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'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.
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.