Chaos - James Gleick [105]
This business of repeating a process indefinitely and asking whether the result is infinite resembles feedback processes in the everyday world. Imagine that you are setting up a microphone, amplifier, and speakers in an auditorium. You are worried about the squeal of sonic feedback. If the microphone picks up a loud enough noise, the amplified sound from the speakers will feed back into the microphone in an endless, ever louder loop. On the other hand, if the sound is small enough, it will just die away to nothing. To model this feedback process with numbers, you might take a starting number, multiply it by itself, multiply the result by itself, and so on. You would discover that large numbers lead quickly to infinity: 10, 100, 10,000…. But small numbers lead to zero: ½, ¼ 1/16…. To make a geometric picture, you define a collection of all the points that, when fed into this equation, do not run away to infinity. Consider the points on a line from zero upward. If a point produces a squeal of feedback, color it white. Otherwise color it black. Soon enough, you will have a shape that consists of a black line from 0 to 1.
THE MANDELBROT SET EMERGES. In Benoit Mandelbrot’s first crude computer printouts, a rough structure appeared, gaining more detail as the quality of the computation improved. Were the buglike, floating “molecules” isolated islands? Or were they attached to the main body by filaments too fine to be observed? It was impossible to tell.
For a one-dimensional process, no one need actually resort to experimental trial. It is easy enough to establish that numbers greater than one lead to infinity and the rest do not. But in the two dimensions of the complex plane, to deduce a shape defined by an iterated process, knowing the equation is generally not enough. Unlike the traditional shapes of geometry, circles and ellipses and parabolas, the Mandelbrot set allows no shortcuts. The only way to see what kind of shape goes with a particular equation is by trial and error, and the trial-and–error style brought the explorers of this new terrain closer in spirit to Magellan than to Euclid.
Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop.
Julia, Fatou, Hubbard, Barnsley, Mandelbrot—these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x2 + y2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind