Chaos - James Gleick [106]
Before computers, even Julia and Fatou, who understood the possibilities of this new kind of shape-making, lacked the means of making it a science. With computers, trial-and–error geometry became possible. Hubbard explored Newton’s method by calculating the behavior of point after point, and Mandelbrot first viewed his set the same way, using a computer to sweep through the points of the plane, one after another. Not all the points, of course. Time and computers being finite, such calculations use a grid of points. A finer grid gives a sharper picture, at the expense of longer computation. For the Mandelbrot set, the calculation was simple, because the process itself was so simple: the iteration in the complex plane of the mapping z→z2 + c. Take a number, multiply it by itself, and add the original number.
As Hubbard grew comfortable with this new style of exploring shapes by computer, he also brought to bear an innovative mathematical style, applying the methods of complex analysis, an area of mathematics that had not been applied to dynamical systems before. Everything was coming together, he felt. Separate disciplines within mathematics were converging at a crossroads. He knew it would not suffice to see the Mandelbrot set; before he was done, he wanted to understand it, and indeed, he finally claimed that he did understand it.
If the boundary were merely fractal in the sense of Mandelbrot’s turn-of–the-century monsters, then one picture would look more or less like the last. The principle of self-similarity at different scales would make it possible to predict what the electronic microscope would see at the next level of magnification. Instead, each foray deeper into the Mandelbrot set brought new surprises. Mandelbrot started worrying that he had offered too restrictive a definition of fractal; he certainly wanted the word to apply to this new object. The set did prove to contain, when magnified enough, rough copies of itself, tiny buglike objects floating off from the main body, but greater magnification showed that none of these molecules exactly matched any other. There were always new kinds of sea horses, new curling hothouse species. In fact, no part of the set exactly resembles any other part, at any magnification.
The discovery of floating molecules raised an immediate problem, though. Was the Mandelbrot set connected, one continent with far-flung peninsulas? Or was it a dust, a main body surrounded by fine islands? It was far from obvious. No guidance came from the experience with Julia sets, because Julia sets came in both flavors, some whole shapes and some dusts. The dusts, being fractal, have the peculiar property that no two pieces are “together”—because every piece is separated from every other by a region of empty space—yet no piece is “alone,” since whenever you find one piece, you can always find a group of pieces arbitrarily close by. As Mandelbrot looked at his pictures, he realized that computer experimentation was failing to settle this fundamental question. He focused more sharply on the specks hovering about the main body. Some disappeared, but others grew into clear near-replicas. They seemed independent. But possibly they were connected by lines so thin that they continued to escape the lattice of computed points.
Douady and Hubbard used a brilliant chain of new mathematics to prove that every floating molecule does indeed hang on a filigree that binds it to all the rest, a delicate web springing from tiny outcroppings on the main set, a “devil’s polymer,” in Mandelbrot’s phrase. The mathematicians proved that any segment—no matter where, and no matter how small—would, when blown up by the computer microscope, reveal new molecules, each resembling the main set and yet not quite the same. Every new molecule would be surrounded by its own spirals and flamelike projections, and those, inevitably, would reveal molecules tinier