Fractal Geometry

computer drawing of the Mandelbrot Set

The Story of Benoit B. Mandelbrot
and 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.

The Geometry of Chaos

Mandelbrot's eclectic research ultimately led to a great breakthrough summarized by a simple mathematical formula: z -> z^2 + c . This formula 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 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 -> z^2 + 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 (shown below) is revealed. The order is of a strange and beautiful kind, containing self similar recursiveness over an infinite scale. This is shown below is the magnification sequence of the Mandelbrot set.

Mandelbrot Set Zoom

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 which 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 choose 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 sometimes even 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, 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.  This happens within the Mandelbrot itself where there are an infinite number of smaller Mandelbrot shapes hidden everywhere within the zig zaggy, spiral edges of the overall form.

A Mandelbrot shape which appears in Seahorse Valley after tens of thousands of magnifications

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
Hath smaller Fleas that on him prey,
And these have smaller fleas to bit 'em,
And so proceed ad infinitum.

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:

Big whorls have little whorls,
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.

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 even in our very blood with its veins, and in our Galaxies with their clusters.

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.

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 and 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 the fractal dimension 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. Chaotician Michael McGuire refers to recent discoveries in brain research which suggests that a fractal structure based on hexagons may be how the receptive fields of the visual cortex are organized. The smallest hexagons correspond to the cells of the retina and perception of fine details, the larger hexagons organize the underlying layers to recognize progressively coarse detail.