The Story of the Geometry of Chaos

The story of Chaos begins in number, specifically in the mathematics and geometry of the fourth dimension. This is the home of Complex numbers and Fractal Geometry. Unlike the other dimensions - the first, second and third dimensions composed of the line, plane and solid - 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. It is an open system where everything is related to everything else. Prior science and math was concerned with closed systems in the first, second and third dimensions. It emphasized "left brain algebra," and ignored "right brain geometry." Since Einstein, however, we know that even the third dimension - solid bodies - is just a model for reality, it does not really exist. We in fact live in the fourth dimension of the space-time continuum. Since Mandelbrot, we know what the fourth dimension looks like, we know the fractal face of chaos. He is the key Chaotician of our times, and before we begin our journey into the geometry of chaos, we must first understand his story.

Benoit Mandelbrot, now both 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 singlehandedly by Mandelbrot. It is now recognized as the true Geometry of Nature. Mandelbrot's fractal geometry replaces Euclidian geometry which had dominated our mathematical thinking for thousands of years.. We now know that Euclidian geometry 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 math world he revolutionized.

Before Mandelbrot, the academic math world was dominated by arithmetics, geometry was relegated to a secondary inferior position. Math 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 young Benoit received no regular formal education.

Benoit 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 a school setting where this was discouraged. He 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 of course 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 was 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 standard bell curves usually found in statistics.

His now famous study in the field of economics concerned the price of cotton, 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.