Table of Contents
Reproduced from: https://schoolofwisdom.com/fractal-wisdom/
Crashes the Wall
At any given moment, life is completely senseless. But viewed over a period, it seems to reveal itself as an organism existing in time, having a purpose, trending in a certain direction.
Aldous Huxley (1894-1963)
New discoveries in the science and mathematics of Chaos research are revolutionizing our world view. They reveal a hidden fractal order underlying all seemingly chaotic events. The fractals are intricate and beautiful. They repeat basic patterns, but with an infinity of variations and forms. The world-view emerging from this scientific research is new, and yet at the same time ancient. With a little thought, and the help of this web, you can better understand the significance of Chaos and Fractals. You can see how to use these insights in your life to create a bridge between Science and Spirituality.
It is possible to apply this new knowledge to better understand your life, to live autonomously, based on freedom, and your own contact with the Source of the Universe, the Infinite. The revolutionary new Fractal mind-set discovered by scientists and mathematicians can help you to see and understand the beauty and meaning that borders all chaos. These fractals are in space, as shown by the graphics on these webs. These fractals also exist in time, as shown by all life, including your own. We are born and die, we wake up and fall asleep, the sun rises and sets, we breathe in and out. There are basic patterns and structure to time and space that provide a unifying coherence. As the sages of long ago put it, “as above, so below.”
This web will explain some of the science and math of Chaos. It will also suggest how you can apply this new knowledge to fashion your own philosophy of life. A philosophy that will help you to cope with and understand disorder, to realize the underlying fractal structure behind the near infinite, often baffling variety of life experiences.
The Ultimate Paradigm Shift
The rapidly accelerating discoveries of Chaos are overtaking our worldview. They teach us that Newton, and indeed almost all of the pre-chaos scientists, were dead wrong in their basic view of the Universe. They thought that there was a predictable cause and effect for everything, and that everything happened according to fixed physical laws. They believed in certainties, not probabilities. Their fundamental image of the Universe was a big clock. The presence of a divine being was only necessary to make the clock and wind it up. After He created the Universe, all God had to do was sit back and watch. The laws would operate in a predictable causal fashion.
Old science actually used to think that if you only knew all of the initial conditions, how the clock worked, you could predict what would happen at any point in time. Science assumed that everything could be known and eventually predicted. The Universe was ruled by a detailed system of unchanging laws. Cosmos and causality reigned supreme. There was no room for chaos and so it was conveniently swept under the rug.
The inevitable outcome of the ordered machine view was the complete winding down of the clock, the end of time in complete entropy – the second law of thermodynamics where everything tends to breakdown, to dissipate. This big picture of science naturally spawned the “God is dead” philosophies, nihilism, the life nausea of existentialism, behavioralism, communism and the like. Now with the Chaos theories this paradigm is itself dead. A whole new scientific view has been born, one much more in accord with an organic view, the common law, and philosophies of hope and spirit.
The cosmic clock image of establishment science first began to crumble at the turn of the century when physicists found that at the nuclear level the causal laws of physics didn’t hold true. The behavior of the atom and individual electron could not be predicted. Still, even in the face of incontrovertible evidence of quantum physics, old ideas die hard. The static civil law mind set would not die easily. Even Einstein could not believe that God would play dice with the Universe. He searched in vain for a unified field theory that would explain away the chance and unpredictability so obvious in the subatomic world. Science struggled to maintain its centuries old view. The belief in a causal cosmos was now on shaky ground because it lacked a subatomic foundation. Still it prevailed because the rest of the world of physics seemed to follow linear, orderly and predictable clock like processes. Besides no one had articulated a different view to replace it. The subatomic world was considered an insignificant anomaly, an exception that proved the rule.
Then along came the Science of Chaos in the last part of this century to show that causality did not apply everywhere else as thought. In fact close measurements revealed that the unpredictable appeared in what was previously believed to be the most ordered and predictable of systems, the swinging of a simple pendulum – the very heart of a clock. As James Gleick’s book Chaos shows the brave early explorers of Chaos found that Science had been fooling itself for centuries by ignoring tiny deviations in its data and experiments. If a number was slightly off what the causal laws predicted, the pre-chaos scientists simply assumed there was an error in measurement in order to uphold the sanctity of the law itself. In order to preserve their pseudo-cosmos, scientists limited their investigation to closed and artificial systems, avoiding the turbulence of open systems like the plague.
Causality was the prime assumption behind all pre-chaos science and it never occurred to anyone to question it. This conceptual bias created a blind spot of enormous proportions. But the reality of open systems, the Chaos lurking behind all order, would not be denied. The charade of perfect order and fudged experimental data could not last forever. By the nineteen seventies it began to crumble, the conceptual blinders were falling from the eyes of more and more scientists. By the nineteen eighties the fly in the ointment, the unpredictable results in what should have been perfect predictability, could no longer be denied. The Science of Chaos was born. Our understanding of the world will never be the same.
After nearly two decades now of work by Chaoticians made up of the leading scientists and mathematicians in a wide variety of fields, the evidence is overwhelming. The world is not a gigantic clock where everything happens in an ordered and predictable manner. The real world is fundamentally disordered, free. Chaos reigns over predictability. Simple, linear systems which are causal and predictable are the exception in the Universe, not the rule. Most of the Universe works in jumps, in a non-linear fashion that can not be exactly predicted. It is infinitely complex. Freedom and free will – the Strange Attractors – prevail over rules and determinacy.
Yet Chaos is no enemy and destroyer of Cosmos, for from out of Chaos a higher order always appears, but this order comes spontaneously and unpredictably. It is “self-organized.” The creation of the Universe is an ongoing process, not just a one time event at the beginning. All and everything – and everyone – is part of this creative process. Over time all systems – from molecules, to life, to galactic clusters – are continually creating new organizations and patterns from out of featurelessness and chaos. The world is not a Clock, it is a Game, a Game of Chance and Choice. In the game random processes – chance and serendipity – allow room for free will, individuality and unpredictable creativity.
The Universe is governed by laws, but the laws are of a different kind than previously thought. Like the common law system, the Laws of Wisdom are inherently flexible. They are not written in stone, they are general. They leave infinite room for creativity within certain general parameters. A few fundamental principals exist to establish the parameters, but the Law governs much more loosely than previously thought. The Laws are subject to changes and modifications over time and depend upon the particular facts. Like the common law, the Laws of Nature appear to have flexibility; many things are decided on a case by case basis. Self organization is the rule, not the exception. Everything is not pre-determined by a rigid and complex system of detailed laws which specify exactly how everything works. There is no detailed blueprint of the universe, just a general set of Laws.
In the words of physicist Paul Davies in his book The Cosmic Blueprint (1988):
There is no detailed blue print, only a set of laws with an inbuilt facility for making interesting things happen. The universe is free to create itself as it goes along. The general pattern of development is “predestined,” but the details are not. Thus, the existence of intelligent life at some stage is inevitable; it is, so to speak, written into the laws of nature. But man as such is far from preordained.
The image of God playing dice with the Universe was threatening and fearful to the old scientists, even the great ones like Einstein, who incidentally grew up in a civil law system. But that was only because they did not understand the order lurking in Chaos, the great beauty inherent in chance. For we now know that it is only through chance that new and unpredictable relationships can be created, entities can self organize to further evolution and create entirely new symmetries and coherence. With the image of the machine clock gone, the insights of relativity can finally be appreciated. Time is not mechanical, it depends on space. Time is flexible, essentially unpredictable from moment to moment, but this does not lead us hopelessly adrift. We can still navigate from the hidden order which appears over time, the statistics from segments of time, from iteration. The order implicit in Chaos is unpredictable on a case by case basis, but still reliable and workable on the long run.
God’s dice liberates us from the prison of determinism, the hopeless tedium of the cosmic clock and the inevitable death of entropy. We have instead an intelligent Universe, where ever new and evolving life forms thrive on Chaos, where negentropy creates higher order from decaying forms. The clock is not winding down as the second law of thermodynamics had thought, it is ever being created anew. God is back in the picture, not just as the creator of the machine who then left — the ghost in the machine — but as the Strange Attractor, the origin of inexplicable and unpredictable order from chance.
This is a new kind of order, a “fractal order,” based on a relatively few basic structural principals from which many transitory laws follow. The Laws of Wisdom we must learn for the journey to self realization are flexible, evolving. Like the common law, they are articulated afresh moment by moment, case by case.The laws are stable, but they do not stand still. Exactly how the basic principals will apply to form governing laws all depends upon the circumstances, the consciousness involved, the entities, the case.
The free will of the individual in connection with the infinite is now primary. All is not determined, everyone has a chance to decide their own fate. The philosophic implications of Chaos are positive and encouraging. The Universe is not a clock, its a game. Enjoy it!
Four Chaos Attractors
The Science of Chaos has discovered four basic Cosmos Attractors: Point Attractor | Cycle Attractor | Torus Attractor | Strange Attractor.
Although known as the four “chaos attractors,” they are really the opposite – they are Cosmos Attractors that balance chaos. The four “Attractors” bring order out of Chaos. They are part of a basic law of four – a “fractal of four.” The Universe has a fundamental pattern of fourfoldness throughout all scales of magnitude. When applied to Nature, including Man, the Law of Four manifests as the four attractors. These attractors balance entropy, providing order from out of chaos. When applied in the microcosmic level “the four” manifests as the four basic energies or forces: electro-magnetic, gravity, and the strong and weak forces. In human consciousness its the four functions of sensing, thinking, feeling and willing. Understanding how the Attractors work in the meso-cosmic world can help us make sense of our world, and make sense of our consciousness functions.
Everywhere we see a hidden order and similarity over scales, such as is that shown geometrically by the Mandelbrot and Julia sets. This hidden order is based on one of the four Attractors, the Strange Attractor. It governs the fourth dimension of space-time reality. The other three attractors, which likewise bring hidden order out of Chaos, follow the first, second and third dimensions, the line, plane and solid. They are called the Point Attractor, the Circuit (or sometimes Cycle) Attractor and the Torus Attractor. As humans living in the fourth dimension we are at our best when we avoid their influences and follow the spontaneity and freedom of the Strange Attractor. Only in this way can we live autonomously in the moment, in tune with what the Chinese call the Tao, the Way, the flow of forces in the fourth dimension.
The four attractors act on all levels of reality to form Cosmos out of Chaos. They make up a newly discovered Wisdom Law fundamental to making sense of what is happening in the real world. The world is not really totally ordered as previously believed. It is fundamentally disordered, chaotic, but it contains forces or attractors of cosmos that create patterns of order over time. They are anchors of order in an otherwise stormy sea. Full understanding of the Attractors requires a new understanding of space and time. As to space, we need to understand how space is the original force – in Sanskrit called Brahman, in Chinese Wu Chi, in Peruvian and Japanese Ki – which creates the world through the point. Real insight into this only comes from direct experience of Ki. A new understanding of time entails realization that time is not really defined by the clock, but by intensity and rhythms. In the fields of the four attractors it is time which makes it possible for order to appear from chaos.
In the computer the electric current automatically creates the iteration. With Man it is not so easy, we ourselves have to return to Zero – Awareness – to start a new iteration. Awareness is sacred space (called “Wakan” by the Native American Lakota tribe; “Mana” by the Polynesians) which you connect with as soon as you attain the center of your true Self, called by the Japanese – Hara. Thus the difficulty which many scientists have in understanding Chaos is not mental, but existential. Their consciousness based on the Cycle attractor of analytical thinking refuses the Strange attractor, which means total individual responsibility. They cannot ground themselves in Zero and experience the true meaning of space and time. As they cannot find their inner core – Awareness/God – they can only see Chaos from the outside. They cannot make the jump from knowledge to Wisdom, to inside the black. Without this anchor they lack the experiential insight – and the confidence and Wisdom this brings – needed to live on the edge where great discoveries are made. They only see isolated Cantor sets and miss the pattern which only comes from awareness of the whole, the Mandelbrot fractal.
The Four Chaos Attractors:
- Point Attractor
- Cycle Attractor
- Torus Attractor
- Strange Attractor
The Point attractor is the simplest way to bring order out of chaos. Its home is the first dimension of the line, a collection of an infinite number of points. With the Point attractor in play a person is invariably drawn to one particular activity, or repelled from another, like the positive or negative poles of electromagnetic energy. There is also sometimes a point just in between attraction and repulsion, the saddle point, when the energies are in balance, just before one force becomes stronger than the other. With the Point attractor there is typically a fixation on one desire, or revulsion, and all else is put aside until it is satisfied or destroyed. With the positive attraction force all roads seem to lead to the same destination; with the negative repelling, all lead from the same place. A positive magnet drawn to negative, a pendulum slowing down with friction and air resistance, or more graphically, a young male dog around a bitch in heat, all demonstrate the workings of the point attractor. It is a black-white, good-bad, single minded attractor, except in the rare instances of the saddle point. Its analogy in consciousness is the feeling function, with its likes and dislikes.
With the Cycle attractor a person is drawn first to one thing and then to another, like a circling magnet, first attracting, then repelling, then attracting again. Its home is the second dimension of the plane, a collection of an infinite number of lines. Under this Attractor you cycle back and forth from a set of two or more activities. Although not as simple and direct as the Point attractor, there is still regularity and simplicity to the cyclic events. An example is a desire to sleep at the end of a day, which when gratified naturally leads to a desire for activity at the beginning of a new day, followed much later by a desire to sleep again, etc. In Nature it can be seen in many ways; for instance, the predator prey systems where the respective predator prey populations cycle up and down in relation to the other.
The Cycle attractor is more complex than the simple attraction or repulsion type Point attractor. Its analogy in consciousness is the thinking function. Like objective thinking the Cycle attractor recognizes both sides and tends to include a third; for example, the synthesis coming out of the thesis and anti-thesis.
With the Torus Attractor there is complex cycling which moves forward and so is different while it repeats itself. Its home is the third dimension of the solid, a collection of an infinite number of planes. With the Torus Attractor there is a high degree of irregularity and complexity in the pattern, particularly when compared to the Circuit or Point attractors. But unlike the Strange attractor, predictions can still be made, the pattern is fixed and finite. Mathematically the Torus is depicted in the shape of a large donut or bagel as shown above. It is made up of a spiraling circle on many planes which may, or may not, eventually hook up with itself after completing one or more full revolutions. Its analogy in consciousness is the sensing function. The complexity of sensations and sensuality.
An example of the Torus attractor at work would be a more complex set of attracting events which occur to a person on many levels over a course of a year, and repeat again, year in and year out. For example, a desire to golf each summer, hike each fall, and eat and drink too much on holidays. In Nature it is shown for instance by the complex interaction of a number of interdependent species: the population of one predator species relates to that of the prey of its prey. For example, the size of the insect population effects the size of the frog population, which effects the size of one of their predators, the trout, which in turn effects their predators, the pike. Unfortunately, most humans are also subject to the complex but predictable influences of the Torus attractor, or the even more simplistic influences of the Cycle or Point attractor.
We need to escape from the deterministic influence of the point, circuit and torus attractors into the unpredictability of the Strange Attractor. This attractor is the basis of Self Organization. There is no apparent order at all to the actions of the Strange attractor. On the surface it appears to be pure Chaos, but nevertheless there is order of a subtle kind which only appears over time when looked at in the right perspective. Its analogy in consciousness is the willing function. Yet, when tied to Awareness – the Zero – it is spontaneous, unpredictable. It appears to be chaotic, yet it has order of a subtle, fractal kind.
The Strange Attractor can take an infinite number of different forms. All of them are fractal and demonstrate infinite self similarity. The best known fractal driven by the unpredictable Zero – the Strange Attractor – is called the Mandelbrot Set. It is shown in static and zoom magnification form all over these webs. Another is the Lorenz Fractal shown above. Still another important Strange Attractor that appears in odd places throughout Nature is called the Feigenbaum fractal. It is shown with one recursive self similarity pointed out.
Reproduced from: https://schoolofwisdom.com/fractal-wisdom/five-dimensions/
Zero Dimension, First Dimension, Second Dimension, Third Dimension, Fourth Dimension.
Number is composed of the nine numerals, basic to all information, uniting geometry and arithmetic, space and time. Their reciprocal relation is clarified through the circle of 5 dimensions, based on the relation of infinity to one. The circle of dimensions summarizes the laws of number and mathematics. The eight points shown on the dimension circle (0 – 7) correspond to the eight directions on the outside of the Wheel, an important Wisdom of the Native Americans.
Including the zero dimension there are five dimensions: 0 – 4. The zero, first second and third dimensions have long been accepted as true, but the reality of the fourth dimension was questioned. Since Einstein, however, the existence of the fourth dimension is now an accepted fact and the first, second and third dimensions are now understood as imaginary. We live only in the fourth dimension, but in order to understand our dimension, our reality, we must also understand each of the other dimensions. Moreover we must realize how infinity permeates each dimension, including the fourth. So we begin our exploration of Number as a Wisdom tool by understanding the dimensions and their relation to the infinite.
The Five Dimensions:
- Zero Dimension
- First Dimension
- Second Dimension
- Third Dimension
- Fourth Dimension
The Zero Dimension is the point, the infinitely small place holder. It exists not in space, but in time only. It is the moment between past and future, the subject, zero. It constitutes potentiality, the four space dimensions constitute actuality. The Zero Dimension is the home of Natural Numbers. The subject point, the moment, is Zero, pure Awareness. Its numbers are the natural numerals.
The nine natural numbers are the basis of quality and invariance. All numbers can be reduced to the nine: for example by addition: 365 = 14 = 5. Becoming aware therefore means deducing or abstracting to the nine fundamental criteria. In the Jewish Cabala this is the nine names of the divine, in Chinese it is the nine forms of the Tao.
The First Dimension is the LINE. It contains an infinite number of points. Temporally, it represents the Future. The moments of Zero Dimension create the future, forming a trajectory.
The first dimension is the home of the Whole or Integer numbers. The points, natural numbers, have no extension. Integer numbers unite positive and negative up to ten and create the number line.
The Whole numbers are created through addition, basic to sensing, and through subtraction, basic to the spirit, the path in the unknown future.
The Second Dimension is the PLANE, such as a square. It contains an infinite number of straight lines. In Time, the Second Dimension represents the Present. The trajectory of moments in the First Dimension (Future) appears like a disk or revolution.
In the second dimension we have the Rational numbers, based on three points, visualized on the plane with a vertical and horizontal axis. The number plane was known to the Pythagoreans and called the Chi:
Rational numbers are produced by division in the positive field and multiplication in the negative field. Zero is the center of the CHI, which for Pythagoras and Plato was the tool of the Demiurge, the Creator. The fields contain only the fractions and products inside the ten numbers. Division is the basis of thinking, multiplication the synergy of the soul.
The Third Dimension is the SOLID, such as a cube. It contains an infinite number of planes or squares. Temporally, the Third Dimension represent the Past. The disc of the Second Dimension (Present) turns one half time around its axis and fills out the sphere of the past.
The third dimension brings out the Real numbers. Real numbers start from zero and connect fractions of the same numerical value, leading to the proportions and functions.
The proportions are the basis of continuity and harmony. They connect fractions of the same value to zero. The functions are the basis of discontinuity. They connect products by which bodies are in relation, as for instance in the atom, where the distances of the electron shells follow the numbers of the central diagonal 1 – 4 – 9 – 16, and the possible number of electrons in each shell, the capacity, follow the diagonal 2 – 8 – 18 – 32. The rational numbers of the second dimension, and the whole and natural numbers of the first and zero dimension, all have a fixed place on the number line. The real numbers in the third dimension are, however, fundamentally different; although they are located somewhere on the number line, they have no fixed place there. To the ancient Greeks who first developed mathematics to a high art in the West, all numbers had to have a fixed location somewhere on the number line. The existence of the Real numbers, with no fixed location, was known only to a few high initiates in the Pythagorean brotherhood who swore to keep it secret. It can be easily understood today by way of the Pythagorean theorem:
The Pythagorean Theorem exemplifies the rational numbers. But what happens if A and B both equal 1? In this case C must equal the square root of 2. But the square root of two is an irrational Real Number. It is a number which goes on and on with no repetition into infinity. 1.41421… . It is a never ending number and has no fixed place on the number line. Unlike an infinite rational number which goes on and on, but repeats, such as a third (.3333333…), where we can know the exact location on the number line, with a Real Number, we can only know its approximate location. There are other examples of Real Numbers, such as Pi (the ratio of a circumference of a circle to its diameter), the square root of any prime number, e, etc. These Real Numbers never end and never repeat.
The Fourth Dimension, space-time continuum, is reality. In the fourth dimension the infinite number of solids in the Universe are in relationship with each other through time and energy. In the Time domain, the Fourth Dimension continues the movement of the Third Dimension (Past) to form a wave, constituting fractally the space-time continuum.
THE HYPERCUBE The Fourth Dimension is portrayed geometrically by fractals and by the Hypercube. The Hypercube is the symbol used in mathematics to try and represent the fourth dimension in two dimensions (a drawing on a piece of paper – a plane).
From the center of the Hypercube through its 8 diagonals the Hypercube is related to everything in the Universe. The infinity in the Fourth Dimension lies in the infinity of relations. This can be expressed in terms of “fractal scaling”, from the infinite small to the infinite big, perpendicular to the other dimensions and including the intervals or fractal dimensions between them. The meaning of fractal scaling is explained later in this Chapter, for now it is sufficient to understand this as scales of magnitude, as for instance from the size of the atom to the size of a galaxy. The Hypercube is cut by 4 diagonals constituting the central point. In consciousness this center point represents the identity or the Self. The number of the diagonals is 4 X 3 = 9, according to the Pythagorean theorem. The four diagonals are 1-5, 2-6, 3-7 and 4-0.
The fourth dimension is the home of the Complex numbers and Fractal geometry. Unlike the other dimensions, the fourth is the real world in which we live. It is the space time continuum of Man and Nature where there is constant change based on feedback. As Mandelbrot recently discovered the fourth dimension includes not only the first three dimensions, but also the gaps or intervals between them, the fractal dimensions.