole journey from its parts.

Newton's method for expressing these many possibilities is often inadequate. He suggests that if we know the cannonball's mass, its initial velocity, and the distance it must travel, then we can compute the time. This is perhaps acceptable for single projectiles cases, and when there are not too many countervailing forces such as friction or atmospheric drag that introduce aberrancies. It is therefore of limited utility.

Hamilton's alternative says that if we know the cannonball's initial energy and its mass and distance, then we can compute its trajectory. The sum of all its energies, which is its hamiltonian, is conserved. This method is good for relative measures and for making system-wide predictions. It is the most useful method in Liouville theorem-like situations such as when there are many objects that we wish to consider together and contrast. It is very good for “if this, then that” analyses. It is less helpful, however, when we need a specific set of values.

Figure 20.84 shows Lagrange's method. This juxtaposes both the potential and the kinetic energies, and then determines actions and trajectories. It neatly summarizes the entire dynamics of any system, and excels where motions are more complex, or when objects move and change more quickly and can adopt a variety of trajectories.

The cannonball's ‘lagrangian’, L, at any point is the difference between its kinetic and potential energies. If we calculate the lagrangian or difference in kinetic and potential energies for each video frame, and then multiply that by the frame's length or duration, then the area under the total lagrangian curve we can draw is that trajectory's total action: i.e. its energy times its time. And since nature most often tends towards the minimum action, then the area or value for action almost always heads towards the minimum possible value.

The lagrangian works for all types of energies, and reveals the most likely state an object will occupy in a given energy field; the moment it will occupy that state; the energy it will have; and its rates of change.

A biological population's equivalent of potential energy is the amount stored in its DNA at any indiscernible point. That is its total stock of Gibbs energy. Its equivalent to momentum is then its Helmholtz energy. This governs its current interaction with the environment in terms of its numbers, masses, and energies. We can put these, respectively, in our tensor's columns and rows. Their sum is their hamiltonian. And since we have a complete equivalency to physical space, their lagrangian is the net statement of their differences.

At its very simplest, a Brassica rapa population that uses 100 joules of energy over 1 second has 100 joule seconds of action. This is exactly the same action as a population that uses 50 joules over 2 seconds … which is the same action as one that uses 25 joules over 4 seconds. But while the actions might be the same, the populations will differ in their numbers, in their masses, and in their chemical bond configurations. A population that disposes 100 joules of energy over 1 second may have the same action as one that disposes 50 joules over 2 seconds and one that disposes 25 joules over 4 seconds, but we can separate them by their lagrangians. Action in internal energy is, therefore, a way in which we can identify similarities and variations amongst populations.

Energy flux/Power
Action, as a first integral, has its converse in power, which is the first derivative of energy with respect to time. It is the energy that passes through an entity or population due to its materials. It is the conversion capability of, and work done by, chemical transformations in watts. It is the energy flux and Wallace pressure, P, for the entire population; or else the biopressure, , per the individual entity. The average flux or power used across an interval multiplied by the time gives us the total energy used across that interval, while the energy itself times the time over which it is held gives us its action.

Genome
For every trajectory the cannonball in Figure 20.84 traverses, many others are always possible. The same overall situation can produce all those different actions, each of which has different lagrangians and the like. The conjoining of energy and time states those more general possibilities. This double integral of energy with respect to time produces the “genome”.

The genome encapsulates the entire suite of actions and trajectories available to any population in that situation. Countless apportionments of number, mass, and energy are always available to any population at the beginning of any circulation between indiscernible points. All the trajectories the genome summarizes are all accessible from the same initial conditions and states. They differ only by the actual energies and times—or actions—upon each specific trajectory, each of which is the genome's derivative in time.

The genome, formally, is the second integral of a population's energy with respect to time. It is measured as kilogrammes metres squared. It holds the entire action possibilities for any viable population. It states the amount of force and energy a viable genome uses to maintain the chemical bonds and structure of that quantity of chemical components. By the Avogadro number, it is a specifiable number of moles of molecules. It is that total mass of chemical components plus sustaining energy that allows a population to deploy its instantaneous survival characteristics by undertaking different actions, through its members, in different prevailing conditions, while all the time remaining equivalent throughout those varied deployments. The genome is the sum total of what a population needs to be viable, and is the summing of all possible trajectories.

If the genomes of all species and populations are now quoted by the mass and the time that forms their equilibrium distributions, then they can immediately be compared. Different populations are using different masses of chemical components—which are their genomes—to do exactly the same thing, which is maintain their equilibrium in the environment. The genome's units, which are kilogramme metres squared, is the quantity of components stated for the putative area of incident energy needed to sustain any population. If we adopt the convention of quoting all populations for 1 metre squared, then all equilibrium distribution populations are equivalent through their masses, numbers, and energies all about the population, for they are all achieving the same genomic equilibrium maintaining purpose.

Physiology
The above conjoining of energy and time that produces the genome of course has its inverse. This is the poloidal directive. It is the first derivative of power, and so is energy's second derivative with respect to time. It is therefore the derivative of the Wallace pressure and the biopressure. We have already met and defined this as physiology.

Physiology, as we here understand it, is more technically called a ‘slew’. The ‘fight or flight’ response is an example of a biological slew. This is the response to a sudden demand for an intense supply of energy and power which must then be directed to certain specific areas, over a limited period of time. The slew therefore measures an energy source's ability to respond to, or to follow, such an input signal. This second derivative is the measure of the responsiveness to variations such as in the lulls, sudden onsets, and peaks and transitions that inevitably occur, over time, in any energy source.

We can better understand physiology and the slew by noting that every morning a large city is more likely to keep increasing its demand for power for longer and at a faster rate, from the national electricity grid, than is a smaller one. The national grid is carefully monitored to predict sudden moments of peak demand, such as in the commercial breaks of popular television programmes when many consumers will choose that specific moment to switch on a kettle to make a cup of tea. The power station must be responsive to such variations in demands, and must slew or adjust its power output accordingly between various towns, cities, and regions.

Biological entities and their populations can therefore be distinguished by the rapidity with which they can slew and respond to changes and to input signals, whether stemming from their own configurations and reconfigurations, or from the environment. This slew or ability to follow an input signal constructs 1-, 2-. and 3-balls by changing the energy content across the current P-1L-1P1L1, and so instantaneously across the prevailing energy gradient.

Inheritance
We can also, of course, distribute energy and time. This is energy's triple poloidal integral. So as an example, trees on windy islands develop roots that support them against ongoing windward stresses. But every once in a while, the wind reverses. Just like building a fight or flight response, any trees that have prepared themselves with suitable roots will survive such reverses. The greater is the imperative a tree can pass on, to its descendants, that, in the face of any ongoing prevailing wind, they must build suitable preparatory roots in expectation of a once-in-a-generation opposing wind, then the greater will be the number of the survivors. This is similar, in principle, to the occasional demand for a fight or flight response that animals must incorporate into their metabolisms and physiologies. The genome's properties include such suites of survival characteristics, which must be properly distributed throughout the population. This need for a distribution throughout succeeding genomes and generations is, formally, the integral of the genome with respect to time, and is called the “inheritance”.

If ten kilogrammes of viable genomic materials can survive for ten seconds, then they have an inheritance of 100 kilogramme square metre seconds. But this is the same inheritance as five kilogrammes of those same genomic materials instead maintaining itself for twenty seconds, or twenty kilogrammes for five. That capability for 100 kilogramme square metre seconds, however it is distributed between materials and time, is the inheritance. Its time derivative at any one juncture is the genome; which permits those genomes to produce their own derivatives in time; which are then the joule seconds of action; which then become one or another lagrangian distribution of DNA and ecological interactions. This is all part of the inheritance.

Convergence
The inverse of the inheritance is the aberrancy of energy, and the rate at which physiology changes. So just as a power station can slew energy to a village when there is a sudden demand, so also does the amount of epinephrine or adrenaline within an entity or population at any moment increase very quickly, even though only small amounts need be continuously maintained. This capability therefore governs the speed and applicability of fight and flight, and the rapidity of building the emergency roots and rootlets, and all such similar activities. But since energy cannot remain stationary, such emergency provisions cannot be stored. Energy's potential for production, delivery, and consumption must always, therefore, be kept in balance through appropriate structures.

The provisioning for slew emergency procedures has the three associated aspects of (a) temporality, (b) intensity, and (c) flow which are analogous to frequency, current, and voltage.

1. Temporality matches the demand for energy to its generation on a second-by-second basis to ensure an entity- and population-wide stability, and to guarantee that every part receives its energy at its appropriate rate. These are, however, biological components with specific structures and capabilities. Temporality establishes a generalized energy delivery and distribution network, with whatever emergency plus maintenance possibilities are pertinent to each part and organism.
2. Flow means that every part always has its lower limit or threshold, below which it is non-functional, and its upper limit, beyond which it burns out or fails.
3. Intensity means that every part, every entity, and every population must receive the precise energy it needs at each moment, whether emergency or non-emergency, and within its specified limits.

While temporality is most usually handled globally, flow and intensity are more generally controlled locally. The parts needed for an emergency response must be built and maintained throughout all non-emergency situations, but they must act as needed and be suitably provisioned, locally, at all times, for all purposes, including the globally announced emergency ones.

The constant monitoring and matching of all such limits and demands in energy is called the “convergence”. It is the triple derivative of energy with respect to time and joins the flounce in anchoring a population. Convergence ensures that each entity's—and each population's—responses are suitably anchored to the inheritance; to the genome; to current energy levels; to physiology; and to the prevailing conditions.

Energy cannot act in time without some force also acting in time. And force cannot act in time without having some effect on some material mass that contains inertia, and which therefore also responds in time. This is the components in our field M0L0 which will respond by transforming in time.

Mass flux
Newton defined force through F = ma: i.e. by stating its link to mass and acceleration. A force exists wherever two objects interact … such as when one mass pushes or pulls another to create an acceleration. A force is any push or pull effect that results from an interaction. When the interaction ceases, then so also does the force. The change in energy is the resulting change in state. The force is then responsible for the transformation. The energy concerned is conserved.

We define a force, in biology, as the torque that carries a given set of components—which is a population—about the circulation. Biological force is that aspect of energy that keeps a given set of chemical components bound into a given number of biological entities, these being the sets of shapes and configurations that allow them to abide by the four laws of biology, the four maxims of ecology, and the three constraints of constant propagation, size, and equivalence.

There must always be a locus of inertia that undergoes the forms of transformation that allow entities and populations to be biological … which is in our case to move toroidally, and so to be carried about a circulation. The torque is applied to a set of chemical components via a mechanical, and so binding, form of chemical energy. We can therefore measure it as the mass flux or Mendel pressure that moves meridionally, and so in kilogrammes per second of biological material maintained. We can measure it either for an entire population as M which is the flux, or else as the mendelity, , which is then the flux density or divergence at each moment. If the mass flux is not changing, then the Helmholtz energy is not changing, and we have a constant velocity field which shows itself in a constant number of kilogrammes per second.

Mendel
The association or first integral of the biological torque with respect to time is its impulse. But if there is an impulse, then there must be some form of biological momentum. This is the mendel, U, which we measure as kilogrammes of biological matter.

Gravity is always active and attractive. A mass in physical space is therefore always looking to move within a gravitational field. If it is stationary, then we look for the surface supporting it. So also, mendels of biological materials can never be without their ongoing transformations. If there are none then it is no longer biological. This is our first law of biology.

The quantity of mendels held at any time, which is the locus of biological inertia, is always replacing its components within its field. The overall quantity of biological material may remain constant, but the molecules and components that comprise it change continuously with physiology and metabolism. We must always eventually search for the activities and substitutions that maintained those mendels of biological materials.

Although mendels are always in a state of transformation, it is nevertheless possible to state how many kilogrammes of biological material are maintained over any given time interval. Those materials are then the mass maintained by the mechanical aspect of internal energy as a response to the average torque applied over that interval. A quantity of biological matter is always looking to proceed about some circulation. It must also always be an expression of some species that provides the centre for its circulation. If we observe a consistent quantity of mendels maintained over any interval, then a consistent torque or force must have been applied to it to equally consistently replace all the components it would otherwise have lost. This is the biological equivalent of gravitational attraction.

Figure 20.85

Figure 20.85 shows us measuring the quantity of mendels or biological materials maintained over two separate intervals ΔtI, and ΔtII. If the time intervals are the same and the mendels are the same, then the torques over the two intervals are by definition the same. However, if either (a) the time intervals are the same but the mendels are different; or (b) the mendels are the same but the times needed to produce them are different, then two different torques were at work. So if we measure two generations or populations of the same species over the same time intervals and get different mendel values in kilogrammes and moles of components, then they are subject to different forces and energies. They faced different biological conditions. And if, as in Figure 20.85, this happens over an entire generation then evolution is very probably at work.

Metabolism
The inverse transform or first derivative of force and time, in ordinary physical space, is the yank, which is mass times jerk, measured as newtons per second. It is the rate at which a force is changing. The biological equivalent is the first derivative of the mass flux.

We have already defined this biological equivalent to the yank. It is metabolism. It is measured in kilogrammes per second per second. It is an acceleration. It is the rate at which the mass flux or mendelity is changing at any moment. It changes in response to ongoing changes in the biological torques or forces as they carry the population toroidally and meridionally—and therefore also poloidally—about the circulation. And since the torque is the derivative of the population's energy with respect to the circulation distance, then if that energy is changing there is a metabolism or change in the mass flux.

This metabolism is a key to evolution. If two entities or populations have different metabolisms, then they are accelerating relative to each other. They are using different mendels of chemical components over time. If they started off the same, then one or the other has varied from their population template. There must be a cause for this difference. Since they form the areas of interaction with the surroundings, they are the 2-ball and the divergences. And since both are moving about a circulation, then the cause is perpendicular to that motion, meaning it must emanate from the surroundings. We must then find the ‘traction vectors’ involved. Those traction vectors are the environment's energy and force but as directly experienced by each population through its three orthogonal surfaces which are its three dimensions. These traction vectors are therefore each population and entity's interaction with its surroundings, stated relative to its bases and units of measure. We carefully measured Brassica rapa's metabolism.

Portage
The poloidal conjoining of force in ordinary physical space is its second integral with respect to time, and is the portage measured in kilogramme meters. It is size and displacement from the mean. We can measure this both relatively and absolutely. It links the mendel, or quantity of biological matter held, to the displacement, or the distance from the mean at that time. Portage states the force applied in terms of the population's responses at that point in the circulation of the generations.

Impulsion
The inverse of the poloidal conjoining, or second integral with respect to time, is the poloidal directive of force, which is its second derivative with respect to time. In ordinary physical space, this aligns mass with the jounce to give the tug which is measured as newtons per second per second. In biology, it is the “impulsion” which is the derivative of metabolism. It is measured as kilogrammes per second3. It gives us yet more information on how the population is responding to the forces and pressures being imposed upon it for it tells us how the metabolism is changing at each instant. It is mass's specific ability, in that entity and population, to follow an input signal.

And for the first part of the final coupling, the distribution of force and time is the triple poloidal integral and the impulsivity. It is the different ways in which portability is distributed. We met this as the distance and the time that a reindeer walks to produce dung. It is measured as kilogramme metre seconds. The biological equivalent is the promenade which enables biological populations to gather the chemical component stocks they need and that allow them to survive in their particular surroundings. If creationism and intelligent design are true, then all promenades and baton twirls must be even, regular, and constant.

Pivot mass
The aberrancy of force and time aligns mass with the fifth derivative of position, which is the crackle, to create the snatch. In biology, this becomes the “pivot mass”. It is that selection of chemical components that the entity must take on, at that rate and quantity, in order to align all values with the template, and to abide by its species definition. It is measured as kilogrammes per second4. The flounce, convergence, and pivot mass are the aberrancies that lead the responses to the environment, and that state the differences in distributions. We again measured this for Brassica rapa.

We now have a complete set of distributions and aberrancies, or triple integrals and derivatives. If creationism and intelligent design are true, then circulations must be equal everywhere about themselves. There must be no aberrancies. All baton twirls must be even, steady, and regular, and have 4-point contacts.

Creationism and intelligent design thus suggest that a first population can differ from another entirely by numerical proportions. Populations can vary a little by numbers up and down, but nothing essential about them changes. The two parallel circles that create the 4-point contacts in Figure 20.59.A similarly imply that two populations can be defined by the eigenvalue or multiplicative relationships we first met in ‘Before We Begin’. The two populations are scaled multiples of each other. But this means that if one population is three times the size of another at one point, then it must be 1/3rd at another or they are not the same population. This is certainly something we can measure.

Figure 20.86

The Ancient Greek philosopher and scientist Anaximander of Miletus produced the first real biology in the Western world. He taught that life originated from the dew that had covered the primordial earth, before being dried up by the sun. The first animals were fish-like creatures with thorny skin, much like the bark of a tree. Human children need much initial protection if they are to survive, and once born from these fish ancestors they were nurtured like viviparous sharks: they developed inside the bodies of their parent units and emerged fully formed and capable.

Anaximander accounted for gravity by placing the earth at the centre of the universe. It then had nowhere to fall. Everything instead fell to it. He was also responsible for the ancient proposal that the planets moved in circles.

As in Figure 20.86, even today, Anaximander's circular proposal for the earth's orbit is acceptable for most practical purposes. If we stick locally, and use the most basic of instruments, it is still very hard to refute. We can pick any random point; make observations; and then truthfully insist, from those local measurements, that since we never see any difference in the two locally, then the entire orbit must indeed be circular. But because of its varying distances, the sun oscillates between about 147 million kilometers, 91 million miles on one axis and 152 million kilometers, 94 million miles on the other. This gives it a mean distance of 150 million kilometers, 92 million miles. The earth therefore varies from a circle by only 5 million kilometres, 3 million miles, in a journey of 939,860,000 million kilometres, 584,000,000 million miles. The earth's eccentricity is currently only about 0.0167. Its orbit is so close to being circular, at every local point, that the circular suggestion and the elliptical reality are indistinguishable everywhere.

If the similar firmament convictions behind creationism and intelligent design are applied to the earth's orbit, then we should be able to scale it using the eigenvalues or multipliers we first met in Before We Begin, with or without its eccentricity, and apply it to Jupiter. Since Jupiter is approximately five times further away from the sun, then we have an eigenvalue of five. If we apply the eccentricity, we should have the shape of Jupiter's proposed orbit. With that eigenvalue in physical space in hand, we can then take measurements and contrast the earth's orbits with Jupiter's to see how closely they match. Creationism and intelligent design are very similar in their proposals about populations and species. It does not matter, so they claim, how much we scale. The essential properties are unchanged..

We need to apply eigenvalues to biology. We want to test Darwin's proposals by doing something similar in our biological space. We want to scale populations and see if they vary. Just as we can measure a planet's eccentricity from a circular orbit, which is its degree of failure, or its aberrancy; then so also do we want to use eigenvalues to measure the exact degree of failure, in biology, for the proposal of the Aristotelian template that undergirds creationism and intelligent design. The more accurate Franklin population that describes a population's circulation of the generations is also exactly like the Aristotelian or creationist and intelligent design one at every local point we care to examine. The two are equally indistinguishable everywhere, locally. It is only when we measure across the entire trajectory that we see the difference. The followers of creationism and intelligent design are therefore very much like the supporters of circular orbits. They can truthfully claim that their description holds, locally. It takes a lagrangian and a careful study of rates of change to tell the difference between them, and to finally refute the circular proposal. We measured Brassica rapa's failure at 0.19 grams of mass, and 1.225 joules of energy. We can easily get those values from our Euler and Gibbs-Duhem equations and our tensors.

Measuring biological entities and populations in only the x, y, and z dimensions of ordinary physical space is not enough to prove evolution. It is also not enough to come up with definitive values for straight lines and circles, and so to show how creationism and intelligent design have failed by departing from one or the other. Measuring only in physical space will give us values for the effects of the mechanical, constant pressure, but will not give us values for the nonmechanical internal energies that cause those.

Since we need to know the nonmechanical, constant volume, reconfiguring Gibbs energy that leads to every population's state and configuration, we must measure them all in the n, , and V directions of our biological space. We must measure them, but in such a way that they can all easily convert each others' values into their own, to determine whether or not reproduction is possible. And since biological entities differ from ordinary physical objects by the constant transformations they undertake, we must measure them all from one indiscernible point to another. We must therefore measure them at each point in their generation lengths. So if, for example, we are interested in our cricket Chorthippus brunneus, which is a semelparous annual, then we must calculate rates and quantities both per the second of absolute clock time, t; and per the year, which is its generational and biological period of interest, T.

Figure 20.87

We need a system of measure that every population can use. Every population must be able to convert its own basis into another's, and conversely. It must also be easy to do. We can measure biological events and gradually prove that things that are equal to the same thing are also equal to each other, using the system of shadows, projection screens, and torches or flashlights shown in Figure 20.87.

We let every population measure both itself and all others, at all times, using its own self as its units of measure. It then has a complete set of the traction vectors that state its interaction with the environment. It then knows exactly what is happening all around it; to it; and to all others. It knows this all about the helicoid. Each can measure poloidally, meridionally, and toroidally. That will be the complete set of forces, curls, and divergences at each point that create the conservation of energy and circulation of the generations from one indiscernible point to another.

Figure 20.87 shows a tree that has established its axes and units of measure. We label them e1, e2, and e3 respectively, for each of the x, y, and z dimensions. These are its bases and unit normals. Every population can independently determine its own set. They have both magnitude and direction within each population's space. Since those three unit normals measure exactly one unit in a given and set direction, then they are all vectors.

The tree has now gathered everything it needs to measure its various displacements, velocities, accelerations, jerks and the like. This space is a scroll. That is true for all. The rulings in the space are a true and accurate reflection of its features and properties.

The tree wants to record how big it is, at this present moment, t0, in all three dimensions. It is going to measure its displacements using the system of three screens placed at the origin, combined with three flashlights at a distance, we see in Figure 20.87.

The tree positions itself close to the origin, and arranges for the flashlights to shine on it from infinitely far away, parallel to each axis. Those three flashlights will split it into three shadows on its three screens. The three shadows will then tell it how big it is in each direction, in terms of its separate displacements or stretches into each. And since those three also have a magnitude and a direction, they are again vectors. We are obviously going to use one set, i.e. the three units e1, e2, and e3, as a ruler, to measure the other set, i.e. the three shadows.

Since the tree wants to know its overall size, we can call its three-dimensional expanse S. It first wants to know its size or displacement, S, in the x dimension. Since that uses e1 as a basis for its measurements, it switches on the flashlight that casts that x-shadow.

The tree can measure the shadow it produces with that x flashlight by using its e1 for a unit. Since finding out how many times longer the one is than the other is to express each in terms of the other, this is a form of ‘direct product’ between e1 and that shadow. The conventional symbol for this shadow-making direct product process between two such vectors is Se1, which roughly means “if you want to measure S using some e1 as a unit, then whatever direction e1 is pointing in, shine a light on your object so it casts a shadow parallel to e1”. Gibbs seems to have been the first to have called this direct product between two vectors their “dot product”, precisely because of the dot placed between them. His student Edwin Bidwell wrote the book Vector Analysis, a Text-Book for the use of Students of Mathematics and Physics Founded Upon the Lectures of J. Willard Gibbs which established it as the current convention.

The tree does not yet know how long its shadow is in the e1 direction because it has not yet been measured. We have so far only isolated the direction. So we can now pick up our e1 unit and lay it out on the shadow to measure the one with the other. And since this process of measuring is again sensitive to direction, then it is another direct or dot product … but this time between e1 and the shadow. So we now have e1 • (Se1). This roughly says “since you want to know how long your object S is in the direction of e1, then first cast a shadow in the e1 direction; and then lay e1 out along the shadow as many times as you need”. And since we have both (a) thrown a shadow in the e1 direction, and (b) measured in that same e1 direction, then we will denote this S11. This says “first cast a shadow in the direction of your first unit, and then measure with your first unit”. We have just created the first component in our tensor.

Unfortunately, not all objects are “cleanly” built. Most things are a little deformed. We therefore switch on the other two lights, one in each of the e2 and e3 directions to pick up any shadows coming from each.

Every population can do the same. Every population will also get a value for measuring in this given e1 direction using the units it establishes specifically for that direction, which is e1 • (Se1), and S11.

We can use exactly the same process for the e2 and e3 directions. We cast a shadow in each direction, which is (Se2) and (Se3), and then measure in each of those e2 and e3 directions. We must be careful to gather up all shadows, including those made by (Se1) in each. This gives us the further six measurements e2 • (Se1), e2 • (Se2), e2 • (Se3), e3 • (Se1), e3 • (Se2), and e3 • (Se3). We can write these more compactly as S21, S22, S23, S31, S32, and S33.

Every population will once again get a value for measuring in each of the e2 and e3 directions using the units specifically established for those directions. They are e2 • (Se2) and e3 • (Se3), which we can write as S22 and S22 respectively.

The shadows are just an aid to understanding. Additionally, since these are all forms of multiplication, then since 3 × (2 × 1) = (3 × 2) × 1 = 6, then the brackets in our various dot expressions are not really needed. We therefore have S11 = e1 • (Se1) = e1Se1, and similarly for all the others. So all we need, to make any tensors work, is to establish our units, and to then make sure that the space we are working in abides by the necessary rules. We can therefore go all around a circulation and build a tensor for our biological space, using our three orthogonal biological dimensions.

We are dealing with biological entities which grow, change, and develop. We are measuring both work and generation times. We must now compute our two rates of change: one for absolute clock time, t, and one across the entire generation length, T. We will then have the values we need for both the Liouville ensemble and our lagrangians at every point about the circulation of the generations.

Biological entities present particular problems, however, because they undertake numerous transformations and changes in shapes right across their generation lengths. Since we want to reckon them across both time and space, we need a simple way to determine whether or not there are distortions.

We will assume that we have been all around the cycle of the generations and got the values for how our biological entities change in the x, y, and z dimensions of physical space. But as the bamboo in Figure 20.88 makes clear, this cannot possibly be enough to give a full biological accounting.

The top of Figure 20.88.A shows a pendulum bob. It is constrained to move up and down in a fashion similar to the Cannabis sativa we saw earlier. I.e. it can do nothing else but move up and down in the z dimension.

The biological equivalent to a restriction in the z dimension is the bamboo growing underneath which goes through its reproductive cycle only ever growing upwards. Its progeny are shorter, and then do the same. They never either broaden or thicken.

As again with the Cannabis sativa we considered earlier, the bamboo may have restrictions, but it still cannot exercise its z options without taking on and then releasing both mass and energy. We can therefore measure it in both of our required temporal units: t and T, or τ. We learn how much it grows in x in absolute clock time; and since we compare it to some previous moment, we also learn how much it grows as a proportion of that prior state, which depends upon its generation length. We are taking poloidal, toroidal, and meridional measurements all about the helicoid.

We then examine Figure 20.88.B, where the upper part shows a pendulum bob that can swing, like a grandfather clock, in both x and z. It does not move in y. The equivalent for our bamboo is to get taller and broader, but never to thicken in y.

These variations still require the taking on and releasing of mass and energy … but since the proteins and DNA must be different from 20.88.A, then the numbers of moles of chemical components and the energies required to configure them must also be different. We again learn how much it grows in those two dimensions both in absolute clock time, and as a proportionate circulation rate.

Figure 20.88.C shows a conical pendulum bob that is free to go around in circles, and so in x and y. It may not not move up and down in z. The bamboo equivalent goes through its cycle without ever getting either taller or shorter. When it reproduces, its progeny have exactly the same height, and then grow and reproduce exclusively by broadening and thickening. The moles and masses of chemical components, plus configuration energies they need, must again be different from the previous two.

And finally, the pendulum bob in Figure 20.88.D is free to move in all three directions. The bamboo plant underneath is now unconstrained, and may freely get taller, and/or thicker, and/or stouter. This again uses different quantities of chemical components and energy.

The quantities and rates of masses and energies used in the above four situations must all be different. If two identical gazelles run across the Serengeti plains and forage in identical ways over a three month period, then their values for masses and energies will be point-for-point the same. If, however, one has a smaller mass than the other; or can run less capably; then their values for mass, and their Gibbs and Helmholtz energies will be different. Every change in the x, y, and z dimensions requires both the Helmholtz and the Gibbs energies, and exchanges between them. No biological entity can move or change in any way, in any dimension in physical space, without also changing in the mass and/or energy, and therefore in the energy density, of our biological space.

Since it is impossible for anything to change in the x, y, and z dimensions of physical space without changing in the Gibbs and Helmholtz energies, and so in the three n, , and V dimensions of biological space, we have no real need to record transformations in x, y, and z. Our biological space abides by all the rules required for any such three-dimensional space. We also have the necessary units it needs for each dimension. Since the tree is measuring all around its generation, it will produce values for both (a) its rates of change in absolute time, t; and (b) its rates of change across the generation length, τ, for all three dimensions. The latter set is its set of displacements, d, all across that generation, which is its sets of forces and energies. The same holds for all other populations whether they measure themselves, or are measured by others.

Our three n, , and V biological dimensions record the entirety of all biological-cellular materials. If two identical trees both grow by one metre in the x direction, then they change by exactly the same quantities of cellular materials and energies. The energies are measured by the forces those masses have exerted over that displacement, which is the same Helmholtz energy and Fd, for both. That Helmholtz energy they need will also be funded by some transfer from the Gibbs energy, which we can measure with both their hamiltonians and their lagrangians.

If two trees are not identical, and are not doing identical things, then their masses and energies expended at any time, along with their masses and energies stored over time, will all be different. We will record all such differences in energy stocks and events in n, M, and V. We will also be able to calculate their different fitnesses, competitions, heredities, genomes, inheritances, flounces, convergences, pilot masses and the rest. Any changes will show in differences in cellular materials and transformations.

Now that we have given every population a way to measure the three Owen tensor diagonals S11, S22, and S33, which are the three dot products e1 • (Se1), e2 • (Se2) and e3 • (Se3), we can use them as eigenvalues to make some A equal to some B equal to some C in our biological space of internal energy.

Every population can take itself as its own basis; but can also use another as its basis to measure its own self from that other point of view. In the same way, the Earth and Jupiter have an eigenvalue of five. Either earth's orbit is 1/5th that of Jupiter's, or Jupiter's is 5 times that of the earth's. It does not matter which way we approach it, we can create accurate values for either one either way, using whichever we choose as a basis.

The earth's eccentricity or deviation from a circle is e = 0.0167. If we want Jupiter to now emulate the earth, then we can calculate exactly where it should be, in its expanded orbit. Once we have calculated its proposed positions, we can then take some measurements to see how it fares. And … we of course find that our eigenvalue predictions do not match the observations. Jupiter's eccentricity turns out to be e = 0.0489: nearly three times the earth's. Not only can we measure that distortion, but we can state the rates at which the two planets approach and depart each other as they express their different eccentricities of orbit.

An eigenvalue only gives us the scaling or rating between the two planets. The combination of magnitudes and directions that go with it, under this largeness proposal, is the set of “eigenvectors” or “characteristic” or “proper” vectors. In other words, if we want to scale the earth by five times in a proper or characteristic way and preserve all its essential properties, we are obliged to preserve certain other characteristic values, in certain prescribed directions, and we must create the eigenvectors that match the eigenvalue. Eigenvalues and eigenvectors allow us to start from a small anything; to scale correctly; and then to produce a smaller or bigger version that is otherwise exactly the same. We can for example suggest that if a first population of human beings wants to have three times as many piano players as a second one, then it must have so many times as much wood and metal to make all the extra pianos. Those eigenvectors state the sets of distances and positions—and so the values for potential and kinetic energies—that match that eigenvalue. We will have the needed differences in hamiltonians and lagrangians, and in speeds and positions all about the new orbit.

If biological populations must stay the same, then they must scale their numbers and/or masses and/or chemical energies by certain eigenvalues, and give us certain predictable eigenvectors. They must also scale their divergences and their curls, and all other dimensionally relevant interactions. We can use these eigenvalues and eigenvectors to make sure that things equal to the same thing are equal to each other, and so to make sure that if A = B, and if B = C, then A = C. We can then determine what it takes to make one generation x times bigger than another, while keeping everything—including their dimensional interactions—otherwise the same.

Eigenvalues and eigenvectors measure the most essential of characteristics as other properties change. They report those essential attributes no matter how much the bases or measures change. If we measure an object with two different sets of units, then its eigenvalues will remain the same and produce the correct interactive and inter-dimensional eigenvectors.

Figure 20.89

The top row of Figure 20.89 shows four objects in physical space. They have lengths, areas, and volumes. Because they have different shapes, they have different eigenvectors.

We now scale the four objects, under some chosen eigenvalue, to become the identical four objects in the second row. The lengths, areas, and volumes must change in characteristic ways to preserve those shapes. We then want to know what the most essential of properties are so we can make sure they are preserved under any such transformations or changes in scale.

Since the only difference between the top two rows in Figure 20.89 is scale, we should be able to calculate the eigenvectors for their lengths, areas, and volumes for every position that the various points, edges, and faces will all occupy. If we want to transform the small cube into the large one, we can align them at their centres.

We first measure to an edge in the x direction in the small one, using e1 as our base unit. We then keep going in that same direction and measure the distance out to the same point in the larger one. We now have our eigenvalue for the scale, and a matching eigenvector to describe the point. We can gradually map lengths to lengths, areas to areas, and volumes to volumes. If the cube becomes the box or the ellipsoid, or the box becomes the cube, sphere or ellipsoid; if there is any other such mutual transformation; then we are not getting the eigenvectors that match the eigenvalue we are working with. Some distortion is taking place, and some length and/or area and/or volume has changed. We can compare the measurements of the transformed objects to the originals and gradually work out what is causing the transformations, and which dimensional interaction is being affected.

If we want to scale the small cube to the larger one, we must repeat an eigenvalue-eigenvector process in every direction, and for every length, area, and volume. We must measure our small cube in the y direction, using our e2 unit. Once we have that y value for our small cube, we apply our eigenvalue to produce the eigenvector, or characteristic magnitude and direction values, that tells us what the similar y measurement should be for that larger cube … assuming it has not been distorted. We then measure. Once we have that value, we compare it to what we calculated as our eigenvector. If the two differ, we will have an exact value for the distortion. We will also know that the space these two cubes are embedded in is curved.

The third row in Figure 20.89 shows a simple increase in numbers. The original objects give us our eigenvalues and eigenvectors, and we can make sure the new objects preserve those lengths, areas, and volumes. If a biological population only increases in its numbers, there must be matching increases in amounts, masses, and energies across the population. However, the eigenvalues and eigenvectors, or characteristic values, will remain the same. There will therefore be an increase in flux amounts to match the increases in numbers, but there should be no change in flux densities, which are characteristic or essential. But if, as in the fourth row, there are also changes in size, we can use eigenvalues and eigenvectors to work out if there have been any accompanying distortions, for those eigenvalues should not change. We can therefore use eigenvalues and eigenvectors to scale populations in similar ways in one, two and three dimensions within our three-dimensional biological space.

As we learned in ‘Before We Begin’, each population's size depends on (a) how far it stretches in each dimension relative to all others; (b) the interactions each of the dimensions have separately with each other; and (c) their three-fold interaction with each other. These determine the body and surface forces and stresses, and therefore also the traction vectors it has through the surface with the surroundings.

When we scale a cube from small to large, we preserve the traction vectors which is the interface with the surroundings. We preserve (a) lengths, (b) areas, and (c) volumes. If the transformed objects are to be the same, then there must be three invariants across those transformations, and as they measure each other:

1. I1 preserves distances and combinations of distances;
2. I2 preserves areas and combinations of areas; and
3. I3 preserves volumes and combinations of volumes.

Eigenvalues and eigenvectors have the same scaling and invariance properties as our tensors. Every magnitude we measure with a tensor also remains itself, no matter how much the basis changes. The three diagonal components in our tensor are S11, S22 and S33. They have a special status. They are the normal pressures.

The three measurements e1Se1, e2Se2, and e3Se3, which are S11, S22 and S33, are the values we get for each unit when we measure in its specific direction. They are the ones that remain true to the object no matter how much we transform or change the basis. This includes when they are scaled. Those three are therefore our principal eigenvalues and eigenvectors.

There is one principal eigenvalue for each direction in space. They are the three invariants that hold good no matter how much we might transform or change our bases of measurement. We can label them λ1, λ2 and λ3. I.e.:

1. λ1 = S11 = e1Se1,
2. λ2 = S22 = e2Se2, and
3. λ3 = S33 = e3Se3.

When any population or object wants to scale to any other, it can now refer to those values. They are certain to be scaled multiples of each other.

When we scale a cube from small to large, we preserve lengths, areas, and volumes which are also I1, I2, and I3 for each. Our three eigenvalues must make sure those invariants are preserved. Every tensor must combine its eigenvalues so it can always produce the correct eigenvectors for every dimension and combination of dimensions. We must preserve all three of lengths, areas, and volumes no matter how much we might try to scale or transform.

Figure 20.90

Our first invariant, I1, preserves sizes and distances in all directions. Figure 20.90 shows an object shifting from Position A to Position B. When it arrives at B, the displacements per each direction, i.e. parallel to each axis, will change. We will get new values for S11 and S22.

Although our two values S11 and S22 change when we move from Position A to Position B, they do so in a coordinated way. As we saw in the Liouville theorem, an increase in one value, in this case in the x dimension, is matched by a proportionate decrease in another dimension, in this case in y. Wherever this object goes on this xy plane, its position values parallel to each axis will retain a concordance based on the orthogonality of those same two axes. There are always matching changes.

The same will hold, pairwise, for all three such axes in any space. Any increase in the amount parallel to one axis is matched by a decrease in another. Therefore, if the characteristic value for any transformation is to be preserved, then the three dot products parallel to each axis must coordinate to give the object the same overall value for S11, S22, and S33 under any scaling, transformation, or change of basis. This suggests our first tensor invariant.

We shall assume that the shadows cast on x and y when it is in Position A are each exactly one unit long. The two values S11 and S22. are therefore e1 = e1SAe1, and e2 = e2SAe2 measured parallel to the x and y axes respectively. When the object shifts to B, whatever is subtracted from one is proportionately added to the other, and conversely.

Our first tensor invariant, I1, must be the sum of the three eigenvalues λ1 + λ2 + λ3. It is the net sum of values parallel to each axis. It must remain constant for any object in any space. It is also the sum of the diagonals along our Owen tensor. It is a value we have already met as both the Ricci scalar and the Liouville constant. It is an overall indicator of the size and dimensionality of any space. And since those dimensions could be anything, then it means that their sum will still be an accurate and invariant descriptor of their space and possibilities.

Now we have successfully used our eigenvalues and measuring systems to preserve all the lengths and sizes in our space, we must preserve the eigenvalue-eigenvector relationships that create our areas and two-dimensional interactions. These are our curls and divergences.

Figure 20.91

As in Figure 20.91, any cube we create in space will have a set of surface areas. The surface we see has a definite area, but the three different xy, xz, and yz perspectives or planes upon the right tell us that it has different shadows and dot products when viewed from each axis. Since they are dimensional interactions then those three areas are λ1λ2, λ1λ3, and λ2λ3. This also means that no matter what the various dimensions are, the sum of their pairwise interactions is an invariant common to them all.

If we rotate the area in space, the projections or views in each direction will change. Each axis may always see a different area to the other two, locally, but they are coordinated, globally. What is not seen from one perspective is seen from the others. As an object presents and disports itself in and to the environment, and to all other objects, its total area is invariant. Our I2 therefore gives information on the overall surface stresses. Since any object has a total area formed by xy, yz, and xz, then another invariant for our tensor, under any transformations or changes in basis, is λ1λ2 + λ1λ3 + λ2λ3.

Figure 20.92

And finally, the third eigenvalue–eigenvector invariant, I3, as in Figure 20.92, is the coming together of all three dimensions. This third invariant indicates the internal and bodily behaviours: the body stresses attributable to the object as its internal parts maintain themselves with respect to each other. It creates a volume interaction. Any cube that preserves its characteristics, under any scaling, must maintain the same relative volumes before and after. Therefore, if objects are to remain the same under a transformation, then the product of their three tensor eigenvalues, λ1λ2λ3, must also remain invariant. Irrespective of what the dimensions represent, they have a triplet or volume interaction that is invariant.

The three invariants I1, I2, and I3 are each constructed from our eigenvalues. Those eigenvalues are themselves invariant. Therefore, any combination will also be invariant. We can for example set all eigenvalues to unity so that λ1 = λ2 = λ3 = 1. We then get 3 for the size of this space from I1 = λ1 + λ2 + λ3 = 1 + 1 + 1. We get 3 for the sum of their surface stresses from I2 = λ1λ2 + λ1λ3 + λ2λ3 = (1 × 1) + (1 × 1) + (1 × 1). And we get 1 as the body stresses of the volume or triple interaction from I3 = λ1λ2λ3 = 1 × 1 × 1. Since these three are all invariants, we can multiply them together to create a population invariant of I1 × I2 × I3 = 9. This holds for all, but with each expressing it in its own units.

If creationism and intelligent design are true, then we should be able to double this cube and have it remain the same. We set all eigenvalues so that λ1 = λ2 = λ3 = 2. We now get I1 = 6, I2 = 12, and I3 = 8. If we multiply them together, we get I1 × I2 × I3 = 576. This is not the same. Doubling all the side lengths increases surface by 12 times, but cubes the volume, which explains the differences. The numbers are all still invariants, but the two cubes have very different descriptions and essential characteristics. The alternatives are instead to double the surface area, which increases the side lengths by √2 and the volume by 2√2; and to double the volume, which increases side lengths by √3 and the area by 9 times.

But doubling an entire cube is of course not the same as only doubling numbers, which is but one of our dimensions. If creationism and intelligent design are true, then a biological population should be able to change its numbers without changing its essential characteristics. Our first eigenvalue changes to give λ1 = 2, while the others remain the same at λ2 = λ3 = 1. The three invariants become I1 = λ1 + λ2 + λ3 = 4; I2 = λ1λ2 + λ1λ3 + λ2λ3 = 5; and I3 = λ1λ2λ3 = 2, which multiply together to give I1 × I2 × I3 = 40. Doubling only numbers has taken us from 9 to 40. It is impossible to double only one dimension in a cube, and expect the other two to stay the same. Area and volume change in coordinated ways, but they do not double when one side doubles. The object is distorted.

Biological interactions are as multi-dimensional as cubes. The relations amongst the eigenvalues and invariants will always change. No population can change numbers and leave other essential properties unaffected. It is impossible to change one eigenvalue without changing the other two. Creationism and intelligent design are impossible because they cannot preserve essential properties.

If creationism and intelligent design are true, then an original population must remain essentially unchanged after a generation. All populations must abide by the identity tensor, I, that we met in The Refutation. Like 3 + 0 = 3 or 3 × 1 = 3, it transforms tensors back into themselves, leaving them essentially unchanged. It has a series of 1's on its diagonal, and 0's in all its off-diagonal or shear positions.

Table 20:5 shows the two required set walks in mass and energy a population must undertake over a generation if it is to be free from changes in numbers:

 m̅’:m̅’ m̅’:p̅’ p̅’:m̅’ p̅’:p̅’

The m̅’:m̅’ and p̅’:p̅’ are the measures we take at the beginning and end of our circulation of the generations, while m̅’:p̅’ and p̅’:m̅’ state the τmass:energy and τenergy:mass sequences that are the DNA interactions covering both mechanical and nonmechanical chemical energies. Their increases and decreases must balance so the population is restored in the time scale stipulated in the template.

Since the identity tensor has 1s on its principal diagonal and zeros everywhere else, we can easily incorporate the above essential development tensor into an identity one to create the 3 × 3 Owen in Table 20:6 that we can use to scale any population:

 1 (n:n) 0 0 λ1 0 1 (m̅’:m̅’) m̅’:p̅’ λ2 0 p̅’:m̅’ 1 (p̅’:p̅’) λ3 λ1 λ2 λ3

The n:n at top left gives the Tnumber:number duration over which the population will restore its BIDE equilibrium distribution. The two eigenvalues λ2, and λ3 remain unchanged no matter what happens to λ1. The tensor can now scale numbers independently of the other two quantities on the principal diagonal. Creationism and intelligent design now claim that this is the reality for all populations.

Creationism and intelligent design are in fact claiming that biological space is homogeneous and isotropic. It is filled with parallel lines and rulings that have zero curvature, and that preserve all shapes. They claim that the summation term in the Gibbs-Duhem equation:

μ = dS = Mdt  + Pdt  -  Σiμi ( dvi - dmi)

is zero; and the two summation terms in the Euler equation:

μ = dS = (
S
U
)V,{Ni} dU  +  (
S
U
)U,{Ni} dV  +  Σi (
S
Ui
)U,V,{Nj≠i} dU  +  Σi (
S
Ui
)U,V,{Nj≠i} dV:

are also zero.

• The first term is the mass flux, stated as the infinitesimal changes in the mendels or kilogrammes of biological matter, dU, while the numbers, N, and the visible presence or energy density, V, hold constant.
• The second term is the changes in the Wallace pressure, which is all infinitesimal changes in visible presence, dV, while while the numbers, N, and the mendels of biological matter, U, hold constant.
• The third term allows numbers to change, and so records all changes in the mendels of chemical components, dU, individually held by each entity as each is introduced into, or else departs from, the population.
• The fourth term also allows numbers to change, recording all changes in nonmechanical chemical energy carried by individual members as each is either introduced into or departs from the population.

These are very simple claims to test. They are largely why we ran our Brassica rapa experiment. All we need is a valid metric that any population can apply.

Figure 20.93

Figure 20.93 shows a population moving about the helicoid of internal energy under the influence of our two fields P-1L-1P1L1 and M0L0. Its DNA and ecology are moving it from t-1 through t0 to t1, as well as through the biochronometric distance τ-1–τ0–τ1 which is some proportion of its generation.

The dot products that measure the population at t0–τ0 are the three lines with arrows formed by e1, e2, and e3 against each of S • e1, S • e2, and S •e3. The units for e1, e2, and e3 are the population means. Our measurements then state how parallel the population is to each axis at each moment. Whichever axis we allocate to number, creationism and intelligent design then insist that it stays invariant no matter how many transformations and permutations the population might go through. That is something we can measure.

The process S • may tell us how much something is pointing in, or is parallel to, the direction of interest, but biological populations do not push neatly in straight lines. They do not stay parallel to one axis. We measure a mass flux, for example, via a dot product against each of our n and axes, which simply means that we always have a value against each. But they do not have to change in the same way to produce any given max flux. If they have different rates of change, then the dot products change in different ways as the mass flux changes. There may be some Se1 measure for S out parallel to e1, but there will be other dot products parallel to the other axes, expressed in those units. Those measures and aspects also produce forces across displacements. They are parts of allowed sets with curls and divergence.

Figure 20.94

As in Figure 20.94, the mutual additions and subtractions on other axes are how the Liouville theorem ultimately creates the circulation. It accounts for all non-parallel pushes or components as they also do work and exert forces in their other directions. Those either add to, or subtract from, the forces native to each direction. The amount of force, or size of area, depends on how they interact.

The area marked e1 & e2 in Figure 20.94.A is formed from a vector force that skews or crosses over from one direction to another. Some of the e1 push has gone in the e2 direction. It is affecting those results. It is formed by the interaction between the e1 and e2 components and directions. Gibbs called it the “cross product”. It is written as e1 × e2.

Figure 20.94.A shows that both e1 and e2 have their means upon their left. The population is not far from its peak at t1 and τ1 while e1 is moving outwards and is just above its mean. The population's position indicates that quite a substantial portion of its net force, as measured by e1, is directed towards e2. By the right-hand rule of vector forces, the area is the force they jointly exert that augments the third dimension of e3, which is upwards in our orientation. The area stretching between them is therefore quite large, relative to e1, and affects e2 by assisting it in whatever it is doing. And since e2 is working on e3 to persuade it to increase above its mean, then the interaction between e1 and e2 is having a net positive, but decreasing, effect upon the torque. This interaction between e1 and e2 therefore gives the overall vector cross product e3 = e1 × e2, which affects the balance of the Gibbs and Helmholtz energies. The cross product therefore measures any interactions a population is undertaking, but measured relative to some other dimension, and that has skewed to that direction. These skewed and cross over forces affect the population through the tensor's off-diagonal elements, and help carry it about the circulation.

Figure 20.94.B shows that e2 and e3 are simultaneously interacting to produce a similar vector cross product. This area of interaction is not so large, meaning that e2 and e3 are not interacting as positively and powerfully as are e1 and e2 at this point. They are more parallel and less orthogonal, and therefore have less of an area. The interaction between e2 and e3 is outwards along the e1 axis and augments it. It gives the vector cross product e1 = e2 × e3.

And, finally, the interaction between e1 and e3 in Figure 20.94.C shows the Liouville theorem at work. The two forces are almost parallel, meaning their area is close to zero. Their parallel components are also in opposition for e1 is decelerating while e3 is accelerating, meaning their net contribution to these cross product forces is close to zero. They will soon reverse. Their vector cross product is e2 = e3 × e1.

These three cross products and areas formed by our three forces and coordinates will now add and subtract all about the population as it maintains an origin at the generation means. They help determine t0–τ0 which is the current moment and the vertex. It is always the current location in the circulation, and the current expression of the Gibbs and Helmholtz energies and their hamiltonian.

Figure 20.95

Figure 20.95 shows the complete interaction between all three forces and dimensions in our biological space of internal energy. Each cross-product area can now be pushed, or shadowed, along the third axis, to create the total energy volume which is the net vector force. It is the population's complete state in internal energy at each point on the helicoid. It is the “triple product” of e1 • (e2 × e3) = e2 • (e3 × e1) = e3 • (e1 × e2). It is the entirety of the forces and the energies the population can muster at any time.

The triple product is the journey from one indiscernible point to another. It is the hamiltonian. It is the generon. It is the complete set of baton twirls. It must hold constant. It must be conserved over the generation.

This three-dimensional energy structure that is the triple product stays the same across the generations. It moves up and down and back and forth negotiating with all three dimensions, constantly creating the triple product that also states the Liouville constant. It extends from the origin to the helicoid and has three parts for the three invariants. The one-dimensional diagonal line of constant length is the 1-ball inside the 0-sphere, and is the population's first invariant, I1, and the sum of the three eigenvalues as I1 = λ1 + λ2 + λ3. This is the sum of its three promenades or required set walks. The cross products are the two-dimensional areas that add and subtract to form the six off-diagonal components in the Haeckel tensor which are the 2-ball inside the 1-sphere, at the same time as they are the 2-sphere surrounding the 3-ball. They are I2 = λ1λ2 + λ1λ3 + λ2λ3 and the divergences and curls. And finally, the three-dimensional statement of an energy volume is the hamiltonian. It is the sum of the Gibbs and Helmholtz energies and the Liouville phase volume. It is the 3-ball inside the 2-sphere and the 0-sphere that surrounds the 1-ball. It is I3 = λ1λ2λ3. These three are very easy to measure.

Creationism and intelligent design propose that their templates, which are a form of ideal, impose external and overriding behaviours upon biological entities. Those templates supposedly tell all entities and populations how to construct themselves, and how to behave, no matter what the environment is doing. It also corrects them if they go awry so they can maintain their essential properties. Those doctrines are now claiming that populations are completely free to interact in any manner they choose … but that variations in numbers will never change the masses or the energies away from template values. They in particular claim that the various two-dimensional areas—which we know as the divergences and the allowed sets—will somehow remain constant both absolutely and relatively, no matter how much numbers vary. We now have eigenvalues and eigenvectors to help state, and then test, those essential properties and behaviours.

If we want to prove that this whole notion of a template is impossible, then we have to resolve those anomalies, and show that neither the firmament nor the templates are viable as options. We shall emulate Galileo and define an ideal motion in our biological space. We shall then show that this ideal is unattainable and that A cannot be equal to B, and that B cannot be equal to C, and so that A cannot be equal to C.

If biological entities wish to reproduce, they must successfully measure each other. Biological interactions require that biological entities and populations judge their relative sizes according to shared metrics and coordinates propagated between predecessors and successors. If a chihuahua and a great dane, or a Pekingese and a St. Bernard, or a horse and a donkey, or a lion and a tiger, or a grizzly and a polar bear wish to reproduce, then they have to behave appropriately relative to each other. They must do so across all three of the dimensions we are using. Given our three sets of eigenvalues, then they must match their eigenvalue-eigenvector interactions. If they are members of the same species, then they must match their mutual lengths, or one-dimensional interactions; their areas or two-dimensional ones; and their volumes or three-dimensional ones. They must also match themselves to the surroundings.

There are also at least two different kinds of interactions that populations and their entities must match across their circulations. There is the set of multiplicative transformations, complete with inverses, symbolized by placements and displacements and their pd = 1 constancy. These match the Helmholtz decomposition theorem. Then there is the set of additive transformations and the mutual conversions as between the Helmhotz and the Gibbs energies. These match the conservation of energy and the Liouville theorem.

Creationism and intelligent design are untenable not only because they violate Galileo's dictum against perpetual motion, but because they do not accept Einstein's much later demonstrations. Newton linked Galileo's discovery to circular motions and planetary orbits. Einstein then showed that space itself is curved. It is affected by the size of the nearest mass. Objects, the spaces they occupy, and the axes and units used to measure them can all change. Since biological populations are the equivalents of regions, then biological space can also vary with population size.

We have given ourselves three sets of axes to work with. We get the first set, i, j, and k, by direct measurements. That is our A population. We get the second set, I, J and K, by measuring many generations of the same species and constructing a Liouville ensemble. That is our B population. As for the third set, Einstein realized that he could only prove his case if he could measure the surrounding space. He therefore used the intrinsic coordinate system that Gauss and Riemann first discovered. They are conventionally called the Tangent, the Normal, and the Binormal. The T, N, and B set is a property of space and is independent of both i, j, and k and I, J and K. That is our C population. If we can prove that A = B = C is impossible, we can prove that creationism and intelligent design are impossible.

Creationism and intelligent design insist that populations can change in numbers while leaving all other properties unchanged. This is a two-dimensional interaction. The essential development tensor that supposedly validates this is also only two-dimensional. It is a 2 × 2 tensor.

Figure 20.96

Physical space is again easier to think with. Any two-dimensional interaction is like an area. And as an area interaction, it is very similar to the parallelograms we see in Figure 20.96.A. Creationism and intelligent design then insist, as in the upper and lower parts of Figure 20.96, that transformations and interactions between larger and smaller do not affect biological essentials. Populations are free to scale back and forth in numbers as often as they wish. They will preserve their essential characteristics no matter how much the coordinates or values have to change to accommodate them. Darwin, however, insists that numbers and scaling will immediately have an effect … which is the three-dimensional interactions we see in Figure 20.96.B.

The two small parallelograms at the top of Figure 20.96.A represent any proposed interaction in two dimensions. It could be reindeer moulting; or our African craftsmen making adungus. It could be anything at all.

The parallelogram skewed rightwards represents some change in our interactions. At least one of the dimensions involved shows a slight change in values. It could be a difference in amounts moulted, or a change in the amount or type of wood used for adungus.

The background grid in Figure 20.96 represents the i and j, and i, j, and k axes. The size of the little squares represents the one unit measure in each direction for population A. The axes standing outside them marked I and J, and I, J and K each have their independent units and scalings. Those represent the more population-wide firmament-style axes. Their measures are independent of any one population or generation and are also independent of the little squares. The T and N axes are not shown in 20.96.A to avoid clutter, but are lightly marked, with dotted arrows, in 20.96.B.

What so surprised Newton, and awoke him from his slumbers, was learning that the areas in the two parallelograms at the top of Figure 20.96.A are the same. That has some very profound consequences. It confirms that creationism and intelligent design are impossible.

If we count the small background squares for the intrinsic i and j grid, we can see the equality of the areas. The entities concerned have maintained whatever area and interaction represents their essential development. And since the area is the same, then it need not be clear, to the population members, that any real change has occurred. An adungu or harp made from a different wood is essentially the same … and … a grizzly might look close enough to a polar bear to encourage reproduction.

If creationism and intelligent design are true, then transformations just like the parallelogram skew in Figure 20.96.A, i.e. exclusively in two dimensions, are eminently possible. And if the doctrines are false, then they are not. Darwinian evolution holds that they are not possible because populations will always show movements in a third dimension, like those in 20.96.B.

The I and J template axes in Figure 20.96.A show the parallelogram's top edge skewing away to the right. Relative to those, the values used on the I-axis to create the areas have changed, even though that area is the same.

The parallelogram's area, as a two-dimensional interaction, is an invariant. This indifference to skews is a way for creationism and intelligent design to insist that the different efforts red-tailed cockatoos must make to reproduce themselves in different conditions are not relevant. It is only relevant that the species reproduce the areal interaction involving itself and its resources. Creationism and intelligent design indeed argue that a population can do a different amount of work, with different resources, while the essential species interaction remains unchanged. Those are still red-tailed cockatoos.

Since there are many possibilities, we can assign initial and final values to each of the two very general axes x and y. We do not care how any population derives its values, nor what coordinate system it uses. Each will have forces acting over distances, and members interacting with resources. These form the sides of a parallelogram. We can express them all as ax + by and cx + dy or anything else of that general type, and completely independently of how we procured those values. If those areas remain the same after some kind of transformation, then we have the required biological compatibility between before and after. It does not matter if it is a monkey or an ant.

Creationism and intelligent design propose that, irrespective of values and coordinate systems, the area in white in Figure 20.96.A will always be formed from the area of its larger surrounding rectangle less that of the two trapezoids and the two triangles. What we add in one place we take away in another, and conversely. If we step around the two figures and suitably add and subtract, we shall have the areal interaction we want. That is the work done and energy expended in our biological space.

The area is an invariant. It is preserved no matter how much the surroundings might change. In spite of differences before and after, we will have our biological compatibility. If the forces and activities truly interact, in any kind of two-dimensional way, to form an area, we can soon find a system that gives us their interaction independently of how we produced them.

We can refer to our four side lengths as a, b, c and d. A little geometry and algebra then tells us that the area we are concerned with is always ad - bc. The numbers might change, but that area always has that value no matter how we measure, nor what basis or coordinate system we select.

Figure 20.97

Figure 20.97 tells us that we can extract our a, b, c and d coordinates and put them in a separate matrix. We now do not care if these are blue whales or mosquitos. Their values are completely independent of how we procured them. The value ad - bc is invariant. That is the value for any two-dimensional property or interaction, no matter what those dimensions might represent.

We have successfully produced a systematic method for handling our essential and invariant transformations. It dates back to ancient China where mathematicians and scribes would solve systems involving exactly such interacting properties by placing bamboo rods on calculating boards to represent the quantities or coordinates they were manipulating. The art then migrated to Japan where it was called fukudai. In 1683 the great mathematician Seki Kowa, also known as Seki Takakazu, wrote Kai-fukudai-no-hō or The Method of Solving Secret Questions, showing how to solve them in a thoroughly modern way.

Leibniz experimented with many different methods for solving these same systems. He used many different terms to describe the processes and results. In 1750, the Swiss mathematician Gabriel Cramer established the first of the general rules for successfully handing such systems. Laplace then developed a more general method. He called the result of such group manipulations the “resultant”. Lagrange further developed them.

Gauss was the first, in his Disquisitiones arithmeticae, 1801, to use the more modern term ‘determinant’ for the resolution of a general system of this kind, although he did not use the term in quite this same way. Cauchy was then the first, in 1812, to show how an array of this kind could be manipulated to give the system's more general solutions: i.e. the resolving of an entire set, all at once, rather than attending to the specific members in a more isolated and piecemeal approach.

The determinant is a single number that represents the process of adding and multiplying a whole set of coordinates in matrix form. It can then be reapplied to the tensor or matrix it originated from to transform it and to state our population's force and energy as the parallelogram's new area and cross product, even after scaling and/or slewing.

That determinant for our matrix turns out to be the same ad - bc we got above using geometry and algebra. We simply multiply together the two values on the matrix's ‘principal diagonal’ to give ad; we then multiply together the two on the ‘anti-’, ‘counter-’, or ‘minor’ diagonal to produce bc; and we then subtract the latter from the former. We can solve the whole system at once, and consider entire populations at once.

This matrix approach greatly enhances the usefulness of our tensors. Both have rows and columns, so they look very similar. However, they are not the same. A tensor is specifically constructed to allow for changes in basis under all possible conditions. A matrix is convenient for, for example, organizing the data collected in an experiment. A tensor is more than just a convenient way of arranging data. Its components include the transformation laws that allow for those changes in basis. Matrices have rules of their own and are useful when we wish to manipulate a tensor independently of its special transformational tensor rules. Matrices can help add, subtract, and manipulate the data from many experiments and certainly give valuable information, but they are not obliged to follow a tensor's special transformational rules.

Since tensors and matrices have very similar structures, we can temporarily represent our tensor with a matrix. This gives us access to all matrix possibilities. We can then use our determinant to scale and to slew our population as much as we want. Whether we think of our population as a tensor or a matrix, we will express its two-dimensional interactions independently of any specific axes or coordinate system. Once it sets its own unit and basis, every population will establish its essential development values as the following 2 × 2 matrix:

 1 0 0 1

The 1s along the matrix's principal diagonal again mean that every species uses itself as its own basis of measure for each dimension at the beginning and end of a generation. The 0s on the minor diagonal mean the values on the principal diagonal are ‘linearly independent’. No matter what the generation length or processes the off-diagonal processes represent, they match each other. All those biological processes and sequences cancel each other out to leave the principal diagonal unaffected.

This biological space is not curving. A measurement due eastwards has no north-south or up-down component and so on and so forth. If we scale in any one direction, we get the appropriate magnitude and do not inadvertently affect any other.

When we apply the ad - bc method to find that essential development determinant, we get (1 x 1) - (0 x 0) = 1. That ad - bc technique for a 2 × 2 matrix means it is multiplicative. It is a two-dimensional interaction and so an area. It is a product of lengths. It therefore matches the multiplicative transformations of the Helmholtz decomposition theorem.

When the physicist Henry Leopold Brose translated Weyl's Raum, Zeit, Materie or Space, Time, Matter from German into English, in 1921, he translated Weyl's use of die Spur—the trail an animal leaves, mainly with its droppings, and cognate to the English “spoor”—with the English “trace”.

A matrix's trace is the sum of the values on its principal diagonal. It tells us the general size of the space. It helps preserve sizes and distances through all transformations. It indicates a pattern of behaviour for whatever that matrix represents. It helps us match the additive transformations for the Liouville ensembles and theorem.

The matrix trace is the same as the sum of the eigenvalues, and so is our first tensor invariant of I1 = λ1 + λ2 + λ3. It therefore helps preserve essential properties.

Since the trace is the sum of the elements upon the principal diagonal, it gives a convenient statement of the general range that space covers. It tells us that the space goes this far in x, this far in y, and this far in z. If some other trace is smaller, then we know it has less space overall.

The trace tells us the relative weightings, densities, and portages for and amongst the dimensions. The component in the trace, and so on the principal diagonal, that has the largest value will also make the biggest contribution to the overall result. It also states the direction from which the greatest changes will come. Since it states the relative changes amongst the dimensions, the trace helps determine (a) the size of the cross product and so their relative joint interactions in space; and also (b) whether or not there is a curl. The trace can therefore give valuable information on the space and its rates of change as we scale. It clarifies the contribution that the independent dimensions make to whatever the determinant characterizes.

Creationism and intelligent design claim that populations preserve all similarities in all dimensions. The relationship between the trace and the determinant also tells us about the space. While the determinant tells us how the area is changing, the trace tells us how the lengths or dimensions change with respect to each other, in their separate ways, to produce that areal or two-dimensional change. This means that traces, and determinants, and all relations between them must stay the same in all possible transformations. But since the trace is additive while the determinant is multiplicative, this is an impossible demand.

A unit square has 1 unit of length in each dimension. Its trace is additive and is the sum of its unit lengths and is 2. Its area is its determinant and is multiplicative. It is 1 square unit. The relationship between trace and determinant is 2:1. It is an essential characteristic for the space.

If we now change one side to 4 units, we get a rectangle. One side has transformed from 1 to 1, the other from 1 to 4. The determinant is 4 and the trace is 5. These two sides do not have the same start and end points on each occasion, and the two dimensions have different rates of change. One has contributed more to the change in area than the other. The trace is therefore telling us how the lengths or dimensions are themselves changing with respect to each other, in their separate ways, to produce any areal or two-dimensional change.The trace can therefore tell us about the curvature. It is telling us that this space is curved.

If we take our original unit square and double it, relations between the two sides do not change. Magnitudes and their rates of change keep pace. They remain equal in each direction and make the same relative contributions at every point. They act in the identical isotropic and homogeneous manner throughout. And since they have made equal and equivalent contributions to the trace, then their essentials have not changed. So whatever their areal or two-dimensional interaction might signify, the two dimensions still make an equal contribution. Whatever properties the essential development matrix signifies, there is no curvature in this space. Creationism and intelligent design would therefore appear to hold for as long as those side lengths remain constant, relatively.

But unfortunately … the determinant in our new square is (2 × 2) - (0 × 0) = 4, while the trace is 2 + 2 = 4. The relationship between trace and determinant started at 2:1, and finished at 4:4. The space might not curve when we double the square, but the trace-to-determinant relationship has changed. We have remained isotropic and homogeneous, yet still had some very important changes in attributes.

We can monitor this relationship between trace and determinant. We can extend our matrices and determinants to three dimensions, and so to the helicoid of internal energy in Figure 20.96.B. Every species can then use its three axes to measure itself, always using its own units and coordinates.

Creationism and intelligent design fail because they insist on restricting populations to two-dimensional interactions. They refuse to accept that the third dimension of numbers is relevant. But if they are true, then there must be no curvature in this biological space.

Figure 20.98

Biological entities and populations face a very special problem which they must resolve, no matter how we choose to measure them. If we establish a mean for the population and index it as the zero point, then every positive value relative to that generation mean must at some time become less than, or negative, relative to that mean; and every positive rate of change that carries a value above that mean must be matched by an equivalent negative one that carries it an equivalent distance below that same mean at some other point in the circulation. Since those two rates must work in opposite directions, then they are opposing vectors. This must hold no matter what the scale; and no matter whether they increase or decrease in size. As in Figure 20.98, populations must go all around an entire circulation and respect their eigenvalues and eigenvectors.

Eigenvalues represent scaling. Once suitably scaled, they produce the eigenvector magnitudes and directions for the space. But our eigenvalues are always positive. The eigenvectors they produce can turn negative relative to their mean, but only because they are establishing a succession of reductions in magnitudes ranging either side of the mean. If we scale by 5 times at one point, then we must scale by 1/5th at another to preserve the mean.

We should not confuse the effects of the base we select—which we can set to zero or unity as is our choice—with what happens to values because we scale. We can put money into a bank account for a positive rate, and withdraw it for a negative one without necessarily going overdrawn. There is a difference between a negative number that simply means “less than the mean” and a “true” negative number. That would in these conditions indicate, for example, a “negative mass”, which is not relevant in biology. The 5 and the 1/5th are inverses which establish movements in contrary directions, but they are still both positive values. So although the eigenvectors can turn negative, relatively, in the sense that they cause the decelerations and the deductions that carry a population below its mean values, the scaling or eigenvalues themselves must always be positive. Eigenvalues must remain positive even though the eigenvectors they produce oscillate in different directions around the mean, and so are sometimes negative relative to that mean.

Each row and column in the Owen tensor represents a dimension in biological internal energy. Creationism and intelligent design demand a principal diagonal entirely composed of 1's, with all off-diagonal elements being zero. Any population with such 1s along its principal diagonal is a statement of the identity tensor. It does what it is told by its template and flawlessly uses its sets of balancing biological sequences to preserve essential properties from generation to generation.

The 2 × 2 essential development and 3 × 3 Owen tensors are both symmetrical. They have equal numbers of rows and columns. The former therefore has four components, the latter nine

If a tensor's eigenvalues must always be positive; but if the eigenvectors must turn negative to complete the cycle; then the essential development matrix and the eigenvectors it proposes must have a set of positive and negative values that properly balance out all its elements, and all their rates of changes, all about the cycle.

Figure 20.99

What must a population do with each of its dimensions to maintain its equilibrium and the species? As on the bottom row of Figure 20.99, we can eliminate one row and column at a time and create a series of submatrices that establish exactly what those mutually balancing processes in a 3 × 3 matrix must be.

Each of those three submatrices is called a ‘minor’. And since each of those minors has a diagonal that lies upon the principal diagonal of its originating matrix, it is called a ‘principal minor’. Each principal minor systematically eliminates an entire dimension from consideration. In the standard three dimensions of space, our three principal minors tell us about the xy, xz, and yz interactions. Each thus exposes the sets of pairwise interactions that creationism and intelligent design demand for each of the two-dimensional, areal, surface stress contribution to the overall space.

All three of our principal minors have positive determinants of 1. Their joint interactions with the surroundings always cancel each other out to leave the generation mean unchanged. All their off-diagonal elements interact to keep them at those values all around the circulation. They always contribute exactly what the template asks them to, no matter what may be happening in the surroundings relative to each other.

We can add all the determinants of all three principal minors together to procure the ‘sum of the principal minors’. The sum of the principal minors tells us the contributions that the various dimensions are making, pairwise, to their joint interaction. It defines any curl, which is the permissible rates of change. It is the cross product contribution. The sum of the principal minors depends upon the values of all the off-diagonal elements.

The sum of the principal minors is another tensor and matrix invariant. It balances all elements and their rates of change. It replicates the Liouville theorem. Its interactions carry the system to stability and equilibrium. If the system is perturbed with an injection of energy, so that some element causes one value in the principal minor to increase, then some other principal minor, or else the two others in combination, will resist. There is a decrease at some other place to preserve their overall sum. This will institute a return to equilibrium. And if the system is perturbed so that energy leaves, then it will again resist by seeking to gain energy through the interactions overseen by the sum of the principal minors, and for the same reasons.

Now we have the sum of the principal minors, we can state the rules that an Aristotelian population must follow. If a given population matrix is symmetrical; and if all its eigenvalues are positive; then it must always be “positive definite”. Thus the identity matrix at top right in Figure 20.99, which has 1s all along its principal diagonal, is positive definite because all its eigenvalues turn out always to be positive. And if a matrix, or the tensor representing it, is positive definite, then its values, and its changes in values, must have certain well-defined properties. They can reduce, but must always remain positive. So also must the population that the positive definite tensor describes. So if something increases by x times in one place, it must decrease to 1⁄x in another; and if it adds y to itself in one place, then it must subtract that same y in another and so forth to maintain the same mean, but without ever becoming actually negative. The multiplicative factor is always adjusted in both magnitude and direction so any similar vector or property always remains in the same direction, no matter how much we might scale. The determinants for the three principal minors ensure that all elements, no matter what the scale of their increases or decreases, remain positive definite throughout.

The sum of the principal minors for our essential development matrix is 3. It is always positive, and always positive definite. All three dimensions therefore interact in specific and determinable values over predictable ranges and rates of change to produce the desired effects.

The sum of the principal minors has exactly the same effect as taking each distinct eigenvalue, multiplying each with all the others by turns, and then summing all those products as in I2 = λ1λ2 + λ1λ3 + λ2λ3. We get 3 for the essential development matrix either way. We have successfully used our matrices to get our second tensor invariant.

Figure 20.100

Figure 20.100 shows us the triple product that our three population force vectors each form relative to our population axes. It is a cuboid structure.

The determinant for a 3 × 3 matrix compares the volume we want to a unit cube. Determining its volume requires an addition and subtraction process very similar to procuring the determinants for a 2 × 2 matrix. Just as we had to add and subtract some side ones to find the true area, and the work done, for a parallelogram, so also do we have to add and remove some side volumes, sandwiched by our axes, before we can determine the true value for a population's energy.

We can express a population's forces and energies with suitable values in x, y, and z as ax + by + cz, dx + ey + fz, and gx + hy + iz. We can then place them in their own 3 × 3 matrix. We place a, b, and c in the top row; d, e, and f in the middle row; and g, h, and i in the bottom row.

We can find the energy volume using a series of minors. We first take any two dimensions, say y and z, and get their determinant, which states that area as yz. We then push that area out along the third dimension to get the volume, which is to multiply by x to give xyz. We now have the volume viewed from that x-orientation. We have got a complete box parallel to the x-axis. We can see from the way the box in Figure 20.100 overhangs some axes, we get some volumes we want, and some we do not want. We can do this for all three dimensions and orientations, adding and subtracting volumes and subvolumes, so we eliminate all those sandwiched outside, against the axes.

The coordinates for y and z form a minor whose determinant and area is given by e, f, h, and i. We therefore get that yz area from that principal minor's determinant, which is (ei - fh). We can therefore multiply that yz area by a, for the x-dimension, to get the contribution it all makes to the overall volume. As at bottom left of 20.100, we therefore multiply the yz determinant by a, to give a(ei - fh). We have xyz as our first stab at the volume.

We now unfortunately have a volume that is that full-sized box, which is rather larger than we really want. It is entirely based on a. As we can see in Figures 20.96.B and 20.100, the amount we have just calculated, using a, overstates the volume by the amount it has intercepted between our desired volume and the axes. We must use the next coordinate to remove that excess. So we turn by 90° to work with the next, y, dimension. It is based on b. We can take up the xz area we view in that orientation, and push it out for the length b to shave away the excess side volumes we do not want.

Our second axis creates another box we can use to whittle away the volume's overstatement. We find the over-volume to be shaved by first creating a minor for the xz area. We use the coordinates d, f, g and i, which gives us the determinant (di - fg). We then multiply that determinant by b to get the volume we must subtract, which is b(di - fg).

We may now have subtracted to remove an overstatement, but we have unfortunately subtracted just a little too much. We took away everything formed by y, but we did so a little indiscriminately. Some of what we took away, as an ostensibly unwanted intercept, was in fact contained in the volume we want. We took away not just the y volume, but the that little bit of it that x and z had contributed between them. We want that portion. We therefore have to restore it.

We then turn another 90° to work with the z dimension. We now use the coordinates d, e, g and h from our last minor and calculate the determinant, for the xy area, which is (dh - eg). We then multiply it by c to get its contribution to the net volume, which is c(dh - eg). We now add that back in. We are done. We have stepped all around the axes and subvolumes, and determined the full volume we seek.

No matter what coordinate system we use; and no matter what values we get; the relation a(ei - fh) - b(di - fg) + c(dh - eg) is the determinant for any 3 × 3 matrix. It always states the three-dimensional interaction, which in physical space would be a volume. But since these are forces and distances, then it is the triple product and the hamiltonian of biological internal energy for a generation.

This 3 × 3 matrix determinant is also the multiplicative eigenvalues identity. It stands in for the eigenvalues product to state the essential development, λ, for any species, again independently of whatever particular axes and coordinates we might have. The determinant is always the highest dimensional element in any particular space. In two-dimensional space it is an area, and in three-dimensional space it is a volume. It gives us I3 = λ1λ2λ3.

When a population is expressed in matrices in unit terms, then every population has a trace of 3; a principal sum of minors of 3; and a determinant or volume of 1. This is true for them all. We also now have the same three invariant values we got using tensors, their eigenvalues, and their eigenvectors.

If we try doubling all side lengths in our unit cube, as a way of doubling the population, we will get a trace of 6, a sum of the principal minors of 12, and a determinant of 8 for the volume or the triple product. However … we are asking our entities to (a) double their numbers; (b) double their masses; and (c) double their energy densities. And … a bacterium, a lily, or a rhinoceros with both twice the mass and twice the energy density has not remained the same in its essential properties.

Creationism and intelligent design insist that we can get the same octupling of a population's energy by octupling numbers and leaving the essential attributes the same. Unfortunately, however, energy is a two-dimensional interaction. It requires both force and distance. If we octuple by extending numbers from 1 to 8 while leaving everything else at the unit, we get a trace of 10 and a sum of the principal minors of 17 to accompany that 8-fold determinant. These tell us that the only way that three interacting dimensions can octuple numbers is to raise each entity's mass by 167%, and each one's configuration energy by 142%. Matrices are confirming what we first learned with eigenvalues: that it is not possible to only change a population's numbers and leave everything else unchanged.

Figure 20.101

Creationism and intelligent design insist that biological populations will always abide by their templates. But this means that, as in Figure 20.101, the two fields P-1L-1P1L1 and M0L0 will produce the sequences of biological events and materials, all about the circulation, that are parallel in the specific sense meant by Helmholtz and Kelvin. If they all follow abstract templates, they will all look just the same. Every population will have rates of change in both mass and energy that constantly exhibit 4-point contacts. No matter how much populations may scale or change in numbers, they must never move faster or slower than these templates dictate.

A constant circulation density means that the generation length must change as numbers change. A human baby born prematurely and introduced early into the circulation means either that that same child, or else some other individual, must live for correspondingly longer to maintain the population's overall rates and balances in mass and energy. And for every super-centenarian who passes his or her 110th birthday, creationism and intelligent design demand that a sufficient number of younger persons must die early to counterbalance them. Eigenvalues must map to exactly the same eigenvectors at all times or else some arbitrary force is changing that population in its most essential properties: i.e. those proposed as independent of any particular axes, coordinates, or spaces … and numbers.

The energetic field P-1L-1P1L1 is always linear, in the same sense that a beam of light is linear. A straight line in the physical world attempts to keep all magnitudes and rates of change constant in all possible dimensions except one. That one is responsible for dispensing the tangential movements that maintain a constant rate of change. It is force exerted in a right line to either maintain a velocity, or else to cause a specific acceleration. It is also all heat, light, electricity, chemical activity and other such radiations … along with gravitational attraction. It always rays in and out of specific points exactly tangential to the circulation and all its entities at all times.

Every point on the helicoid of internal energy is indiscernible. Every time point and location t0τ0 has infinitely many others like itself ranging before and after it, all the way from minus infinity to plus infinity. In the same way, gravitational attraction follows Newton's inverse square law. It diminishes steadily out towards infinity, away from any mass. But even though gravity's potential range is infinity, the Helmholtz decomposition theorem sets a boundary about every object and region that makes its totals of mass and energy finite. No matter how large or small any time interval or circulation might be, the field P-1L-1P1L1 always has very definite limits and a measurable quantity of mass and energy that can be assigned to a very definite number of entities. Thus we are always able to identify one or another biological entity as the direct source for any given ecological transactions.

The material field M0L0 is very different. It is curved. Biological matter cannot be identified independently of some specific allocation of P-1L-1P1L1. Biological entities can only be known through their specifically biological transactions, all of which involve energy. But even though we never find biological materials absent of biological energy, those materials can nevertheless be uniquely characterized by their divergences.

Divergences—i.e. flux densities—generally have sources. We will invariably notice that the materials entering the volume elements that transport any flux or physical property in space and time tend to be a little denser one side than the other. If we follow that varying flux density or divergence into itself, in the getting-denser direction, we will eventually find whatever is responsible, which is its source. We will find the tap or spout generating it. And if we follow it in the less dense direction, we will usually find the sinkhole responsible for dissipating it.

Fluxes and their divergences may follow a general pattern, but Maxwell pointed out that magnetism is very unusual. He described magnetic behaviour by saying “magnetic field cannot diverge from any source” (quoted in Leon N. Cooper, Physics: Structure and Meaning (New Edition), Brown University, 1992, University Press of New England, pp. 228–237). This was his famous discovery that magnetic field lines loop continuously. Magnetic fields do not move in straight lines. They go round in circles. They never look as if they have discrete origins. They do not ray in or out of discrete points. If we cut a magnet in half, we simply get two smaller magnets. We never get separate north and south poles.We will never find an isolated magnetic source-point.

If we examine both sides of any magnetic volume element, we will always find exactly the same amount of flux coming in on one side as we do going out on the other. And since the amount of flux entering one side is always the same as that departing on the other, then there is never a sign of any source. We will just go around in circles.

Our biological space has the three dimensions of number, mass, and energy. Number, n, is a distinct dimension. If numbers must stay constant, then its divergence must stay constant, and the amount of number in every direction must stay the same. Number must also always stay on the same flat plane. We must always see the same numbers of entities in a population, no matter where or when we look. And if the divergence in numbers must stay constant, then number must participate in the same kinds of repeating circles as magnetism.

If mass and energy can indeed undertake their required set walks while number remains constant, then they must each automatically create loops and circles with respect to number. Those two circles formed, respectively, by n and and n and (or V) are each one's allowed set. Numbers must therefore interact with both mendelity and biopressure to create the mass and energy fluxes, M and P, through n and n. But they will each also use the inverse of /n and /n to establish the rates at which the flux oscillates between its minimum and its maximum values.

Every biological volume element—i.e. cells and entities—has two sides. Those two sides are both (a) moments in time, as t-1 and t1; and (b) locations or positions in the circulation as τ-1 and τ1. But if the numbers remain constant, then the divergences in mass and energy must increase and decrease using that constant divergence in numbers as their radii of execution.

As infinitesimal volume elements, entities and populations are always changing their flux amounts or sizes. They first fill; and then empty. They have both divergences and convergences. So when a biological divergence is overall positive so that mendelity is increasing, then there is a net gain in entity size across that population. But that net gain must ultimately be followed by a reversal in which biological divergence is overall negative, so that mendelity is decreasing and net entity size decreases. Biological entities are always either growing; reproducing; or dissipating. Biological volume elements contain first progenitors and then progeny constantly looping around each other as their divergence in number remains constant.

Magnetic and biological volume elements are similar in that both loop continuously. But biolgical populations do so by oscillating between their positive and negative divergences. Biological materials are therefore characterized by an ever-imminent change in direction that is an oscillating rate of change. The net result is a generational mean that sits at the centre of a dynamic circulation.

The field M0L0 differs from P-1L-1P1L1 because instead of delivering and removing energy, it gathers and returns resources to and from the surroundings. It also differs because although the amount contained in any region is finite, it is itself without limits. It extends infinitely in its forwards and backwards directions, looping continuously. It goes all around the circulations of the generations increasing and decreasing all about its mean to return to its indiscernible point.

When there is a positive biological divergence, then entities act as if they are the sources from which all biological materials emerge. They tend steadily away from a minimum size and towards a maximum one. But they will eventually reach a maximum size. They will then reproduce. That progeny means the population will undertake a negative divergence or convergence so they can head to a minimum size. The emerging entities act as if they are the sinks or destinations for biological mass and energy. Old and tired components and entities are sloughed off. But those smaller entities that result will eventually repeat the process; reverse that divergence; and act as if they are the new sources.

Figure 20.102

If creationism and intelligent design are true, then as in Figure 20.102.A, both the mechanical and the nonmechanical chemical energies must have certain very specific relationships with respect to numbers and each other, as they oscillate between their distinct minima and maxima. Every population will have both a smallest and a largest viable size as mass and energy undertake their promenades or required set walks, either side of their means.

Both pairings of population mass flux, M, and mendelity, , and Wallace pressure, P, and biopressue, , must increase and decrease at the same rates. As mechanical chemical energy converges into sperm, seeds, and other such constructs, in acts of reproduction, the individual and the population fluxes will keep step with each other. The same holds for the energy. They must remain in the same plane that defines n as constant and may not move in that third dimension of number.

Both M and and P and must describe perfect circles with respect to numbers, which is the right helicoid. All populations and generations must therefore have the 4-point contacts shown in Figure 20.102.A. They may not undertake the uneven promenades or required set walks shown in 20.102.B or C, which are representations of what we measured for Brassica rapa. Those lines of action are not parallel. Their axes and distributions are uneven either side of the normal. They display aberrancies and jerks.

If creationism and intelligent design exist, then both mechanical and nonmechanical energy must abide by certain rules. They will each have positions of least, mean, and most for their quantities; and transformations of least rapid, mean, and most rapid. These must coordinate.

When the population is at Stage II in Figure 20.102.A, its displacement or position vector for mass, which is its quantity of mechanical chemical energy is at its mean. It must have finished moving towards that mean at II. Its rate or velocity must be carrying it to III. However, its displacement or position vector for its Wallace pressure, which is its quantity of nonmechnanical chemical energy, must be at its minimum at that time.

When the population was at Stage I, it was at its minimum in mass. Its transform or rate of activity between I and II must have been inclining it away from I, and towards II. Its velocity when it gets to II must therefore be at 90° to what it was when it was at I … which is in its turn at 90° to what it will be when it eventually arrives at III. Therefore, the velocities or rates of change at all four points—I, II, III and IV—must be mutually orthogonal. These relative positions of least, mean, most, and mean are mutually at 90°; and their velocities must also be exchanged in regular order, but between mean, most, mean and least. The population's velocity vector, or sense of direction in its movement, is always at 90° to its displacement.

There is also the acceleration. If the population's rate of transformation, or velocity, is to change in these coordinated ways, then its acceleration when it is at I is changing its velocity to what it will be when it is at II … meaning its acceleration vector is constantly at 90° to its velocity vector … and so is immediately at 180° to its displacement vector. In other words, when one is least, another is at the mean, and another is at its most. And since jerk changes acceleration, then the population's jerk vector is both (a) constant; and (b) constantly at 90° to the acceleration vector … meaning it is always at 180° to the velocity vector, and 270° to the displacement one. This relationship between means and maxima must hold all around the circulation for all possible populations.

Creationism and intelligent design are therefore demanding that we recreate Aristotle's ideas on circular planetary motion, but within biology. Every population must have a minimum for its mechanical chemical energy that occurs at I, which is the end of a promenade or required set walk and is the maximum depression of that see-saw. And if something has reached its minimum, then it has actualized its potential to tend towards that minimum. It must instead begin actualizing its potential to head to its maximum. So the population is moving in one direction at t-1; is temporarily stationary in its value at t0; and is turning so it is moving in the reverse direction by t1.

The minimum point in mass, at t0, has another important property. It is a point of maximum differential between all the various generations and populations. It is the point where the dot products cast by mass show their greatest potential differences in values. They have all reached their respective minima and are all about to head back in to the mean. The circulation location at I is therefore the maximum down swing of all see-saws, and so the maximum possible gradient, where gradient is given the symbol ‘∇’.

Since I is the minimum point for mass in the circulation, it is switching from becoming increasingly negative to becoming increasingly positive. We have t0 as its furthest point away from the mean. Since it is changing direction, then mass flux is travelling in one direction between t-1 and t0; and in the opposite one between t0 and t1. The sum of those changes in direction must be zero, so that +∇MM - ∇MM = 0. If we do not have a net change of zero then the population has not yet reached the point of greatest gradient, and t0 is not the furthest away or maximum differential point.

The gradient tells us how steeply we have to climb, so we can get from one point within that space to another. All populations when at their minimums in Helmholtz energy align themselves so their gradients have their maximum possible slopes back to the generation mean. They look to increase masses to the maximum possible degree, and therefore experience the greatest force to do so. We can express this as ∇minM.

If mechanical chemical energy is to change from being infinitesimally negative to being infinitesimally positive between t-1 and t1, and so it sums to zero, then the sum of the nonmechanical energies must also be zero. We must therefore have ∇PP = 0, which means ∇P = 0. But there is only one way in which that gradient can be zero. Nonmechanical energy experiences no force to change, which means it must be at its mean value. We must therefore have +∇MM - ∇MM = ∇P = 0.

The vectors for mechanical and nonmechanical energy must be mutually orthogonal. When one is at its minimum or its maximum the other is at its mean and so on and so forth, and so that when one is at its maximum gradient the other is flat and level.

The mechanical and nonmechanical aspects of biological internal energy must coordinate not just here, but all around the cycle. Once the entities pass I, they must absorb mass and energy at rates that allow them to arrive at II with the appropriate properties. Energy density continues to decline to its minimum, while mass increases to its mean. Energy density at II is at the end of its required set walk, and therefore emulates mass at I. It reverses course across its own t0, and will begin heading back to its own mean, which it will arrive at at III. Mass and mechanical energy will support it by being at their own mean at II, which is their zero point. We have ∇M = 0, which is a flat twirling baton. We must have +∇PP - ∇PP = ∇M = 0.

These patterns will repeat as mass heads to its maximum at III. But since energy is at its minimum, we have ∇minP. The two continue to be 90° apart. to compete the cycle with promenades of oscillating gradients relative to the mean.

If creationism and intelligent design are to be true, then we must move regularly amongst the gradient values in 4-point contacts so that ∇minM + ∇meanP → ∇meanM + ∇minP → ∇maxM + ∇meanP → ∇meanM + ∇maxP → ∇minM + ∇meanP. The jerk, acceleration, velocity, and displacement vectors are all then displaced by 90°. Those 4-point contacts are then the template that drives every population from I to IV.

Our biological forces are biological sequences … and our biological sequences are biological forces. Every force can be measured and has an impulse. Every impulse has a definite beginning and end. No forces are instantaneous. They all take time—i.e. portions of the circulation—to either institute or remove their effects.

Creationism and intelligent design demand that all increases and decreases in forces and impulses, either side of any and all t0s have matching rates both before and after. All intervals t-1t0t1 are the same for all generations. They must all measure the same circulation spans τ-1–τ0–τ1.

If creationism and intelligent design are true, then all these forces and events must have 4-point contacts. They must have the ranges, means, and distributions to match. Those distributions are the increases and decreases that create a circulation. But this means that, for example, the man's pushing on the shuttlecock in Figure 20.12 must be a precisely slowed down version of the woman's racquet hit. They must both apply and remove their forces at exactly the same relative moments either side of each of their points of maximum contact, which are point-for-point and frame-by-frame equivalent over those different lengths. They must impose yanks, tugs, and snatches that meet all defintions of Helmholtz-Kelvin parallel. All times, distances, masses, and activities must match so that all population configurations—c-1, c0 and c1—and their Weyl and Ricci tensors are smooth and even. If this is not so, then they cannot maintain 4-point contacts over either distance or time.

Figure 20.103

As in Figure 20.103.A, the template's interactions must always ray in instantaneously. The field P-1L-1P1L1 must push M0L0 directly and orthogonally about the circle. Populations may not follow Figure 20.103.B, which has changed orientations, and is a variation. The sums of the derivatives either side of t0 are not equal. That is immediately a jerk. The 4-point contacts are being destroyed, and the circulation length and density are changing.

Figure 20.104

We must make it as likely as possible for creationism and intelligent design to be true. We must create a flat biological universe that is as close to Figure 20.104 as possible. It exhibits 4-point contacts everywhere. The divergences it allows run along the lines of force radiating out from its centre. They are all orthogonal relative to that mean. All curls and circulations go about its surface and are parallel while being orthogonal to all those divergences at all points. This holds across all three dimensions. The former provide the nonsolenoidal and rotational mechanical energy, while the latter provide the rotational and solenoidal nonmechanical variety.

Each circulation in Figure 20.104 has the length T. Each is a journey about a complete surface. Each involves rates of change over all dimensions. All populations and all species will maintain an even and constant circulation density relative to all others.

Each combination of divergence and circulation in Figure 20.104 has the same sequences of biological events in any given interval relative to both its resident entities and others. Every population measures a constant increment not just in its own circulation, but in those of all others. Thus predators interact with prey at the appropriate locations in each of their circulations, and always with appropriate quantities of mass and energy. Every population both experiences its own invariant template, and observes all others following each of theirs. Eigenvalues and eigenvectors match because any time interval Δt in any population always maps to some specific generation length, Δτ, in all others. We have TΔτ = Δt everywhere, and the biological events we observe at any point will allow us to deduce not just that circulation, but those of all others.

Figure 20.105

Creationism and intelligent design are two-dimensional theories. They claim that we will always measure a flat circle. The third dimension of number is irrelevant. Darwin, however, insists that we will always move in that third dimension. We will always be pulled around and hug the outside of a sphere.

Gauss proved, as on the left of Figure 20.105, that it is impossible to flatten a sphere, or sector of sphere, into a circle without tearing or breaking it. He showed that if we measure any circle's radius, and then walk all around its perimeter, we can both (a) carefully measure its circumference and its area, and (b) calculate what they should be. Spheres and circles have different rates of change for their surfaces and areas. They both have the minimum possible radiuses for their respective surfaces and the maximum areas for their respective dimensions: one in two, and the other in three. If the circle's radius is r, then its area is πr2, while a sphere of the same radius has a surface area of 4πr2. So for every infinitesimal increment in radius, dr, the sphere's surface area increases by 4r, which is four times as quickly as the circle of the same radius. Any sphere increments its area more quickly than any circle. We can even calcuate the underlying sphere's curvature from the result. If a circle wishes to become a sphere then it must diverge outwards. It must grow in area. And similarly, any sphere that wants to become a circle must converge so it loses in area. If the area we measure for what we think is a flat circle ends up larger than the one we calculate, then we know that the surface we have walked on is curved. We are in fact on a sphere. We have effectively tried to fit a sphere into a circle. It is impossible to go simultaneously about both a circle and a sphere and have zero divergences from the one with respect to the other. If the divergence is zero, there is no tendency to flow either towards or away from that point. Net equilibrium forces are at work. Only a sphere of infinite radius can increment its surface as slowly as any circle. We will know from our measurements whether or not our surface is curved—i.e. is extending into a third dimension. Only a sphere of infinite radius with a completely flat surface can possibly fit into any finite circle without tearing or distorting.

Creationism and intelligent design are two-dimensional theories. They claim that the third dimension of number is irrelevant. They claim that we will always measure a flat circle. Darwin, by contrast, insists that numbers are relevant, and that we will always measure a sphere.

If number is truly irrelevant to biological populations then we must always measure the curls and the divergences associated with a flat and two-dimenional circle. However, a flat and two-dimensional circle will also be a region upon some sphere of infinite radius. If the field vectors tend to point towards or away from a point, then the field has a divergence at that point, which is its rate of change of area with respect to distance. If the divergence is changing, forces and vectors are at work, pointing directly towards some source or sink. We can measure the divergence that depends on numbers.That is the basis of our Brassica rapa experiment.

Figure 20.106

The gradient tells us how quickly something is changing with respect to something else. It is a vector, because it also tells us the direction in which that change is occurring. It therefore tells us about the object's structure. Figure 20.106 shows what the gradient can achieve in terms of stating a population or a landscape's deviations about its surface. We are currently assessing the gradient at P where the three screens meet. We are assessing the surface, S, which has a gradient at every point, even if it is flat in some direction.

We have one screen for each dimension. They help us cast the shadows which tell us the gradient's three components and dot products. We can therefore declare any gradient in terms of our three dimensions in space.

If we were standing at P, our head would be pointing into space along the arrow marked ∇S which is the landscape's gradient in S at that point. The direction our head would be pointing in would be telling us both the direction and the magnitude of the steepness underneath our feet, which is the forces and the potential energies working at that point to pull us directly inwards to the centre. Everything that happens on the surface is because of those forces.

But the three screens tells us that finding the gradient is not all we can do. We can bring our three axes plus their unit vectors to bear. We can express that gradient relative to them: i.e. in terms of the units in each direction. As again in Figure 20.106, we can determine the gradient's magnitude and direction relative to our three separate directions. We can accurately describe this space and its forces using any basis and coordinates we wish. We straight away get the three gradient dot products of ∇Sx, ∇Sy, and ∇Sz. They are technically called the terrain's “directional derivatives”, and can also be symbolized by the function ƒ(∂S⁄∂x, ∂S⁄∂y, ∂S⁄∂z). They tell us how rapidly the terrain is sloping relative to each direction, and again relative to the planetary centre. They tell us how any ball is going to roll, which is a response to the forces flowing through that space, once again relative to that centre of mass.

We can now express these gradient magnitudes, in each separate direction, in terms of our chosen unit vectors in our usual three ways:

1. Our first option is to regard the terrain as fixed in space. We then have the three population axes i, j, and k. We can walk physically all around the landscape measuring its three directional derivatives at every point. This would be like using a compass, the magnetic north pole, and the gravitational-magnetic attractions embedded in the terrain. East-west, north-south, and up-down would then be our three i, j, and k axes, along with their units of measure e1, e2, and e3. Our directional derivatives per each direction will be e1 • ∇Se1, e2 • ∇Se2, and e3 • ∇Se3, giving us our principal diagonal of S11, S22, and S33. This set will be our A.
2. We have the three firmament and more general species axes I, J, and K from the Liouville ensemble. So our second option is to turn to the firmament. We can use the fixed stars to guide us. Every time our head shifts relative to those fixed stars, we know the terrain has changed underneath our feet. Those will be the measurements relative to I, J, and K, which have the units of measure E1, E2, and E3. Our three directional derivatives will be E1 • ∇SE1, E2 • ∇SE2, and E3 • ∇SE3. This new S11, S22, and S33 will be our B.
3. Or, finally, we have the surface axes T, N, and B. We can use Gauss and Riemann's methods. We can carefully measure the terrain itself, drawing parallelograms upon it and measuring those in the way they described. We then get the intrinsic surface axes T, N, and B, all about the circulation. We will let its three units of measure be m1, m2, and m3. We will now have m1 • ∇Sm1, m2 • ∇Sm2, and m3 • ∇Sm3. This latest S11, S22, and S33 set will be our C.

Creationism and intelligent design mean that both the gradient and the rate of change in gradient for number must always be zero. They insist that nothing to do with gradients or divergences in numbers relative to either of the mass or energy fluxes have any discernible effect on any population. But these are specific sequences of acrtivities. It is the insistence that n • ∇M, n • ∇P, • ∇n and • ∇n all be zero. They are the four shear components τ12, τ13, τ21, and τ31 that are the outside column and row in the 3 × 3 Haeckel tensor. They are the no-entry signs.

If we express the average number over the generation as n’, then the claim is, more specifically, that the two gradients in numbers n’ • ∇n and n • ∇n’—i.e. the gradients in number towards the generation mean, and the gradient from the generation mean towards the current numbers—are also always flat and zero. That is again something we can easily measure for that gradient, ∇, is also a partial derivative, ∂. We have our equations from the refutation to test for all this.

Whichever of the three systems we use, we measure the same gradient every time. We can then use both our tensors and our matrices to convert freely from one perspective and/or coordinate system to another so we can find out what is going on.

The gradients and changes in gradients are surface—and therefore also tensor—invariants. If creationism and intelligent design are true, then those three sets of dot products are the A, B, and C that—barring only differences in bases and values—will be equal.

There are innumerable difficulties with the creationist and intelligent design firmament proposal. Force, displacement, and velocity are directionally sensitive. Work—given by W = Fd—is not. Since work is independent of direction, it can easily be conserved even as energy landscapes shift and alter gradients so that forces change directions and distances. A waterfall can maintain the same potential energy through retaining the same overall drop even as a mountain changes its profile so that gradients point in different directions. These quantities and their rates of change are very simple to measure and to compute and are the traces, sums of principal minors, and determinants. This was the basis of our Brassica rapa experiment.

Our gradient structure has the four dimensions of n, , V … and t. That t is responsible for the changes in gradients, portages, and forces all about the circulation. It goes around, as τ, to create the indiscernible point. It incorporates distance and time; the biological and the nonbiological; and the transforms or velocities from one to the other. The dot and cross products will tell us how these various dimensions and properties interact to create biological entities and their circulations.

Figure 20.107

We can see the four portages we need to create our biological space in Row 3 in Figure 20.107. They form the gradient structure that are the forces that unlock evolution and show to us whether the equality that creationism and intelligent design demand for their 4-point contacts and isotropic and homogeneous spaces is or is not possible.

We can walk along the required sets or promenades that are our gradients, and find out if they form two-dimensional circles, as creationism and intelligent design claim; or if they form three-dimensional spheres as Darwin insists. Since we are using the fourth dimension of time, we can assess any ongoing changes in all of them over both distances and times.

Row 1 in Figure 20.107 has the complete set of eight points that establish our biological space of internal energy's size and dimensionality. Since ordinary physical space is much easier to think with, we have labelled our eight points Left, Right, Back, Front, Up and Down … and Start and End. Since they are all points, they are 0-spheres and indiscernible. They are called 0-spheres because they are points, and have zero dimensions.

We have labelled our points Left-Right, Back-Front, Up-Down, and Start-End because they are easier to visualize. But we could just as easily have labelled them as Few-Many for the n dimension; Large-Small for the dimension; Poor-Rich for the or V dimension; again keeping Start-End for the fourth for the generation.

Three of the sets of points have the potential to create the standard three dimensions of physical space. The Start/End pair can create a fourth. We can think of it as similar to what Hot/Cold would be. It is equally as capable of being distributed throughout our space.

Our eight points help create the 4 × 4 trace of S00, S11, S22 and S33, which is the principal diagonal for the Haeckel tensor. They are the four 1-balls, gradients, twirling batons, see-saws, and promenades.

As in Row 2, we can lay our 0-spheres and points down upon each other in pairs. They may now be in the same location and indistinguishable, but the identity of indiscernibles is still not the same as the indiscernibility of identicals. Our points are still separable. We can push them apart so we can begin creating our biological space and its gradients and forces.

As in Row 3, we now stand ourselves in between these pair-points that we have laid upon each other. We create the space we need by gently pushing them apart to separate them. We link them with a line we create. These are the 1-balls and batons for our promenades or required set walks. They are called 1-balls because they are lines with points of 0-spheres at their ends. Each end-point describes a point in our biological space, while the 1-ball describes the distance, and eventually gradient, between them. It can twirl as a baton to help create our internal energy space.

We can also think of this pushing apart to create a separation or dimension between points as integrating them … with differentiation then being, for our limited purposes, the attempts to discover the ingredients that compose things. Our promenades are always the 1-balls or batons twirling within the 0-spheres.

As in Row 4, we can now take any two of our 1-balls or promenades and associate them with each other, which is to integrate them with respect to each other. We select two each of the Left-Right and Back-Front pairings. In order that we can distinguish them, we label the centres of the Left-Right ones. We have given one the label Up, the other Down.

When we associate a Left-Right with a Back-Front, as we do on the fourth row, we move to two dimensions. This is again their first integral with respect to each other. It is an area and a divergence. We create our allowed set, which is a 2-ball and a disc. They are called 2-balls because they are two-dimensional, and so are areas.

Our two 2-balls are currently both without a boundary. Apart from their distinguishing labels, they are identical. We lay them on top of each other. The middle of the first disc still has the label Up, the middle of the second one, which is now underneath it, is Down.

Since our two 2-balls have no boundary, we weld or stitch their edges together. They are now both a 1-sphere and a 2-ball. They are called 1-spheres because their bounding objects are lines, which are one-dimensional.

Although they are otherwise identical, the two 2-balls are separable inside their common 1-sphere boundary. All promenades across this 2-ball and area have a gradient with respect to each of those centres. We establish the convention we see in the diagram: a journey from Front to Back and Right to Left is Up relative to the centre, while a journey from Back to Front and Left to Right is Down.

We then walk to the middle of the two 2-balls or discs. We stand there between them and push them apart. This is to integrate for a second time, which is to conjoin. They become the 3-ball we see far upon the right of Row 4. They are called 3-balls because it takes three coordinates or numbers to specify them, which makes them a volume object.

The part that was labelled Up in our 1-sphere and 2-ball moves from the centre. It becomes the 3-ball's top; while the part that was labelled Down becomes its bottom. From this perspective, Back becomes invisible, although we know that we will see it if we turn the 3-ball around. All required set walks are now possible, and we have successfully moved to three dimensions. This is how we created the globe we met in Figure 20.39, which is our biological internal energy space.

We have a 4 × 4 Haeckel tensor to contend with. We need to master the fourth dimension. We therefore repeat the process, as in Row 5. We go to the centre of a first 3-ball. We label it Start. We go to the centre of a second. We label that End.

We then place those two labelled 3-balls directly upon one another so their interiors interpenetrate. We weld a common boundary. We now have a 2-sphere surrounding our 3-ball. Its centre has the joint label Start/End. It is called a 2-sphere because it has a two-dimensional and areal expanse acting as its surface.

We now walk into the middle of our 2-sphere and position ourselves between its two 3-balls. We gently push them apart. This is to integrate yet again, which we have called a distribution. The process is exactly the same, although we have an extra dimension. Since we labelled one Start, and the other End, we are creating a generational gradient across our sphere as our fourth dimension. This is no more mysterious than a temperature distribution would be, if we had labelled them Hot/Cold.

Whatever our fourth dimension might be, we create the crude diagrammatic representation of the hypersphere or 4-ball we see at bottom right in Row 5 of Figure 20.107. It is called a 4-ball because it now takes four numbers or coordinates to specify its interior. We are making one of those measure time across a generation from Start to End, so we can distribute that time and those sequences evenly throughout our space.

The entire 3-ball that was previously at the centre, and that bore the label End has gone to the back. It would be invisible, from this perspective, in 4-space. The 3-ball that carried the label Start has come to the front. The Start sphere in the diagram is somewhat larger to suggest this. Since we will not be going to any higher dimensions, we give our 4-ball its 3-sphere boundary. It is called a 3-sphere because (a) it acts as surface, and (b) it takes three coordinates to specify any location upon it as a surface. It is a 3-sphere covering to the 4-ball interior, which instead takes four coordinates to specify it.

The diagram we provide makes its best attempt to show the general bulbousness and convexity of a 3-sphere and its contained 4-ball. It shows only a few of the more surface 3-spheres. Just as we can place two-dimensional 1-sphere elliptical shapes inside equally two-dimensional circles to suggest a 2-sphere or globe, so also do the few spheres we have drawn try to suggest a limited form of four-dimensional hypersphere surface perspectivity. There are also lines within the frontmost sphere to suggest the convergence to the centre typical of all spheres, no matter what the dimension. And just as the elliptical lines upon a circle can suggest the steady increase in a sphere's diameter as we approach its centre, which eventually embrace the great circle that encompasses its diameter; the similar spheres in the diagram suggests the steady increase in the diameters of the spheres as we approach the hypersphere's centre, eventually including the great sphere centred upon the similar four-dimensional diameter. It is, however, impossible to indicate both the surface and the body spheres on a two-dimensional device monitor or piece of paper. Four dimensions cannot be properly displayed in three.

We need to create a biological space of internal energy that has all the properties we need, including behaviours in time. The gradients we produce across each surface—which is to differentiate them—are all invariant properties for each surface. And since the gradients for a two-dimensional circle are different, in each direction, from those for a three-dimensional sphere; which are different yet again from those for a four-dimensional hypersphere; then we can easily take measurements, in all directions, to see what the world we live in is really like. If our various gradients, including those in time, match those for a flat, two-dimensional, isotropic and homogeneous space, then we live in that kind of a world and creationism and intelligent design are correct. But if the measurements for gradients do not match those for a flat circle, then we live in a three-dimensional one. That is the simplicity of an experiment.

When we pushed our two 3-balls apart to create our hypersphere, we freed up the hypersphere's centre. It is also the centre of the generation or circulation. We can label it t0’–τ0’. All negatives relative to it tend towards the start of a generation on one side, while all positives relative to that same centre tend towards the end of a generation on the other.

We can now stand at our midpoint of t0’–τ0’ with our arms outstretched; touch each far edge; turn ourselves about; and go through a circulation. We will deposit all properties and features needed at each point as we do so. We create the walk that lasts a generation.

If we now switch our axes and coordinates from Left-Right, Back-Front, Up-Down, and Start-End or x, y, z and to Few-Many, Large-Small, Poor-Rich, and Start-End for n, , or V, and t, we will produce all the portages, promenades, transforms, directives, aberrancies, associations, conjoinings and distributions that are the complete panoply of biological events. We create the gradients, and rates of change of gradients, by stating everything relative to the centre of t0’–τ0’.

The process of pushing objects apart from a centre is to integrate them. It changes points to distances, distances to areas, areas to volumes, and volumes to distributions. It establishes gradients, and rates of change of gradients, every time. To take a gradient is then to step down a dimension and to differentiate. It is to find a derivative: the rate at which the something that makes up the higher dimensional objects is changing relative to some other property.

• A line, which is one dimensional, can act as the 1-sphere whose rate of change tells us how quickly an area, which is the 2-ball inside it, is growing or changing.
• An area, which is two-dimensional, can act as the 2-sphere that tells us how quickly a volume, which is 3-ball inside it, is growing or changing.
• And a volume, which is three-dimensional, can act as the 3-sphere that tells us how quickly a hypersphere and 4-ball inside it is growing or changing.

The hyperdistances we move through in four dimensions follow the same general rules and patterns as distances in three. We will label our hypersphere radius ‘’ (the Sanskrit letter ‘ra’ (pronounced more as if it rhymes with the ‘u’ in ‘cup’)) to show that it has elements of all four dimensions, x, y, z, and t; or else n, , V, and t. As we push it around, we can measure its dot and cross products against each of the four dimensions. We then know how it is changing with respect to each of them.

Figure 20.108

We conventionally label a point in 3-dimensional space with the x, y, and z axes and dimensions that we see on the left of Figure 20.108. We get a whole succession of duples—which are complete xy planes—for each infinitesimal push outwards into z.

By convention, a point in 4-dimensional space gets the quartet of coordinates x, y, z, and w we see upon the right. We therefore get a whole succession of triples—which are complete xyz volumes—for each infinitesimal push along w.

We are now free to move infinitesimally about. We can report all its dot products and distances against all four coordinates, which is how far away from the axis or generation midpoint they each are. These values will be our displacements, placements, presements, and the like at every point in the circulation. We have then located every point in our 4-dimensional hypersphere—and so in the generation—relative to that generation centre. We can then state the sequence of three-dimensional objects—which are values for n, , V—within that generation.

Since there are equal amounts of generation time-distance before and after that midpoint of t’0–τ’0, then our four-dimensional hypersphere, centred upon that hyperorigin, circles through the beginning and endpoints of the generation, which are t-1τ-1 and t1τ1. All points are now either before or after the midpoint. At the generation midpoint itself, the time-distance gradient is flat. Every property increases and decreases relative to that centre.

Time-distance now moves from a minimum to a maximum. Its steepness depends upon both the time and the distance to t’0–τ’0 which is the hypersphere's midpoint. Creationism and intelligent design insist that the gradients in number have no relevant dot products in any direction. But this is merely a whole set of three-dimensional volumes we can slough off our four-dimensional space. We simply define a set of values for Large-Small, Poor-Rich, and Start-End, leaving Few-Many unchanged at the origin to give σ(n, , V) = c. Creationism and intelligent design insist that that is where all populations exist all about all possible generations. We can now investigate that claim in any real case.

Figure 20.109

Two dimensions are much easier to think with. If, as in Figure 20.109, we hold a bucket and whirl it around, or simply walk about in a circle as the woman is doing, then we go about the resulting circle's perimeter or circumference. We create an allowed set or 2-ball from a 1-sphere surround as the 1-ball moves with a given velocity.

As we walk forwards in Figure 20.109, we are pushing the 1-ball radius, r, ahead of us by an infinitesimal value, dr. The area will increase, equally infinitesimally, by πdr2. The 1-sphere surround, which is the derivative, is therefore measuring the area's rate of increase.

But since we are pushing the area, that circumference or 1-sphere surround is the displacement. We are also doing work. We have had to push out into the surroundings. That distance we walk about the 1-sphere boundary also tells us the work. It tells us what is happening to the area or 2-ball we are surrounding with that walk. We have found a rate of change for our area by stepping down a dimension from the area to the line that is creating it.

The area we cover will increase steadily as we push the radius. The area will grow to 90°, and then to 180°, where it is at its maximum. It will then diminish to 270°, and gradually come back to zero from behind. That is a generation. We have now used a 1-sphere to circumscribe a 2-ball.

Our moving 1-sphere tells us how quickly the area—which is the 2-ball and the divergence and the Helmholtz pushing-out energy—is changing from zero, where we begin; out to its maximum, when it is diametrically opposite; and then back again to zero. The circumference is therefore the transform or first derivative of the area.

The populations on the right of Figure 20.109 are also moving about their circulations. That 1-ball for the radius moves by the similar amount dt, which is a movement in linear and absolute time as measured by the linear planimeter … but it also moves by dτ, which is an areal movement about the circulation as measured by the polar one. The area is both the work done and the amount of Helmholtz thrusting out energy. It increases. But since that movement is also a gradient, ∇, in time, it is a force doing work. That gradient changes with respect to both t and τ. And whereas it is the hand holding the bucket that does the work and thrusting out upon the left, it is the population's metabolism and physiology that does the work that carries the population about the circulation. The distance from the beginning of the generation tells us the work done and the energy expended.

Figure 20.110

When we similarly push our hyperradius, , forwards by the infinitesimal amount d, we push through from the beginning of a generation towards the end. Exactly as in Figure 20.109, we trace out an infinitesimal “hyperarea” which grows steadily; reaches its maximum; and returns to zero. But since we are working in four dimensions we produce the hyperarea we see in Figure 20.110 … for a hyperarea in four-space is represented by three numbers, and is therefore a volume in ordinary three-space. We can see the effects of that volume. It is the Helmholtz energy thrusting out in relative generational time from a prior moment and location in the circulation which is the beginning of the generation at t-1τ-1, through the present moment and location of t0τ0, to the end of the generation location at t1τ1. It is the entire triple product of energy the population uses over a generation. It is our internal energy that configures its DNA.

We can express the appearance of a 3-ball in this reality as the hyperarea of a 3-sphere more technically as “every non-empty intersection of a 3-sphere with a 3-space is a globe (unless it merely touches, in which case it is a point)”. That 2-sphere surface and its 3-ball interior that we detect in this reality, as we measure a population, is our four-dimensional hypersphere's derivative. The hypersphere therefore holds the population's entire energy on its surface in a timeless four-dimensional equilibrium. Entities appear, grow, transform, reproduce, die, and are replaced by others as our hyperradius turns courtesy of our polar planimeter.

Creationism and intelligent design demand a perfect exchange of biological forces and energies, all across the circulation. There must be zero aberrancy. But since we have given ourselves a fourth dimension to work with, this is nothing else but the request for a completely even distribution of all properties across that fourth dimension, so they can manifest that way in three.

The moon's mass is smaller than the earth's. It therefore has much less gravitational intensity. Its gravitational field is flatter. It is more isotropic and homogeneous. We can create a similarly flatter and more homogeneous biological space, for our biological internal energy, by taking thinner and thinner slices of our four-dimensional hypersphere. The thinner the slice, the more even the accompanying distribution hyperarea, and the closer it gets to being a sphere of infinite radius. In the same way, the smaller a mass, the more even the gravitational field.

We can now produce a three-dimensional space with a completely flat and even gradient, along with all necessary tensor values. The thinner the slices we take from our four-dimensional hypersphere, the more convincing the impression that it is centred upon infinity. The earth's gravitational field is so weak it is easy to believe it is flat, even, and parallel everywhere. So also, the weaker the biological field, the more exactly does dt = dτ.

The force a population uses to go about its circulation now has a very simple source. It is the energy the population holds in its chemical bonds at each point on its hypersphere. Its derivative is the necessary force and torque. Creationism and intelligent design then propose that all those lines of force and energy are perfectly parallel for the field P-1L-1P1L1, with all material responses then being perfectly circular in and for M0L0. The hypersphere's slices are also then perfect through being infinitesimally thin.

We now have a Riemannian manifold complete with metric that we can use to determine whether or not neighbourhoods are identical in different directions; and whether or not they have aberrancies. If creationism and intelligent design hold, then we will measure the necessary circles and parallelograms. They will all be perfect. But if those proposals fail, we will measure the extent of that failure. It will be the curvature. It will also be a surface and tensor invariant that is as characteristic or essential as any similar two-dimensional property.

Figure 20.111

It might be true that the long succession of even and infinitesimally thin slices creationism and intelligent design demand is impossible, we cannot prove it in an experiment without taking the accurate measurements that shall prove this. We have our set of i, j, and k axes that we see in Figure 20.111.A. These are always “live”. They are what we actually measure in real time in any population we select, as it goes about its circulation and interacts with its environment.

The second set of axes, which we also see in Figure 20.111.A, is I, J, and K. These are the multi-generational comparative Liouville ensemble values. They stand at the helicoid's centre and form its axis. We get those proposed template values from the population's Liouville values of n’, m̅’, and p̅’.

We can compare each distinct population and generation we measure as an i, j, and k to the Liouville I, J, and K ones we can also measure and calculate. If the population we are measuring belongs to the species or ensemble indicated by the Liouville ensemble, then these two sets of values should coincide. This is our test for A = B.

The Liouville ensemble and the population we measure will both undertake sequences of biological events. Those sequences are strictly biological, and are therefore independent of both. Any population will undertake them at some rate, simply by being biological. They are the sequences we create by pushing the hyperradius around.

We now need a direct method of reckoning those sequences that are embedded in this space, and that are intrinsic to biological internal energy. We see a set in Figure 20.111.B. It is being pulled by the point t0τ0. We want to describe them independently of any biological inertial frame of reference, or any specific biological vantage point. We see that final set of space-based axes as T, N, and B in Figure 20.111.B. They are our C.

Those T, N, and B axes were discovered, independently, by the French mathematicians Jean Frédéric Frenet and Joseph Alfred Serret. Frenet published first. His important idea was that we can think of every point in space as the nexus of a moving frame based on its tangent, which we can think of as the 1-ball. The 1-sphere is the distance bounding any area, and that is often the circuitous route a curve describes. There is frequently a 1-ball that would be a shorter path on that surface. There is always a pair of 0-spheres bounding any 1-ball, while the 1-sphere describes any area they might contain.

As the moving frame moves along the curve of interest, we can measure both the net distance it has moved, along the curve, in the style that Gauss and Riemann taught; as well as the rate at which it might have leaned, twisted, and turned about over that distance, and as the 1-sphere has differed from the 1-ball. This compares the curve's changes to its own net displacement over the distance. It is therefore intrinsic to the surface.

As in Figure 20.111.B, we can easily calculate a tangent, T, to our line, and so to our surface, at any point. As is always the case, the tangent measures the curve or surface's tendency to depart from a straight line. That tangent is the derivative. It points in the direction of motion.

We can get a value for our tangent by measuring the distance between whatever two points we have chosen. It is simply the integral between those two points. It is always exactly one unit of length.

If an insect takes a circuitous route between A and B, we can always determine both (a) the 1-ball of its straightest short line path, which is the direct line to its final displacement, and (b) the 1-sphere of the curved distance it has actually travelled. We can then take that straight-line displacement as its actual traversal, and so compare the tangent to the distance to determine the tangent's magnitudes. The tangent is now that path's derivative or rate of change.

Now we have a tangent, we can repeat the process and take the tangent's derivative. This further derivative measures an acceleration. It measures the rate at which the tangent itself is changing over the distance we have measured it. The tangent's rate of change of course depends upon the nature of the underlying surface.

This derivative of the tangent has another property. As a tangent to a tangent, it is an acceleration to a velocity, and so is guaranteed to be orthogonal or at 90° to the tangent itself. This is therefore the normal, N.

The tangent and the normal, T and N, lie within the same plane. They form an entire plane that is tangential to the original curve. Since it just touches the original surface or curve at every point, it is called the (kissing or) “osculating plane”.

We now have an osculating plane that contains both the tangent and the normal. We need a value for how rapidly the tangent is actually changing within that plane. That curvature is the acceleration or rate at which the tangent is curving or changing towrds the normal at that point. This is the surface's curvature, κ.

We now want the tangent's dot product relative to the normal, which is dTdsN. That curvature and dot product is always some proportion of the distance over which we have measured. That distance is an absolute value given by ‖dTds‖. If we set the one over the other, we will have the surface curvature, towards the normal, we are looking for. It is κ = (dTds) ⁄ ‖dTds‖.

This curvature we have measured within the osculating plane states the rate at which the tangent is turning towards the normal in that plane. The tangent's complete change, over the entire distance, is therefore the curvature, which is a rate, multiplied by the normal: κN.

That curvature is always very straightforward to measure. It is an intrinsic property of any surface. We now have the purely meridional component of any movement about the circulation.

The tangent and the normal also of course have a cross product, T × N. This states how orthogonal the two are. We now have the movement into the third dimension, which is the binormal, B. Since it is a cross product, it is immediately orthogonal to both the tangent and the normal.

Since the normal and the binormal, N and B, are orthogonal both to each other and to the tangent, they form a plane: the normal plane. We then go through a similar process to find the rate at which they curve with respect to each other on their mutual normal plane.

The rate at which the normal and the binormal curve towards each other tells us how much the curve is lifting up, into the third dimension, out of the osculating plane. We in other words measure the rate at which the binormal is changing relative to the normal, or dBdsN. This is its torsion, τ.

The torsion measures the degree to which the plane fails to be a true plane by flexing up and down, creating hills and valleys. It is the rate at which the binormal is turning. And since the derivative of this binormal is perpendicular to both the binormal and the tangent, it is proportional to the normal. We are getting both the divergence and the curl.

We now have three mutually orthogonal axes, all defined by the surface itself. The tangent, normal, and binormal appear by dint of measuring the surface and its properties. They tell us everything happening to it.

We also now have have the toroidal, poloidal, and meridional aspects of our biological sequences. The torsion tells us how rapidly a population is moving in time, as it also moves meridionally in space. It gives us the poloidal value, dt.

By convention, if an observer sees the osculating plane turning counterclockwise as the point moves along the curve, i.e. in the direction of the increase from binormal to normal, then the torsion is considered greater than zero, and the curve is moving upwards through the osculating plane. It is negative otherwise. The torsion is therefore measured as -τN relative to its left-right behaviour regarding the tangent. We can therefore establish a convention for when a population is accelerating or decelerating into and along the generation in time.

And, finally, since the tangent and the binormal are also orthogonal, they form a joint plane: the rectifying plane. This is where the curvature, κ, and the torsion, τ, combine forces. If the tangent, T, turns more towards one axis, then it becomes more parallel to that axis, but is simultaneously becoming more orthogonal to the other. Thus if both the curvature and the torsion are considered positive with respect to the tangent, then they must each be negative with respect to the normal. The rectifying plane therefore always has positive torsion for the binormal, τB, and negative curvature for the tangent, -κT, or else vice versa.

We now have a moving Frenet-Serrat trihedron that tells us everything we want to know about the underlying surface behaviours:

 Tangent Normal Binormal Tangent 0 κN 0 Normal -κT 0 τB Binormal 0 -τN 0

The Frenet-Serrat formula and its curvatures establish a ‘skew-symmetric’ matrix. Frenet gave six of the nine formulae required. Serrat gave the final three some four years later. The advantage of the Frenet-Serrat trihedron is that no matter how much the surface may change directions and orientations, all measurements made of it are accurate.

The zeros on the principal diagonal give a trace of zero meaning it does not change sizes or dimensionalities. The shears or off-diagonals are matching opposites. There is also a line of zeros on the anti-, counter-, or minor diagonal. The skew symmetry comes from the matching pairs of opposite shears that accompany it. We have -κ and +κ, and -τ and +τ placed symmetrically either side. Therefore the sums of all the principal minors are zero. There is no skew or curl. This tells us of any differences populations might exhibit between their absolute and linear times measured in one dimension, and their relative and generational times measured in the other, each with its appropriate meridional aspect. Finally, the determinant is zero, so the volume stays the same. Everything we measure belongs to what we are measuring, and has not been arbitrarily introduced by the frame or our method of measuring.

The Frenet-Serrat trihedron allows us to examine an entire circulation. We can view it as a straight line, locally, that always keeps its origin to its left; yet we can still measure its global curvature. The frame itself tells us exactly how the surface, curve, or object is moving through space. It tells us the degree of bending and twisting in and out of all three planes. There is no intrinsic distortion. Everything we measure belongs to the object itself. It has been in no way altered either by the space, or by our measurement process.

The zeros on the principal diagonal are effectively the S11, S22, and S33 measurements. They mean that the tangent, the normal, and the binormal always measure themselves exactly. They are always fully parallel to themselves. They have no mutually orthogonal components. They do not make their own selves curve. They have zero effects on each other. Any population whose members are always fully themselves will reflect the Liouville ensemble properties.

The tangent and the binormal also do not affect each other. They are defined as cross products. By definition, therefore, they are always at 90°. They are never any more or less parallel or orthogonal than they already are. Since they have zero effect upon each other, they have zeros on their minor diagonal.

The normal is the only axis that both the tangent and the binormal can affect with their curvatures. The four non-zero entries of κN, -τN, -κT and -τB are in all the off-diagonal positions. The Frenet trihedron is skew-symmetric because those four entries are symmetrically located either side of each position, and are each others' negatives. As hard as the tangent pushes the normal away, just so hard does the normal fly away from the tangent. They therefore have opposite signs for the curvature. And as fast as the binormal pushes the normal away, just so hard does the normal seek to fly away from the binormal, again giving them opposite signs, but this time for the torsion. The curvature and torsion are mirror images symmetrically located either side of the principal diagonal, meaning all terms in all off-diagonal positions balance out, so that the frame itself introduces no curls, no skews, and no rotations. Nevertheless, if the normal is positively affected by the normal's curvature, it is negatively affected by the binormal's torsion; and if it is negatively affected by the tangent's curvature, it is positively affected by the binormal's torsion. The precise degrees of effect are different … and measurable. They belong, however, to the surface.

Figure 20.112

Shortly after he learned about it, the French mathematician Jean Gaston Darboux realized that the Frenet-Serrat trihedron was still not quite accurate or complete. As in Figure 20.112, there is a difference between the frame twisting because the surface underneath it is twisting, and the frame twisting because, as at the bottom of Figure 20.112, a rotational force is being directly applied to that frame. If the thumb, which represents the binormal, is directly torqued, then some kind of angular momentum or curling force exists. The tangent and the normal, upon their osculating plane, as represented by the index and middle fingers, will turn to point in different directions. Darboux had thus realized that although the trihedron's three axes accurately describe all translational velocities and movements, there is a potential difficulty because we do not properly assess any rotations there might be.

Darboux resolved this issue by pointing out that if we indeed grab hold of the thumb and rotate, then although the index finger might now point in a different direction, it is still orthogonal to its partner. As fast as the index finger is rotated round to point at a different location, just so fast is the middle finger dragged around, by an exactly corresponding amount, to point in an associated direction. The one might not be as parallel as it was before, but it has substituted an orthogonality, and conversely for the other. The two have exchanged matching quantities of the parallel and the orthogonal, which is their dot and cross products. The rotations Darboux has discovered might give us different measures on each axis, but they coordinate to give the same overall value, which is again a surface invariant. It is now a simple geometrical manipulation to make the various triangles and parallelograms lie in equivalent planes, and then to measure their areas using the same unit vectors. We can now suffer any rotations, and the trihedron is still correct.

The Darboux vector says that since the dot and cross products over all three of the osculating, normal, and rectifying planes will coordinate, then the total amount of rotation, w, that a frame experiences is the sum of the separate rotations in each direction. They give w = w1T + w2N + w3B where w1 is a ‘roll’ or a rotation about the x-axis, w2 is a ‘pitch’ or rotation about the y-axis, and w3 is a ‘yaw’ or rotation about the z-axis. (N.B. this rotational ‘pitch’ within the Darboux vector should not be confused with the helicoid pitch, which is instead a rise in height over a generation. If we do not specify, then we mean the helicoid's generational or time measure). We can therefore measure the net angular velocity, w, of all those rotations by measuring the resulting areas, and changes in areas, upon their respective planes. Those areal velocities then tell us about all the curls and rotations.

Our TNB frame, suitably amended with the Darboux vector, is now completely reliable. We can use it to measure everything going on in any neighbourhood of any biological entity and accurately determine its spaces.

We have now replicated Einstein's repudiation of the firmament. We measure things-in-themselves. Everything we measure belongs directly to the surface. If we measure our biological space of internal energy as curved, then it is curved. The Frenet-Serrat trihedron measures eactly what is there and introduces nothing.

If creationism and intelligent design are true, then the TNB path a population is on should always give the same measure as both of the population ijk and the species IJK values. Neither ijk nor TNB should register any change with respect to the IJK sitting at the centre of the helicoid. And since a helicoid has both a constant curvature, κ, and a constant torsion, τ, then every species will be defined by a suitable set of values that are easy to measure and to predict. This is the basis of our Brassica rapa experiment.

We are at last ready to resolve the anomaly of biological time and dt dτ. We simply emulate Galileo and define an ideal motion. In our case, the more infinitesimal the increment in our hypersphere radius, , the more isotropic, even, and homogeneous the resulting space. Its dot products upon the four axes will become ever more flat, and even, and will approach this as a limit. When dt = dτ, all biological masses, energies, and events, flow completely evenly, and we have ijk = IJK = TNB.

Since a biochronometric measure, τ, is both a time measure and a distance measure, then an ideal motion keeps all those increments even. And since the generation's midpoint establishes gradients to and away, then an ideal motion keeps the ratio between absolute time, t, and generation time, τ, constant so that dt and dτ are even either side of any possible interval. We then have the same distance being covered in the same unit of time. The velocity both across the generation and through time is constant. We move about the circulation at a completely even speed. The acceleration inwards towards the centre is constant. There is no jerk. The zero point, which is the mean, is maintained. There is no aberrancy.

We have now described the biological firmament that creationism and intelligent design demand for every species. Even though it is impossible, we have shown that the firmament exists when we have an infinitesimal slice so that ijk = IJK = TNB. And that is what we can look to measure to see if if those things are so. We have made this very simple. We need only measure in our three dimensions of a population's mass, energy, and numbers.

Unfortunately for creationism and intelligent design, Einstein's relativity theory has demolished the very idea they rely on for their proposed validity: that absolute clock time can move at a constant speed. There is therefore no reason why that globe in Figure 20.39 should ever rotate evenly; nor why generation lengths should be even, in the way they require. As a population struggles with reality in the form of its surroundings, its shear components state how far removed from the creationist and intelligent design ideal both it and its circulation are. And since time does not flow evenly, then the helicoid of internal energy is not obliged to facilitate even movements through time, which is both (a) even rates of change in clock time, dt; and (b) even rates of change all about the generation, dτ.

If creationism and intelligent design are true, then the number dimension is always zero relative to all others. There must be no mutual gradients or changes with respect to number. The two divergences τmass:energy and τenergy:mass must act together to produce the entire sequence of events, on the flat plane, that is the two-dimensional area that establishes Tmass:mass and Tenergy:energy for the species template. If those doctrines hold, then no population should ever exhibit a change in its generation time. This is very easy to test in any experiment.

If the ratio between absolute and biological time is even so that dt = dτ at all times, then the six off-diagonal biological events in the Haeckel tensor of τmass:energy, τenergy:mass, τnumber:mass, τmass:number, τnumber:energy, and τenergy:number must create and reflect this. They must be perfectly matched. Their shears must sum to zero. Each will act as a divergence or force per unit area. And since one of the dimensions is time, then all rates and generation distances must match. The weighted generational averages Tnumber:number, Tmass:mass, and Tenergy:energy will also be constant and invariant. They will come together to produce a generation time, T, that must also be constant and invariant.

No matter which axes we use, the Owen tensor's principal diagonal will be S11, S22, and S33. Those three sets of measures will have no components, dot products, or values interposing from any other coordinates or angles. They express each directional measure in terms of that direction's unit, no matter what the axes. If we use the axes ijk, IJK, and TNB, we will have our A, B, and C respectively. If creationism and intelligent design are true, then those three sets of measures will be the same. They will differ only in their combinations of bases and values. They will be scaled multiples of each other. But … this can only happen if they are each a multiple of the identity tensor, which also only has values on its principal diagonal, and zeros everywhere else.

If, however, Darwinian evolution is true, then there will be uneven movements across our various dimensions … including in time, and across the generation length. There must then be a force. That uneven force must in its turn be the derivative of some uneven flow of energy across both absolute clock time, and relative generational distance. We must then determine the aberrancy and the source of the curvature poloidally, toroidally, and meridionally all about the helicoid. There is only one dimension that could exercise such a force. That dimension is numbers as fitness; as competition; and as Darwinian evolution which is changes in numbers over time. We have already established that these are integrals and derivatives with respect to generational distance. We now have a way to measure them. And that is what we did in our Brassica rapa experiment.

We have now described the biological firmament that creationism and intelligent design demand for every species. It is very simply defined. It is when ijk = IJK = TNB. And that is what we can look to measure to see if the claims the doctrines make are true.

We are approaching these issues of Darwinian fitness, competition, and evolution by imagining a traditional African hunter-gatherer community leaving and then returning to its base camp. We extend our imagination by proposing that the community perambulation lasts exactly one generation: i.e. that the community returns to its specific camp just as the cohort of individuals born there begets progeny as the next generation, with their parents becoming grandparents and so forth. There is then a midpoint—some t’0τ’0—to both the generation and the perambulation. We can describe both the physical and the biological journey relative to that midpoint.

As is well known, the best place for a population to feed is not necessarily the best place to breed. Salmon undertake their heroic displays of fortitude to reach their breeding grounds a few miles upstream, where they promptly die; monarch butterflies flutter some 2,000 miles, 3,219 kilometres, between southern Canada and central Mexico; humpback whales migrate some 5,000 miles, 8,047 kilometres; and the the bar-tailed godwit flies 6,835 miles, 11,000 kilometres, in a scant eight days. The times for journeys, and the times to reproduce may be distinct, but the issues raised by ijk = IJK = TNB are easier to apprehend when they are considered together so we can determine a midpoint for both the journey through physical space and the generation.

We must also confront Dawkins' claim, in The Blind Watchmaker, that ‘If you want to understand life don’t think about vibrant, throbbing gels and oozes, think about information technology’ (Dawkins, 1986). Unfortunately, information entropy is an intrinsic part of information technology. The one does not exist without the other.

Perhaps the easiest way to approach these matters is to consider the distinction between body and surfaces stresses, and body and surface forces. We can then think of body stresses and forces as the internal processes bodies must go through to produce the messages that emerge from their surfaces. Those are then absorbed across the surfaces of other bodies, which then go through transformations as those messages are absorbed.

Information technology exists because information theory achieved the remarkable task of demonstrating something that now seems so obvious, it is difficult to imagine a time when nobody knew that information is physically and quantitatively measurable:

Before 1948, there was only the fuzziest idea of what a message was. There was some rudimentary understanding of how to transmit a waveform and process a received waveform, but there was essentially no understanding of how to turn a message into a transmitted waveform (Gallagher, Robert G. “Claude E. Shannon: A Retrospective on His Life, Work, and Impact.” IEEE Transactions on Information Theory, November 2001, pp 2681-2695).

In 1977 NASA launched the Voyagers 1 and 2 spacecraft as part of a proposed “Grand Tour” to study the outer reaches of the solar system and the interstellar medium (JPL, 2012). The two Voyagers had the fastest possible rates of information transmission: some 21.6 kilobytes per second. But information technology improved so dramatically that by the time they each reached Neptune, some 4.4 billion miles, 7.2 billion kilometres away, information transmission rates had skyrocketed to around 130 kilobytes per second, making the two spacecraft seem rather pedestrian. To put this another way, when the Voyagers took off in 1977 the body processes that produced information would only allow that information to emerge across their surfaces relatively slowly.

As we learned in Before We Begin, information entropy, which deals with conveying messages in a message space, is very different from the thermodynamic entropy that deals with temperature, pressure, volume, and other such parameters in physical space. I.e. it has an impact on those ‘vibrant, throbbing gels and oozes’ that instead occupy physical space, and so have distinct body stresses and forces. Those ‘gels and oozes’ must, after all, constantly shape and reshape themselves, generation after generation, so they can transmit their information across their surfaces so they are received across the surfaces of other gels and oozes, which then reformat their bodies so they can do the same across their own surfaces.

Human DNA is encoded in some 6 ×109 base pairs of DNA. If each base pair is represented as either a 0 or a 1, then the entire genome can be stored in 1.5 × 109, or 1.5 gigabytes. This is approximately two CDs. But if it is to be of any use, that information must be encoded in the correct format, which means the correct body stresses and forces. At the present time, this means a humanoid biological entity. That information must then be transmitted and retransmitted across a suitable surface at a rate of that many bits per each generation … but always encoded in that specific form, and with that specific transmitter–receiver and body-surface combination.

Figure 20.113

In spite of Dawkins' exhortation, information theory—at least, when considered alone, and in the way Dawkins has suggested here—is incomplete. We cannot consider information without also considering its storage medium. Figure 20.113, which is a repeat of Figure 0.9 from Before We Begin, again tries to make these distinct issues clearer by presenting the body and the surface forces and stresses. Those books represent the knowledge, or memes, that a first generation must pass on to a second across a suitable surface—in this case, those pages—in biological space. But those books have definite body forces and stresses. They must be handled and read. The message must cross their surfaces and be understood in the way the encoders intended. This can only happen if there are actions in the surroundings. This requires a transfer back and forth across some suitable surfaces.

Harps and adungus are also methods of encoding memes and information. They cannot just sit around unused. Their shapes and bodies must be engaged with so that their surfaces can transmit the desired encoding back and forth between the bodies that find them meaningful. The relevant skills and appreciation must be taught and instigated, and the harps and adungus must be both made and played. All this must then be similarly passed on to a succeeding generation. Even preserving them in museums takes force, mass, and energy. Both that information and its meaning, as knowledge, must therefore be absorbed by the first generation. It must then be re-recorded into a new set of books, magazines, instruments, and concerts, and then transmitted to the second, which must then do the same for a third and so forth. It may also be hard to measure, but just as we can say that a child is “half-way through school” and discuss their entire development relative to that half-way point, so also can we search for a way of quantifying transfers of information across surfaces, and from one body to another.

If one of the successor generations involved in this genes and memes transmission through biological spacetime sets the same knowledge and information in a slightly larger font, then its members must either use more books as in 20.113.B; use larger books, as in 20.113.C; or else some combination of both. One or another generation can also leave the font-and-book configurations the same, yet put different and/or more information of the same general kind into those books. But as we also learned earlier, we cannot preserve any information of any kind, not even as the choreography of a honey bee dance or a set of hieroglyphs, without using force, mass, and energy that is deployed in the surroundings by some entity to both act on and enshrine that information. The relationship between the body and surfaces forces and stresses, and so to the information, can easily change. The half-way transfer point will also change to reflect this.

Genes and their memes are not the only examples of discrete physical objects that influence the environment through mass, force, energy, and information transmitted into and received from the surroundings through a juxtaposition of body and surface stresses and forces. So also are the electricity and the magnetism used to do that encoding. These two sets embody similar concerns and principles in their respective spaces.

Genes and memes, electricity and magnetism, all require domains. They require the deployment of bodily forces and stresses which encode force and energy in and through their domains. The biological domains are the distinct biological entities. But each domain, whether biological or electromagnetic, transmits the required information across its surfaces, and through a set of distinct stresses and forces that arise in its partitioned domains. Again in each case, the behaviours and the forces that arise across those surfaces occur because of the behaviours within those domains; and, just as importantly, between those domains. Those forces in each case arise because of given internal molecular structures.

Both electricity and magnetism and genes and memes conduct ongoing transactions under energy that affect the behaviour of their respective domains. They are both examples of forces exerted in the surroundings as a function of the surface stresses that arise from their given domains. Those domains are accurately described by Coulomb and Poisson. They arise from body forces, and the internal configurations behind those surfaces. Those body stresses and forces lead to molecular modes and vibrations imposed upon the surroundings, always via surfaces.

The domains are in each case affected by their body and surface stresses and behaviours. No magnet, for example, is truly permanent. When one is heated, its individual electron spins become more highly energized. They start to point in different directions. The domain walls or boundaries between regions that were at one time aligned begin to slide around and rearrange. The total magnetic effect is reduced, and less force is exerted in the surroundings. The greater the heat energy applied, the less the average alignment. At the well-defined Curie temperature for every magnetic substance, the domains completely collapse and the magnetic effect disappears. Iron's Curie temperature is 1,043 kelvins. This is therefore the temperature at which iron's entropy increases sufficiently to change its configuration and mode of vibration, and destroys its magnetism. If we keep putting energy into iron until it reaches 1,811 kelvins then it melts, which is a further increase in its entropy. In between its Curie and its melting temperatures, it has not yet heated enough to melt, but is hot enough to be non-magnetic. For many magnetic substances, the domains will reform as the substance cools back down from the Curie temperature, but unless an external magnetic or electromagnetic field is applied, they will all continue to point in different directions. Thus the magnet's ability to exert force in the surroundings, and across its surface, is a direct product of its domains and its internal body stresses and states, and so of its molecular modes of vibration.

Heating a magnet to its Curie temperature is only one of various methods of inputting energy that can raise a magnetic substance's entropy enough to alter its molecular vibrations and configurations, and so that of its molecules, to affect its external magnetic capabilities, and so its ability to exert force in the surroundings across its surfaces. Applying alternating currents is another method. The magnet's polarity then flips back and forth, becoming weaker and weaker. This is called degaussing. The energy inputs have affected the energy outputs by affecting the domains.

Magnets made from steel are sensitive to mechanical shocks. Their domains can be realigned and demagnetized simply by hitting them with a hammer. Many horseshoe magnets can be demagnetized by bending them into a circle so there are no explicit poles. Grinding a magnet up will also demagnetize it because the pieces align so their north and south poles abut and cancel out, and there is no net external force. All these methods involve energy and increase entropy, and thus destroy not just the magnetism, which is the behaviour of discrete domains, but the ability to affect the surroundings. The energy input therefore affects the energy output.

Some, but by no means all, of the electrons in a ‘paramagnet’ such as gold, copper, or aluminum, spin in the same direction. However, their thermal motions usually keep them relatively randomly oriented. As paramagnets, they are therefore usually non-magnetic … until placed in a magnetic field, at which point they demonstrate a very small amount of polarity, and act as if they themselves are magnetized. However, their magnetic capability drops back to zero when the applied field is removed. Their resulting field strength is therefore due entirely to the presence of some other field, and is directly proportional to that external field strength. But their attractive force is only ever extremely weak, and can only be measured with sensitive instruments. It nevertheless exists. In all such cases, therefore, the forces that can be imposed in the surroundings are a function of, and are not independent from, the energies and configurations within those surroundings. They are therefore an expression of surface stresses and forces, and are a result of the complementary effect those surroundings can have upon the object's body stresses and forces. The energy input again affects the energy output.

We can now define a biological population as a biological field. The individual biological entities are the domains of activity within those fields. The domain behaviours change with the addition and removal of energy. We can then measure those transformations.

We then define genes, within this biological geometry, by declaring that they are a manifestation of the mechanical chemical aspect of the biological internal energy enshrined within the field. They are therefore meridional, but are carried toroidally about the circulation from one moment to the next, and so from indiscernible point to indiscernible point about the helicoid of internal energy. They are the base pairs, the chromosomes, and other such structures formed from the genetic code of adenine, thymine, cytosine and guanine, ATCG, and all their alleles or allelomorphs such as Bb. They have a locus, often a name, and so forth. Their transmission and flow from one generation to the next, as kilogrammes of mendels, are their body stresses and configurations. They are one aspect of the inertias and the impulses within each domain of internal energy in our biological field.

We define memes, as another aspect of biological internal energy, by saying that they are all those interactions with the environment as arise from the energies, the forces, and the configurations of the body surface molecules, and so that emerge through its surface, into the surroundings, and so from the same domains. They are therefore the collective traits, behaviours, and informations arising from vibrating genes in interaction with the environment, in each moment in time, and as their integrals and differentials. They are the energetic poloidal pressure that the domains or biological entities exert upon those same surroundings.

If a first generation of genes codes for a great deal of autism or musicianship, then that will be recorded in both the genes and the memes transmitted toroidally to a second generation, and that are then re-recorded by the work done upon those molecules and genes both meridionally and poloidally; and if they do not move toroidally, which can only happen with accompanying meridional and poloidal components, then they are not re-recorded. Those are the forces and energies arising from the domains, and as the statement of the force reciprocally exerted in the environment. That is a second aspect of the inertias and the impulses associated with each domain of internal energy.

The three constraints of constant propagation, size, and equivalence are simply a means of cajoling population domains and entities to behave in given ways so they can travel, toroidally, from one indiscernible point to another. That is the torque they apply relative to each of their given generation lengths of Tnumber:number, Tmass:mass, and Tenergy:energy. Each domain of mass and energy imposes that torque about itself through its matching biological sequences in its shear components, and as the propagating current-elements that seek to impose that same journey on everything in the neighbourhood. Those torques and energies are the hamiltonians as the sum of the Gibbs and Helmholtz energies at any point.

We have defined genes and memes more rigorously with our three constraints of constant propagation, constant size, and constant equivalence. We have the Frenet-Serrat trihedron and can quantify them all meridionally, poloidally, and toroidally. They are quantified as  ∫ T0dP < P’ = N’p̅’ = ∫ T0dM < M’ = N’m̅’  = ∫ T0dS < S’ = N’s̅’ = 0 respectively, along with their verbal renditions. They are the backbone that govern all relations for both their body and their surface behaviours, which interpenetrate. They are in their turn governed, respectively, through the four laws of biology and the four maxims of ecology. These things are all true by definition.

Just as magnets and paramagnets can change their behaviours and switch their forces and activities on and off with variations in masses, energies, and configurations, then so also can genes and memes as an expression of a biological field and the constraints. Genes are therefore the circulating field M0L0. They create body forces and stresses. They are hereditary and move meridionally over distances. Memes are their energizing field P-1L-1P1P1. They are the surface forces and stresses. They ray in and out poloidally from the surroundings and so they are—as Dawkins intended them—non-hereditary over times. The two then interact toroidally over the circulation through their respective differentials and integrals, including with their distributions of heredity, inheritance, and impulsivity, ranging to their matching aberrancies of flounces, convergences, and pivot masses.

We measure a population's body stresses—which are its biological field structure as arise from its molecular dispositions—by measuring its mass, its mendelity, and so its mechanical chemical energy at each moment. We measure its surface stresses by measuring its energy, its biopressure, and so its nonmechanical chemical energy at those same moments. We measure them as functions of both absolute clock time and relative biochronometric generation length; and both individually, as well as per the population, and so using both our polar and linear planimeters. This is done through our three constraints, which provide the bedrock for our accompanying four laws and four maxims whose attributes we can also measure. These are again the definitions of this biological field and its spaces and forces. Since we have tensors, we can express them in terms of each other, and in terms of their mean values, which are always the unit.

Since body and surface forces and stresses are inseparable, we can easily add and subtract the thermodynamic form of entropy by injecting and removing mass and/or energy to any system. As with memes, the system boundaries do not have to be well-defined. Thermodynamic entropy states every system's direct interaction with the environment, eventually manifest as the absolute zero of temperature. It is therefore more concerned with uncontrolled degrees of freedom, albeit measurable from within each system as its ongoing interaction relative to its current thermodynamic state of molecular vibrations, and so as joules per kelvin.

Information and information entropy are somewhat different. We add and remove information to whatever is our storage medium—be it DNA or otherwise—in ways that are distinctly different from those in which we add and remove mass and/or energy to any system. The addition and removal of information is more concerned with well-defined systems with controlled degrees of freedom in definite message spaces. Each distinct message source and recipient will have an interaction with the surrounding cosmos, but information entropy is concerned with their specific interactions across their surfaces as messages, and as a result of the body interactions that transmit and receive those messages. It is therefore more limited in its measurement scope: i.e. in terms of the properties we need to measure to derive it.

If we toss a coin 10 billion times and store our results on a hard drive, then the information entropy we have stored is 1.2 × 1011 bits. If we then erase or reformat that hard drive, the disk returns to its previous simple state, information-wise. Its information entropy has been reduced back to zero. However … we cannot either fill or erase that hard drive without expending energy. This immediately means that the thermodynamic form of entropy increases when we first fill the hard drive … and … increases yet again when we erase our data. That brand of entropy will therefore increase even as our hard drive's specifically information entropy decreases. Both the forms of entropy affect both body and surface stresses, but their forms of reckoning are as different as their scopes. This important difference affects the nature and character of all biological entities.

Information entropy began with Shannon's concern about telegraphing. He was concerned with the accuracy of messages transmitted. That is certainly an issue in genetics when seen, as Dawkins prescribes, as the transference of information. Both genetics and information are concerned with bodies and with surfaces, but their domains of reference are distinct.

Unfortunately, the word “information” in ordinary language has a somewhat different meaning from “information” as used in Shannon's information theory. This latter information should more fully and correctly be called “Shannon information”. In the same way, “Shannon entropy” is very different from Clausius' entropy. As the US physicist Edwin Jaynes expressed it:

This is an unfortunate terminology, which now seems impossible to correct. We must warn at the outset that the major occupational disease of this field is a persistent failure to distinguish between the information entropy, which is a property of any probability distribution, and the experimental entropy of thermodynamics, which is instead a property of a thermodynamic state as defined, for example by such observed quantities as pressure, volume, temperature, magnetization, of some physical system. They should never have been called by the same name; the experimental entropy makes no reference to any probability distribution, and the information entropy makes no reference to thermodynamics. Many textbooks and research papers are flawed fatally by the author’s failure to distinguish between these entirely different things, and in consequence proving nonsense theorems (Jaynes, 2003, p. 351).

The information theory Shannon produced concerns a transmitter, a receiver, and a channel of communication. He introduced four important ideas:

1. Every communication system has an architecture, or design, that separates (a) the source, (b) the channel, and (c) the receiver. This is easily understood, within genetics, as a communication system involving a progenitor, an act of reproduction, and the progeny. We therefore seem to have our genes and our memes.
2. No matter what the message's content, whether it be text, sound, an image, or anything else, the channel always “sees” that message as the answers to a specific series of yes-or-no questions, such as ‘are you the letter “A”?’, ‘are you the colour red?’, ‘is it raining right now?’, or ‘are you this base pair?’. For this reason, information of whatever kind can therefore be rendered as a series of 0s and 1s. All modes of communication can be encoded in ‘bits’ of some specific kind, which was a term used in print for the first time, in that way, in Shannon's original paper. The message therefore has the potential to be transmitted across the channel so the receiver can potentially regenerate it without error through that sequence of questions or 0s and 1s.
3. Every communication channel has a speed limit. It is theoretically possible, below that limit, to send any message, no matter how faint or indistinct. The message might have to be made longer and more complex, and so take longer to be transmitted, but the probability of error can be brought as low as desired within that speed constraint. Error free communication can be made increasingly likely below that limit, but it is impossible to be error free above it.
4. Given all the above, then the main issue in sending information is encoding it, at source, in an efficient manner, so as to increase the chances of accurate reception. This means minimizing the message's length and complexity to increase the chances of success, and to decrease the uncertainty in the message.

A transmitter of any kind, in information theory, is a source for a finite number of possible message conveyances. Their explicit content is again irrelevant:

… the fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages (Claude E. Shannon and W. Weaver; The Mathematical Theory of Communication; 1949; University of Illinois Press, Urbana, Ill).

Information theory is only concerned with how much information is sent in a string as its symbols, and not with what that information or those symbols represent. The founder of information theory has made it very clear that the proposed meaning is beyond the theory's scope.

Shannon's information theory is only concerned with the likelihood of the message being received in the same state as it was sent: i.e. with it being deciphered as the sender had intended. Since the recipient has to identify a given message as one amongst many possible others, then the more messages, or symbol numbers, or combinations, that the message source is capable of transmitting, then the greater is the initial uncertainty, at the receiving end, regarding which particular message was sent. This is again independent of whether or not the original and intended message was ‘gibberish’, ‘shibberig’, or ‘biferishg’. It is perfectly possible for the recipient to conclude that the intended message was ‘gibberish’, when the sender in fact deliberately meant either of the other two options. That is separate from the engineering problem of transmitting the message accurately.

What the recipient concludes about what was meant by any message, or what should instead have been sent, is outside the scope of information theory. The theory is only concerned with accurate transmission. This is particularly important to remember when considering reproduction. There is a difference between “this is exactly what I mean”, and “make the most of this message”. What matters is the genes as they are received by the progeny … not what the progenitors might have “intended” to send as those genes … if they even had a “meaning” in mind. Creationism and intelligent design insist that there is always an accurate meaning, and that it is always accurately conveyed. This is the very thing that Darwinian evolution denies.

We can now make sense of the apparent tautology we met earlier: We establish, by definition, that every biological population takes exactly one generation,, T, to circulate the events of a generation, τ.

Figure 20.114

The dot in Figure 20.114 is a current biological event, along with all pertinent information in mass, number, energy, and time. It occurs at a definite point t0τ0 in the circulation of the generations. It is a given biological entity or entities with specific amounts of mass and energy. The meanings of the directions, relative to those two axes, are as follows:

1. Horizontal. These are all properties and entities at the same time, t0, and irrespective of their different sizes, states, and conditions. Those distinct points result from their different locations in the circulation at that time. Those parallel to the horizontal axis, and so directly to right and left, exhibit different sizes and configurations at that time. They are therefore moving across the circulation with different magnitudes, and so at different rates. Since these have all received different amounts of mechanical and/or nonmechanical energy at that same moment, then they have the different spatial values of τ-1, τ0, and τ1, as arranged on the bottom axis at that same time. They therefore establish the population's range between bigger and smaller, and more or less energy rich, again at that time. The entities concerned are all therefore at different points in the circulation at that time. If we consider the helicoid, then the larger and more energy rich members are closer to the outside of the step or spiral, while the smaller and less energy rich ones are closer to the axis. The former have larger spatial values and are taking longer to go about the circulation. They have dt < dτ. The latter do so more quickly and with smaller values and so have dt > dτ. We therefore have dt dτ. All those in the grey area remain reproductively accessible to each other in spite of their different states and times. They pursue the same events at the same rates toroidally about the entire circulation. They share the same mean values, stresses, and forces. They have dt = dτ.
2. Vertical. These are all properties and entities that share the same sizes, states, and conditions, and so are at the same point in the circulation, τ