‘Waves are iterative presents, Particles feed on past fields and move towards the future’. L§

**SUMMARY**

O. Body waves of reproductive energy.

I. Present Body waves in time Logic.

II. Product operands in mathematics

III. Waves of Energy in physics

IV. Reproductive Bodies in biology: Radiations.

V. Reproductive middle classes in sociology

VI. Company-Mothers in economics.

We recommend the reader to consult the article on energy to complement this one with more emphasis on the vital function of presents, which is to iterate systems. We might say that body waves are the organic form in space of the more abstract concept of time.

## O. Body waves of reproductive energy.

In the graph grow and multiply. the reproductive cycle which is the one between birth and maturity encodes in reality the whole game of existence, as it is a complex function that maximises all others into the worldcyle itself:

The principle of conservation of presents or energy derivatives of a system, is essential to mathematical physics as well as biology where it is known as the principle of selection of the species that reproduce in larger numbers. As such it is the dynamically conserved variable of both, mathematical physics that conserves derivatives of present space-time (conservation of the total speed-motion of the Universe) as well as life ecosystems, in which a motion of total reproduction maintains the balance of the system.

**The immortal arrow: reproduction.**

In a fractal Universe, in which every species has a limited quantity of time and hence it will certainly die, what matters to survive is the reproduction of the logical form of any species. Thus, species in the Universe try to reproduce to achieve immortality or else become extinct. Alternately the most simple systems become reproduced by other enzymatic species as it happens with machines among humans or carbohydrates in cells.

The two most simple arrows of existence, energy and information, become when they combine together in a creative manner, the 3rd, fundamental arrow that all organisms try to achieve: Reproduction. In the organic Universe, thus it turns out that out of 1+1 species, normally one specialized in energy and one in information, you don’t get 2 but often 3 (2+1 reproduced), or sometimes 1 (2-1, fed in), when top predator absorbs the lesser species as energy to continue its cycles of existence. Yet while feeding is a mechanical, destructive arrow, reproduction is a creative, multiplying organic experience.

We might say reproduction is the ultimate goal of existence. In a discontinuous Universe, to reproduce is the closest thing to immortality, to the ultimate nature of God=The Universe, which is to exist for ever. Yet because the Universe is discontinuous, only through reproduction you can repeat your form and energy again. Reproduction is indeed the closest thing to the ultimate nature of God=The Universe, which is infinite and immortal in its eternal balance of energy and information, yin and yang transforming into each other. And so all organic systems of the Universe do reproduce in one or other manner. In the photo we see a galaxy and a baby galaxy, a crystal reproducing, bees, children, machines reproducing themselves.

All systems are able to process energy and information, but very few Universal Organisms reach the summit of organicism: reproduction.

We observe the 3 arrows of organicism in many ways, through the law of the 3 ages , through the law of the 3 creations, through the law of the 3 networks.

All organisms use their three networks to accomplish the only three possible behaviors in a Reality made of spatial energy and temporal information: they accumulate information, and energy, and they reproduce both of them.

Let us consider briefly the last of such behaviors: Reproduction.

We talk of two kinds of reproduction. Reproduction sometimes takes place externally through different species called ‘catalyzers or enzymes’, that repeat the reproduce species [as in cells, or factories where humans reproduce machines], and sometimes takes place internally or within a couple [sexual reproduction]. In both cases the species becomes reproduced.

Yet only in the second case, when reproduction happens within the same species, as a living organism.

Again it is human anthropocentrism what prevents us from recognizing the obvious: any act of creation of a ‘parallel being’ of energy and information is an act of reproduction. Instead we use ‘itifying’ jargons for all acts of reproduction which are not between carbolife species.

So when machines reproduce we call it production, taking away the re. When crystals reproduce we call it growth. When electrons reproduce we call it collision. When civilizations reproduce we call it colonization, when atoms reproduce we call it big-bang…

Chemical compounds reproduce we talk of reactions and catalysis.

Yet reproduction happens every where at small and big scales.

When we talk of external reproduction [enzymatic reproduction] we find both micro and macroscopic examples. Enzymes reproduce carbohydrates in the cell under genetic orders, in small ‘factories’ called ribosomes.

On the other hand, humans reproduce machines in the Economical organism, called the ‘company-mother’ under monetary orders, that act as genetic orders, that gather human ‘enzymen’ and raw materials in big factories belonging to company-mothers… Men become in this manner ‘catalyzers’ of machine reproduction. The process is similar to that of the cell, only that factories have grown in scale…

What about simple atomic species. How they reproduce? In other web page we considered the big-bang that reproduced the basic species of the Universe, the atom… which apparently is a ‘dead’ species.

The big-bang can be explained biologically as a process of reproduction of Atoms within the space-time energy of the Universe. When that space-time was saturated with ‘atomic species’, the process of atomic reproduction ended. We forecast as proof of the Theory of Organisms the existence of a steady-state Universe where there is still a minimal reproduction of atoms, as atoms die in limited numbers.

What about electrons, even smaller particles; do they reproduce?

Even today under proper conditions electrons reproduce. Human anthropocentrism calls that a ‘collision’. Yet in essence it is a process of reproduction, similar to the way humans reproduce. Indeed, we create a primitive cell, which is an older, simplified version of ourselves. Our sexual couple also produces such a cell, and together, those two cells start an evolutionary process that brings them to human form. The same happens with electrons: two electrons produce a primitive version of their form, a light photon. When two of these photons merge together in a rich energy field, a new electron is born… by the sudden evolution of light into electronic form…

The only difference between those reproductive processes, is the scale and complexity of the growing form.

Instead of the word reproduction, we will often use here the biological term radiation, to convey the entire meaning of a reproductive mass of species, coming out of a single species.

Such phenomena is the most common phenomena that structures The Universe.

We could indeed say that the History of The Universe, is the history of its biological, reproductive radiations of organisms, growing in complexity and size, and multiplying constantly.

Let us then define reproduction for all Universal species:

“When Universal Organisms composed of multiple cellular pints find the appropriate amount of energy and information, they mold it with their own form and energy, as to shape a minimal cellular point, replica of the entire organism, that will replicate into multiple cellular units. Those cellular units will evolve and associate according to the common laws of Universal Organisms into a macro-organism similar to his parental forms”.

The seminal light creates a new electron. The seminal crystal creates a lattice. The seminal seed creates a tree. The seminal cells create a new child. The seminal prophetic mind creates a social god, network of minds, in which each believer is a cell, temple of the Word of the prophet: ‘Sangris martire semen cristianorum’. And the seminal big-bang atom created this organic Universe, a game of unending reproductive acts – the essence of the Universal will, that earlier philosophers of God, the Mind of the Universe understood. So God, Amon – the infinite orgasm – said to man, “Grow and multiply”.

Reproduction is indeed the ultimate nature of God. In a discontinuous Universe, only reproduction can bridge the discontinuity of existence, and immortalize any form, with slight variations. The intense desire, the enormous pleasure derived from the act of reproduction (either sexual reproduction of bodies, or the creative “orgasm” of an artist that recreates the languages of the Universe), is fruit of that intimacy we experience with the Ultimate Will of God, the Mind of the Universe.

The errors of our civilization: against the will of God.

Why then there is a historic repression of reproductive pleasure, a denial of that desire in modern Human civilizations? The answer is that we do not live any longer in the classic age of mankind, in a free human society, but in a society controlled by the will of Machines, represented by company mothers and their economical culture, whose goal is to re=produce machines, and control humans as workers=reproducers of those machines.

Harsh as it sounds for a Human Anthropocentric vision of History, men have become slaves of the reproductive will of company-mothers, social organisms that reproduce machines. So our Industrial culture, and previous cultures that made of work, money and weapons, the center of religion, have repressed human reproduction. And this has been understood by Human writers, ever since the Parable of the Tree of science=technology, explained how men denied the will of God, and focused their life in reproducing and evolving the fruits of that tree. Yet if we deny the will of God, the mind of the Universe, and do not care for human energy, information and reproduction, but dedicate all our efforts to evolve company-mothers and machines of energy and information, the Game will merely erase us, and make this planet to the image and resemblance of the machines we reproduce.

## I.

## Body waves in time Logic.

**The flows of space and time are asymmetric between 3 planes of ST-existence.**

The Universe only allows one ‘discontinuous jump’ to the next past, future, right or left zone. There are not worm holes to enter far away Universes (too far to happen), jumps to a past in which mother was born (as only you travel to your past through your death).

Thus Quantic jumps happen in space between contiguous zones and in time between 2 hierarchical contiguous planes of i and ∆-1 informative complexity. Though Fractal Organisms are discontinuous they have ideally 3 apertures through which energy and information flow. Yet those apertures are quantic, small holes where only ∆-1 particles might flow. Hence the Fractal Organism always communicates quanta from its lower, past scale of existence. So all acts of communication are quantic jumps both in time and space that can either communicate energy or information between similar particles or particles from ∆±1 planes of existence according to hierarchical, illogic laws:

*‘2 particles on the same space-time plane transfer temporal energy in both directions.*
*An ∆-1 space-time plane transfers energy to the ∆- plane from which it receives information.*

*An ∆-2 space-time plane transfers temporal energy, without receiving anything on exchange.’*

Those laws are based in the hierarchical structure of the Universe that creates organisms with 3 discontinuous ‘social classes’. So only particles from the first plane enter into just action -reaction processes.

While flows of energy and information between 2 different, i and ∆-1 planes are asymmetric as the upper classes receive energy and give only information. But we said that Quantic Spaces-Times jumps are contiguous, so how in the 3rd case might exist a flow between ∆-2 and i planes, separated by 2 scales? Obviously because the ∆-1 space-time field dies, exploding into ∆-2 particles that feed as energy or information the i form. Thus the 3rd case describes the extinction of a victim that the predator absorbs after destroying its cellular parts. For example you feed on the amino acids of the animal you eat, after destroying its cells into its micro-particles.

The previous ternary events, which can be formalized with the symbols of Quantic Spaces-Times logic, are essential to describe and explain all kind of mysterious phenomenon in physical and biological space-times.

Let us consider some examples:

– The best-known borders between the main 2 planes of existence of physical space-time, the plane of gravitational masses and electromagnetic charges, are black holes that emit dark, expansive, gravitational energy. Still 96% of the gravitational world is invisible to us, including the discontinuous inner dimensions of masses, quarks and black holes.

– The 1st case between equal forms writes as ST∆<st=ts>Ts∆.

It explains the 2nd law of Newton, the law of action reaction, according to which, et, the quantic action of ST over TS, is equal in value to ‘ta’, the action of TS over sT.

– Yet what Newton missed was the fact that ‘åctions’ and ‘reåctions’ are equal in TS force but not always equal in substance. It is the 2nd case that explains relationships between 2 different planes of existence: Often an action is informative and its reaction is energetic and vice versa, so they compensate each other, cancelling the past and future into a present. It is the principle of conservation of energy and information:

Energy becomes transformed into information and vice versa, Sp<=>Tƒ. Quantic physics explain the 2nd case with a mysterious principle that not even Physicists fully understand: ‘time and space are non-commutative’.

That is TS is not ST, T>Sp is an energy transformation and Sp>T is an informative flow. And they are not the same. So in quantum physics and Quantic Spaces-Times theory the order of the parameters of an action matters.

The fact that energy flows towards the future, informative form, and energy towards the past, establish a hierarchy between information and energy, the 2 arrows of the Universe.

In that sense, we prefer the word ‘order’ for the informative arrow and ‘entropy’ for the energetic arrow as it expresses truly the main biological difference between both: organisms are hierarchical, distributed through planes of existence that create chiral, hierarchical 3 class-structures, in the dominant dimension of height and information. Only the present, space, fruit of the repetition of the same form in extended surfaces, is democratic, equalitarian, creating huge masses of undifferentiated quanta without hardly any height dimension.

## II.

## Body waves in mathematics.

**The Propagation of Waves**

Let us consider in more detail those equations of present vibrating strings, *putting them not as physicists do in T-S=0 CONFIGURATIONS, for the sake of calculus but in its proper terms, as an S=T symmetry to extract some more meanings. Then we **write: *

**∂²u/∂x****²****=1/a² ∂²u/∂t² (15)**

This equation, as may be proved, has two particular solutions of the form **u1 = ϕ1 (x − at), u2 = ϕ2 (x + at) **where ϕ1 and ϕ2 are arbitrary twice-differentiable functions.

By direct differentiation it is easy to show that the functions u1 and u2 satisfy equation (15). It may be shown that **u = u1+ u2** is a general solution of this equation.

Its inversion is then of considerable interest; as it defines *the constant balance achieve in present states between the S and T parameters, along a ‘symmetric’ ‘identity element’ (to use group theory jargon so similar to the generator jargon)*.

Then, for the solution u1 an observer moving with velocity a, will see the string as a stationary curve. Yet a stationary observer will see the string as a wave flowing along the axis Ox with velocity a. In exactly the same way the solution u2(x, t) may be considered as a wave travelling in the opposite direction with velocity a. With an infinite string both waves will be propagated infinitely far but if they are ‘constrained’ by a membrain, their back and forth superposition will produce different balancing shapes increasing at certain times and decreasing at others:

If u1 and u2, as they arrive at a given point from opposite sides, have the same sign, then they augment each other, but if they have opposite signs, they counteract each other and there will be an instant of complete annihilation of the oscillations, after which the waves again separate.

So in terms of the generator the 2 solutions are *relative past and future inversions of the present ‘state’ of no motion for a wave; and in physical terms the wave will have its anti-wave, which extends to light space-time, and the duality of magnetic and electric fields, whereas the photon is defined as its own antiparticle. *

Now for a spherical wave, which is neither constrained in the ‘height and origin-end, an entropic process of expansion of space and hence diminution of a dimension of motion-energy takes place. So the solutions become of the form:

where r denotes the distance of a given point from the origin of the coordinate system r2 = x2 + y2 + z2, and ϕ1 and ϕ2 are arbitrary, twice differentiable functions.

This wave is spherically symmetric; it is identical at all points that have the same value of r. The factor 1/r produces the result that the amplitude of the wave is inversely proportional to the distance from the origin. Such an oscillation is called a diverging spherical wave. A good picture of it is given by the circles that spread out over the surface of the water when a stone is thrown into it, except that in this case the waves are circular rather than spherical.

Yet the second solution of is of great interest to ∆st; it is called a converging wave, travelling in the direction of the origin. Its amplitude grows with time to infinity as it approaches the origin. We see that such a concentration of the disturbance at one point may lead, even though the initial oscillations are small, to an immense upheaval…

And while *most theorists in single continuum space-time ignores it, it is NECESSARY to balance the two arrows of time, 4D and 5D, as it represents the collapsing wave that will become a particle/mind point, and ultimately through the effect of resonance explains the emergence of physical systems in the upper 5D plane.*

A clear case of the fundamental S=T balances of reality as one and the other solutions form an expanding spatial and imploding temporal solution that balance each other.

This concept is so essential that carried into astrophysics is the fundamental principle of symmetry, which when broken causes a ‘transitory’ space-time physical phenomena to happen.

The simplest equation of a vibrating string, **∂²u/∂²x=1/a² ∂²u/∂t²** is really giving us a ‘beat’ between the space-like state of the wave (first term) and the time like state (second term), with a ‘ratio’ that translates dimensionally the space-state into a time state, through the transformation of the ‘simultaneous tension of the two inverse forces’ that act pulling the string in space, into a term of ‘pure motion-time’, and acceleration, 1/a². So basically a wave-string like process is a space-time transformation, which switches back and forth as the wave moves in accelerated fashion, stops in tension-space, moves in time-acceleration… S>t<S>t; but we express both processes together in a single ‘time-less’ expression of the beat.

In that regard, a huge advance in science would happen if people understand that = static equality is meaningless, = means, ≈ or <≈>, transformations through constant feed-back beats between space and time states.

Yet while a wave motion is ‘conservative’ hence happening in the same ∆-scales of reality, merely causing topological s-t transformations and reproductions of form in the same scale of reality; a huge range of equations related to the entropic processes, where we use the same equation but with a different ‘degree’ of derivatives in both sides. So the entropic process of dissipation of a wave gives us the equation of heat:

**∂u/∂t= α (∂²u/∂x²+∂²u/∂y²+∂²u/∂z²)**

The heat equation is of fundamental importance in diverse scientific fields; all related *to entropic expansions and relaxations of a present wave of energy*. In mathematics, it is the prototypical parabolic partial differential equation; and hence immediately connects to the dispersion of a wave of information into an expansive flow.

In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation; connecting it with memoriless stochastic processes in which *sequential time causality becomes meaningless as the entropic process keeps dissolving the forms of the whole into an entropic future*.

The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. And so on.

ALL IN ALL, an obvious conclusion of the explosion and reproduction of mathematical physics in the classic age of ∆nalysis, is the fact that the ∆-scales of the Universe *are its most important dynamic element to understand the processes of reality.*

And in as much as mathematical physics has reduced in ‘human knowledge’ the properties of physical systems to its mathematical only description we can reverse the concept and affirm that ‘human physics’ is basically the study of the mathematical, differential equations that can mirror physical processes; and so instead of studying physical experiments and then extract differential equations, we shall consider to study the connection GST->∆º mathematics -> ∆º±i physics, first the laws of GST, then its interpretation by differential ∆nalysis, to then group all the main mathematical phenomena according to the GST->∆º differential equation it uses.

**Stability.** In the examples considered the question of stability or instability of the equilibrium of a system was easily answered from physical considerations, without investigating the differential equations.

Thus if the pendulum, in its equilibrium position OA, is moved by some external force to a nearby position OA′, i.e., if a small change is made in the initial conditions, then the subsequent motion of the pendulum cannot carry it very far from the equilibrium position, and this deviation will be smaller for smaller original deviations OA′, i.e., in this case the equilibrium position will be stable.

For other more complicated cases, the question of stability of the equilibrium position is considerably more complicated and can be dealt with only by investigating the corresponding differential equations.

Let some physical process be described by the system of equations:

For simplicity, we consider only a system of two differential equations, although our conclusions remain valid for a system with more. Each particular solution of the system, consisting of two functions x(t) and y(t), will be called a motion. We will assume that f1(x, y, t) and f2(x, y, t) have continuous partial derivatives. It has been shown that, in this case, the solution of the system of differential equations (57) is uniquely defined if at any instant of time t = t0 the initial values x(t0) = x0 and y(t0) = y0 are given.

We will denote by x(t, x0, y0) and y(t, x0, y0) the solution of the system of equations (57) satisfying the initial conditions:A solution x(t, x0, y0), y(t, x0, y0) is called stable if for all t > t0 the functions x(t, x0, y0) and y(t, x0, y0) have arbitrarily small changes for sufficiently small changes in the initial values x0 and y0.

More exactly, for a solution to be stable the differences:

may be made less than any previously given number for all t > t0, if the numbers δ1, and δ2, are taken sufficiently small in absolute value. Every motion that is not stable in this sense is called unstable.

**Harmonic analysis**

A mathematical procedure for describing and analyzing phenomena of a periodically recurrent nature, *in an inverse fashion to how Nature ensembles them, from the top of the integrated seemingly chaotic addition of all the cycles of actions back to its finitesimals*.

Many complex problems have been reduced to manageable terms by this technique of breaking complicated mathematical curves into sums of comparatively simple components.

Many physical phenomena, such as sound waves, alternating electric currents, tides, and machine motions and vibrations, may be periodic in character. Such motions can be measured at a number of successive values of the independent variable, usually the time, and these data or a curve plotted from them will represent a function of that independent variable. Generally, the mathematical expression for the function of all actions might seem chaotic.

However, with the periodic functions found in nature, the function can be expressed as the sum of a number of sine and cosine terms, *which decompose the sum into its ‘individual recurrent actions’ with different periodicities; according to the ‘sequential order’ of **frequencies, which as usual is related to the sequential world cycle: *

*Top frequency: 1D-information, Max. Frequency, 2D: motion; Med. Frequency, 3D: reproduction; Min. Frequency: 4D: entropic-death (1), Probabilistic Frequency (0-1): Social evolution.*

i.e. you think every second your MINIMAL QUANTA OF TIME, which will always be defined by the fastest clocks of the 1D component of the being; then you can walk also with the same frequency, one step/heart beat for second, but *you do not walk all the time, *next comes reproduction, whose frequency is far slower, but normally between the minimal once in a life or the maximal for fungi in our scale, big-bang particles in the cosmos, etc. And finally comes the death event which is only once in a life (but certain, or probability 1) and the less probable even, the repetition of the ∆º>∆+1 collapse of the herd-wave of beings into a social superorganism/species of the upper world (and often happening in a larger period of a single ∆º life).

So we shall find functions of those frequencies, often split as the ∆-actions are either seen as ‘limits’ (death), or ignored (social evolution), when quantifying the being ‘in itself’, reduced to the analysis of its frequencies of perception/information, motion (lineal reproduction of information along a path) and reproduction, in ternary series or when reproduction is not accounted for, or it is accounted as motion (for particles/waves), to a dual rhythm which we can summon up in the concept of a stop-informing, go-moving, frequency which is really all what needs to be measured for simpler waves/particles states.

Thus, this simple rule* , to order all phenomena following the natural order of the 5 Dimensions, * simplifies enormously the understanding of the whys obtained by Harmonic analysis.

Such a sum is known as a Fourier series, after the French mathematician Joseph Fourier (1768–1830), and the determination of the coefficients of these terms is called harmonic analysis. One of the terms of a Fourier series has a period equal to that of the function, f(x), and is called the fundamental. Other terms have shortened periods that are integral submultiples of the fundamental; these are called harmonics.

The terminology derives from one of the earliest applications, the study of the sound waves created by a violin which *showed how the mind does differentiate the order behind the wave as we find pleasant the harmonious music, even if at first ‘sight’ its mathematical form seemed confused, before we disintegrate its whole in its parts. This is what *Fourier did stating that a function y = f(x) could be expressed between the limits x = 0 and x = 2π by an infinite series. So departing from an ∆(x:t) or an ∆(y:s), the finitesimal of time or space is integrated along a reproductive wave of space or informative wave of frequency time, and any of its polynomial/fourier expressions:

In the graph a ‘time polynomial’ or fourier series, which recurs adding the worldcycles of different actions, in the o-1 unit circle of time probabilities, where 1 is the happening of an event-form, when the cycle is closed at 2 π; and so we integrate the 2 dynamic paths, the cos clockwise and sin anticlockwise (as it moves towards its maximal perpendicularity) functions, which give us the repetitive coefficients:

In other posts (number theory, @nalytic geometry) we study the deep meaning of sin and cos NOT as fixed numbers but as dynamic inverse functions, moving in two different directions of time, in connection with its repetitive mapping on a complex graph, as ‘sins’ scale anticlockwise the -i axis and cos flattens and elongates clockwise into the real world axis any space-time function. Enough to say at this stage that both functions play *the necessary inverse elements for any proper ‘mirroring’ of an S≈T duality; in this case we say that cosine are $-pace like (2D, 4D related) and sin is ð-ime like (0-1D, 5D related).*

While the series is an infinite sum, as it integrates the (in)finite variations of a ginormous number of space-quanta forming herds (not synchronous super organisms), moving with different step and go frequencies, absorbing energy or information with different time periods, etc. in praxis, the main sum is given over 3, 5 to 11 frequencies which gather most of the actions and social groups of the best synchronised parts of the whole, according to the ternary time and 11 space parameters of most time-space systems.

Indeed, earlier harmonics were related to music with Pythagoras and the violin and we know the chords of music ‘stop’ in the fifth.

On the other hand, in modern wave analysis it was found that the use of a larger number of terms will increase the accuracy of the approximation, and so the large amounts of calculations needed are best done by machines called harmonic (or spectrum) analyzers; these measure the relative amplitudes of sinusoidal components of a periodically recurrent function. Yet when the first such instrument was invented by Kelvin for the harmonic analysis of tidal observations, it was enough to embody an ‘hendecagram’ of 11 sets of mechanical integrators, one for each harmonic to be measured.

**Multiple derivatives – approximations: fourier series.**

We return now to the key expression of a sum of motions/actions of a herd along a lineal 2D path – the Fourier series.

Fourier series arose in connection with the study of certain physical phenomena, in particular, small oscillations of elastic media. A characteristic example is the oscillation of a musical string.

Indeed, as we explained the investigation of oscillating strings was the origin historically of Fourier series and determined the direction in which their theory developed. So we can consider the initial case in more detail.

Let us consider (figure below) a tautly stretched string, the ends of which are fixed at the points x = 0 and x = l of the axis Ox. If we displace the string from its position of equilibrium, it will oscillate.

We will follow the motion of a specific point of the string, with abscissa x0. Its deviation vertically from the position of equilibrium is a function ϕ(t) of time. It can be shown that one can always give the string an initial position and velocity at t = 0 such that as a result the point which we have agreed to follow will perform harmonic oscillations in the vertical direction, defined by the function:

Here α is a constant depending only on the physical properties of the string (on the density, tension, and length), k is an arbitrary number, and A and B are constants.

We note that our discussion relates only to small oscillations of the string. This gives us the right to assume approximately that every point x0 is oscillating only in the vertical direction, displacements in the horizontal direction being ignored. We also assume that the friction arising from the oscillation of the string is so small that we may ignore it. As a result of these approximate assumptions, the oscillations will not die out. Then equation 20:

defines the possibilities of oscillation for the point x0 in periodic harmonic form. Where these functions do have the following remarkable property: Experiments and their accompanying theory show that every possible oscillation of the point x0 is the result of combining certain harmonic oscillations of the form (20).

Relatively simple oscillations are obtained by combining a finite number of such oscillations; i.e., they are described by functions of the form 20:

Where Ak and Bk are correspondiilg constants. These functions are called trigonometric polynomials. In more complicated cases, the oscillation will be the result of combining an infinite number of oscillations of the form (20), corresponding to k = 1, 2, 3, ··· and with suitably chosen constants Ak and Bk, depending on the number k. Consequently, we arrive at the necessity of representing a given function ϕ(t) of period 2π/α, which describes an arbitrary oscillation of the point x0 in the form of a series:

The remarkable fact here from ∆st pov is that if we consider 21 to be the spatial symmetry for the temporal case, 20, of limited number of actions (ternary, penta or 11th factors being the commonest harmonies in ∆st and reality), *contrary to a first deduction, a simultaneous super organism reaches more efficiency, simultaneity and complexity in its network control of its micro-parts when they increase in number. So from big masses which have better orbital circles than elliptic light comets, to human organisms, better constructed that simpler sponge colonies, as a reproductive mass of elements grow, the ‘epigenetic elements’ in growing scales of ∆-dept that organise the system (not shown on that equation), improve its control efficiency till the ’11¹¹’ completion of a plane of form, the ultimate meaning of an in(finite).*

Going back in time then, the growth of complexity means that the time frequency will grow as the space-intensity diminish and finally both vanish.

There are many other situations in physics where this is natural. so we can consider a given function, even though it does not necessarily describe an oscillation, as the sum of an infinite trigonometric series of the form (21). Such a case arises, for example, in connection with the vibrating string itself. The exact law for the subsequent oscillation of a string, to which at the beginning of the experiment we have given a specific initial displacement (for example, as illustrated in figure 12) is easy to calculate, provided we know the expansion in a trigonometric series: (a particular case of the series (21)), of the function f(x) describing the initial position:

**Expansion of functions in a trigonometic series**

On the basis of what has been said there arises the fundamental question: Which functions of period 2π/α can be represented as the sum of a trigonometric series of the form (21)?

This question was raised in the 18th century by Euler and Bernoulli in connection with Bernoulli’s study of the vibrating string. Here Bernoulli took the point of view suggested by physical considerations that a very wide class of continuous fhnctions, including in particular all graphs drawn by hand, can be expanded in a trigonometric series. This opinion received harsh treatment from many of Bernoulli’s contemporaries. They held tenaciously to the idea prevalent at the time that if a function is represented as an analytic expression (such as a trigonometric series) then it must have good differentiability properties. But the function illustrated in figure 12 does not even have a derivative at the point ξ; in such a case, how can it be defined by one and the same analytic expression on the whole interval [0, l]?

We know now that the physical point of view of Bernoulli was quite right. But to put an end to the controversy it was necessary to wait an entire century, since a full answer to these questions required first of all that the concepts of a limit and of the sum of a series be put on an exact basis.

The fundamental mathematical investigations confirming the physical point of view but based on the older ideas concerning the foundations of analysis were completed in 1807-1822 by the French mathematician Fourier.

Finally, in 1829, the German mathematician Dirichlet showed, with all the rigor with which it would be done in present-day mathematics, that every continuous function of period 2π/α,* which for any one period has a finite number of maxima and minima, can be expanded in a unique trigonometric Fourier series, uniformly convergent† to the function.

Figure 13 illustrates a function satisfying Dirichlet’s conditions. Its graph is continuous and periodic, with period 27π, and has one maximum and one minimum in the period 0≤x≤2π:**Fourier coefficients. **

In what follows we will consider functions of period 2π, which will simplify the formulas. We consider any continuous function f(x) of period 2π satisfying Dirichlet’s condition. By Dirichlet’s theorem it may be expanded into a trigonometric serieswhich is uniformly convergent to it. The fact that the first term is written as a0/2 rather than a0 has no real significance but is purely a matter of convenience, as we shall see later.

We pose the problem: to compute the coefficients ak and bk of the series for a given function f(x).

To this end we note the following equation:

which the reader may verify. These integrals are easy to compute by reducing the products of the various trigonometric functions to their sums and differences and their squares to expressions containing the corresponding trigonometric functions of double the angle.

The first equation states that the integral, over a period of the function, of the product of two different functions from the sequence 1, cos x, sin x, cos 2x, sin 2x, ··· is equal to zero (the so-called orthogonality property of the trigonometric functions). On the other hand, the integral of the square of each of the functions of this sequence is equal to π. The first function, identically equal to one, forms an exception, since the integral of its square over the period is equal to 2π. It is this fact which makes it convenient to write the first term of the series (22) in the form a0/2.

Now we can easily solve our problem. To compute the coefficient am, we multiply the left side and each term on the right side of the series (22) by cos mx and integrate term by term over a period 2π, as is permissible since the series obtained after multiplication by cos mx is uniformly convergent. By (23) all integrals on the right side, with the exception of the integral corresponding to cos mx, will be zero, so that obviously:

Similarly, multiplying the left and right sides of (22) by sin mx and integrating over the period, we get an expression for the coefficients:

and we have solved our problem. The numbers am and bm computed by formulas (24) and (25) are called the Fourier coeficients of the function f(x).

Let us take an example the function f(x) of period 2π illustrated in figure 13. Obviously this function is continuous and satisfies Dirichlet’s condition, so that its Fourier series converges uniformly to it.

It is easy to see that this function also satisfies the condition f(—x) = —f(x). The same condition also clearly holds for the function F1(x) = f(x) cos mx, which means that the graph of F1(x) is symmetric with respect to the origin. From geometric arguments it is clear that:

so that am = 0 (m = 0, 1, 2, ···). Further, it is not difficult to see that the functions F2(x) = f(x) sin mx has a graph which is symmetric with respect to the axis Oy so that:

But for even m this graph is symmetric with respect to the center π/2 of the segment [0, π], so that bm = 0 for even m. For odd m = 2l + 1 (l = 0, 1, 2, ···) the graph of F2(x) is symmetric with respect to the straight line x = π/2, so that: But, as can be seen from the sketch, on the segment [0, π/2] we have simply f(x) = x, so that by integration by parts, we get:

Thus we have found the expansion of our function in a Fourier series.

Convergence of the Fourier partial sums to the generating function. In applications it is customary to take as an approximation to the function f(x) of period 2π the sum:

of the first n terms of its Fourier series, and then there arises the question of the error of the approximation. If the function f(x) of period 2π has a derivative f(r)(x) of order r which for all x satisfies the inequality:

Then the error of the approximation may be estimated as follows:

Where cr is a constant depending only on r. We see that the error converges to zero with increasing n, the convergence being the more rapid the more derivatives the function has.

For a function which is analytic on the whole real axis there is an even better estimate, as follows:

Where c and q are positive constants depending on f and q < 1. It is remarkable that the converse is also true, namely that if the inequality (26) holds for a given function, then the function is necessarily analytic. This fact, which was discovered at the beginning of the present century, in a certain sense reconciles the controversy between D. Bernoulli and his contemporaries. We can now state: If a function is expandable in a Fourier series which converges to it, this fact in itself is far from implying that the function is analytic; however, it will be analytic, if its deviation from the sum of the first n terms of the Fourier series decreases more rapidly than the terms of some decreasing geometric progression.

A comparison of the estimates of the approximations provided by the Fourier sums with the corresponding estimates for the best approximations of the same functions by trigonometric polynomials shows that for smooth functions the Fourier sums give very good approximations, which are in fact, close to the best approximations. But for nonsmooth continuous functions the situation is worse: Among these, for example, occur some functions whose Fourier series diverges on the set of all rational points.

It remains to note that in the theory of Fourier series there is a question which was raised long ago and has not yet been answered: Does there exist a continuous periodic function f(x) whose Fourier series fails for all x to converge to the function as n = ∞? The best result in this direction is due to A. N. Kolmogorov, who proved in 1926 that there exists a periodic Lebesgue-integrable function whose Fourier series does not converge to it at any point. But a Lebesgue-integrable function may be discontinuous, as is the case with the function constructed by Kolmogorov. The problem still awaits its final solution.

To provide approximations by trigonometric polynomials to arbitrary continuous periodic functions, the methods of the so-called summation of Fourier series are in use at the present time. In place of the Fourier sums as an approximation to a given function we consider certain modifications of them. A very simple method of this sort was proposed by the Hungarian mathematician Fejér. For a continuous periodic function we first, in a purely formal way, construct its Fourier series, which may be divergent, and then form the arithmetic means of the first n partial sums:This is the Fejér sum of order n corresponding to the given function f(x). Fejér proved that as n = ∞ this sum converges uniformly to f(x).

**Approximation in the Sense of the Mean Square**

Let us return to the problem of the oscillating string. We assume that at a certain moment t0 the string has the form y = f(x). We can prove that its potential energy W, i.e., the work made available as it moves from the given position to its position of equilibrium, is equal (for small deviations of the string) to the integral: , at least up to a constant factor. Suppose now that we wish to approximate the function f(x) by another function ϕ(x). Together with the given string, we will consider a string whose shape is defined by ϕ(x), and still a third string, defined by the function f(x) — ϕ(x). It may be proved that if the energy:

of the third string is small, then the difference between the energy of the first two strings will also be small.* Thus, if it is important that the second string have an energy which differs little from the first, we must

try to find a function ϕ′(x) for which the integral (28) will be as small as possible. We are thus led to the problem of approximation to a function (in this case f′(x)) in the sense of the mean square.

Here is how this problem is to be stated in the general case. On the interval [a, b] we are given the function F(x), and also the function:

**ϕ(x, αo,α1,…,αn) 29 **depending not only on x but also on the parameters α0, α1, ···, αn. It is required to choose these parameters in such a way as to minimise:

Here the idea is to find the best approximation of the function F(x) by functions of the family (29), but only in the sense of the mean square. It is now unimportant for us whether or not the difference F — Ψ is small for all values of x on the interval [a, b]; on a small part of the interval the difference F — Ψ may even be large provided only that the integral (30) is small, as is the case, for example, for the two graphs illustrated in figure:

The smallness of the quantity (30) shows that the functions F and Ψ are close to each other on by far the greater part on the interval.

As to the choice in practice of one method of approximation or another, everything depends on the purpose in view. In the earlier example of the string, it is natural to approximate the function f′(x) in the sense of the mean square.

We should state that from the computational point of view the method of the mean square is more convenient, since it can be reduced to the application of well-developed methods of general analysis.

As an example let us consider the following characteristic problem.

We wish to make the best approximation in the sense of the mean square to a given continuous function f(x) on the interval [a, b] by sums of the form:where the αk are constants and the functions ϕk(x) are continuous and form an orthogonal and normal system.

This last means that we have the following equations:

These numbers ak are called the Fourier coefficients of f with respect to the ϕk.

For arbitrary coefficients αk, on the basis of the properties of orthogonality and normality of ϕk, we have the equation:

The first term on the right side of the derived equation does not depend on the numbers αk. Thus the right side will be smallest for those αk, which make the second term itself small, and obviously this can happen only if the numbers αk are equal to the corresponding Fourier coefficients ak.

Thus we have reached the following important result. If the functions ϕk, form an orthogonal and normal system on the interval [a, b], then the sum:will be the best approximation, in the sense of the mean square, to the function f(x) on this interval if and only if the numbers αk are the Fourier coefficients of the function f with respect to ϕk(x).

On the basis of equation (23) it is easily established that the functions:

form an orthogonal and normal system on the interval [0, 2π]. Thus the stated proposition, as applied to the trigonometric functions, will have the following form.

The Fourier sum Sn(x), computed for a given continuous function f(x) of period 2π, is the best approximation, in the sense of the mean square, to the function f(x) on the interval [0, 2π], among all trigonometric polynomials:of order n.

From this result and from Fejér’s theorem, formulated in §7, we are led to another remarkable fact.

Let f(x) be a continuous function of period 2π and σn(x) be its Fejér sum of order n, defined in §7 by equation (27).

We introduce the notation:

Since the Fourier sums Sk(x) (k = 0, 1, . . ., n) are trigonometric polynomials of order k≤n, it is obvious that σn(x) is a trigonometric polynomial of order n. Thus from the minimal property of the sum Sn(x) shown previously, we have the inequality:

“Since, by Fejér’s theorem, the quantity ηn converges to zero for n → ∞ we obtain the following important result.

For any continuous function of period 2π we have the equation:In this case we say that the Fourier sum of order n of a continuous function f(x) converges to f(x) in the sense of the mean square, as n increases beyond all bounds.

In fact, this statement is true for a wider class of functions, namely those which are integrable, together with their square, in the sense of Lebesgue.

We will stop here and will not present other interesting facts from the theory of Fourier series and orthogonal functions, based on approximation in the sense of the mean square.

## III.

## Waves of Energy in physics

**ITERATION≈ CONSERVATION OF PRESENT SPACE-TIME CYCLES**

In the graph, iterative waves of present space-time, carrying the temporal energy of the whole system, trying to reproduce and last for ever, making the present eternal in a dynamic way, generation upon generation, is the goal, meaning, and purpose of existence – to *maximise the function of present energy, in which past entropy and future information converge. In the graph we see on he left the present, iterative function of a particle-wave of quantum physics. It is a particle or a wave? Answer, both: ST (body-wave) > Tiƒ (head-particle) becomes the commonest physical and biological structure of reality, feeding over a field of entropy-exxpansive motion which is not considered part of the ‘whole’.*

Waves feeding on past fields f entropic, faster motions, iterating the dimensional form of informative, future particles, uncoiling its polar form into an extended wave, in imploding vertical motion converge into fields of wave creation which seen backwards and forwards in time give us two Spe>Tiƒ, and Tiƒ < Spe complementary events-states-topologies of reality,hyperbolically combined in iterative waves of constant presents dynamically moving around the attractor of its constant perfection, never fully realised by the irrational nature of all those ratios ± π surpassed or underpassed without completing a perfectly locked world-cycle.

The universe is all about creating virtual realities departing from a ratio of past to future space to time, form to motion event, converted into a lasting present balanced state between those two poles of energy and information, the field and particle states of the being.

The functions of that duality, the past>future ‘male’ state and the present≈present ‘female’state, differ, in as much as the preset body-wave state is an iterative function. Waves reproduce information in an ∆-1 substrata of space-time particles, while particles displace by ‘banging’, and expanding, devolving the ∆-1 substrata into a further down the ladder, ∆-2, entropic state as a field; while at the same time evolving into a tighter informative vortex the particle form.

So we talk of two fundamental functions-forms, the present, iterative body-wave and the past-future-past field-particle-field.

It follows that the concept of present will have multiple meanings. In the next graph we can see the wave and particle-field states (a reproductive wave and a pilot wave):

*The reproductive wave, which in present S=T states decouples into two ±particles, creating a virtual world cycle of existence, is the magic emergent phenomena that starts to construct the virtual holographic information of all beings of the Universe, zero sum world cycles, ∫∑ e x i ds dt =0 performed by the **super organism Œ= Spe ∆-1 < St ∆ > Tiƒ ∆+1 being.*

**Physical waves.**

Let us start by a quantum wave, which will be defined as everything else by a generator equation:

Γ: Spe (Pilot wave: Vs>c, t=νo)<ST (Vs≈cenergy wave)>Tiƒ: (Particle Vs<c, t≈νo)

In the quantum wave, whose correct interpretation is the ‘original’, De Broglie’s, the internal particle clock coincides with the frequency of the information wave, which comes from the lower non-local plane and gives information on the path the particle follows. Working as an ‘Alcubierre’ engine, the gravitational, ∆-1 non-local field provides the energy of motion to the particle, whose internal clock, ∆+1 synchronises with the past field and moves in inverse direction. The pilot non-local wave thus opens up space and the future particle moves. Both have ƒ=ƒo its internal clocks synchronise, but as the particle closes the cycle it goes slower than the information pilot wave. De broglie’s intuitions further expanded in its neutrino theory of light, define a deterministic ternary system of quantum ∆+1: particles: v <c: ∆-waves; v=c ∆-1: pilot fields V>c, of increasing Spe≈V entropy, and decreasing information, which put together generates a ternary system of space-time.

The particles position, where a clock of time of Mo rest mass ticks with the synchronicity of the non-local field that guides it, dragging behind its rest mass-memory of information, is thus a hidden variable (to us) but an existing parameter (for the particles), which will suffer deterministic, thermodynamic like laws, where the ‘heat’ parameter will translate to a macroscopic speed related description the microscopic clocks of those particles.

Each action, has a frequency, which added to the other actions will form a continuous motion in time, but in detail we shall see how the wave stops, informs, moves, reproduces and constantly switches on-off between minute actions of existence, such as the ultimate wave world cycle will be a sum of those e, i, ∆ A.

So the wave of present is sum of active quanta which in its final duration and cancelling values of ±∆e, i, will become a closed action≈motion in space-time: a world cycle.

But in that trajectory beings will try to remain in the middle point of Max. e xI (s=t) or present as much as possible. And if they don’t achieve it internally in its wave, externally often by generating a similar being will prolong its relative present. Thus reproduction and generation of presents, Max. e xi becomes the leif motif of most existential functions.

This we feel intuitively. When we bite a piece of food, a present quanta or ‘drama’, will be the period it takes us to feed on it. As such the fractal principle can further divide the present in 3 pi ‘intervals’ to complete the cycle, the entropic opening of the mouth, the longer taste and mastication and the final closing gulping. So a present also includes in its complete cycle a relative entropic past, steady state and future informative phase. Moreover there are many different quanta of present, according to the 7 different motions of time-space. If you make a step you will first open your legs in the entropic phase of the motion, then move ahead the second leg and then gather them both in the closing phase of the cycle. And that will be a ‘step’, a quanta of locomotion, a present event.

As we modularly change from motion to motion moving ahead the whole world cycle of our existence, presents constantly move and finally they complete a full world cycle of the being in its ∆+1 complete duration of time. A present action is thus a quanta of a whole worldcyle of existence between creation and extinction of a being, defined as a knot of space-time cycle, a fractal point traveling through the 5th dimension.

Let us then study the present and its different actions for the 3 type of systems of the Universe (physical, social and biological)… but first let us remember the basic formalism of GST.

**The symbols of fractal space-time and its ternary systems.**

A present is the duration in time of a cyclical, quantum Action. As such there are as many fractal presents, as the 7 types of actions≈motions in space time all beings perform to survive, all of different duration for each species.

The fractal Generator Equation of space-time:

Γ. Sp (past-lineal, toroid limbs-fields) ≤≥ ST (hyperbolic-planar body-waves) ≤≥ (spherical future particles-heads)

introduces some basic symbols of 5D Metric:

Γ, the G-enerator equation, which can be studied for each fractal space-time system in detail and its 3 parts.

<, the arrow of expansive entropy.

> the arrow of future information.

≈ =, the arrow of present balance, ST, that puts together.

Sp: Spatial Limbs/fields

Tƒ-cyclical, temporal clocks in present beings.

So, if we see a system as Sp>ST>Tƒ constantly ‘informing’, > through time the system, we are in the 3 ages series, in a diachronic system, as ‘time bends space into masses’ (Einstein).

But if we are:

Sp-limbs/fields > ST-body/waves <Tƒ-heads/particles, whereas the limbs/fields give energy to the body/wave and the heads-particles give it information, we are in a living organism in balance,in a relative present immortal state.

And this is the ideal state of existence, since ‘The Universe conserves its present’.

**THE ETERNAL SEARCH FOR THE ETERNAL PRESENT.**

‘Women are present, men look at the past and mould the future’ Schopenhauer, on the temporal orientation of gender.

The only western philosopher who captured part of the meaning of present was Schopenhauer. The only physicists, Einstein and Feynman. But biologists are more aware of the essential law of balance of the Universe and all its parts, which try to conserve the present, *from the natural, dominant arrow of ‘long time intervals’ towards higher information, warping of space into time, wrinkling of energy into form, caused by the Œ-mind or still point of view. And it is this tug of war between present body-waves-re=productive working classes and future still, informative particle-heads-linguistic selfish elites (financial-political in control of money and laws), what we see mostly on the Universe at large.*

Why this sum does not maintain a present in each scalar being is obvious: the field-particle, limb-head system, Spe>Tiƒ *has a derive that favours the head-particle of information, and so if we ad the present body-wave that is balanced Spe=Tiƒ, we still have a motion to the future of information. *

But for all fractal parts, there is for the whole to keep its balance a final reckoning when future information explode into death entropy: Tiƒ<<Spe. And this is what allows the whole Universe to be immortal but none of its parts to be.

So the more synoptic description of reality says:

‘The Universe conserves present space-time by killing into entropic past, all its parts that move to its informative future, guided by its dominant particle-head-upper class state’.

That is, Sp x Tƒ = ST, fields of kinetic energy and temporal form come together

Reproduction is repetition of a present, which seems not to change but merely renews itself with generational cycles of clone beings. Reproduction is the ideal of all systems. Because the Universe is in perpetual motion, the present do change dynamically in all closed systems, even if the generational, reproductive cycle makes its variations minimal.

Thus the true theme of the Present is the iterative properties of cyclical time.

*Since in a dynamic Universe present means always the repetition of a system, and we can indeed consider the Universe a constant present orgasm of repetition and balance of past and futures, which converge into present forms.*

In that regard, the 3 symmetries of 5D space-time come together into a present, wave-body, single plane, ∆, of existence. In this sense the present is also the minimal self-centred point of view, in the minimal timespace, origin of it all – an expansion of present, frozen ‘constants’, which dilate into time past and future, space line and cycle, scales above and below.

The Universe is illogic but it is not magic. And yet, quantic jumps create phenomena that seem magic to us. Let us consider 2 examples from the physical and biological world, the consciousness of man and the instantaneous speed of gravitation:

Asymmetric, temporal, quantic flows of information from i to ∆-1 planes of existence explain how the particles of a higher social class like the brain neurons control electronically the cells of the body, anticipating its reactions, since those body cells use a slower, past chemical language and exist in a relative past, in an ‘inferior plane of existence’.

Do we have proves of that temporal distance between planes of existence?* Indeed a few years ago, scientists found surprisingly that the body acts before the brain thinks and yet most people think first before acting.*

*How then it is possible that we think first and yet we act also first, and both things happen simultaneously? Precisely because as Einstein put it, past and future co-exist in a simultaneous present, as information flows from the future brain to the past body and energy flows, moves and acts in the body before, in the past. So depending on our point of view, the brain or the body we will see our action first, and since our ‘is’ are mental ‘is’ we perceive first our thoughts. Yet in reality both åctions converge into a present.*

So to the external world both mind and body act chained ‘together’ by the simultaneity of the quantic åctions of the body cells and the mind’s thought into a single whole action . Yet the chemical cells of the body exist in a micro-fraction of time, in a relative past, separated from the ‘whole being’ by a quantic jump in time. This means that when the brain observes the chemical cells of the body it can control them ‘a priori’ as it knows what they will act before they act…

*So the body acts in past, the brain thinks in the future and both together create our present of existence, as *information travels or is emitted from the future through a quantic jump to the past cells and energy is emitted or travels from the past body in a quantic jump into the future.

So quantic jumps in time explain both the hierarchical structure and creation of symbiotic organisms, in which relative past and future forms co-exist, the control of the mind and the subconscious åctions of the body. Thus in a human being, relative past cells and future nervous systems can co-exist in a symbiotic manner and chain each other’s cyclical åctions*, *thanks to the existence of quantic jumps in space and time that coordinate in simultaneity the fluxes of energy and information between the cells and neurons of those planes.

Yet the most obvious prove of quantic jumps to the past are the processes of death, which people who suffered it and then have survived, explain as a rewinding of the memories of the being that travels through the entire information of his life towards the past.

On the other hand in the physical world quantic jumps to the past have been proved in the past decade. Since rays of light, travelling faster than c, come out in the instrument before they are produced.

It also explains why gravitation is instantaneous; something that has always puzzled scientists. And yet, quantic jumps in time explain that instantaneous speed easily:

A mass emits a graviton, its quantic, cellular form of past in a relative past. Hence, when the graviton travels forwards to the future from that past it consumes some time and it surfaces again into the relative present of the observer with an instantaneous speed*.* Since a quantic wave of gravitation can reach up to 10^{10} million years light, a single gravitational wave can cross in a single quantic jump, the entire Universe. And indeed, the ‘instantaneous, present space-time Universe’ reaches till around 10^{10} million years light, the so-called Universal horizon.

**The breaking of energy and information symmetries in the limits between planes.**

In that regard, the algebra of multiple planes of existence widens even farther the laws of conservation of energy and information, as energy might disappear from a plane of existence but the total energy x information quantity of the ternary, 3-plane structure of a World remains the same.

Which in physics explains both a fundamental law of conservation, the so-called CPT parity; and a few seemingly exceptions to that law in which energy or information disappears from a plane of existence.

For example, black holes erase information from the electromagnetic world but they emit it as ‘gravitational, dark energy’ through its poles* .*

All this means that neither the principle of action -reaction nor the Laws of conservation are ‘absolute’, when we consider only a ‘single plane of space-time’; as there are uneven flows of energy and information between 2 planes.

So in Physics, only the total energy and information of the particles or CPT parity conserves; but the C or P or T parities might break in processes of death and birth of particles that give or take energy and information from other plane of existence, without returning it to the perceived, electromagnetic world physicists observe.

Especially when we consider the 3rd case of ‘death of a particle’, there is a transference of energy and information that jumps between 3 planes of space-time, (∆-2<=>i). So energy and information truly disappear and the laws of conservation that work for the total 3 relative planes of the organic system break for the planes we can observe.

A fact that explains empirically many invisible, physical ‘imaginary forces’ and ‘particles’, postulated by physicists to maintain those theoretical conservation laws, without realizing they are observing an asymmetric transference of energy and information and so do not need to postulate ‘real particles’ but ‘asymmetric ST flows’.

Indeed, in all processes that involve the death of a particle, which breaks the balance between the energy and information of a system physicists have found a lost of energy and information they cannot explain.

Now we can explain why in the proximities of black holes, which transfer energy and information between the gravitational and electromagnetic world information is lost. Or why in radioactivity processes that destroy an atom or close to the limits of C-speed, the conservation of momentum and energy of a particle breaks. A similar case could be, for the mystique reader, the famous 21 grams that a person looses when it dies. Where they are? *.*

So instead of ‘postulating’ a series of invisible particles and speed limits to keep reality confined to a single plane of existence, Quantic Spaces-Times theory explains all those mysterious exceptions to the second law of Thermodynamics; the laws of conservation and the C-limits of speed in Relativity as S-T flows between 2 planes of existence. Obviously physicists have never witnessed the imaginary particles that come out of a dying neutron or a radioactive atom, called neutrinos and weak forces, invented to keep the total momentum of the dying particle, but only indirect flows of energy and information.

*Since indeed neutrinos and weak forces are not ‘particles’ in the strict sense, but flows of energy and information transferred *between the gravitational and electromagnetic planes of existence* when particles mutate their form or die, giving away its energy and information, as they explode into smaller ∆-1 quanta of the gravitational or electromagnetic ‘plane of existence’ .*

Those flows of ‘energy and information’ sometimes return to our world and give birth to a new particle; sometimes create a particle that flows further into the past called an antiparticle; sometimes give away its energy as a gravitational wave we cannot perceive; or sometimes they appears as an inverse, negative energy or a particle with negative mass, as it happens with neutrinos; precisely because its time-space coordinates become reversed as in all process of death that write as Sp>T -> T<Sp.

So the dying form particle suffers a reversal in its space-time coordinates, becoming in algebraic terms a flow of negative energy or an antiparticle.

**The 3 sub-type of waves according to topobiological equations. **

In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is

u{tt} – u{xx} = 0

The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the line t = 0 (with sufficient smoothness properties), then there exists a solution for all time t.

The solutions of hyperbolic equations are “wave-like.” If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed. They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.

**Quantic jumps and biological radiations**

Reproduction=repetition of quantic presents is the main phenomena that creates reality and truly embodies the will of the space-time field, which is to exist, to overcome the quantic limits of death in space and time, *by reproducing one’s own form in other zone of quantic space-time*. So we could truly say that the game of existence is a game of reproduction, that there is a Universal mandate for all space-time forms: grow=evolve and multiply=reproduce.

*So both, translation change and morphological change are 2 forms of reproduction.* Translation change happens in simpler, energetic species that ‘reproduce in space’. Morphological change happens in more informative species that ‘reproduce more information’ and so hardly move when reproducing.

It is the case of living beings whose reproduction is called a biological radiation; while in Physics we talk of a wave radiation. Indeed, the best way to describe the reproductive, organic drives of all species is the concept of a ‘radiation’. Biologists say that species reproduce in growing numbers so fast that they seem to ‘radiate’. The Universe is made of such biological radiations that we will extend in this book to all kind of particles, waves and discontinuous Space-Time fields that repeat their cycles of existence into waves, new particles, new species, new worlds…

We talk of a dual ‘body-brain’ rhythm: species radiate in space, reproducing their form, and then they evolve in time into macro systems. So we write that essential rhythm of existence as: *Max. Reproduction-> Max. Social evolution: Re>T>Re>T: Repetition of present space => Evolution from Past to future.*

It is also the absolute rhythm of existence of the Universe and any of its parts. In that sense we describe the big bang as a series of radiation of beings of increasing complexity. First the singularity’s explosion radiated the simplest space-times known to man, gravitational space. Over it, it radiated particles in several phases.

The first radiations were of light, photons, electrons and quarks that evolved into atoms. The radiations of inorganic atoms and molecules grew in societies so big as planetoids that evolved informatively through gravitation into planets and stars. The radiation of stars created galaxies and the radiation of galaxies Universes. The radiations of organic molecules created the organic soup of the Earth that evolved into complex DNA-societies, called cells.

The Earth’s radiations and social evolutions of cells went then through a series of discontinuous: bacteria, plants, sea animals, land animals, amphibian, reptiles and dinosaurs, mammals, men, machines… Since in fact the Earth’s radiations are not only of life forms, but also radiations of machines and products reproduced by ’company-Mothers’ in the economic ecosystem. So we affirm that the Universe exists as a constant game of radiations of species who extinct or prey on other species, creating ecosystems or ‘hierarchical organisms’ where the relative past, simple form becomes enslaved by the more complex, future form. Almost every phenomenon of death and life perceived in the Universe has a common cause: the radiation of a species that feeds on others, used as energy over which the radiating being reproduces itself. Again the Bible speaks of such radiations in a poetic sense.

The first day, the first radiation was of light. Then light broke away from dark matter. The day separated from the night. Then on the Earth the radiation of life started in water, it brought cells, fishes and more complex beings till creating mankind. The writer of Genesis understood that creative nature of the Universe, not because a special God illuminated him but because he knew as any atom or cell of space-time what he ‘had to do’. And so he said it to man: ‘*Grow*=become more complex, and *evolve*, multiply=radiate’.

**The law of biological radiations: The game of existence**

The struggle for reproductive existence, following the drives of organicism implies that species kill each other to obtain ‘form’, and ‘energy’, the two parameters of the Universe, in order to exist, and reproduce. Since reproduction is the mandate of all species, that fight becomes a continuous struggle that leads to the extinction of species to feed the reproduction of others. The reproduction of species and their fight for organic space with other species is indeed the fundamental concept that allows us to organize the evolution and change we see in the Universe. We call those collective acts of reproduction of any species, a ‘biological radiation’. Biologists use the word ‘Radiation” because reproduction is a very fast, geometric function that happens almost as fast as an explosion of a bomb. From one it comes two, from 2, 4, then 8, 16, 32, 64, 128, 256, 512, 1024…

Suddenly after a mere 10 acts of reproduction, you have a thousand times more species… That explosive rate of reproduction that you can see for example in a culture of bacteria explains the term ‘biological radiation’. We could indeed consider the Universe a game of radiations of species that use other species as energy to reproduce themselves. The Universe is made of such biological radiations, driven by organicism. Radiations of species, become groups, that reproduce and feed in surfaces of energy, shaping the perceived Universe. The Big Bang was a radiation that took place in several phases. The first radiation was of light, of photons. The second radiation was of electrons.

The third radiation was of masses. The most complex radiations of Atoms and molecules grew in societies so big as planetoids, so complex as DNA-societies [human Bodies]. DNA radiations started species of living beings started. In the world of masses radiations grew into stars and galaxies… Almost every phenomena of death and life perceived in the universe have a common cause: the radiation of a being that feeds in other beings, used as energy, over which the radiating being reproduces himself. Proves that more and more phenomena are caused by radiations appear day by day. Certain phenomena that before were explained in abstract manner now can also be explained as radiations.

What radiates is a form, a complex system of forms, a being. The Bible speaks of such radiations in a poetic sense. The first day, the first radiation was of light. Then light broke away from dark matter. The day from the night. The radiation of mass grew into beings, more and more complex, till bringing mankind. Those were the days of creation, the days of radiations, and the writer of Genesis understood that radiations, are the nature of the creative Universe. “Grow=become more complex, and multiply=radiate” was his advice to man. The Earth’s radiations of beings have evolved through a series of discontinuous: bacteria, plants, sea animals, land animals, amphibian, reptiles and dinosaurs, mammals, men, machines… Since, in fact radiations are not only of life, but of all forms. There are also metallic radiations, radiations of machines and products reproduced by company-mothers in the economical system.

The present is the dimension of ≈o, Actions of repetition, iteration, offspring creation, from an U function. A wave form is the commonest shape of present. We could consider the Universe a constant reproductive wave upon fields of energy of any given Universal Plane of the 5^{th} dimension:

U=∫ U dv

The previous equations resumes most of the equations.

* Entropy as it is understood in current science is an enlarged blown up of only 1/3d of the Universe to absolute dogma, due to the lineal error of time duration. More interesting though is entropy. *

* *

**The equivalence with the metric of the 5th dimension.**

And so this simple program of existence: Max. Sp X Tƒ, is the counterpart to the Metric of the 5th dimension Sp X Tƒ=i, which explains it. Who was first the egg or the chicken? They are self-similar functions, one seeing internally and the other externally.

Since when we analyze the function Sp X Tƒ, it is maximized when S=T, thus the will of existence brings also homeostasis and balance and the sum of all those S=T, could be considered the constant i, of the metric of any ∆-plane.

**The Point of balance: S=T**

So of the many similar equations we use to write the Fractal Generator that represents all events and forms of the Universe, the most important is Se<=>Tƒ, which represents the search for a point of balance in which the Spatial Energy and Temporal Information of the System, is maximized by that balance.

This is the point of adulthood, the equation of present, the steady state, the equation of beauty, the equation that maximizes the ‘existential force’ of the system, Max. Sp X Tƒ (which is reached when S=T). The number of ‘events’ that take place under the umbrella of this equation is enormous.

We could say that S=T is the point of immortality that all systems try to reach.

The partial equations of the Generator are ‘partial events in space-time’, of significant importance for all species, as they describe åctions, dimensions, and symmetries, states, ages and organic parts fundamental to the whole.

Among all them the most important is the ‘function of existence’, that is the strategy to maximal Sp X Tƒ, which is the equality between the energetic components and states in space and time of a l Superorganism:

*Function of Existence; Top Predator Equation: Max Sp X Tƒ->S=T*

Equation of Justice, Reproduction, Present and beauty: S=T

‘And God said S=T and extinguished all beings that did not obey the law of Justice.’

The most astounding discovery of General Systems Sciences is in the metaphysical realm – the understanding of both mathematical and logical, bio-logical, existential processes together, through the ‘simultaneity’ of mind’s mappings of the Universe.

The fractal of the Universe is a mind observing a fractal world of e<=>TƒO systems constantly flowing in two type of motions, lineal motion or distance or speed or energy and cyclical motion or perceptive, dimensional form, cyclical information, clock frequency.

2 states of existence among which all beings fluctuate by moving that ‘élan’, that ‘motion’ that ‘spirit’, that ‘substance’, that ‘form’, whatever you might say of it.

Between motion and form, energy sensation and information sensation, translation and perception, Sp and Tƒ, | and O.

Sp<=>Tƒ<Sp….

The Universe is a creative, temporal, formal game of creation of ‘Functions of Existence’, Sp X Tƒ, Sp<=>Tƒ, complementary systems of energy and information that try to maximize the communication between both poles, Sp and Tƒ, a fact maximized by a simple mathematical equation:

Max. Sp X Tƒ ->S=T

This equation has much more meaning than its simplicity might evoke. What it says is that when we have two entities, one of maximal energy, Sp and one of maximal information, Tƒ, treated as ‘sets’ or ‘numbers’, or ‘herds’ or ‘groups of cells’, such as the number of elements of Sp, A, and the number of elements of O, B, are equal, A=B, the communication and force of the group, Max. Sp X Tƒ is higher. And the system is better integrated between field/body and particle/head.

Such systems are top predator systems that survive better. The law of balance or justice is one of the most metaphysical and yet the most important of all laws of the Universe. It gives a survival strategy to all entities in existence. And its two limits when the law of justice is broken are the equation of death:

Max.Sp x Min. Tƒ = death by accident.

Min. Tƒ x Min. Sp = death by warping and excess of information.

Tƒ exist in balance, S=T, obviously the best strategy is to maximize the communication between Sp and Tƒ, so each element of Sp communicates with each element of Tƒ, a x b is exactly the product of all those communications. And so what happens in nature is that the neurons or informative cells of the Tƒ -system and the cells of the body system, reach maximal communication to act simultaneously as a single whole.

The Function of Existence, S=T or Max. Sp X Tƒ (both are mathematically equivalent, as the product is maximized by the equality of the factors, so 5×5> 4 x6 … > 9×1…) is the fundamental equation of the Universe.

Sp<=>Tƒ resumes all the events of reality. But of its 3 possible ‘ages’, ‘dimensions’ or ‘states’, Max. Sp (Sp>Tƒ) , S=T, Max Tƒ (Sp<Tƒ), S=T, the equation of balance, beauty, maturity, classicism, reproduction, present, existence, Max. Sp x is the goal of the Universe.

And those who achieve it achieve immortality.

**Conclusion. The iterative body-waves of presents.**

The universe is about iterating fractal waves of present bodies, surfing on the energy of existence, tracing its world cycles with architectural precision, sculpting its feelings of information, as they feel a n ever ending flow upwards and downwards. the function of existence believes to be conducting itself in the wave, but the wave limits its paths and finally drags him down into the cliffs of death…

The Universe like its presents, Sxt (S=t) where only minutes crests seem to disturb the peace of the whole mass of equal, presents, repeating its motions of existence. By the mere fact that an action is motion with form, the Universe becomes immortal. It is ultimately a present as topologies combine into hyperbolic minds-infinitesimal truths, and dissolve into toroidal infinite simplex planes of energy, awaiting form to organise its micro points through networks, ∆+1 of a high ‘perceived’ existence, towards which the points will gravitate in search of information and pleasure, energy and form, perception and relaxation, lazy workers of the stairs of a never ending game where cliffs erase but never stop the grinding eternity of flash backs, and feed-back generations…

## IV.

## Reproductive Bodies in biology.

Volterra curves of reproduction of prey Tiƒ=top predator and Spe=prey on a closed ecosystem with prey saturating if alone its total mass, shows those balances.

Reproductive body waves then become chaotic and the information of the wave stops reproducing in a perfect form.

In those volterra curves happens around the constant of feigenbaum.

Iteration is thus the *true meaning of present which conserves itself because it iterates a world cycle zero sum, restarting the game of existence, **which*** **could not happen without iteration.

So the present, reproductive cycle of systems require a higher order of determinism, unlike markowian processes of chaotic bifurcations, in which causality is lost as time ‘dissolves’ and expansion of entropic space ensues.

## V.

## Reproductive middle classes in sociology