Chapter Six of Laws
of Wisdom
The outer world can often seem as chaotic as our inner world - our stream of consciousness. Coherence can all too easily elude us. We usually experience a convoluted flow of happenings and events. The fragmented, fractal nature of everyday reality, and people, is one of our basic problems. To use thinking to sort things out - to start making sense of it all - we must first find the basic structure to reality. The structure reveals the order underneath the chaos.
We have seen the fourfold nature of consciousness; the laws concerning the four functions and four brain waves. An equivalent fourfold law applies in the material world. This was recently discovered by scientists working in the new field of Chaos. They found that seemingly-chaotic, lawless actions in the outer world actually followed a hidden order. The order they discovered was fourfold. They found that all outer phenomena are governed by what they call the four "attractors". The attractors are forces which bring order out of disorder. They are called the point attractor, the cycle or circuit attractor, the torus attractor and the strange attractor. The attractors are in accord with the four functions: torus-sensing; cycle-thinking; point-feeling and strange-willing. They form a basic Constitutional Law of the outer world of nature.
- The Four Attractors
The Point attractor depicted on the previous page is the simplest way to bring order out of chaos. With the Point attractor in play, an animal or thing is invariably drawn to one particular activity, or repelled from another, like the positive or negative poles of electromagnetic energy. 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. Like the feeling function it is a black-white, good-bad, pleasure-pain mechanism.
The Cycle or Circuit attractor depicted above creates order in a bipolar fashion. An animal or thing is drawn first to one thing, and then to another. An example is a circling magnet, first attracting, then repelling, then attracting again. 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. Another example is the predator prey systems where the respective predator prey populations cycle up and down in relation to the other. 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 shown on the previous page there is complex cycling which moves forward. Thus while it repeats itself it is also different. 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, albeit quite complicated. Its analogy in consciousness is the sensing function - the complexity of sensations and sensuality.
The Torus attractor is seen 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 affects the size of the frog population, which affects the size of one of their predators, the trout, which in turn affects 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.
The famous Mandelbrot fractal shown above is the perfect exemplification of the Strange Attractor. This attractor is the basis of all Self Organization. We will explore its meaning, and the case of its discoverer, the mathematician Benoit Mandelbrot, later in this chapter . There is no apparent order at all to the actions of the Strange attractor. On the surface it appears to be pure Chaos. It is spontaneous; a-causal. But nevertheless, there is still order of a subtle kind which is created by this attractor. It is an order which only appears over time when looked at in the right perspective. Its analogy in consciousness is the willing function. For more information of the Attractors and their basis in geometry, see my book with Keyserling Chance and Choice. For general scientific information on the attractors and chaos see the best selling book by journalist James Gleick entitled Chaos: making a new science(1987).
- LAW AND DISORDER
To better understand the attractors and how they fashion order from chaos, we have to understand the laws of disorder. This allows us to appreciate the inherent flexibility of Law. The new discoveries from the Science of Chaos make it clear that the Laws of nature are not rigid as once thought, they are flexible.The Laws of Wisdom, the Laws of consciousness and nature, are closer to the system of common law in effect in the United States and Great Britain, than the more rigid civil law systems in effect for the rest of the world. The civil law system, originating from the Napoleonic Code, is based on statutes - on static written rules. The common law on the other hand, although it includes statutes, is primarily based on case law - on decisions made by judges considering unique facts, interpreting the statutes. As Judge Aldisert puts it, "The heart of the common law tradition is adjudication of specific cases". For this reason the common law isinherently flexible and changes with time and circumstance. As the great American jurist Roscoe Pound put it: "Law must be stable, and yet it cannot stand still". The common law flows from the facts of particular cases. From the cases come narrow rules of law, then slowly over time, broader principles of law are fashioned from the rules of many cases. In the often-quoted words of law professor, Munroe Smith, in Jurisprudence (1909),
The rules and principles of case law have never been treated as final truths, but as working hypotheses, continually retested in those great laboratories of the law, the courts of justice. Every case is an experiment; and if the accepted rule which seems applicable yields a result which is felt to be unjust, the rule is reconsidered. It may not be modified at once, for to attempt to do justice in every single case would make the development and maintenance of general rules impossible; but if a rule continues to work injustice, it will eventually be reformulated. The principles themselves are continually retested; for if the rules derived from a principle do not work well, the principle itself must ultimately be re-examined.
Common law is not etched in stone, it is continually created anew. In fact, above the entrance to Yale Law School is the engraving: ๎The law is a living growth, not a changeless code.๎ The particular "hornbook" laws may vary and be modified as facts mold the law, demand exceptions or even the creation of new laws. The "Law" is a subtle, flexible thing which defies certainty and absolute predictions. As the great jurist Cardoza put it in his essay, Growth of Law (1924), "When uniformities are sufficiently constant to be the subject of prediction with reasonable certainty, we say that law exists". Cardoza recognized that certainty of prediction was never absolute, that in any one case therule of law could err. For Cardoza, as for today's modern physicist, Law is a matter of probabilities, not certainties.
The new discoveries of the Science of Chaos, and the Scientists in this field - "Choaticians" as the movie "Jurassic Park" calls them - are revolutionizing the world of science. The discoveries of Chaos 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.
- PARADIGM SHIFT IN SCIENCE
Old science actually used to think that if you only knew all of the initial conditions and 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, behaviorism, 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 the common law, and the 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 of certainty.
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 at all costs.
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 1970s it began to crumble, the conceptual blinders were falling from the eyes of more and more scientists. By the 1980s 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 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. As the southern jurist Logan Bleckly said in 1879: ์... it is always probable that something improbable will happen.๎
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 principles 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 blueprint, 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 in 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 principles from which many transitory laws follow.
The Laws of Wisdom we must learn for the journey to self realization are akin to 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 principles 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, it is a game. Enjoy it!
The story of Chaos begins in number and geometry. A key player in this story is Benoit B. Mandelbrot. Our journey into the heart of chaos necessarily begins with his case. Benoit Mandelbrot, an IBM scientist and Professor of Mathematics at Yale, made his great discoveries by defying establishment, academic mathematics. In so doing he went beyond Einstein's theories to discover that the fourth dimension includes not only the first three dimensions, but also the gaps or intervals between them, the fractal dimensions. The geometry of the fourth dimension - fractal geometry - was created almost single-handedly by Mandelbrot. It is now recognized as the true Geometry of Nature.
- THE CASE OF: BENOIT B. MANDELBROT v. ACADEMIC MATH
Mandelbrot's fractal geometry replaces Euclidian geometry which had dominated our mathematical thinking for thousands of years. We now know that Euclidian geometry - with its perfect forms - pertained only to the artificial realities of the first, second and third dimensions. These dimensions are imaginary. Only the fourth dimension is real. More on this later; first, a little on the man behind the Laws and the mathematical world he revolutionized.
Before Mandelbrot the academic math world was dominated by arithmetic. Geometry was relegated to a secondary, inferior position. Mathematics prided itself in its detached, abstract isolation, completely apart from the real world - particularly nature - breathing instead the refined and pure air of its own self-contained universe of number. In the last century it even divorced itself from physics, its sister science for centuries. The elite world of mathematicians became very isolationist, very remote from nature.
Then along came Benoit Mandelbrot to change math forever. An unlikely revolutionary, he was born into the atmosphere of academic math. His uncle, Szolem Mandelbrot, was a member of an elite group of French mathematicians in Paris known as the "Bourbaki". Benoit Mandelbrot was born in Warsaw in 1924 to a Lithuanian Jewish family. His parents foresaw the geo-political realities and moved to Paris in 1936. They picked Paris because Szolem Mandelbrot was well-established there as a mathematician. The Mandelbrot family, a necessarily tight knit group, survived the War in Tulle, a small town south of Paris, where youngBenoit received no regular formal education.
It is amazing but true, that one of the great math geniuses of all times was never taught the alphabet, and never learned multiplication tables past fives! Even today he claims not to know the alphabet, so that it is difficult for him to use a telephone book. Still, he had a special genius, and after the war Benoit enrolled in elite Paris universities, and started to follow in his Uncle's mathematical footsteps.
He had a tremendous gift in math, but it proved to be quite different from his uncle's, in fact quite different from anything seen before in academia. He had a visual mind, a geometric mind. In the Parisian school setting of that time this talent was discouraged. Still he defied convention and solved problems with great leaps of geometric intuition, rather than the "proper" established techniques of strict logical analysis. For instance, in the crucial entrance exams he could not do algebra very well, but still managed to receive the highest grade by, as he puts it, translating the questions mentally into pictures. Benoit was clever and hid his gifts until he had obtained his doctoral degree in math. Then he fled academia and his uncle's "bourbaki" math and began to pursue his own way.
His journey took him all the way to the United States, far from academia, eventually in 1958 leading to the shelter of IBM's research center in Yorktown Heights, New York. His choice of the world's most successful computer company as employer proved to be quite fortuitous. The young genius from the French math establishment was allowed free reign to pursue his mathematical interests as he wished. They proved to be more diverse, eclectic and far-reaching than anyone could have imagined.
His intellectual journey took him far from the beaten roads of academic math into many out of the way disciplines. For instance, he became expert in certain areas of linguistics, game theories, aeronautics, engineering, economics, physiology, geography, astronomy and physics. He was also an avid student of the history of Science. Importantly, he was also one of the first mathematicians in the world to have access to high speed computers. In his words,
Every so often I was seized by the sudden urge to drop a field right in the middle of writing a paper, and to grab a new research interest in a field about which I knew nothing. I followed my instincts, but could not account for them until much, much later.
The seemingly random pursuit of knowledge from a variety of unrelated fields was unheard of at the time. All of academia and science were heading in the opposite direction towards ever greater specialization. His concern with a broad spectrum made him an unpopular maverick in establishment circles, and generally unwelcome in the fields he would visit. Still, he was a brilliant mind, and wherever he went he left behind intriguing insights, and managed to stay in the good graces of his employer. It was Mandelbrot, for instance, who when investigating economics first discovered that seemingly random market price fluctuations can follow a hidden mathematical order over time, an order which does not follow the standard bell curves usually found in statistics. (ps - I am working on possible applications of this now (1998-2000) to the U.S. stock market. see: www.moneywisdom.com)
His now famous study in the field of economics concerned the price of cotton. This is the commodity for which we have the best supply of reliable data going back hundreds of years. The day to day price fluctuations of cotton were unpredictable, but with computer analysis an overall pattern could be seen. Patterns in statistics are nothing new, but in economics they are quite elusive. Moreover, the pattern that Mandelbrot found was both hidden and revolutionary.
Mandelbrot discovered a pattern wherein the tiny day to day unpredictable fluctuations repeated on larger, longer scales of time. He found a symmetry in the long-term price fluctuations with the short- term fluctuations. This was surprising, and to the economists - and everyone else - completely baffling. Even to Mandelbrot the meaning of all this was still unclear. Only later did he come to understand that he had discovered a "fractal" in economic data demonstrating recursive self similarity over scales. The explanation of this key Law requires more background into the geometry of chaos.
Mandelbrot's eclectic research ultimately led to a great breakthrough summarized by a simple mathematical formula: z -> z² + c. Mathematicians are actually embarrassed as to how ridiculously simple this breakthrough equation really is. The arrow symbol -> means ์iteration๎. Iteration is the feedback process where the end result of the last calculation becomes the beginning constant of the next. In other words the result of z² + c becomes the z in the next repetition. So if say z=2 and c=3, the result of the first iteration would be 7 (2² + 3). Then the next time z=7, and c remains the same, 3. So the result is 52 (7² + 3). And so the process continues. Like life it is a dynamic equation, existing in time, not a static equation. For more details on this equation, and exactly how it works with zero as the starting value of Z to create the geometric Mandelbrot forms, see Chance and Choice.
- THE GEOMETRY OF CHAOS
This famous formula - z -> z² + c - is now named after its inventor and is called the Mandelbrot set. It is significant to understand that this formula - and the Law of Wisdom which it represents - could not have been discovered without computers. It is no accident that his discovery, which many say is the greatest in twentieth-century mathematics, occurred in the research laboratories of IBM.
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. For many color pictures of the Mandelbrot set and other fractals, see Chance and Choice, and many other fine books available on fractals.
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 which 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 (1983): 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.
Before Mandelbrot, mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically. But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers. This fractal geometry is capable of describing mathematically the most amorphous and chaotic forms of the real world. As Mandelbrot said, Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.
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". Mandelbrot coined the word fractal based on the Latin adjective "fractus". He chose this word because the corresponding Latin verb "frangere" means "to break", "to create irregular fragments". 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.
In this sense of the word fractal, it is now easy to see how our "natural consciousness", our consciousness before we complete the individuation process, is inherently fractal. It is fragmented, broken up into irregular fragments. Our task is to realize the higher, hidden order of the fractal, to bring out a continuity of consciousness in our very being. For a fractal as a geometric figure not only has irregular shapes - the zig zag world of nature - but there is lurking in the disorder a hidden order in these irregular shapes. The irregular patters are self similar over scales. The overall pattern of a fractal is repeated, with similarity, and sometimeseven with exactitude, when you look at a small part of the figure. It is recursive. For instance, if you look at the irregular shape of a mountain, then look closer at a small part of the mountain, more often than not you will find the same basic shape of the whole mountain repeated again on a smaller scale. When you look closer still, you see the same shape again, and so on to infinity. Strange but true! The animated graphic below gives you a pretty good idea of fractal recursiveness as you look more closely at detail to the Mandelbrot set.
As Mandelbrot points out this idea of "recursive self similarity" was originally developed by the philosopher Leibniz, and popularized by the writer Johnathan Swift in 1733 with the following verse:
So, Nat'ralists observe, a Flea
Mandelbrot notes that this same verse was followed in 1922 by Lewis Richardson, a mathematician studying weather prediction, who coined the following widely-known (among scientists) quote concerning "turbulence", the chaotic condition of liquids and gases:
Hath smaller Fleas that on him prey,
And these have smaller fleas to bit 'em,
And so proceed ad infinitum.
Big whorls have little whorls,
The ideas of self similarity and scaling embodied in these verses are critical to understanding the Laws of Chaos. Wherever we look in nature we find fractals with self similarity over scales. It is in every snow flake, every bolt of lightening, every tree, every branch. It is in our very blood with its veins, in our Galaxies with their clusters; it's even in our brocolli. 1
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.
Thanks to Mandelbrot and other recent insights of Chaoticians, we now have a mathematical understanding of some of the heretofore secret workings of Nature. We understand for the first time why two trees growing next to each other in the forest, at the same time, from the same stock, with the same genes, will still end up unique. They will be similar to be sure, but not identical. Just so every snow flake falling from the same cloud at the same time under identical conditions is still unique, different from all of the rest. This is only possible because of the infinity which lies in the dimensions and the interplay of chance - the unpredictable Chaos.
An understanding of how the fourth dimension includes the infinity of intervals between the other dimensions can be gained by visualizing a few of the better-known fractal dimensions (sometimes called Hausdorff dimensions by mathematicians). One of the most famous fractal dimensions lies between the zero dimension and the first dimension, the point and the line. It is created by "middle third erasing" where you start with a line and remove the middle third; two lines remain from which you again remove the middle third; then remove the middle third of the remaining segments; and so on into infinity. What remains after all of the middle third removals is called by Mandelbrot "Cantor's Dust". It consists of an infinite number of points, but no length. An example of the process (not exactly to proportion) is shown here.
The Cantor's Dust which remains is not quite a line, but is more than a point. The dimension is calculated to have a numerical value of .63. It was discovered by mathematician George Cantor in the beginning of the Twentieth Century. It was considered an anomaly and was avoided by most mathematicians as a "useless monstrosity". In fact this fractal dimension is a part of the real world of the fourth dimension and corresponds to many phenomena of Man and Nature. For instance, Mandelbrot cracked a serious problem for IBM by discovering that the seemingly random errors which always appeared in data transmission lines in fact occurred in time according to a fractal dimension similar to the one illustrated by Cantor's Dust. Knowing the hidden and mathematically-precise order behind the apparently-random errors allowed IBM to easily overcome this natural phenomena of data transmission by simple redundancies in the transmission.
Another well known fractal dimension lies between a line and a plane, the first and second dimension. It is called the Sierpiniski Gasket after mathematician Waclaw Sierpiniski and has a fractal dimension of 1.58. Create it by starting with an equilateral triangle and remove the open central upside down equilateral triangle with half the side length of the starting triangle. This leaves three half size triangles. Then repeat the process on the remaining half size triangles, and so forth ad infinitum. The remaining form has infinite lines but is less than a plane.
Fractal forms are also found in the body. The best known example are the arteries and veins in mammalian vascular systems. The bronchi of the human lung are self similar over 15 successive bifurcations. This area of biological research is just beginning, but the early results are promising and may lead to breakthroughs in understanding how the body functions.
Fractal geometry and the insights of the science of Chaos are based on Complex Numbers. Unlike all other numbers, such as the natural numbers one through nine for instance 1.2.3.4.5.6.7.8.9, the Complex Numbers do not exist on a horizontal number line. They exist only on an x-y coordinate time plane. Here regular numbers on the horizontal grid combine with so called "Imaginary Numbers" on the vertical grid. This x-y plane is shown below with an accurate depiction of the location of the primary forms of Mandelbrot fractal on that plane.
- THE MATHEMATICS OF CHAOS
Imaginary Numbers are simply numbers where a negative times a negative creates a negative, not a positive, as is the rule with all other numbers. In other words, with imaginary numbers -2 multiplied times -2 = -4, not +4. The Complex Numbers when iterated - subject to constant feedback - produce Fractal Scaling.
With Imaginary numbers a negative times a negatives creates a negative, not a positive. Is that so illogical? Without Imaginary numbers the complex dynamics and turbulence of the real space/time world could not be described mathematically. Imaginary numbers combine with real numbers to create Complex numbers. Complex numbers are the basis of much of higher math. They allow mathematicians to see many essential connections and relationships in mathematics which would not otherwise be possible. Complex numbers allow an algebraic understanding of the hidden unity in the ideal world of numbers. They also provide a geometric description of the fractal beauty of the real world, the zig-zag world of nature and other very complicated systems. This is not possible with the other, non-complex numbers, that exist alone without Imaginary numbers.
For more information on complex numbers, the mathematics of chaos, and the meaning of all this in our everyday lives, see my book with Keyserling Chance and Choice. It emphasizes these subjects and includes many full color illustrations of the Mandelbrot set and other fractals.
One of the most important characteristics of Strange attractors is their great sensitivity to initial conditions. Mild mannered Ph.D., Edward Lorenz, a mathematician and research meteorologist, discovered this important Law of Chaos quite by accident. Again, the computer made it all possible. He had access to one of the first computer systems in 1960 called a "Royal McBee" at the Massachusetts Institute of Technology. Although incredibly crude by today's standards, it was the marvel of its day, in spite of all of the noise and smoke it made. He was playing with a computer simulation that modeled a simple weather system. It was an early attempt to use computer power to try to predict the weather.
- THE CASE OF EDWARD LORENZ v. THE WEATHER MAN
At that time accurate long-term weather prediction was felt to be possible. Everyone agreed that all you needed to do that was a computer powerful enough to be programmed to simulate the lawful interactions of all of the variables involved in the weather, the temperature, pressure, wind speed, etc. This was in accord with the old pre-chaos scientific view of the Universe as a mechanical clock.
It was Lorenz who found out that this quest was impossible, that accurate weather prediction beyond a few days is inherently impossible. According to his discoveries the weatherman will always be doomed to failure from time to time, on all but short-term and simple predictions. With his early computer at M.I.T. Lorenz quite by accident discovered that complex open systems like the weather have an enormous sensitivity to initial conditions. The sensitivity is so great that prediction is inherently impossible. He found out that the slightest rounding off of the initial numbers in the weather formula would lead to entirely different results. In other words, extremely tiny differences in the initial numbers would quickly lead to huge variations in the calculations. In time the slightest little thing could end up making a huge difference in the end result.
Edward Lorenz' important discovery is now exemplified by the famous example which he gives to explain his discovery: "the wing movements of a butterfly in Peru may later through an extremely complex series of unpredictably-linked events magnify air movements and ultimately cause a hurricane in Texas". This Law is now nicknamed the "Butterfly Effect" and is now known to apply to all chaotic systems, not just the weather. It is an inherent characteristic of all Strange attractors. The "Butterfly Effect" thus applies in all complex open systems which change over time. It applies to all dynamic systems such as the weather, so that the smallest of changes triggers a chain reaction of unexpected exponential consequences.
Like most basic Laws, the recent discovery and scientific proof of the Butterfly Effect is really a rediscovery and verification of long-held folk wisdom. As James Gleick points out in his book Chaos, sensitive dependence on initial conditions is an old idea that can even be found in nursery rhymes:
- THE NEW CONTOURS OF A CHAOS PHILOSOPHY
For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost!Thanks to Lorenz we now know this rhyme is truly profound. In our world of complex systems and turbulence there is always extreme sensitivity to initial conditions. Even the smallest effort can unexpectedly multiply and have a great impact. It is beyond our capacity to predict what will happen, what little action might, or might not, lead to profound change.
The insights of Chaos, where it has been proven that negligible changes in chaotic systems can produce significant unexpected results, stands as an inspiration to all individuals. If you are in the world starting something new, you might make a big difference. No one can know for sure - their straw, no matter how small, might just be the one to break the camel's back. Your nail might save the kingdom. Your beginning efforts of personal transformation, may be important to the entire world. Like the butterfly's wings, if the timing and connections are just right, your new little work may well lead to a hurricane of change. History is replete with examples of this, both for good and bad.
We know that when we are subject to the first three attractors we are manipulated, predictable. Only with the Strange attractors can we be free, can we participate in the great flow, adding our butterfly wind to the great weather of history. We must learn how to base our life on the Strange attractors. According to Keyserling this means "to exist at the meeting point between the real and the possible". It means to have consciousness from out of Awareness. As Don Juan explained to Castenada, it means to live in "the crack between the worlds".
When we find the flow, we are then within the force of the Strange attractor. We can then see the hidden order behind the seeming Chaos. It is a time of no time, a flowing peak experience where all seems to go right by itself, effortlessly. It is a time when dreams and wishes are fulfilled that you did not even know you had.
All fractals with their infinite recursiveness portray in two dimensions - the infinity between zero and one, the potential and the actual. Mandelbrot's formula goes even further to provide a mathematical map to navigate in the crack between the worlds, to cope with Chaos and bring our potential into actuality. Mandelbrot's inspiratio- z -> z² + c brings order from Chaos in the fourth-dimensional world of Complex numbers in time, in iteration. We too live in the complex world of the fourth dimension, in a space-time continuum of turbulence and chance. Thus Mandelbrot's discovery should also apply in a fractal recursive manner to provide us with the formula for coherent living in a chaotic world. The philosophy suggested is a dynamic and pragmatic process of feedback, experimentation, detachment and grounding in Awareness for constant renewal.
If an action - Will - goes astray, does not work, falls off into infinity, then stop that activity. Once the mistake is obvious and certain, choose to let go of the failure. Then choose again to take a chance - to return to Zero, pure Awareness - God - where inspiration for a new action will come again. When the idea comes, go for it, don't wait for certainty or you may never act at all and life will surely pass you by. Seldom does inspiration come with the certainty of a burning bush. Choose to take a chance, try it without attachment or preconceptions.
Then travel the new vector - the new complex number - the new activity - to see whether or not it is part of a larger order, or part of Chaos - whether or not it is within the black of the Mandelbrot fractal. Only time - constant iterations - will tell. You can only discover the force of the Strange attractor by doing - trial and error with feedback - learning from your mistakes and always beginning anew from Awareness. If the new action is successful - leads to greater order and coherence - continue to follow it. It may be perfectly stable, well within the black, and like a successful business after it is well started, the activity can eventually go on with others, or by itself, without your attending to it. So again the activity may be finished for you. There are probably many other things remaining for you to do. Keep experimenting, changing, try many things at once. Otherwise you may stagnate in success - drown in black ink - and never see the big picture. You may lose the opportunity to again experience the fractal beauty and excitement of living on the edge. Thus an established success, like failure, should lead to freedom, to a new activity into the unknownfuture. It is the destiny of actualizing Man to pioneer the frontiers of Cosmos in the midst of Chaos.
Sometimes a new activity starts off ordered and successful, but later falls apart and iterates into Chaos. Don't be too attached to something just because it starts off working. If it later stops working, recognize the facts and let go. Conversely, many times things start off working poorly. Don't give up at the first sign of difficulty, because the beginnings are always hard. But when you are sure that it will not work, that it is definitely on its way to nowhere, then leave it. Re-center yourself in Awareness to find a new direction, and try again with something different. Don't be frustrated, eventually even Edison found the light.
Even when a person is not consciously participating in this process - is uncentered and unaware - the actions of the Strange attractor can still manifest in the desire to do the unexpected, the wild hair, fluke decision. When under its influence the pull seems to be towards disorder and serendipity. The hidden order lurking behind the Strange attractor may only appear much later, or through synchronicity with other events. An example might be a desire to make a career change which makes no sense at the time, but later in life is recognized as an essential step to a larger order; or perhaps you are seized by a sudden eccentric desire to go to a place never seen before and there meet your future wife who is also there by chance. Mandelbrot's own life is a living example of this with his impulsive change from one field of study to another, leading eventually to his great discovery. Examples of this hidden order in Chaos abound in life and nature, and even in man made things such as fluctuations in the price of cotton.
A word of warning here about the superstition trap. Attractors are real and can be experienced directly. This is scientific fact. Chaos is ordered by the attractors. When we act with the attractors to bring order, even with the Strange attractor, we are, like Nature herself, using constrained chance, structured chance. There is a fundamental and important difference between the reliance upon constrained chance and Strange attractors, and reliance upon blind luck and superstition. The lazy will take the latter course and convince themselves they are living on a higher plane. These same people will complain to God when their luck turns. Do not confuse Chance and Choice with chance and more chance. There is a fine line here; gambling and superstition are a real danger for some and must be avoided. Chaos for the hell of it is a dead end, so are superstitions.
Strange attractors are very real phenomena. They can be observed over time and are precisely, mathematically calculable. They are a part of everyone's life, not the private domain of Chaoticians. Try out these theories for yourself. Realize that things will always go wrong, that nothing can remain perfectly ordered forever, and plan accordingly. The only stability in life is constant change. Work to be prepared to ride the chances as they come. Make your own luck. Look for the chances, know where and how to look. Know how to make a choice and when. Chaos is a philosophy of self reliance and inner coherence based on reason, but it goes beyond the limits of reason to embrace the whole world - to know Chaos.
As shown in the last chapter, the Constitutional Laws are holistic. We can now see how they are also fractal. They repeat in different fields and scales with self similarity. They interrelate and complement each other in a fractal recursive manner. The basic principles appear again and again wherever we look, like Mandelbrot's fractal in the world of complex numbers. For instance, the Law of three appears in all phenomena in different forms. Everything has a material, energetical and spiritual component. Each gene in your body is made of matter, has energy, and stores the blueprint - spirit - for the construction of the entire body.
- THE CONSTITUTIONAL LAWS ARE FRACTAL
Each constitutional principle is like that gene, it contains the pattern of the whole Universe within itself as if it were a giant hologram. But whereas a hologram is uniform - each point of the hologram contains the same pattern as every other point - in a fractal slight variations are possible. Although some fractals have identical repetitions of the exact same patterns, just like a hologram, with other fractals there is only self similarity, not identity. It is not uniform like a hologram. It does not repeat the same pattern, but it still contains and reflects the whole like a hologram. It is like Ezekiel's "wheels within wheels", each spinning in a different direction. There is general uniformity, and yet there is also variety and uniqueness. We are all part of one being, one holistic hologram. Yet since that one being is near infinite, each of us is, potentially at least, a "one of a kind" individual. So in a manner of speaking when God made each of us She threw away the mold. Still, each unique mold is made using the same basic patterns.
As the great Roman lawyer and emperor Marcus Aurelius said in his book Meditations:
Remembering always what the World-Nature is, and what my own nature is, and how one stands in respect to the other - so small a fraction of so vast a Whole ๑ bear in mind that no man can hinder you from conforming each word and deed to that Nature of which you are a part.
Since the Universe is a fractal, not a hologram or a machine, the coherence of the Universe still allows for infinite diversity. There is no complete determinism in the lawfulness. Spontaneity and freedom are possible. As seen before if the Universe is a giant jigsaw puzzle, then each bit of the puzzle ultimately fits with every other, each in its own place and time. The Universe and its Laws are holistic. Moreover, the puzzle is infinitely large - we can never finish putting it all together. Although it looks vaguely similar wherever we look, each part of the puzzle is quite different. The Universe and its Laws are also fractal.
We can know the Constitutional Laws which are few in number, but, thank God, we can never even come close to knowing all of the Laws. They are limitless. Complete wisdom - knowledge of the constitutional principles - is possible, but not complete knowledge. This is because Wisdom is limited, whereas knowledge is limitless. The wisdom tools, the keys, are limited in number and can be mastered, but the doors to be opened by these keys are without number. There will always be more to learn, more new and unexpected ways for the few basic principles to combine, bend and move to create ever new and unexpected phenomena.
The most we can hope for to learn is how to learn, to learn the principles of coherence. We may ultimately comprehend the parts of the Universe we come to know, but we will never be bored by the Universe and there will always be more to know. Within the basic parameters - the given structure - the content is unpredictable. Only the overall pattern and statistics can be predicted. That is the great joy, freedom and beauty of our fractal Universe. The rules of the game may be given and limited, but the plays we choose to make within the confines of those rules are limitless, and the outcome of each game is unknown.
1. See for instance Michael McGuire's book of photographs and drawings of fractals An Eye For Fractals (1991), or Briggs and Peat's Turbulent Mirror (1989), or Fractal Forms (1991) edited by Guyon and Stanley, or Symmetry In Chaos (1992) by Field and Golubitsky.
- Detour to learn more about Fractals and Chaos
- at the School of Wisdom Fractal Webs.