Chaos - James Gleick [112]
In Barnsley’s technique, however, chance serves only as a tool. The results are deterministic and predictable. As points flash across the computer screen, no one can guess where the next one will appear; that depends on the flip of the machine’s internal coin. Yet somehow the flow of light always remains within the bounds necessary to carve a shape in phosphorous. To that extent the role of chance is an illusion. “Randomness is a red herring,” Barnsley said. “It’s central to obtaining images of a certain invariant measure that live upon the fractal object. But the object itself does not depend on the randomness. With probability one, you always draw the same picture.
“It’s giving deep information, probing fractal objects with a random algorithm. Just as, when we go into a new room, our eyes dance around it in some order which we might as well take to be random, and we get a good idea of the room. The room is just what it is. The object exists regardless of what I happen to do.”
The Mandelbrot set, in the same way, exists. It existed before Peitgen and Richter began turning it into an art form, before Hubbard and Douady understood its mathematical essence, even before Mandelbrot discovered it. It existed as soon as science created a context—a framework of complex numbers and a notion of iterated functions. Then it waited to be unveiled. Or perhaps it existed even earlier, as soon as nature began organizing itself by means of simple physical laws, repeated with infinite patience and everywhere the same.
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* A Mandelbrot set program needs just a few essential pieces. The main engine is a loop of instructions that takes its starting complex number and applies the arithmetical rule to it. For the Mandelbrot set, the rule is this: z→z2 + c, where z begins at zero and c is the complex number corresponding to the point being tested. So, take 0, multiply it by itself, and add the starting number; take the result—the starting number—multiply it by itself, and add the starting number; take the new result, multiply it by itself, and add the starting number. Arithmetic with complex numbers is straightforward. A complex number is written with two parts: for example, 2 + 3i (the address for the point at 2 east and 3 north on the complex plane). To add a pair of complex numbers, you just add the real parts to get a new real part and the imaginary parts to get a new imaginary part:
To multiply two complex numbers, you multiply each part of one number by each part of the other and add the four results together. Because i multiplied by itself equals –1, by the original definition of imaginary numbers, one term of the result collapses into another.
To break out of this loop, the program needs to watch the running total. If the total heads off to infinity, moving farther and farther from the center of the plane, the original point does not belong to the set, and if the running total becomes greater than 2 or smaller than – 2 in either its real or imaginary part, it is surely heading off to infinity—the program can move on. But if the program repeats the calculation many times without becoming greater than 2, then the point is part of the set. How many times depends on the amount of magnification. For the scales accessible to a personal computer, 100 or 200 is often plenty, and 1,000 is safe.
The program must repeat this process for each of thousands of points on a grid, with a scale that can be adjusted for greater magnification. And the program must display its result. Points in the set can be colored black, other points white. Or for a more vividly appealing picture, the white points can be replaced