Chaos - James Gleick [108]
Peitgen shared little of the mathematicians’ unease with the use of computers to conduct experiments. Granted, every result must eventually be made rigorous by the standard methods of proof, or it would not be mathematics. To see an image on a graphics screen does not guarantee its existence in the language of theorem and proof. But the very availability of that image was enough to change the evolution of mathematics. Computer exploration was giving mathematicians the freedom to take a more natural path, Peitgen believed. Temporarily, for the moment, a mathematician could suspend the requirement of rigorous proof. He could go wherever experiments might lead him, just as a physicist could. The numerical power of computation and the visual cues to intuition would suggest promising avenues and spare the mathematician blind alleys. Then, new paths having been found and new objects isolated, a mathematician could return to standard proofs. “Rigor is the strength of mathematics,” Peitgen said. “That we can continue a line of thought which is absolutely guaranteed—mathematicians never want to give that up. But you can look at situations that can be understood partially now and with rigor perhaps in future generations. Rigor, yes, but not to the extent that I drop something just because I can’t do it now.”
By the 1980s a home computer could handle arithmetic precise enough to make colorful pictures of the set, and hobbyists quickly found that exploring these pictures at ever-greater magnification gave a vivid sense of expanding scale. If the set were thought of as a planet-sized object, a personal computer could show the whole object, or features the size of cities, or the size of buildings, or the size of rooms, or the size of books, or the size of letters, or the size of bacteria, or the size of atoms. The people who looked at such pictures saw that all the scales had similar patterns, yet every scale was different. And all these microscopic landscapes were generated by the same few lines of computer code.*
THE BOUNDARY IS WHERE a Mandelbrot set program spends most of its time and makes all of its compromises. There, when 100 or 1,000 or 10,000 iterations fail to break away, a program still cannot be absolutely certain that a point falls inside the step. Who knows what the millionth iteration will bring? So the programs that made the most striking, most deeply magnified pictures of the set ran on heavy mainframe computers, or computers devoted to parallel processing, with thousands of individual brains performing the same arithmetic in lock step. The boundary is where points are slowest to escape the pull of the set. It is as if they are balanced between competing attractors, one at zero and the other, in effect, ringing the set at a distance of infinity.
When scientists moved from the Mandelbrot set itself to new problems of representing real physical phenomena, the qualities of the set’s boundary came to the fore. The boundary between two or more attractors in a dynamical system served as a threshold of a kind that seems to govern so many ordinary processes, from the breaking of materials to the making of decisions. Each attractor in such a system has its basin, as a river has a watershed basin that drains into it. Each basin has a boundary. For an influential group in the early 1980s, a most promising new field of mathematics and physics was the study of fractal basin boundaries.
This branch of dynamics concerned itself not with describing the final, stable behavior of a system but with the way a system chooses between competing options. A system like Lorenz’s now-classic model has just one attractor in it, one behavior that prevails when the system settles down, and it is a chaotic attractor. Other systems may end up with nonchaotic steady-state behavior—but with more than one possible steady state. The study of fractal