The entropy of the universe causes the universe to be random leading to chaos but chaos is not random it's transformation it is consciousness evolving to a higher state/level it is evolving to become an infinite fractal.

## What is Chaos Theory?

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.

Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Many natural objects exhibit fractal properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit complex, chaotic behavior. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.

### Principles of Chaos

- The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and fractals will be?
- Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the World, accurate long-range weather prediction will always remain impossible.
- Order / Disorder Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.
- Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighboring water molecules may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.
- Feedback: Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.
- Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

### Chaos as a spontaneous breakdown of topological supersymmetry

In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, which is an intrinsic property of evolution operators of all stochastic and deterministic (partial) differential equations. This picture of dynamical chaos works not only for deterministic models, but also for models with external noise which is an important generalization from the physical point of view, since in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics (e.g., the butterfly effect) is a consequence of the Goldstone's theorem—in the application to the spontaneous topological supersymmetry breaking.

### Quantum chaos

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

In seeking to address the basic question of quantum chaos, several approaches have been employed:

- Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
- Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).
- Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
- Direct application of the correspondence principle.

In 1917 Albert Einstein wrote a paper that was completely ignored for 40 years. In it he raised a question that physicists have only, recently begun asking themseves: What would classical chaos, which lurks everywhere in our work do to quantum mechanics, the theory describing the atomic and subatomic worlds? The effects of classical chaos, of course, have long been observed - Kepler knew about the motion of the moon around the earth and Newton complained bitterly about the phenomenon. At the end of the 19th century the American astronomer William Hill demonstrated that the irregularity is the result entirely of the gravitational pull of the sun. So thereafter, the great French mathematician-astronomer-physicist Henri Poincare surmised that the moon's motion is only mild case of a congenital disease affecting nearly everything. In the long run Poincare realized, most dynamic systems show no discernible regularity or repetitive pattern. The behavior of even a simple system can depend so sensitively on its initial conditions that the final outcome is uncertain. At about the time of Poincare's seminal work on classical chaos, Max Planck started another revolution, which would lead to the modern theory of quantum mechanics. The simple systems that Newton had studied were investigated again, but this time on the atomic scale. The quantum analogue of the humble pendulum is the laser; the flying cannonballs of the atomic world consist of beams of protons or electrons, and the rotating wheel is the spinning electron (the basis of magnetic tapes). Even the solar system itself is mirrored in each of the atoms found in the periodic table of the elements. Perhaps the single most outstanding feature of the quantum world is its smooth and wavelike nature. This feature leads to the question of how chaos makes itself felt when moving from the classical world to the quantum world. How can the extremely irregular character of classical chaos be reconciled with the smooth and wavelike nature of phenomena on the atomic scale? Does chaos exist in the quantum world'? Preliminary work seems to show that it does. Chaos is found in the distribution of energy levels of certain atomic systems; it even appears to sneak into the wave patterns associated with those levels. Chaos is also found when electrons scatter from small molecules. I must emphasize, however, that the term 'quantum chaos' serves more to describe a conundrum than to define a well-posed problem.

Considering the following interpretation of the bigger picture may be helpful in coming to grips with quantum chaos. All our theoretical discussions of mechanics can be somewhat artificially divided into three compartments [see illustration] although nature recognizes none of these divisions. Elementary classical mechanics falls in the first compartment. This box contains all the nice, clean systems exhibiting simple and regular behavior, and so I shall call it R, for regular. .Also contained in R is an elaborate mathematical tool called perturbation theory which is used to calculate the effects of small interactions and extraneous disturbances, such as the influence of the sun on the moon's motion around the earth. With the help of perturbation theory, a large part of physics is understood nowadays as making relatively mild modifications of regular systems. Reality though, is much more complicated; chaotic systems lie outside the range of perturbation theory and they constitute the second compartment. Since the first detailed analyses of the systems of the second compartment were done by Poincare, I shall name this box P in his honor. It is stuffed with the chaotic dynamic systems that are the bread and butter of science. Among these systems are all the fundamental problems of mechanics, starting with three, rather than only two bodies interacting with one another, such as the earth, moon and sun, or the three atoms in the water molecule, or the three quarks in the proton. Quantum mechanics, as it has been practiced for about 90 years, belongs in the third compartment, called Q. After the pioneering work of Planck, Einstein and Niels Bohr, quantum mechanics was given its definitive form in four short years, starting in 1924. The seminal work of Louis de Broglie, Werner Heisenberg, Erwin Schrodinger, Max Born, Wolfgang Pauli and Paul Dirac has stood the test of the laboratory without the slightest lapse. Miraculously. it provides physics with a mathematical framework that, according to Dirac, has yielded a deep understanding of 'most of physics and all of chemistry" Nevertheless, even though most physicists and chemists have learned how to solve special probleins in quantum mechanics, they have yet to come to terms with the incredible subtleties of the field. These subtleties are quite separate from the difficult, conceptual issues having to do with the interpretation of quantum mechanics. The three boxes R (classic, simple sytems), P (classic chaotic systems) and Q (quantum systems) are linked by several connections. The connection between R and Q is known as Bohr's correspondence principle. The correspondence principle claims, quite reasonably, that classical mechanics must be contained in quantum mechanics in the limit where objects become much larger than the size of atoms. The main connection between R and P is the Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theorem provides a powerful tool for calculating how much of the structure of a regular system survives when a small perturbation is introduced, and the theorem can thus identify perturbations that cause a regular system to undergo chaotic behaviour. Quantum chaos is concerned with establishing the relation between boxes P (chaotic systems) and Q (quantum systems). In establishing this relation, it is useful to introduce a concept called phase space. Quite amazingly this concept, which is now so widely exploited by experts in the field of dynamic systems, dates back to Newton. The notion of phase space can be found in Newton's mathematical Principles of Natural Philosophy published in 1687. In the second definition of the first chapter, entitled "Definitions", Newton states (as translated from the original Latin in 1729): "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly." In modern English this means that for every object there is a quantity, called momentum, which is the product of the mass and velocity of the object. Newton gives his laws of motion in the second chapter, entitled 'Axioms, or Laws of motion.' The second law says that the change of motion is proprotional to the motive force impressed. Newton relates the force to the change of momentum (not to the acceleration as most textbooks do).

Momentum is actually one of two quantities that, taken together, yield the complete information about a dynamic system at any instant. The other quantity is simply position, which determines the strength and direction of the force. Newton's insight into the dual nature of momentum and position was put on firmer ground some 130 years later by two mathematicians, William Rowan Hamilton and Karl Gustav-Jacob Jacobi. The pairing of momentum and position is no longer viewed in the good old Euclidean space or three dimensions; instead it is viewed in phase space, which has six dimensions, three dimensions for position and three for momentum. The introduction of phase space was a powerful step from a mathematical point of view, but it represents a serious setback from the standpoint of human intuition. Who can visualize six dimensions? In some cases fortunately phase space can be reduced to three or even better, two dimensions. Such a reduction is possible in examining the behavior of a hydrogen atom in a strong magnetic field. The hydrogen atom has long been a highly desirable system because of its simplicity. A lone electron moves around a lone proton. And yet the classical motion of the electron becomes chaotic when the magnetic field is turned on. How can we claim to understand physics if we cannot explain this basic problem?

Under normal conditions, the electron of a hydrogen atom is tightly bound to the proton. The behavior of the atom is governed by quantum mechanics. The atom is not free to take on any arbitrary energy, it can take on only discrete, or quantized, energies. At low energies, the allowed values are spread relatively far apart. As the energy of the atom is increased, the atom grows bigger, because the electron moves farther from the proton, and the allowed energies get closer together. At high enough energies (but not too high, or the atom will be stripped of its electron!), the allowed energies get very close together into what is effectively a continuum, and it now, becomes fair to apply the rules of classical mechanics. Such a highly excited atom is called a Rydberg atom. Rydberg atoms inhabit the middle ground between the quantum and the classical worlds, and they are therefore ideal candidates for exploring Bohr's correspondence principle which connects boxes Q (quantum phenomena) and R (classic phenomenal). If a Rydberg atom could be made to exhibit chaotic behavior in the classical sense, it might provide a clue as to the nature of quantum chaos and thereby shed light on the middle ground between boxes Q and P (chaotic phenomena). A Rydberg atom exhibits chaotic behaviour in a strong magnetic field, but to see this behavior we must reduce the dimension of the phase space. 'The first step is to note that the applied magnetic field defines an axis of symmetry through the atom. The motion of the electron takes place effectively in a two-dimensional plane, and the motion around the axis can be separated out; ornly the distances along the axis and from the axis matter. The symmetry of motion reduces the dimension of the phase space from six to four. Additional help comes from the fact that no outside force does any work on the electron. As a consequence, the total energy does not change with time. By focusing attention on a particular value of the energy, one can take a three-dimensional slice-called an energy shell-out of the four-dimensional phase space. The energy shell allows one to watch the twists and turns of the electron, and one can actually see something resembling a tangled wire sculpture. The resulting picture can be simplffied even further through a simple idea that occurred to Poincare. He suggested taking a fixed two-dimensional plane (called a Poincare section, or a surface of section) through the energy shell and watching the points at which the trajectory intersects the surface. The Poincare section reduces the tangled wire sculpture to a sequence of points in an ordinary plane. A Poincare section for a highly excited hydrogen atom in a strong magnetic field is shown on the opposite page. The regions of the phase space where the points are badly scattered indicate chaotic behavior. Such scattering is a clear symptom of classical chaos, and it allows one to separate systems into either box P or box R.

What does the Rydberg atom reveal about the relation between boxes P and Q? I have mentioned that one of the trademarks of a quantum mechanical system is its quantized energy levels, and in fact the energy levels are the first place to look for quantum chaos. Chaos does not make itself felt at any particular energy level, however; rather its presence is seen in the spectrum, or distribution, of the levels. Perhaps somewhat paradoxically in a nonchaotic quantum system the energy levels are distributed randomly and without correlation, whereas the energy levels of a chaotic quantum system exhibit strong correlations [see illustration]. The levels of the regular system are often close to one another, because a regular system is composed of smaller subsystems that are completely decoupled. The energy levels of the chaotic system, however, almost seem to be aware of one another and try to keep a safe distance. A chaotic sytem cannot be decomposed; the motion along one coordinate axis is always coupled to what happens along the other axis.

The spectrum of a chaotic quantum system was first suggested by Eugene P. Wigner, another early master of quantum mechanics. Wigner observed, as had many others, that nuclear physics does not possess the safe underpinnings of atomic and molecular physics: the origin of the nuclear force is still not clearly understood. He therefore asked whether the statistical properties of nuclear spectra could be derived from the assumption that many parameters in the problem have definite, but unknown values. This rather vague starting point allowed him to find the most probable formula for the distribution. Oriol Bohigas and Marie-Joya Giannoni of the Institute of Nuclear Physics in Orsay France, first pointed out that Wigner's distribution happens io be exactly what is found for the spectrum of a chaotic dynamic system.

Chaos does not seem to limit itself to the distribution of quantum energy levels, however, it even appears to work its way into the wavelike nature of the quantm world. The position of the electron in the hydrogen atom is described by a wave pattern. The electron cannot be pinpointed in space; it is a cloudlike smear hovering near the proton. Associated with each allowed energy level is a stationary state, which is a wave pattern that does not change with time. A stationary state corresponds quite closely to the vibrational pattern of a membrane that is stretched over a rigid frame, such as a drum. The stationary states of a chaotic system have surprisingly interesting structure, as demonstrated in the early 1980s by Eric Heller of the University of Washington. He and his students calculated a series of stationary states for a two-dimensional cavity in the shape of a stadium. The corresponding problem in classical mechanics was known to be chaotic, for a typical trajectory quickly covers most of the available ground quite evenly. Such behavior suggests that the stationary states might also look random, as if they had been designed without rhyme or reason. In contrast. Heller discovered that most stationary states are concentrated around narrow channels that form simple shapes inside the stadium, and he called these channels "scars" [see illustration]. Similar structure can also be found in the stationary states of a hydrogen atom in a strong magnetic field [see illustration] The smoothness of the quantum wave forms is preserved from point to point, but when one steps back to view the whole picture, the fingerprint of chaos emerges. It is possible to connect the chaotic signature of the energy spectrum to ordinary classical mechanics. A clue to the prescription is provided in Einstein's 1917 paper, He examined the phase space of a regular system from box R and described it geometrically as filled with surfaces in the shape of a donut; the motion of the system corresponds to the trajectory of a point over the surface of a particular donut. The trajectory winds its way around the surface of the donut in a regular manner, but it does not necessarily close on itself. In Einstein's picture, the application of Bohr's correspondence principle to find the energy levels of the analogous quantum mechanical system is simple. The only trajectories that can occur in nature are those in which the cross section of the donut encloses an area equal to an integral multiple of Planck's constant, h (2pi times the fundamental quantum of angular momentum having the units of momentum multiplied by length). It tums out that the integral multiple is precisely the number that specifies the corresponding energy level in the quantum system. Unfortunately as Einstein clearly saw, his method cannot be applied if the system is chaotic, for the trajectory does not lie on a donut and there is no natural area to enclose an integral multiple of Planck's constant. A new approach must be sought to explain the distribution of quantum mechanical energy levels in terms of the chaotic orbits of classical mechanics. Which features of the trajectory of classical mechanics help us to understand quantum chaos? Hill's discussion of the moon's irregular orbit because of the presence of the sun provides a clue. His work represented the first instance where a particular periodic orbit is found to be at the bottom of a difficult mechanical problem. (A periodic orbit is tike a closed track on which the system is made to run: there are many of them, although they are isolated and unstable.) Inspiration can also be drawn from Poincare, who emphasized the general importance of periodic orbits. In the begining of his three-volume work, "The New Methods of Celestial Mechanics" which appeared in 1892, he expresses the belief that periodic orbits "offer the only opening through which we might penetrate into the fortress that has the reputation of being impregnable." Phase space for a chaotic system can be organized, at least partially around periodic orbits, even though they are sometimes quite difficult to find.

In 1970 I discovered a very general way to extract information about the quantum mechanical spectrum from a complete enumeration of the classical periodic orbits. The mathematics of the approach is too difficult to delve into here, but the main result of the method is a relatively simple expression called a trace formula. The approach has now been used by a number of investigators, including Michael V. Berry of the University of Bristol, who has used the formula to derive the statistical properties of the spectrum. I have applied the trace formula to compute the lowest two dozen energy levels for an electron in a semiconductor lattice, near one of the carefully controlled impurities. (the semicondoctor, of course, is the basis of the marvellous devices on which modern life depends; because of its impurities, the electrical conductivity of the material is half-way between that of an insulator, such as plastic, and that of a conductor, such as copper.) The trajectory of the electron can be uniquely characterized by a string of symbols, which has a straightforward interpretation. The string is produced by defining an axis through the semiconductor and simply noting when the trajectory crosses the axis. A crossing to the "positive" side of the axis gets the symbol +, and a crossing to the 'negative" side gets the symbol -. A trajectory then looks exactly like the record of a coin toss. Even if the past is known in all detail even if all the crossings have been recorded-the future is still wide open. The sequence of crossings can be chosen arbitrarily. Now, a periodic orbit consists of a binary sequence that repeats itself; the simplest such sequence is (+ -), the next is (+ -), and so on (Two crossings in a row having the same sign indicate that the electron has been trapped temporarily.) All periodic orbits are thereby enumerated, and it is possible to calculate an appropriate spectrum with the help of the trace formula. In other words, the quantum mechanical energy levels are obtained in an approximation that relies on quantities from classical mechanics only. The classical periodic orbits and the quantum mechanical spectrum are closely bound together through the mathematical process called Fourier analyis. The hidden regularities in one set, and the frequencies with which they show up, are exactly given by the other set. This idea was used by John B. Delos of the College of William and Mary and Dieter Wintgen of the Max Planck Institute for Nuclear Physics in Heidelberg to interpret the spectrum of the hydrogen atom m a strong magnetic field. Experimental work on such spectra has been done by Karl H. Welge and his colleagues at the University of Bielefeld, who have excited hydrogen atoms nearly to the point of ionization where the electron tears itself free of the proton. The energies at which the atoms absorb radiation appear to be quite random [see illustration], but a Fourier analysis converts the jumble of peaks into a set of well-separated peaks. The important feature here is that each of the well-separated peaks corresponds precisely to one of several standard classical periodic orbits. Poincare's insistence on the importance of periodic orbits now takes on a new meaning. Not only does the classical organization of phase space depend critically on the classical periodic orbits, but so too does the understanding of a chaotic quantum spectrum.

So far I have talked only about quantum systems in which an S electron is trapped or spatially confined. Chaotic effects are also present in atomic systems where an electron can roam freelly, as it does when it is scattered from the atoms in a molecule. Here energy is no longer quantized, and the electron can take on any value, but the effectiveness of the scattering depends on the energy. Chaos shows up in quantum scattering as variations in the amount of time the electron is temporarily caught inside the molecule during the scattering process. For simplicity the problem can be examined in two dimensions. To the electron, a molecule consisting of four atoms looks like a small maze. When the electron approaches one of the atoms, it has two choices: it can turn left or right. Each possible trajectory of the electron through the molecule can be recorded as a series of left and right turns around the atom until the particle finally emerges. All of the trajectories are unstable: even a minute change in the energy or the initial direction of the approach will cause a large change in the direction in which the electron eventually leaves molecule. The chaos in the scattering process comes from the fact that the number of trajectories increases rapidly with path length. Only an interpretation From the quantum mechanical point of view gives reasonable results; a purely classical calculation yields nonsensical results. In quantum mechanics each classical trajectory is used to deftne a little wavelet that finds its way through the molecule. The quantum mechanical result follows from simply adding up all such wavelets. Recently I have done a calculation of the scattering process for a special case in which the sum of the wavelets is exact An electron of known momentum hits a and emerges with the same momentum. The arrival time for the electron to reach a fixed monitoring station varies as a function of the momentum and the way in which it varies is so fascinating about this problem. The arrival time fluctuates over small changes in the momentum but over large changes a chaotic imprint emerges which never settles down to any simple pattern [see illustration].

A particularly tantalizing aspect of the chaotic scattering process is that it may connect the mysteries of quantum chaos with the mysteries of number theory. The calculation of the time delay leads straight into what is probably the most enigmatic object in mathematics, Riemann's zeta function. Actually it was first emploed by Leonhard Euler in the middle of the 18th century to show the existence of an infinite number of prime numbers (integers that cannot be divided by any smaller integer other than one). About a century later Bernhard Riemann, one of the founders of modem mathematics, employed the function to delve into the distribution of the primes. In his only paper on the subject, he called the function by the Greek letter zeta. The zeta function is a function of two variables, x and y which exist in the complex plane). To understand the distribution of prime numbers, Riemann needed to know when the zeta function has the value of zero. Without giving a valid argument, he stated that it is zero only when x is set equal to 1/2. Vast calculations have shown that he was right without exception for the first billion zeros, but no mathematician has come even close to providing a proof. If Riemann's conjecture is correct, all kinds of interesting properties of prime numbers could be proved. The values of y for which the zeta function is zero form a set of numbers that is much like the spectrum of energies of an atom. Just as one can study the distribution of energy levels in the spectrum so can one study the distribution of zeros for the zeta function. Here the prime numbers play the same role as the classical closed orbits of the hydrogen atom in a magnetic field: the primes indicate some of the hidden correlations among the zeros of the zeta function. In the scattering problem the zeros of the zeta function give the values of the momentum where the time delay changes strongly. The chaos of the Riemann zeta function is particularly apparent in a theorem that has only recently been proved: the zeta function fits locally any smooth function. The theorem suggests that the function maydescribe all the chaotic behavior a quantum system can exhibit. If the mathematics of quantum mechanics could be handled more skilfully, many examples of locally smooth, yet globally chaotic, phenomena might be found.

TWICE in 20th-century physics, the notion of unpredictability has shaken scientists' view of the Universe. The first time was the development of quantum mechanics, the theory that describes the behaviour of matter on an atomic scale. The second came with the classical phenomenon of chaos In both areas unpredictable features changed scientists understanding of matter in ways that were totally unforeseen. How ironic then, that these two fields, which have something so fundamental in common, should end up as antagonists when combined. For by rights, chaos should not exist at all in quantum systems- the laws of quantum mechanics actually forbid it. Yet recent experiments seem to show the footprints of quantum chaos in remarkable swirling patterns of atomic disorder. These intriguing patterns could illuminate one of the darkest corners of modern physics: the twilight zone where the quantum and classical worlds meet. The quantum theory is one of the most successful theories in modern science.

Developed in the 1920s, it accounts for a vast range of phenomena from the nature of chemical bonds to the behaviour of subatomic particles, making predictions that have been tested to unprecedented levels of accuracy. But at its core there are troublesome features: Prominent among them is Heisenberg's uncertainty principle-if you know the speed of a quantum particle, for instance, you can never know its exact location. The notion that some aspects of nature are simply unknowable has caused sleepless nights for more than a few physicists. Chaos is a younger discipline. Although some of its conceptual elements had already been appreciated by Leibnitz in the 17th century and Poincare in the 19th century, chaos theory did not become fashionable until the 1980s when scientists began to realize that the phenomenon is widespread in the natural world. It arises when a system is unusually sensitive to its initial conditions so that a small perturbation of the system changes its subsequent behaviour in a way that grows exponentially with time. Chaos has been observed in, among other things, pendulums, the growth of populations, planetary dynamics, and weather systems. Probably the most famous example of chaos is the so-called "butterfly effect" in which, in theory, the tiny air disturbance from the flapping of a butterfly's wings can ultimately lead to a dramatic storm. of course, although both these theories place fundamental limits on what we can know about the world, the unpredictabilities in quantum theory and chaos are different in kind. But the particular problem with quantum chaos is that in quantum mechanics small perturbations generally only lead to small perturbations in subsequent states. Without the exponential divergence in evolutionary paths, it is difficult to see how there can be any chaos. This behaviour of quantum systems is often attributed to a special property of the quantlani equations: their linearity.

An everyday example of linearity can be seen in a rubber band. When it is stretched a little the extension is proportional to the force. Nonlinearity steps in when you pull too far and the band reaches its limit of elasticity. Stretch even further and it snaps. Because nonlinearity is known to be a crucial ingredient in chaotic systems. it is often said that quantum mechanics cannot be chaotic because it is linear. But according to Michael Berry, a leading theorist in the study of quantum chaos at the University of Bristol, this issue of linearity is a red herring. "This is one of the biggest misconceptions in the business," he says. Berry's preferred explanation for the difference between what happens in classical and quantum systems as they edge towards chaos is that quantum uncertainty imposes a fundamental limit on the sharpness of the dynamics. The amount of uncertainty is quantified in Heisenberg's uncertainty principle by a fixed value known as Planck's constant. "In classical mechanics, objects can move along infinitely many trajectories," says Berry. "This makes it easy to set up complicated dynamics in which an object will never retrace its path - the sort of beliaviour that leads to chaos. But in quantum mechanics, Planck's constant blurs out the fine detail, smoothing away the chaos."

This raises some interesting questions. What happens if you scale down a classically chaotic system to atomic size? Do you still get chaos or does quantum regularity suddenly prevail? Or does someting entirely new happen? And why is it that macroscopic systems can be chaotic, given that ultimately everything is made out of atoms and therefore quantum in nature? These questions have been the subject of intense debate for more than a decade. But now a number of experimental approaches have begun to offer answers.

#### Scrambled spectra

One of the earliest clues came from investigations of atomic absorption spectra. If an atom absorbs a photon of light it is possible for one of its electrons to be kicked into a higher energy state. Normally, an atom's energy levels are spaced at mathematically regular intervals, accounted for by an empirical formula given 19th century physicist Johannes Rydberg. If an atom absorbs photons with different energies, electrons are kicked into different levels, and the result is a nice tidy absorbtion spectrum whose details are characteristic of the chemical element involved. But when the atom is subjected to a magnetic field the line structure of the spectrum becomes distorted. When the field is sufficiently intense the spectrum becomes so scrambled it looks pretty much random at higher energies. The phenomenon is easier to understand in classical rather than quantum mechanical terms. Viewed classically, atomic electrons move in orbits around the nucleus rather like planets round the Sun. A magnetic field, though, introduces an additional force which causes the electrons to swerve from their normal trajectories. It's rather like a stray star encroaching upon the Solar System. If it got sufficiently close the gravitational pull would at some point become comparable to the pull between the Earth and our sun. At this moment the earth would find itself in a tug-of-war between the sun and the interloping star. Such a system would very probably be unstable, with the Earth switching critically between orbits around the sun and the other star. The result would be a chaotic orbit. In the case of excited atoms, for small fields and lower energy states. The electromagnetic swerving is small compared with the electrostatic pull towards the nucleus and the electron continues to follow a stable orbit. But for strong fields and highly excited states where the electron is on average very much further away from the nucleus, the swerving force becomes comparable to the inward pull of the nucleus In this situation, according to vclassical predictions, the motion ought to be chaotic. The effect was first studied back in 1969 by two astronomers Garton and Tonkins of Imperial College, London, who wanted to find out how the spectra of stars would be affected by their powerful magnetic fields. Their experiments on barium atoms produced one of the first surprisesbecause their resulting spectrum still displayed considerable regularity. A group at the University of Bielefield in Germany repeated the experiments in the 1980s using higher resolution equipment. Although the randomness was more apparent in their spectra, it was still clear that quantum mechanics was in some strange way superimposing its own order on the chaos.

#### Quantum billiards

More recently, signs of quantum suppression of chaos have come from anotheianother experimental approach to quantum chaos: quantum billiards. On a conventional billiard table it is quite common for a player to pot a ball by bouncing the cue ball off the cushion first. In the hands of a skilled player, such shots are often quite repeatable. But if you were to try the saine shot on a rounded, stadium-shaped table, the results are far less predictible: the slightest change in starting position alters the ball's trajectory drastically. So what you get if you play stadium billiards is chaos. In 1992 at Boston's Northeastern University, Srinivas Sridhar and colleagues substituted microwaves for billiard balls and a shallow stadium-shaped copper cavity for the table. Sridhar's team then observed how the microwaves settled down inside the cavity. Although their apparatus is not of atomic proportions (a cavity typically measures several millimetres across) the experiment exploits the precise similarity between the wave equations of quantum mechanics and the equations of the electromagnetic waves in this two-dimensional situation. If microwaves behaved like billiard balls, you would not expect to see any regular patterns. The experiments, however, reveal structures known its "scars" that suggest the waves concentrate along particular paths. But where do these paths come from? One answer is provided by theoretical work carried out back in the 1970s by Martin Gutzwiller of of the IBM Thomas Watson Center in Yorktown Heights near New York. He produced a key formula that showed how classical chaos might relate to quantum chaos. Basically it indicates that the quantum regularities are related to a very limited range of classical orbits. These orbits are ones that are periodic in the classical system. If, for example, you placed a ball on the stadium table and hit it along exactly the right path, you could get it to retrace its path after only a few bounces off the cushions. However, because the system is chaotic these orbits are unstable. You only need a minuscule error and the ball will move off course within a few bounces. So classically you would not expect to see these orbits stand out. But thanks to the uncertainty in quantum mechanics, which "frizzes" the trajectories of the balls, tiny errors become less significant and the periodic orbits are reinforced in some strange way so that they predominate. Sridhar's millimetre-sized stadium was a good analogy for quantum behaviour, but would the same effects occur in a truly quantum-sized system? This question was answered recently by Laurence Eaves from the University of Nottingham, and his colleagues at Nottingham and at Tokyo University. Eaves conducted his game of quantum billiards inside an elaborate semiconductor "sandwich". He used electrons for balls, and for cushions he used a combination of quantum barriers and magnetic fields. The quantum barriers are formed by the outer layers of the sandwich, which gives the electrons a couple of straight edges to bounce back and forth between, The other edges of the table are created by the restraining effect of the magnetic field, which curves the electron motion in a complicated way. As in Sridhar's stadium cavity, the resulting dynamics ought to be chaotic.

#### Number crunching

To do the exeriments, Eaves needed ultra-intense magnetic fields, so he took his device to the High Magnetic Field Laboratory at the University of Tokyo, which is equipped with some of the most powerful sources of pulsed magnetic fields in the world. Meanwhile his colleagues in Noitingham, Paul Wilkinson, Mark Fromhold and Fred Sheard, squared up to a heroic series of calculations, deducing from purely quantum mechanical principles what the results should look like. In a spectacular pape that made the cover of *Nature* last month, the team produced the first definitive evidence for quantum scarring, and precisely confirmed the quantum mechanical predictions. Sure enough, the current flowing through the device was predominantly carried by electrons moving in certain 'scarred' paths. Quantum regularity was lingering in the chaos rather like the smile of the Cheshire cat in Alice's adventures in wonderland.
In case these ideas seem academic it is worth noting that quantum chaos could play an important role in the design of future seniiconductor devices. At the moment, transistor devices on silicon chips are still large enough for the electrons to move through them diffusively like molecules in a gas. But as chip manufacturers squeeze ever more logic gates onto silicon, says Eaves, in the next is years transistors may become so small that electrons will instead flow through them more like quantum billiard balls. "At this point, we may well need the principles of quantum chaos to understand how these devices will work," he says. But where does that leave the problem of how quantum mechanics turns into the classical world on larger scales? One way of looking at the problem is to investigate how a quantum chaos system actually evolves with time. Last December, Mark Raizen and his colleagues at. the University of Texas managed to do just that, using an experimental version of a quantum kicked rotor. The idea is to couple two oscillating systems to produce chaos. Imagine pushing a child's swing. If you time your pushes in rhythm with the swing, then it simply rises higher and higher. if you push at a different frequency, the swing will sometimes be given a boost and sometimes slowed down. if this is done too vigorously, the oscillations become chaotic. In Raizen's quantum version, ultra-cold sodium atoms were subjected to a special kind of pulsed laser light. The laser beam was bounced between mirrors to set up a short-lived standing wave - a periodic lattice of light that remains motionless in space rather like the acoustic nodes on a violin string. Depending on their precise location in the standing waves, the sodium atoms are pushed around by the magnetic fields in the lattice. According to classical calculations, the result is that the atoms should be kicked chaotically along an increasingly energetic random walk. Raizen's results confirmed a long-standing prediction of the quantum theoretical descriptions of these systems. The atoms did indeed move in a chaotic way to begin with. But after around 100 microseconds (which corresponds to around 50 kicks) the build-up in energy reached a plateau.

#### Break time

In other words. quantum mechanics does suppress the chaos but only after a certain amount of time known as the 'quantum break time'. This turns out to be the crucial feature that distinguishes between quantum and classical predictions of chaotic systems. Before the break time, quantum systems are able to mimic the behaviour of classical systems by looking essentially random. But after the break time, the system simply retraces its path, it is no longer random, but akin to a repeating loop, albeit of considerable complexity. But if this is right, how can classical systems exhibit chaos? Macroscopic objects such as pendulums and planets are, after all, made out of atoms and are therefore, ultimately, quantum systems. it turns out that classical systems are in fact behaving exactly like quantum systems. The only difference is that for classical systems, the quantum break times of macroscopic systems are extraordinarily long-far longer than the age of the Universe. If we could study a classical system for longer than its quantum break time, we would see that the behaviour was not chaotic but quasi-periodic instead. Thus, quantum and classical realities can be reconciled, with the classical world naturally embedded in a larger quantum reality. Or, as physicist Dan Kleppner of ttie Massachusetts Institute of Technology puts it, "Anything classical mechanics can do, quantum mechanics can do better". Since much of the experimental work on quantum chaos has agreed with theoretical predictions, it could be tempting to say "So what?". We already knew that quantum theory was right. Well, research on quantum chaos does hold out the promise of some remarkable discoveries. Berry is excited by what appears to be a deep connection between the problem of finding the energy levels of a quantum system that is classically chaotic and one of the biggest unsolved mysteries in mathematics: the Riemann hypothesis. This concerns the distribution of prime numbers. If you choose a number n and ask how many prime numbers there are less than *n* it turns out that the answer closely approximates the formula: *n*/log *n*. The formula is not exact, though: sometimes it is a little high and sometimes it is a little low. Riemann looked at these deviations and saw that they contained periodicities. Berry likens these to musical harmonies: "The question is what are the harmonies in the music of the primes? Amazingly, these harmonies or magic numbers behave exactly like the energy levels in quantum systems that classically would be chaotic."

#### Deep connection

This correspondence emerges from statistical correlations between the spacing of the Riemann numbers and the spacing of the energy levels. Berry and his collaborator Jon Keating used them to show how techniques in number theory can be applied to problems in quantum chaos and vice versa. In itself such a connection is very tantalising. Although sonictimes described as the Queen of mathematics, number theory is often thought of as pretty useless, so this deep connection with physics is quite astonishing. Berry is also convinced that there must be a particular chaotic system which when quantised would have energy levels that exactly duplicate the Riemann numbers. "Finding this system could be the discovery of the century," he says. it would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos. Until we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It is a tantalising thought. Out there is a physical structure waiting to be discovered. if we find it, the remarkable experiments that we have recently witnessed in this discipline would be crowned by an experimental apparatus that could do more than anything to unlock the secrets of quantum chaos.

## Prime numbers

### Prime 7

The following information is from thejaxis:

The multiversal language of mathematics is literally composed of an endless matrix of possible combinations within numbers and equations. All of these possibilities share connection to the whole, as is the Law of One.

Using fourth dimensional mathematics (seemingly breaking laws of math although patterns appear), any numerical form can be broken down by using prime 7. This occurs through scale-symmetry of overlapped digits within a numerical form. (See my highlights for numerical symmetries). Prime 7 accomplishes this by appearing within multiples, 6-digit or less, in every complex numerical form or sequence.

Prime 7 acts as a mathematical key in which to use 4D math to begin breaking any number down to one. Like color to light, we have 7 becoming 1. We literally use 7 as a key to the mathematical matrix, which is funny because major and minor keyes in music are composed of 7 notes.

#### Light and color

The decimal form of 1/7 = mathematical light refraction, just as one light becomes seven colors. Light and color cannot be defined by length, width, nor depth as why I believe them to be fourth-dimensional. Matter, as we perceive it in it's physicality is 3D, but all matter has some sort of 4D color which overlaps the 3D shape of the matter.

#### Padovan-Fibonacci spiral forms dimensions and prime 7

Both the Fibonacci and Padovan Sequence create the same spiral mathematically. The Fibonacci Sequence uses squares, and the Padovan Sequence uses equilateral triangles. I believe these sequences are giving us insights into how higher dimensions behave and exist.

In 2D, we see a hexagon of 120°angles, or triangles of 60° angles. In 3D, we can see cubic shapes of 90° angles, as well as equilateral pyramids of 60° angles. Depending on our perspective of the shape(s) these angles can alternate and exist simultaneously.

We can see how the Fibonacci and Padovan sequences are composed of 6-digit or less multiples of 7, using scale-symmetry to overlap these multiples as they exist simultaneously. I believe this is a 4D expression of geometry using arithemetic, which is supported by the math mimicking symmetry. The 6-digit or less multiples represent the six cardinal directions of 3D geometry based on axes, and prime 7 is the seventh direction of within as concerned with 4D geometry, where multiple perspectives all exist within one shape.

I believe that the 4th dimension does not continue to grow outwards as the first three dimensions, but rather uses the already created space and fills it with infinite possible perspectives. We already understand this 4th dimensional nature in our understanding of tesseracts. A tesseract is a cube which is mathematically and geometrically perfectly balanced within the lines and angles of its outer surrounding cube. The cube uses the first 3Ds to exist, but the tesseract uses the space within the cube to find balance.

When finding the volume of a shape which lies within, we need all measurements of the 3Ds to create the 4D measure of volume or space within. This is what makes the 7th direction of 4D geometry to be within, towards the center. Higher dimensions are not beyond us, but amongst us as they overlap the 3D matter of our reality. This 7th direction of possible geometry behaves the same as gravity, a 4D measure that is measured through and within the center.

#### The breaking of symmetry

The breaking down of numbers by using prime 7 reminds me of quantum atom theory which says the breaking of symmetry forms sacred geometry and now that I think about it this could link into chaos as a spontaneous breakdown of topological supersymmetry. Prime 7 could therefore be the hexagonal yod/cubistic matrix. 7 appears within multiples, 6-digits or less, this could explain how 7 expresses itself which is why it forms the 6-fold flower of life.

### A prime case of chaos(Quantum chaos)

#### Prime numbers are like musical chords

This is an interesting quote from this PDF: "Prime numbers are a lot like musical chords, Berry explains. A chord is a combination of notes played simultaneously. Each note is a particular frequency of sound created by a process of resonance in a physical system, say a saxophone. Put together, notes can make a wide variety of music. In number theory, zeroes of the zeta-function are the notes, prime numbers are the chords, and theorems are the symphonies."

### quasicrystals connection to prime numbers

About a year ago, the theoretical chemist Salvatore Torquato met with the number theorist Matthew de Courcy-Ireland to explain that he had done something highly unorthodox with prime numbers, those positive integers that are divisible only by 1 and themselves.

A professor of chemistry at Princeton University, Torquato normally studies patterns in the structure of physical systems, such as the arrangement of particles in crystals, colloids and even, in one of his better-known results, a pack of M&Ms. In his field, a standard way to deduce structure is to diffract X-rays off things. When hit with X-rays, disorderly molecules in liquids or glass scatter them every which way, creating no discernible pattern. But the symmetrically arranged atoms in a crystal reflect light waves in sync, producing periodic bright spots where reflected waves constructively interfere. The spacing of these bright spots, known as “Bragg peaks” after the father-and-son crystallographers who pioneered diffraction in the 1910s, reveals the organization of the scattering objects.

Torquato told de Courcy-Ireland, a final-year graduate student at Princeton who had been recommended by another mathematician, that a year before, on a hunch, he had performed diffraction on sequences of prime numbers. Hoping to highlight the elusive order in the distribution of the primes, he and his student Ge Zhang had modeled them as a one-dimensional sequence of particles — essentially, little spheres that can scatter light. In computer experiments, they bounced light off long prime sequences, such as the million-or-so primes starting from 10,000,000,019. (They found that this “Goldilocks interval” contains enough primes to produce a strong signal without their getting too sparse to reveal an interference pattern.)

It wasn’t clear what kind of pattern would emerge or if there would be one at all. Primes, the indivisible building blocks of all natural numbers, skitter erratically up the number line like the bounces of a skipping rock, stirring up deep questions in their wake. “They are in many ways pretty hard to tell apart from a random sequence of numbers,” de Courcy-Ireland said. Although mathematicians have uncovered many rules over the centuries about the primes’ spacings, “it’s very difficult to find any clear pattern, so we just think of them as ‘something like random.’”

But in three new papers — one by Torquato, Zhang and the computational chemist Fausto Martelli that was published in the Journal of Physics A in February, and two others co-authored with de Courcy-Ireland that have not yet been peer-reviewed — the researchers report that the primes, like crystals and unlike liquids, produce a diffraction pattern.

“What’s beautiful about this is it gives us a crystallographer’s view of what the primes look like,” said Henry Cohn, a mathematician at Microsoft Research New England and the Massachusetts Institute of Technology.

The resulting pattern of Bragg peaks is not quite like anything seen before, implying that the primes, as a physical system, “are a completely new category of structures,” Torquato said. The Princeton researchers have dubbed the fractal-like pattern “effective limit-periodicity.”

It consists of a periodic sequence of bright peaks, which reflect the most common spacings of primes: All of them (except 2) are at odd-integer positions on the number line, multiples of two apart. Those brightest bright peaks are interspersed at regular intervals with less bright peaks, reflecting primes that are separated by multiples of six on the number line. These have dimmer peaks between them corresponding to farther-apart pairs of primes, and so on in an infinitely dense nesting of Bragg peaks.

Dense Bragg peaks have been seen before, in the diffraction patterns of quasicrystals, those strange materials discovered in the 1980s with symmetric but nonrepeating atomic arrangements. In the primes’ case, though, distances between peaks are fractions of one another, unlike quasicrystals’ irrationally spaced Bragg peaks. “The primes are actually suggesting a completely different state of particle positions that are like quasicrystals but are not like quasicrystals,” Torquato said.

According to numerous number theorists interviewed, there’s no reason to expect the Princeton team’s findings to trigger advances in number theory. Most of the relevant mathematics has been seen before in other guises. Indeed, when Torquato showed his plots and formulas to de Courcy-Ireland last spring (at the suggestion of Cohn), the young mathematician quickly saw that the prime diffraction pattern “can be explained in terms of almost universally accepted conjectures in number theory.”

It was the first of many meetings between the two at the Institute for Advanced Study in Princeton, N.J., where Torquato was spending a sabbatical. The chemist told de Courcy-Ireland that he could use his formula to predict the frequency of “twin primes,” which are pairs of primes separated by two, like 17 and 19. The mathematician replied that Torquato could in fact predict all other separations as well. The formula for the Bragg peaks was mathematically equivalent to the Hardy-Littlewood k-tuple conjecture, a powerful statement made by the English mathematicians Godfrey Hardy and John Littlewood in 1923 about which “constellations” of primes can exist. One rule forbids three consecutive odd-numbered primes after {3, 5, 7}, since one in the set will always be divisible by three, as in {7, 9, 11}. This rule illustrates why the second-brightest peaks in the primes’ diffraction pattern come from pairs of primes separated by six, rather than four.

Hardy and Littlewood’s conjecture further specified how often all the allowed prime constellations will occur along the number line. Even the simplest case of Hardy-Littlewood, the “twin primes conjecture,” although it has seen a burst of modern progress, remains unproved. Because prime diffraction essentially reformulates it, experts say it’s highly unlikely to lead to a proof of Hardy-Littlewood, or for that matter the famous Riemann hypothesis, an 1859 formula linking the primes’ distribution to the “critical zeros” of the Riemann zeta function.

The findings resonate, however, in a relatively young research area called “aperiodic order,” essentially the study of nonrepeating patterns, which lies at the intersection of crystallography, dynamical systems, harmonic analysis and discrete geometry, and grew after the discovery of quasicrystals. “Techniques that were originally developed for understanding crystals … became vastly diversified with the discovery of quasicrystals,” said Marjorie Senechal, a mathematical crystallographer at Smith College. “People began to realize they suddenly had to understand much, much more than just the simple straightforward periodic diffraction,” she said, “and this has become a whole field, aperiodic order. Uniting this with number theory is just extremely exciting.”

The primes’ pattern resembles a kind of aperiodic order known since at least the 1950s called limit periodicity, “while adding a surprising twist,” Cohn said. In true limit-periodic systems, periodic spacings are nested in an infinite hierarchy, so that within any interval, the system contains parts of patterns that repeat only in a larger interval. An example is the tessellation of a strange, multipronged shape called the Taylor-Socolar tile, discovered by the Australian amateur mathematician Joan Taylor in the 1990s, and analyzed in detail with Joshua Socolar of Duke University in 2010. According to Socolar, computer experiments indicate that limit-periodic phases of matter should be able to form in nature, and calculations suggest such systems might have unusual properties. No one guessed a connection to the primes. They are “effectively” limit periodic — a new kind of order — because the synchronicities in their spacings only hold statistically across the whole system.

For his part, de Courcy-Ireland wants to better understand the “Goldilocks” scale at which effective limit-periodicity emerges in the primes. In 1976, Patrick Gallagher of Columbia University showed that the primes’ spacings look random over short intervals; longer strips are needed for their pattern to emerge. In the new diffraction studies, de Courcy-Ireland and his chemist collaborators analyzed a quantity called an “order metric” that controls the presence of the limit-periodic pattern. “You can identify how long the interval has to be before you start seeing this quantity grow,” he said. He is intrigued that this same interval length also shows up in a different prime number rule called Maier’s theorem. But it’s too soon to tell whether this thread will lead anywhere.

The main advantage of the prime diffraction pattern, said Jonathan Keating of the University of Bristol, is that “it is evocative” and “makes a connection with different ways of thinking.” But the esteemed number theorist Andrew Granville of the University of Montreal called Torquato and company’s work “pretentious” and “just a regurgitation of known ideas.”

Torquato isn’t especially concerned about how his work will be perceived by number theorists. He has found a way to glimpse the pattern of the primes. “I actually think it’s stunning,” he said. “It’s a shock.”

#### Quasicrystal metamaterials and spacetime structure

Quasicrystal metamaterials interact with spacetime and light(as I will go over in a future post which will be linked here) and this could link into the rhombic hexeconta geometry of fractal fields and this could possibly link into Rhombic triacontahedron and this all suggest spacetimes structure is a fractal what fractal? the E8 tetrahedron grid fractal which I have already proved and it shows that quasicrystal geometry is encoded in these fractals. These crystals squeeze light causing negative mass fields and this could link into negative mass fields and imaginary mass fields.

### Big Question About Primes Proved in Small Number Systems

On September 7, two mathematicians posted a proof of a version of one of the most famous open problems in mathematics. The result opens a new front in the study of the “twin primes conjecture,” which has bedeviled mathematicians for more than a century and has implications for some of the deepest features of arithmetic.

“We’ve been stuck and running out of ideas on the problem for a long time, so it’s automatically exciting when anyone comes up with new insights,” said James Maynard, a mathematician at the University of Oxford.

The twin primes conjecture concerns pairs of prime numbers with a difference of 2. The numbers 5 and 7 are twin primes. So are 17 and 19. The conjecture predicts that there are infinitely many such pairs among the counting numbers, or integers. Mathematicians made a burst of progress on the problem in the last decade, but they remain far from solving it.

The new proof, by Will Sawin of Columbia University and Mark Shusterman of the University of Wisconsin, Madison, solves the twin primes conjecture in a smaller but still salient mathematical world. They prove the conjecture is true in the setting of finite number systems, in which you might only have a handful of numbers to work with.

These number systems are called “finite fields.” Despite their small size, they retain many of the mathematical properties found in the endless integers. Mathematicians try to answer arithmetic questions over finite fields, and then hope to translate the results to the integers.

“The ultimate dream, which is maybe a bit naive, is if you understand the finite field world well enough, this might shed light on the integer world,” Maynard said.

In addition to proving the twin primes conjecture, Sawin and Shusterman have found an even more sweeping result about the behavior of primes in small number systems. They proved exactly how frequently twin primes appear over shorter intervals — a result that establishes tremendously precise control over the phenomenon of twin primes. Mathematicians dream of achieving similar results for the ordinary numbers; they’ll scour the new proof for insights they could apply to primes on the number line.

#### A New Kind of Prime

The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want.

Alphonse de Polignac posed the conjecture in its current form in 1849. Mathematicians made little progress on it for the next 160 years. But in 2013 the dam broke, or at least sprung major leaks. That year Yitang Zhang proved that there are infinitely many prime pairs with a gap of no more than 70 million. Over the next year other mathematicians, including Maynard and Terry Tao, closed the prime gap considerably. The current state of the art is a proof that there are infinitely many prime pairs with a difference of at most 246.

But progress on the twin primes conjecture has stalled. Mathematicians understand they’ll need a wholly new idea in order to solve the problem completely. Finite number systems are a good place to look for one.

To construct a finite field, start by extracting a finite subset of numbers from the counting numbers. You could take the first five numbers, for instance (or any prime number’s worth). Rather than visualizing the numbers along a number line the way we usually do, visualize this new number system around the face of a clock.

Arithmetic then proceeds, as you might intuit it, by wrapping around the clock face. What’s 4 + 3 in the finite number system with five elements? Start at 4, count three spaces around the clock face, and you’ll arrive at 2. Subtraction, multiplication and division work similarly.

Only there’s a catch. The typical notion of a prime number doesn’t make sense for finite fields. In a finite field, every number is divisible by every other number. For example, 7 isn’t ordinarily divisible by 3. But in a finite field with five elements, it is. That’s because in this finite field, 7 is the same number as 12 — they both land at 2 on the clock face. So 7 divided by 3 is the same as 12 divided by 3, and 12 divided by 3 is 4.

Because of this, the twin primes conjecture for finite fields is about prime polynomials — mathematical expressions such as x^{2} + 1.

For example, let’s say your finite field contains the numbers 1, 2 and 3. An polynomial in this finite field would have those numbers as coefficients, and a “prime” polynomial would be one that can’t be factored into smaller polynomials. So x^{2} + x + 2 is prime because it cannot be factored, but x^{2} − 1 is not prime: It’s the product of (x + 1) and (x − 1).

Once you have the notion of prime polynomials, it’s natural to ask about twin prime polynomials — a pair of polynomials that are both prime and that differ by a fixed gap. For example, the polynomial x^{2} + x + 2 is prime, as is x^{2 }+ 2x + 2. The two differ by the polynomial x (add x to the first to get the second).

The twin primes conjecture for finite fields predicts that there are infinitely many pairs of twin prime polynomials that differ not just by x, but by any gap you want.

#### Geometries of primes

Finite fields and prime polynomials might seem contrived, of little use in learning about numbers in general. But they’re analogous to a hurricane simulator — a self-contained universe that provides insights about phenomena in the wider world.

“There is an ancient analogy between integers and polynomials, which allows you to transform problems about integers, which are potentially very difficult, into problems about polynomials, which are also potentially difficult, but possibly more tractable,” Shusterman said.

Finite fields burst into prominence in the 1940s, when André Weil devised a precise way of translating arithmetic in small number systems to arithmetic in the integers. Weil used this connection to spectacular effect. He proved arguably the most important problem in mathematics — the Riemann hypothesis — as interpreted in the setting of curves over finite fields (a problem known as the geometric Riemann hypothesis). That proof, along with a series of additional conjectures that Weil made — the Weil conjectures — established finite fields as a rich landscape for mathematical discovery.

Weil’s key insight was that in the setting of finite fields, techniques from geometry can be used with real force to answer questions about numbers. “This is part of the thing that’s special to finite fields. Many problems you want to solve, you can rephrase them geometrically,” Shusterman said.

To see how geometry arises in such a setting, imagine each polynomial as a point in space. The polynomial’s coefficients serve as the coordinates that define where the polynomial is located. Going back to our finite field of 1, 2 and 3, the polynomial 2x + 3 would be located at the point (2, 3) in two-dimensional space.

But even the simplest finite field has an infinite number of polynomials. You can construct more elaborate polynomials by increasing the size of the largest exponent, or degree, of the expression. In our case, the polynomial x^{2} − 3x − 1 would be represented by a point in three-dimensional space. The polynomial 3x^{7} + 2x^{6} + 2x^{5} − 2x^{4} − 3x^{3} + x^{2} − 2x + 3 would be represented by a point in eight-dimensional space.

In the new work, this geometric space represents all polynomials of a given degree for a given finite field. The question then becomes: Is there a way to isolate all the points representing prime polynomials?

Sawin and Shusterman’s strategy is to divide the space into two parts. One of the parts will have all the points corresponding to polynomials with an even number of factors. The other part will have all the points corresponding to polynomials with an odd number of factors.

Already this makes the problem simpler. The twin primes conjecture for finite fields concerns polynomials with just one factor (just as a prime number has a single factor — itself). And since 1 is odd, you can discard the part of the space with the even factors entirely.

The trick is in the dividing. In the case of a two-dimensional object, such as the surface of a sphere, the thing that cuts it in two is a one-dimensional curve, just as the equator cuts the surface of the Earth in half. A higher-dimensional space can always be cut with an object that has one fewer dimension.

Yet the lower-dimensional shapes that divide the space of polynomials are not nearly as elegant as the equator. They are sketched by a mathematical formula called the Möbius function, which takes a polynomial as an input and outputs 1 if the polynomial has an even number of prime factors, −1 if it has an odd number of prime factors, and 0 if it has only a repeated factor (the way 16 can be factored into 2 × 2 × 2 × 2).

The curves drawn by the Möbius function twist and turn wildly, crossing themselves in many places. The places where they cross — called singularities — are especially difficult to analyze (and they correspond to polynomials with a repeated prime factor).

Sawin and Shusterman’s principal innovation was in finding a precise way to slice the lower-dimensional loops into shorter segments. The segments were easier to study than the complete loops.

Once they cataloged polynomials with an odd number of prime factors — the hardest step — Sawin and Shusterman had to determine which of them were prime, and which were twin primes. To do this, they applied several formulas that mathematicians use to study primes among the regular numbers.

Sawin and Shusterman used their technique to prove two major results about prime polynomials in certain finite fields.

First, the twin primes conjecture for finite fields is true: There are infinitely many pairs of twin prime polynomials separated by any gap you choose.

Second, and even more consequentially, the work provides a precise count of the number of twin prime polynomials you can expect to find among polynomials of a given degree. It’s analogous to knowing how many twin primes fall within any sufficiently long interval on the number line — a kind of dream result for mathematicians.

“This is the first work that gives a quantitative analogue of what is expected to be true over the integers, and that is something that really stands out,” said Zeev Rudnick of Tel Aviv University. “There hasn’t been anything like this until now.”

Sawin and Shusterman’s proof shows how nearly 80 years after André Weil proved the Riemann hypothesis in curves over finite fields, mathematicians are still energetically following his lead. Mathematicians pursuing the twin primes conjecture will now turn to Sawin and Shusterman’s work and hope that it, too, will provide a deep well of inspiration.

### Pascals triangle, Prime numbers, Music and Gas

A musical note travels through space in the form of longitudinal waves, the result of billions of collisions of gas particles. Plichta demonstrates that the mechanism of control of the transmission of a musical note through space "could only be caused by the sequence of the reciprocal prime numbers". The sequence of the reciprocals of the prime numbers obeys a certain pattern that we find in fractal geometry. How does fractal geometry relate to the transmission of a musical note in space? Plichta applies probability theory to gas filled space and showing that the collisions of particles in a gas medium creates a distribution which follows a pattern that conforms to the numbers in Pascal’s triangle. The pattern emerges as a result of billions of collisions of particles in a gas medium which reduce to yes/no decisions.

## Base 10 problem

Everything is based on grids and in our modern world the common grid system is based upon the geometry of the square and this square-based system forms the cartesian coordinate system and this system works with the numbers 0-9 and higher, this counting system governs everything we order and therefore the way we perceive reality.

This system underlies all of science since we have been using it, this system is great at separating things but as we shift into a quantum paradime we must break out of the box, from the quantum perceptive the limits of the cartesian coordinate system becomes painfully obvious the cartesian coordinate system is the biggest problem in physics and limits physics, this system dominates all of maths and physics.

The cartesian coordinate system is very linear and as a result, it governs us and our entire way we think into a box it limits our ability to think holographically and to see the whole organism(The whole universe as one).

"Physical law should refer primarily to an order of undivided wholeness similar to that indicated by the hologram. Rather than to an order of analysis into separate parts." - David Bohm

This system is more about static systems then our living dynamic organic and interconnected reality. This system is the relationship between our consciousness and spacetime and is deeply embedded in us.

### Pythagorean numerology

The system originates from the Pythagorean numerology which does have some true magick and some aspects of the Pythagorean model(based on the tetractys) are good but the use of the numbers 0-9 keeps us stuck in a limited way of being because 9 is an ending. The numbers 0-9 are important as archetypes, they are on the tree of life they are eminations from source each number is a ray of creation they were never meant to be used as a counting system.

### reforming base 10 and using base 9

A better system to use is the Chaldean numerology system which uses numbers 0-9 but the counting system only uses 1-8 it is more musical and this system is seen in the octagon, the I Ching and it is also seen in the 3D tree of life when it forms an octagon when it rotates.

#### 168(1.68) and 1.618(phi)

An accidental discovery when I was looking at numbers in base 9, 1.68 which is Robert Grant's constant, 168 is an important number in Stephen M. Philips work, is 1.61(phi) in base 9!

## Chaos theory and numbers in Xen Qabbalah

### 8(4+4 fold) and 5

When placing the Fibonacci spiral over the 10/12 tree of life(Kathara grid) the spiral goes through the 5th point and curves around a circle contained within a square where the top point of the square is the 8th point in the 10/12 grid.

8 and 5 both form 13 when added together and the number 13 is important because there are 13 spheres that make up the 3D seed of life which is 12 surrounding one which would be the 12 points of the 10/12 grid and the 13th being the center of the grid AKA the zero-point.

The circle joining the 5 and 8 points lines up with the center circle making up the 2D seed of life which is the hexagonal yod.

There are 15 pathways and 12 sephirot making up the 10/12 tree of life and 15+12=27 so the 10/12 grid corresponds to three to the power of three, the tetractys/seed of life corresponds to three to the power of two.

These diagrams also show the Krystal Fibonacci spiral(Doubling sequence+Fibonacci sequence) where the spiral is more circular encodes pi therefore encoding the tripling sequence.

### 4-fold geometry Hunab Ku

The Hunab Ku is a 4 fold geometry which forms the 441 cube matrix another 4 fold geometry. The Hunab Ku is the view of the 3D tree of life from above and corresponds to the 4 walls around Yantras such as the Sri Yantra which corresponds to the I Ching(64 tetrahedron grid) which corresponds to the Ying Yang of the Hunab Ku. The I Ching also corresponds to binary/doubling sequence which corresponds to the Kathara grid spiral(Also something random to say 4 Kathara grids can be placed into a cube).

The Chaldean numerology system uses numbers 0-9 but the counting system only uses 1-8 it is more musical and this system is seen in the octagon, the I Ching and it is also seen in the 3D tree of life when it forms an octagon when it rotates.

#### Plato's lambda and trinity sequences

The tetrahedron grid fractal which is formed out of the doubling sequence forms the cosmic tree of life in the form of the powers of 3 showing base 3 comes out of base 2. If we place the doubling sequence and powers of 3 on the outside of a triangle we form Plato's lambda.

##### Pascals triangle tetractys

The 10 spacetime dimensions can be repersented in the form of the pascals triangle tetractys to form the 4 large spacetime dimensions and the hexagonal yod(Cube) encoding the 12 vibrational dimensions. Pascals triangle is a membrane that originated from the zero-point field which is beyond the 12/15 vibrational dimensions and is pure energy which is also known as the God worlds.

##### Plato's lambda tetractys trinity

The Zero-point is formed out of 3 sequences as shown on the diagram and two of them form Plato's lambda.

- The doubling sequence corresponds to the number E which is related to growth and this is therefore related to the growth of light.
- The Fibonacci sequence is related to phi as we already know.
- The tripling/powers of 3 sequence would therefore correspond to pi talked about below.

##### Trinity Fibonacci expansion of light

If we repeat this fractal to infinity we would have all, infinite, Fibonacci numbers and as shown the Fibonacci numbers repeat and I said they double but how does that work? The amount of each Fibonacci number is one of the numbers in the doubling sequence which corresponds this infinite fractal to the infinite doubling vesica pices.

##### multidimensional polygons and tetractys

Pascals triangle and tetractys levels both encode multidimensional polygons which makes sense because multidimensional polygons correspond to numbers and vibrational dimensions.

### Plato's lambda

In *Timaeus*, Plato’s treatise on Pythagorean cosmology, the central character, Timaeus of Locri (possibly a real person), describes how the Demiurge divided the World Soul into harmonic intervals. Having blended the three ingredients of the World Soul — Sameness, Difference and Existence — into a kind of malleable stuff, the Demiurge took a strip of it and divided its length into portions measured by the numbers forming two geometrical series of four terms each: 1, 2, 4, 8 and 1, 3, 9, 27, generated by multiplying 1 by 2 and 3 (Fig. 1). This became known as "Plato’s Lambda" because of its resemblance to Λ, the Greek letter lambda. Then, according to Timaeus:

“he went on to fill up both the double and the triple intervals, cutting off yet more parts from the original mixture and placing them between the terms, so that within each interval there were two means, the one (harmonic)* exceeding the one extreme and being exceeded by the other by the same fraction of the extremes, the other (arithmetic) exceeding the one extreme by the same number whereby it was exceeded by the other. These linksgave rise to intervals of 3/2 and 4/3 and 9/8 within the original intervals. And he went on to fill up all the intervals of 4/3 (i.e., fourths) with the interval of 9/8 (the tone), leaving over in each a fraction. This remaining interval of the fraction had its terms in the numerical proportion of 256 to 243 (semitone). By this time, the mixture from which he was cutting off these portions was all used up.”

These numbers line but two sides of a tetractys array of ten numbers from whose relative proportions the musicians of ancient Greece worked out the frequencies of the notes of the now defunct Pythagorean musical scale.

The three numbers missing from the Lambda are shown in red in Figure 2. The sum of the 10 integers is 90 and the sum of the integers 1, 8 & 27 at the corners of the tetractys is **36**. The seven integers at the centre and corners of the grey hexagon shown in Figure 2 with dashed edges add up to 54. Hence, the 36:54 division displayed by the Lambda Tetractys differentiates between its corners, which correspond to the Supernal Triad of the Tree of Life, and its seven hexagonal yods, which correspond to the seven Sephiroth of Construction. Historians of science and musical scholars have focussed upon the integers as "number weights" whose significance is that their ratios are the tone ratios of the notes of the Pythagorean musical scale. They paid no attention to the magnitude of the numbers themselves because, influenced by the connotation given by Plato that their ratios determined the more important tone ratios, they seemed to have no relevance to music or anything else. The author's research into various sacred geometries and holistic systems indicates that this is untrue. The number 90 is always present in such systems as a defining parameter, as this section will demonstrate. The Lambda Tetractys has been regarded as nothing more than a heuristic device for generating the tone ratios of musical notes. Instead, it has profound, fundamental meaning, for — together with its tetrahedral generalisation to be described shortly — it characterizes the very nature of holistic systems. It is the arithmetic counterpart of the holistic pattern that sacred geometries embody.

#### Binary-Trinary Fibonacci Platonic Lambda

Regarding the Platonic Lambda in theTimaeus, Plato states that God created the Cosmic Soul using two mathematical strips of 1, 2, 4, 8 and 1, 3, 9, 27. These two strips follow the shape of an inverted “V” or the “Platonic Lambda” since it resembles the shape of the 11th letter of the Greek alphabet “Lambda”.

- 1,2,4,8-Doubling Sequence.

- 1,3,9,27-Tripling Sequence.

The 1,2,4,8 follows the same 1,2,4,8,7,5 patterning of the doubling circuits in our device created from the 24 reduced Fibonacci numbers.

The 1,3,9,27 tripling sequence also brings to mind the 3,6,9 vector generated by the 24 reduced Fibonacci numbers.

Plato states; “Now God did not make the soul after the body, although we are speaking of them in this order; for having brought them together he would never have allowed that the elder should be ruled by the younger… First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [i.e. between 1, 2, 4, 8] and the triple [i.e. between 1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals”

The Platonic Lambda diagram was first attributed to Crantor of Soli (335-275 BC). It is shown in Cornford’s commentary on the Timaeus, as well as references^{.} but not in the references of Jowett, Thomas Taylor and the other commentaries. While the even (double) series of 1, 2, 4, 8, and odd (triple) series of 1, 3, 9, 27 are cited often, none of these commentators mention the sum of the two series adds up to 55 as shown below:

The Soul of the Universe is the sum of the two series (Timaeus 35b):

- Sum of the double interval series (powers of 2) = 2
^{0}+ 2^{1}+ 2^{2}+ 2^{3}= 1 + 2 + 4 + 8 = 15

- Sum of the triple interval series (powers of 3) = 3
^{0}+ 3^{1}+ 3^{2}+ 3^{3}= 1 + 3 + 9 + 27 = 40

Sum of the double & triple interval series (Timaeus) = 15 + 40 = **55**

#### The three laws of resonance

The Real Pythagorean Triangle as I have researched it is a Hermetic/Chinese/Essenic resonance code essential to tha practical application of tha harmonic mathematics transmitted to Pythagoras through tha Essenes who were then still practicing tha musical secrets of Solomon. We are just now (2600 years later) beginning to understand tha possibilities of using this extremely simple resonance formula using a Pythagorean Kanon which I believed was secretely used in the Pythagorean Ceremonies and possibly many thousands of years before Pythagoras according to recent information concerning tha great ancient civilizations that predate Greece. Pythagoras’s musical secrets were not discovered by him but recovered by him- connecting tha still living Hermetic secret traditions of his time.

My personal opinion is that his practical music teaching was at tha source of all his cosmic knowledge. Pythagoras was a sort of musical avatar with a knowledge kept so secret that until now we are finally ready as an evolving civilization to receive this special gift of quantum resonance or practical string theory.

One of tha undisputed laws of Pythagoras was his Law of Silence. Pythagoreans never wrote anything down. His disciples were required to participate in a a 4 year practice of absolute silence before being considered to be allowed entrance to tha Inner Temple of Mysteries where I believe tha Universal Laws and Harmonic Principles were demonstrated by Pythagoras to his Inner Temple Disciples; not utilizing the written word but revealed on the musical instrument known as tha Kanon, or Pythagorean Harp, and since they (the Pythagoreans) never wrote anything down, tha Kanon was in effect their secret living Bible, expressing tha 3/2 Yin/Yang Cosmic Law of 3 limit mathematical acoustic nature which never changes within an ever changing constellation by altercating the many small individual movable bridges and by also altercating the tension of each string either up or down in frecuencies matching exact Pythagorean harmonics. Tha result is a radiant cosmic sonic creation used in tha Pythagorean ceremonies. That is to say tha music spontaneously composed and intuitively created on the Kanon always changes (similar to jazz improvisers and world music virtuosos today) yet uniquely with the 3/2 spiraling Pythagorean scale where tha underlying unchanging principle of harmony are the 3 laws: tha monadic principle of a vibrating string ,tha law of doubling( the octave), and tha sacred 3/2 interval which uniquely transforms source energy into an infinite spiral at tha quantum and cosmic level of creation.

After a vow of 4 years of silence in order to join Pythagoras’s inner circle I believe his disciples actually could keep a secret hence the confusion in our contemporary theoretical body of historical music relating to Pythagoras. Perhaps he actually understood and could practice an ancient Hermetic sonic formula relating to a spiritual science which reveals access for humanity to tha portal of spherical time, as well being a primary catalyst to practice lucid dreaming, galactic and inter-dimensional travel, experiencing life after death, realizing perfect health, abundance, and divine miracles.

THE REAL SONIC PYTHAGOREAN TRIANGLE IS EQUALATERAL, REPRESENTING THE TRINITY WHICH DESCRIBES UNIVERSAL CREATION IN MUSICAL TERMS. We now may have found tha First Piece of tha Quantum Puzzle: In The Beginning was tha Word: “Resonance”

##### The law of one

The Law of One is referred to by tha Pythagoreans as tha Monad. The Chinese referred to it as tha Tao or the Way of Silence whence the Resonance of Creation or Pythagoras’s idea of tha Universal Monochord comes from. The Physicists now calls it tha Dark Energy which is mathematical proof of what tha Chinese knew thousands of years ago, that tha nothing in space is actually something (actualy 96% of the energy in our universe): in other words tha Tao or Source Field of energy itself. Tha Monad expresses Tha Cosmic All and Everything. It is most effectively represented by tha Monochord , a one stringed instrument used by tha Pythagoreans which sonically mirrors tha source field of multi/universal creation. In musical scale terminology it represents tha Tonic or Principal Key Note of tha Harmonic Pythagorean Modes.

##### The law of two

Tha Law of 2 Yin (Tha Law of Doubling) or in common musical terms Tha Law of tha Octave. This is tha phenomena we have known about throughout tha ages. Any frequency, for example, tha musical note A at 222hz ,will be lower in pitch and at the same moment be equal to A 444 and therefore also equal to A 888 and so on (doubling in one direction or halving in the other direction) to infinity: The psychological and cosmological implications to this concept are vast. It implies 2=1 which our solidly accepted real world perceived by our other non-auditory senses does not compute. 2=1 is tha musical solution to the enigma of quantum mechanics: ”is tha smallest microcosmic reality a wave? or a particle? or both”. Music does show us the real truth, and if only used correctly can reveal to us that separation is unity. Applying tha Law of the Octave we can access any supersonic frecuency or even minute waves and particles on the infinite spiral of creation by using the formula 1=2=4=8=16=32=64=128=256 to infinity and applying that theory backwards from whatever sonic or even super/sonic frequency (or forward from a slower than sound frecuency like an electromagnetic field for example) , until it returns to us becoming an audible listening frequency that we can apply on tha Pythagorean Harp.

The Duad expresses the number 2, The Yin Principle. It represents the Bridge of the Universal Monochord which divides the monadic string in two equal parts, becoming the twin octaves of the original monadic key note. These twin vibrations demonstrate the harmonic Law of the Octaves, that tha number 2, the Cosmic Doubler and Separator is actually and paradoxically equal to tha number 1. Therefore tha proportional points on the monochord which demonstrate tha law of the Duad are 1/1 (as previously explained), 2/1, 4/1, 8/1, and inversely 1/2, 1/4, 1/8. Any of these points intuitively chosen by setting a bridge on selected strings of a Kanon will provide harmonic grounding when creating a constellation. This sonic phenomena relates biologically to tha negative charge (also charged with dark energy because of its special cosmic relationship with the One ) and also the grounding end of the axis of our bodies toroidal electromagnetic biofield. We can receive that ground straight from the exact center of the earth and on through our first 3 chakras and settling into our heart chakra which is located in the exact center of our electromagnetic biofield.

Hence The # 1 and the #2 set the stage for tha multitude of creation to begin.

##### The law of three

Thus enters the Law of 3 (Yang) ... interacting with Yin and initiating tha Infinite Spiral of tha Sonic Rainbow, mirroring creation: pure yin/yang rainbow of resonance gestated by utilizing a continuous series of perfect 5ths 3/2 and a continuous series of perfect 4ths ( fifths backwards) 2/3 and so on till infinity. It represents the energy axis of our toroidal electro bio-magnetic field. It is tha place where we receive light from above through our crown chakra and settling in to the heart chakra where we spontaneously resonate our personal and unique song always in tha moment – Tha Now, never performing a premeditated melody or harmonic sequence.Tha important thing to understand here is that by simply using this resonant source energy formula utilizing a Pythagorean Kanon we can access a genuine holistic and balanced creative sonic consciousness, in contrast to the intellectual left brain belief system that music education is currently employing. Harmonic practice could provide the bridge between Science and Spirituality and potentially reactivate all our 12 human DNA strands if we choose to evolve.

One of our most important limitations today is how contemporary civilization has been thoroughly indoctrinated with our current 12 tone equal temperament tuning system which is non harmonic . By non harmonic I mean in reference to tha common known fact that our tempered scale tones do not coincide with tha natural overtone and undertone series - including and even beyond tha Pythagorean 3 limit. Our modern equal temperament tuning system fails to purely activate these Pythagorean frequencies or even any of the natural just harmonic series of our clasical sacred tradition. In fact the continual indoctrination of tonal control in some form or another as applied to occidental music history dates back at least to tha time of Aristotle.

The True Pythagorean Scale generates a thirteen note scale, and since it actually only uses 12 of the 13 notes it in many ways resembles our 12 note chromatic scale used today. It also may be reconstructed using whole steps (9/8) obtained by tha formula (3/2x3/2)/2 and half steps at (256/243) obtained by tha formula (1x2x2x2x2x2x2x2x2/1x3x3x3x3x3 in order to construct the Rainbow Modes which resemble our diatonic scale and the 7 Greek modes with 7 notes each. A Pythagorean 13 note scale is created by spiraling 6 perfect 5ths (3/2 up from the fundamental tonic) and 6 perfect fourths (2/3 down from tha tonic). Tha two end notes of this spiral cycle of 13 notes (with tha tonic in the center) are our near equivalents to tha augmented 4th and the diminished 5th in our tempered scale which in this case are no longer tempered and equal. That difference describes an interval which is not normally played and is referred to as tha "Pythagorean Comma" which is 23.46 cents (almost 1/4th of our equal tempered 1/2 step. By theoretically and symbolically continuing this spiral we create a resonant road to infinity. This illusive number which is precisely 531441/524288 should be celebrated as the perfect geometric portal to infinity and studied as profoundly or even more so than tha numbers of pi and phi, the golden mean ratio and other special harmonic numbers. There are many secret and special qualities of this magic ratio which I believe will be revealed to us in time. However almost all musical experts past and present have mistakenly claimed that this number is and always has been an enigma.They insist that the Pythagoreans were terrorized by this comma. I am sure that neither Pythagoras nor his followers were ever terrorized by this incredible fact of nature. Tha beauty of this comma is hidden in that very mystery. It contains the secret of our resonance underlying biology, electromagnetism and tha quantum field.This Spiral doesn’t close to make tha circle of fifths as does our equal tempered chromatic scale. Reaching tha13th interval in the cycle of Pythagorean fifths it instead spirals on to infinity just like nature herself. It is tha formula of creation itself. The numbers 1, 2 and 3 generate the geometric pattern we call the Flower of Life.

##### To Recapitulate

The Monad, The Duad, and the Triad then express the Trinity, The Active Yang Principle in Action. The Triad symbolically represents The Many strings of a Harmonic Kanon, a multi stringed instrument utilizing strings of equal thickness and length, and numerous moveable bridges. This was the musical instrument used in secret by the Pythagoreans that I am now attempting to reactivate today. The Pythagorean Harp is capable of activating all of tha Pythagorean Rainbow Modes, utilizing exponents of the number 3 Yang as it relates proportionally to tha exponents of tha number 2 Yin (the pure double of the cosmic 1 Monad) thereby creating a spiraling sonic rainbow with an infinite creative capacity which models the cosmic principles of yin and yang. In other words, the Pythagorean Kanon is a creative harmonic laboratory mirroring cosmic creation. By programming positive emotional intentions the harmonic points of the strings are activated on the Kanon. Our constellation is created by positioning the movable bridges on selected strings by intuitively choosing any one the harmonic points: 3/2, 3/1, 6/1, 2/3, 1/3, 1/6 for each chosen string.

##### The natural tendency for expansion of creation

The natural tendency for expansion of creation is represented by this tenfold Pythagorean triangle of which the secret of the music of the trinity is amplified and reveals the potential harmonics of cosmic creation.

Every diagonal starting from any of the 10 numbers leaning to the left is yin or the law of doubling. Every diagonal starting from any of the 10 numbers leaning to the right is yang or the law of tripling. Theoretically it continues in a similar fashion on to infinity.

##### The Pythagorean Three Limit Lambdoma

The sonic harmonic numbers in actuality expand infinitely. This chart includes all the harmonic intervals which apply to universal creation utilizing the formula 2= yin and 3= yang.

##### Sacred Geometry, Numerology, and Astrology

Sacred Geometry, Numerology, and Astrology are then gestated by these resonant laws. Water is its conduit. Gravity and electromagnetism describe it. The harmonic laws of tha True Pythagorean Scale represent these creative kanons and when activated on tha Pythagorean Harp transform into a representative model of tha sonic spiral of tha space/time/multi-dimensional continuum.

All laws and harmonic principles were demonstrated by Pythagoras on tha musical instrument known as tha Kanon, or Pythagorean Harp ,and since they (tha Pythagoreans) were to never write anything down, the Kanon was in effect their secret Living Bible, expressing tha 3/2 Cosmic Laws of mathematical acoustic nature which never change within an ever changing radiant cosmic sonic creation by always initiating new constellations. In past times and as reactivated in tha present, tha music spontaneously and intuitively created on the Kanon always changes - yet tha underlying principles of 1,2 and 3,that is tha Tao, Yin and Yang never change. It is tha sacred 3/2 interval which uniquely transforms source energy represented by the number 1 into an infinite spiral at the quantum and cosmic levels of creation.

#### Robert Grants Platos Lambda and tetractys

72 = 2^3x3^2......Male to the Power of Female MULTIPLIED by Female to the Power of Male; Square to the Power of Circle EXPANDED by Circle to the Power of Square....Past to the Power of Future TIMES Future to the Power of Past = ETERNAL NOW. Light to the Power of Dark MAGNIFIED by Dark to the Power of Light = The Universe. Divine Balance through the non-judgement and merger of opposites is expanded awareness, gratitude, empathy and love....When opposites merge each not only complements but EXPONENTIALLY EXPANDS the other. 432hz Harmonic Tuning is based on the exponential powers of the numbers 2 and 3—in Perfect “Pythagorean JUST TUNING”. This Grid series determines all TIME numbers, these “Super” Numbers are also the dimensional references of all SpaceTime measurements using the ancient Imperial systems and are fundamental to the scaling of each celestial body within our Solar System. These include the mile diameter measurements for the Sun, Moon, Earth, all planets, distances between orbits of each as well as the Great Year’s Precession of Equinox. Each is derived as simple ratios of the number 72. Interestingly, the Square Root of the Slope Angle of the Great Pyramid of Giza is 7.2° (Slope Angle = 51.84°). All of this latest research comes as a result of the recent decryption of the precise proportions of ONE Circle (=π and Trinary) and ONE Square (e and Binary). (Incidentally, the base length of the Square DaVinci gave us is 7.2 inches while the top length of the Square is 7.071 inches (= 1/Root2) and represents Euler Expansions, while the base length is 7.2 inches representing Binary Trinary expansions of 72 at the point where the Vitruvian Man has his feet planted firmly on ‘Terra Firma’. 3^2(2^3) = 72, the perfect mirror symmetry of both Binary and Trinary expansions infinitely. This Squaring can be depicted infinitely in each direction as multiples of 12 and also as a Cube and Cuboctahedron. 72 is also the ONE number representing the merger of the Hexagon and Pentagon as 2+3=5 versus 2x3=6; and 72° is the Arc Length of a Pentagon; whereas the Sum of Angles of the Hexagon is 720°. Furthermore, 72 years is precisely ONE Degree of the Earth’s Precessional Cycle (72 x 360° = 25,920 years). Oh, and you may want to Google just how far our Solar System is from the Center of the Milky Way Galactic “Disc”: “Approximately 26,000 Light Years”?.........what do you want to bet it’s actually 25,920 Light Years? Do you think that the Giza Plateau may actually be ‘The World’s Oldest and Largest Clock’, trying to teach us, in an age of darkness, TIME’s True nature? The Tetractys.....It’s always been about the Harmonic Series.....now we are understanding that it is about the convergence of both Binary (2^n) and Trinary 3^n) Numbers.....culminating in the formation (and ultimately of the number 72 and by extension all TIME numbers)

### Fibonacci, Mandelbrot set and more

The Mandelbrot set is the set of complex numbers and the julia set is basically something similar which looks just like the dragon curve fractal which is like a fractaling curve superstring membrane and I suggest that the membrane face of the higher-dimensional shape that our universe is on is like this dragon curve fractal. The fractaling membrane curve that our universe is, is shown above in the pascals triangle, trinity sequences and Plato's lambda diagram.

#### Logistic map, Mandelbrot/Mandelbuble, packing of spheres and Fibonacci

The Logistic map is basically a representation of the doubling sequence and it is a fractal and whats interesting is the Logistic map is apart of the Mandelbrot Set. The Mandelbrot Set has a connection to phi which is shown in the picture and the doubling sequence has a connection to the Fibonacci sequence and there are also connections between the Fibonacci sequence and Mandelbrot Set which is important to understand for the next bit because the Platonic solids are formed out of the Fibonacci sequence.

The Mandelbulb is a 3D manifestation of the Mandelbrot Set and starts as a sphere. It iterates outward from there to form the Mandelbulb in its various complexities. Interestingly, the Platonic solids can be found in the 3D Mandelbrot set. You can play connect the dots with the bumps on the shape and get the Platonic solids. Nested Platonic solids fractalize out to create the spheres within spheres (nested spheres) that create the 3D Mandelbulb. This shows how the Mandelbrot Set actually describes a 3D spherical structure. The Platonic solids as close-packed spheres is the big secret of the Mandelbrot/Mandelbulb fractal. From the side the Mandelbulb looks like a series of close-packed spheres. Going inside the structure you find the fractalized geometry – the tendrils of the universe.

The Mandelbulb looks very similar to the fractaling flat torus, the torus expands out of the tetrahedron grid fractal and if you place a sphere around each tetrahedron you get the flower of life/packing of spheres which as shown is formed out of the Mandelbrot/Mandelbulb fractal too. The Platonic Solids faces can fractalize out in infinite iterations until they create perfect spheres. The Platonic Solids are fractal structures and the Platonic solids and the Mandelbrot set are inextricably connected.

The flat torus is constructed out of a square showing another connection between the torus and one of the faces of the flower of life packing of spheres and this face can correspond to a magic square and the diagram above in Robert Grants work and it can also correspond to The Theory of Everyone diagram which I will go over in a later post.

#### Mandelbulb

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.

A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers.

##### Tachyonic field

#### Fibonacci and doubling sequences

##### Kathara and vortexes

### Powers of 3 and Pi

As we know time numbers are multidimensional polygons and they divide numbers into 12 and 12 as a unit of time is related to the circle and 360 degrees and 360 is related to the 8 tetractys which encodes the 64 tetrahedron grid and the 64 tetrahedron grid forms multidimensional polygons in many ways but we will look into its relation to the 8 tetractys:

- 8 tetractys=1+2+3+4+5+6+7+8=36
- The 8 tetractys encodes the 7 tetractys so:
- 8 tetractys+7 tetractys=64=64 tetrahedron grid
- If each point in the 8 tetractys was a yod then the value of the 8 tetractys would be 360

As we know the 64 tetrahedron grid and tetrahedron grid fractal forms the powers of 3 cosmic tree of life which encodes multidimensional polygons.

There are 6 vortexes as explained in Kathara and vortexes but there are actually 3 since two are one, now these 3 vortexes each correspond to one of the trinity sequences and they are all vesica pices fractals one of them showed in the first diagram under Kathara and vortexes.

The first 5 powers of 3 form 11 and this relates it to 11:33 as shown in the diagram and this relates it to the trinity and as shown in the diagram this trinity forms the circle and pi.

#### Pi and the pentagram

#### Flower of Life

##### Pentaflower of life

The pentaflower of life is one of the faces of the dodecahedron which makes up the 120 cell(cubistic matrix) these pentaflowers are vortexes and the pentaflower is made up of 5 Fibonacci spirals this corresponds to the vibrational dimensions which uses 6 Fibonacci spirals.

### 1133, Pentagram and dimensions

The connection between the number 11. pi, the trinity, and the pentagram is shown above now we will go deeper into this and look more into the trinity and its connection to 11:33, the 11:33 trinity forms 1331 the last row of pascals triangle tetractys and this line is equal to 8 and above in the picture, the 3rd line forms 7 and when this is added to the 8 it forms the 7:8 correspondence and forming the 64 tetrahedron grid.

The pentagram is equal to 11 as shown and therefore it is related to En**light**enment. The number 11 also shows a 5:6 correspondence which is also formed out of the 64 tetrahedron grid and the number 1133 through the 7 star tetrahedron tetractys. This also relates to the 3D seed of life which encodes phi through 8(44 or 4+4).

#### Kathara and vortexes

The bottom right image shows the expansion of the squaring of the circle/star tetrahedron(Hexagram) which encodes the Krystal Fibonacci spiral and this expansion forms the tetrahedron grid fractal and the left bottom image shows the expansion of the unicursal hexagram in the 64 tetrahedron grid and its relation to the Fibonacci numbers and this relates it to the enneagram and vortex maths which relates to phi as shown and it also relates it to the 10 spacetime dimension/12 vibrational dimension tree of life which grows based on the Krystal Fibonacci spiral which lines up with the center of the double seed of life.

### prime numbers and 1/7

All numbers are comprised of prime numbers and all numbers(other than numbers in the 3 times table) reduce to 1, 2, 4, 8, 7 and 5 which is equal to the doubling sequence which can then be reduced to the number 2 since it is the prime that all the numbers in the doubling sequence can be reduced to, the doubling sequence then corresponds to the (fractaling)tetrahedron/tetrahedron grid fractal and this corresponds it to different cubistic matrix levels which already corresponded it to the doubling sequence and the basic structure of the cubistic matrix is 4 and 2×2=4 is the first composite number and it corresponds to the faces of the cubistic matrix and this corresponds it to the 4 points of the prime number cross(PNC).

As shown above 7 forms the basic structure of the cubistic matrix(Hexagonal yod) in 2D and 7 is related to 1/7 and 1/7=0.142857 which is 1, 2, 4, 8, 7 and 5, therefore, encoding the doubling sequence and 1, 2, 4, 8, 7 and 5 also forms the torus as shown in vortex maths and new layers can be added to the torus making it bigger and at 24 layers it can then encode the PNC and 24 repeating Fibonacci numbers plus as we know the PNC can be turned into a torus. The torus which is the shape of (electro)magnetic fields forms out of the tetrahedron grid fractal, E8 is a torus and is made out of tetrahedron in 3D so it is apart of the tetrahedron grid fractal so it is the 64 tetrahedron grid. (more information above: Flower of life and electromagnetism)

We have now explained the hexagonal yod as 1/7 and now lets talk about pascals triangle tetractys. As we know the last line of pascals triangle tetractys forms the 64 tetrahedron grid/231 gates/E8 and each line in pascals triangle is equal to a number in the doubling sequence this shows a correspondence between pascals triangle and the tetrahedron grid fractal(This also shows another way the tetractys corresponds to the tetrahedron grid fractal) and as we know the tetrahedron grid fractal is made up of black holes(Negative volume ones which shows a connection between -1/12(infinite tetractys) and the infinite tetrahedron grid because this shows proof and that the infinite tetrahedron grid is a negative volume singularity/black hole) and this shows the numbers in pascals triangle are black holes so pascals triangle is the tetractys and this is also shown in my X and Y equation.

## Extra information

- The tetractys/tetragrammaton/pascals triangle/platos lambda which is formed out of binary-trinary is the triangle which encodes all harmonics which are the vibrational dimensions.

### Prime number cross

### Alpha and Omega(Pi and E)

Doubling sequence(e) joining with Tripling sequence(pi) is shown above and the pi and e correspondence to the sequences is also shown above and in this post.

### Higher-dimensional numbers

#### Time number

Time numbers encode multidimensional polygons and are lower dimensional forms of multidimensional polygon numbers.

### The Flower(s) of Life conclusion

### Walter Russell Cosmogony

#### The Revelatorium and Robert Grant

#### Explaining the universe with Electromagnetism

##### Fractaling universe

"The entropy of the universe causes the universe to be random leading to chaos but chaos is not random it's transformation it is consciousness evolving to a higher state/level it is evolving to become an infinite fractal."

The entropy leads consciousness to evolve into more complex structures leading to more advance consciousness and the most advance consciousness in the universe is actually the universe because its structure is like a brain.

### The Hierarchical Pattern of Mindspace and more

### Tetractys, number theory and more

### Kabbalah Holons

### 12, 21, 28, 64, 144, 168 and 441 and there relationship to the pentagram and phi

### The Octonion Math That Could Underpin Physics

### Source

- en.wikipedia.org/wiki/Chaos_theory
- en.wikipedia.org/wiki/Quantum_chaos
- empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/quantumchaos.html
- fractalfoundation.org/resources/what-is-chaos-theory/
- A prime case of chaos
- www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/
- www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-20190926/
- vismath5.tripod.com/metz/
- Stephen M. Phillips
- tombedlamscabinetofcuriosities.wordpress.com
- thakanon.org/pythagorean-music-theory.html
- en.wikipedia.org/wiki/Mandelbulb