Chaos - James Gleick [104]
Many fractal shapes can be formed by iterated processes in the complex plane, but there is just one Mandelbrot set. It started appearing, vague and spectral, when Mandelbrot tried to find a way of generalizing about a class of shapes known as Julia sets. These were invented and studied during World War I by the French mathematicians Gaston Julia and Pierre Fatou, laboring without the pictures that a computer could provide. Mandelbrot had seen their modest drawings and read their work—already obscure—when he was twenty years old. Julia sets, in a variety of guises, were precisely the objects intriguing Barnsley. Some Julia sets are like circles that have been pinched and deformed in many places to give them a fractal structure. Others are broken into regions, and still others are disconnected dusts. But neither words nor the concepts of Euclidean geometry serve to describe them. The French mathematician Adrien Douady said: “You obtain an incredible variety of Julia sets: some are a fatty cloud, others are a skinny bush of brambles, some look like the sparks which float in the air after a firework has gone off. One has the shape of a rabbit, lots of them have sea-horse tails.”
AN ASSORTMENT OF JULIA SETS.
In 1979 Mandelbrot discovered that he could create one image in the complex plane that would serve as a catalogue of Julia sets, a guide to each and every one. He was exploring the iteration of complicated processes, equations with square roots and sines and cosines. Even after building his intellectual life around the proposition that simplicity breeds complexity, he did not immediately understand how extraordinary was the object hovering just beyond the view of his computer screens at IBM and Harvard. He pressed his programmers hard for more detail, and they sweated over the allocation of already strained memory, the new interpolation of points on an IBM mainframe computer with a crude black and white display tube. To make matters worse, the programmers always had to stand guard against a common pitfall of computer exploration, the production of “artifacts,” features that sprang solely from some quirk of the machine and would disappear when a program was written differently.
Then Mandelbrot turned his attention to a simple mapping that was particularly easy to program. On a rough grid, with a program that repeated the feedback loop just a few times, the first outlines of disks appeared. A few lines of pencil calculation showed that the disks were mathematically real, not just products of some computational oddity. To the right and left of the main disks, hints of more shapes appeared. In his mind, he said later, he saw more: a hierarchy of shapes, atoms sprouting smaller atoms ad infinitum. And where the set intersected the real line, its successively smaller disks scaled with a geometric regularity that dynamicists now recognized: the Feigenbaum sequence of bifurcations.
That encouraged him to push the computation further, refining those first crude images, and he soon discovered dirt cluttering the edge of the disks and also floating in the space nearby. As he tried calculating in finer and finer detail, he suddenly felt that his string of good luck had broken. Instead of becoming sharper, the pictures became messier. He headed back to IBM’s Westchester County research center to try computing power on a proprietary scale that Harvard could not match. To his surprise, the growing messiness was the sign of something real. Sprouts and tendrils spun languidly away from the main island. Mandelbrot saw a seemingly smooth boundary resolve itself into a chain of spirals like the tails of sea horses. The irrational fertilized the rational.
The Mandelbrot set is a collection of points. Every point in the complex plane—that is, every complex number—is either in the set or outside it. One way to define the set is in terms of a test for every point, involving some simple iterated arithmetic. To test a point, take the complex number; square it; add the original number; square the result; add the original