Great Throughts Treasury

"Although knowledge is without bound, Wisdom has a limited number of criteria. The criteria follow a basic mathematical structure, a core fractal upon which all of reality, and knowledge, are built. The criteria of Wisdom can be represented by a geometric image: the Wheel , as shown here. This symbol is found in all anthropological traditions of the world. It is the key, the Rosetta Stone, to understand these traditions, to comprehend how they all speak to the same human condition. The Wheel allows you to think universally and act globally, in tune with history and your own inner destiny. It is the basic teaching of any School of Wisdom. Today, for the first time, we have information about the Wheel from cultures throughout the world. Professor Arnold Keyserling has created a new holistic, global synthesis of The Wheel as his life's work. The modern Wheel is based on all cultures of the world, including the scientific and mathematic."

"Complex numbers and fractal geometry are the most important to Man, yet they were the last to be discovered by reason, and the most difficult to grasp. The full significance of the mathematics of the fourth dimension could not be realized until the downfall of the Newtonian-Euclidian mind-set and its replacement in the 1970's and 1980's with the Science of Chaos. The Chaos insights were led by Benoit Mandelbrot, an IBM scientist and Professor of Mathematics at Yale, with many other scientists close behind. Computers helped Mandelbrot realize the full significance of the formula which now bears his name."

"Life is organized fractally by scaling (cell division) and number generation (cell multiplication). Matter, the basic structure of which is the atom, also follows the numerical-fractal system. Matter is ordered into ten types of elements, in correspondence to the nine numerals and zero: I. ALKALI METALS: 3Li, 11Na, 19K, 29Cu, 37Rb, 47Ag, 55Cs, 79Au, 87Fr. II. CALCIUM GROUP:4Be, 12MG, 20Ca, 30Zn, 38Sr, 48Cd, 56Ba, 80Hg, 88Ra. III. ALUMINUM GROUP: 5B, 13Al, 21Sc, 31Ga, 39Y, 49In, 57La, 81Tl, 89Ac. IV. CARBON GROUP: 6C, 14Si, 22Ti, 32Ge, 40Zr, 50Sn, 72Hf, 82Pb. V. NITROGEN GROUP: 7N, 15P, 23V, 33As, 41Nb, 51Sb, 73Ta, 83Bi. VI. OXYGEN GROUP: 8O, 16S, 24Cr, 34Se, 42Mo, 52Te, 74W, 84Po. VII. HALOGENS: 1H, 9F, 17Ci, 25Mn, 35Br, 43Tc, 53I, 75Re, 85At. VIII. IRON GROUP: 26Fe, 27Co, 28Ni, 44Ru, 45Rh, 46Pd, 76Os, 77Ir, 78Pt. IX. RARE EARTHS: 58Ce, 59Pr, 60Nd, 61Pm, 62Sm, 63Eu, 64Gd, 65Tb, 66Dy, 67Ho, 68Er, 69Tm, 70Yb, 71Lu, 90Th, 91Pa, 92U. 0/X. NOBLE GASES: 2He, 10Ne, 18Ar, 36Kr, 54Xe, 86Rn."

"Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension. He proved that the fourth dimension includes the fractional dimensions which lie between the first three. He calls the in between or interval dimensions the "fractal dimensions". He has shown mathematically and graphically how nature uses the fractal dimensions and what he calls "self-constrained chance" to create the complex and irregular forms of the real world."

"Fractal geometry and the insights of the science of Chaos are based on Complex Numbers - the numbers of the fourth dimension which are capable of modeling the dynamics of chaotic systems. Unlike all other numbers, the Complex Numbers do not exist on the number line at all, even with an approximate location like the Real Numbers. The Complex Numbers only exist on an x-y time plane involving the so called Imaginary Numbers. They have only indirect reference to the number line."

"Since Einstein we know without a doubt that reality is a space-time continuum where matter and energy are interchangeable. Science and mathematics have learned a great deal about space, but very little about time. What we do know is strange. The passage of time is not objective, it relates to the velocity of matter. Thus as matter approaches the speed of light (the speed limit of matter), time slows to a near stop. A person traveling in a space ship moving at the speed of light thus would have no time and would not age. Time is flexible and personal. Moreover empirical studies show that most events in nature happen in some kind of time cycle. They repeat, increase and decrease, move in and out, in regular, albeit sometimes quite complex cyclic intervals. Beyond knowing that time is personal and cyclic, modern science has just begun to explore the nature and psychodynamics of time."

"The Circuit or Cycle attractor corresponds to Consciousness, the function of Thinking and realm of Soul. With this attractor a person is drawn first to one thing and then to another, like a circling magnet, first attracting then repelling then attracting again. Here there is a cycling back and forth from a set of two or more activities. There is some 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. Like thinking it sees both sides and tends to include a third; for example, the synthesis coming out of the thesis and anti-thesis."

"The development from animal man to human man presupposes the understanding and transformation of brain and mind which the religions called spiritual rebirth. To understand this rebirth, and consciously participate in it, you need only look to the structure of the human brain. It is made up of three parts: neocortex, limbic system, and cerebellum/brain stem."

"Semiotics shows the development of speech power, numbers and music the systemic frame of awareness; together they form the original code of the mind. But matter also follows the same structure by mathematical necessity."

"The Enneagram and The Wheel are the bridge between awareness and consciousness, sense and meaning. They constitute the original Cabala. Out of the nine parts of speech, and twelve components of the mind, all possible meaning arises. The structure of the inner world of mind and language mirrors that of the outer world of matter and vibration. The outer world, including music and color, follows the same laws of number which govern the inner world. The laws of number bridge the inner and outer worlds. By understanding number we understand ourself. We gain a powerful Wisdom tool with which we can make sense of our world."

"The Torus attractor corresponds to three-dimensional Matter, the function of Sensing and the realm of Body. This is a more complex cycling which moves forward and so is different while it repeats itself. There is a high degree of irregularity and complexity in the pattern of the Torus attractor, particularly when compared to the Circuit or Point attractors. But unlike the Strange attractor, a pattern can still be found and predictions made. Mathematically the Torus is three dimensional and is shaped like a large donut or bagel. 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."

"The Wheel allows the "Myth of Science" to be used as a tool for self liberation, a function in the past filled only by religion. With the Wheel the meaning and significance of science, as well as the major religions and philosophies of the world, can be understood and applied for fulfillment. The philosophy of the Wheel is naturally one of tolerance because the Wheel allows you to see the common denominators underlying the different religions and spiritual traditions of the world. It is opposed only to dogma - where one person's dream and meaning of life is imposed on everyone else. You make sense of the Universe, you do not find sense that is already there, or sense made by someone else. You make it. Only then is true freedom possible, when you yourself make sense of your own life, when you explain life to yourself and do not take another's explanation of your life."

"The Wheel is the primal archetype which makes possible an understanding of the Wisdom of all cultures of the World. This basic symbol unites the differing traditions by their underlying mathematical criteria and conscious experience. All human Wisdom can be organized by and has a place in the Wheel. It represents the basic structure underlying all Wisdom. The Wheel is an open, holistic system with which a person can fashion their own philosophy and life meaning, and so reach a state of profound Awareness or Wisdom. From out of Awareness-Wisdom, you can know what chances to choose, and can live your life as a free, autonomous individual, yet integrated with the Cosmos and the infinite. This is the Path of Wisdom."

"The basis of evolution is self-organization. There is no master plan in nature or civilization. God is not playing dice with the Universe, but He-She-We does make a good deal of it up as we go along. Self-organization develops without predetermination. There is free will, a chance to choose. But the self-organization is structured and shaped by the four scales or dimensions. Self-organization can be understood as the capacity to create information based on the Mandelbrot vector 0 : z z? + c considered as a continuous creative process. The key is spontaneous improvisation in the moment - acting in the Tao - returning to Zero. This whole field has been thoroughly explored for millennia by Chinese thinkers and so we will use their terminology."

"The Five Dimensions: 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. 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 imaginary dimensions. Moreover we must realize how infinity permeates each dimension, including the fourth."

"The geometric construction of the Wheel follows a simple mathematical sequence corresponding to the five dimensions of space and time: The construction of the Wheel begins with a point in space. . The point is the Zero Dimension, the infinitely small place holder. It is described with the Natural Numbers. From out of the point two lines are drawn whereby the four chaos attractors form a cross with the 10 numbers. The line is in the First Dimension , made up of an infinite number of points next to each other in one direction. It is described with Whole Numbers. The Plane of the Second Dimension, made up of an infinite number of lines and described with Rational Numbers - multiplication and division inside of the radius 10 - generate the Pythagorean CHI, consisting of the fractions born out of division and the products of multiplication. The end of the 2 diagonal of the two dimensional CHI forms the center of two circles, one going up to 10, the other through the zero point. These circles spin along the center as a fixed axis to create a Sphere with Three Dimensions. The sphere and the Third Dimension is made up of an infinite number of planes connected to each other. This is described by the Real Numbers. In the third dimension the Wheel is divided into twelve equal sections as formed by the sound waves of the normal musical scale - the temperated cycle of fifths which make up the twelve tone scale. The twelvefold division inherent in tonal scaling corresponds with the twelve moon cycles or months of the solar year. The archetypal symbols of the zodiac shown below are used in many cultures of the world to represent the twelve sectors of the Universe surrounding the Earth and the basic components of time. This is not astrology, although it does have some components in common with ancient astrology as developed in Babylonia in 2340 B.C. In the third dimension the twelve fields of the Wheel also constitute the light - pigments - color circle. In the Fourth Dimension of Complex Numbers and Fractal Geometry, the real world of Man and Nature in which we live, the numerals organize the 9 criteria of Wisdom shown by the Enneagram inside the Wheel. Each of these nine represents a fractal - self similar, but not identical - repetition of the basic theme applied in a different field of knowledge. The basic theme is the underlying order behind all chaos and diversity - The Wheel - which makes possible a coherent philosophy of the World. This is a philosophy without walls, only foundations, underlying structures."

"The Mandelbrot set is a dynamic calculation based on the iteration (calculation based on constant feedback) of complex numbers with zero as the starting point. The order behind the chaotic production of numbers created by the formula z -> z2 + c can only be seen by the computer calculation and graphic portrayal of these numbers. Otherwise the formula appears to generate a totally random and meaningless set of numbers. It is only when millions of calculations are mechanically performed and plotted on a two dimensional plane (the computer screen), that the hidden geometric order of the Mandelbrot set is revealed. The order is of a strange and beautiful kind, containing self similar recursiveness over an infinite scale. See the graphic below of the Mandelbrot set, and the others which follow in this chapter. These graphics provide an accurate right brain glimpse into this world. Mandelbrot's formula summarizes many of the insights he gained into the fractal geometry of nature, the real world of the fourth dimension. This contrasts markedly with the idealized world of Euclidian forms of the first, second and third dimensions. These forms had preoccupied almost all mathematicians before Mandelbrot. Euclidian geometry was concerned with abstract perfection almost non-existent in nature. It could not describe the shape of a cloud, a mountain, a coastline or a tree. As Mandelbrot said in his book The Fractal Geometry of Nature: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.""

"The New Age is based on computer technology and the global information network -- the Internet. They are the hardware and software of the noosphere. Conscious programming starts from a scientific hypothesis, but beyond that is Awareness. Awareness has in it all of the criteria explained up to now. Awareness is Zero dimension, peak experience, satori, samadhi, nirvana, rapture, bliss. It is the key which allowed Mandelbrot to discover the Geometry of nature, the order of the strange attractor in the fourth dimension. 0 : z z? + c. You cannot stay in Zero. It is the beginning. The problem is how to link Awareness with Consciousness in everyday life. Normal discursive consciousness hinders this linkage. In the past irrational methods of connection were used, such as in Zen Buddhism and Sufism. But there has also always been a rational way, as for instance with the Spanish-Jewish cabbalists, or the Taoists of the Golden Flower. They knew how to link up awareness with consciousness using number and symbol. Their method employed reason in a holistic system, but went beyond reason to include all of existence. Their thinking systems in essence mirrored all of reality in the same sense that the cubic Kaaba in Mecca is said to contain all criteria. In today's technical world, the way of thinking as a bridge between Awareness and Consciousness is emphasized, and the irrational methods are becoming secondary."

"The Path of Wisdom essentially has four steps: (1) know your body, (2) create your energy body, (3) create your meaning in life, and (4) participate in a global network of friends where your meaning can be fulfilled in history. These steps are not sequential, they all happen at once, with a change in emphasis over time."

"The Point Like attractor corresponds to Energy, the function of Feeling and the realm of Spirit. With this 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."

"The Wheel is the translation key to all spiritual knowledge. It allows a holistic understanding of the Wisdom teachings of all cultures of the World. This basic symbol unites the differing traditions by their underlying mathematics and consciousness. All human Wisdom can be organized by, and has a place in the Wheel. It represents the basic structure underlying all Wisdom. The Wheel is an open, holistic system. With it a person can fashion their own philosophy and life meaning. The Wheel empowers a person to reach a state of profound Awareness, or Wisdom, without depending on a Guru or religion to tell them what to do. From out of Awareness-Wisdom, you can know what chances to choose. You can live your life as a free, autonomous individual, yet still be integrated with the Cosmos and the infinite. When properly understood The Wheel can serve as a thinking tool to bridge Science and Religion, the left and right brains. The full significance of otherwise dry scientific knowledge can then be felt in the right brain. Conversely, otherwise mystical religious-spiritual experiences can be understood rationally in the left brain. This book explains the dominant belief system of our time, the Myth of Science, and shows how this myth can be used in a practical and spiritually fulfilling way. The mathematical structure of the Wheel is used to integrate Science into world culture and the great myths of the past."

"Wisdom is the ability to live coherently in a chaotic world. It requires the "knowledge behind the knowledge". This means an understanding of the hardware of the mind, its structure, as opposed to the mind's software - the particular languages, sciences, and religions which occupy the mind. This is knowledge of chance and chaos, and how to find the hidden order which lies behind chaos. The order is found in structure, number, geometry and music. They allow you to make sense of chance, to understand the meaning of your life. Understanding the mind's hardware empowers the individual to make the appropriate choices in the chaotic world of chances."

"We need to escape from the deterministic influence of the point, circuit and torus attractors into the freedom and spontaneity inherent in the Strange Attractor. The Mandelbrot fractal is one of an infinite number of forms which the Strange Attractors assumes. The Feigenbaum fractal shown above is another geometric form found hidden in time throughout nature which is based on the Strange Attractor. Being fractal, all of these geometric forms are recursively self-similar over infinite scales of magnitude. 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. It is an order of self-similarity, sometimes even idenity, where a geometric shape allows for comprehension. It is an order where infinity is constrained, but never contained. The Strange Attractors correspond to Self-Organization, Willing, Awareness and the fourth dimension of fractal forms where chance is inherent. Feigenbaum and Mandelbrot Although these Strange Attractors may look very different from each other, they frequently have a hidden relationship with each other. This is shown for instance, in the diagram to the left, comparing the Feigenbaum and Mandelbrot fractals. The relationships can take near infinite forms and are not always as obvious as the proportional one shown here."

"The whole spectrum of human knowledge is synthesized to reveal the spiritual significance of the dominant belief system of modern times, the Myth of Science. Myth is not used in a derogatory sense, but in its original meaning as a true belief system which gives meaning to life, and links with the great belief systems of the past."