Chaos - James Gleick [111]
The chaos game made use of a fractal quality of certain pictures, the quality of being built up of small copies of the main picture. The act of writing down a set of rules to be iterated randomly captured certain global information about a shape, and the iteration of the rules regurgitated the information without regard to scale. The more fractal a shape was, in this sense, the simpler would be the appropriate rules. Barnsley quickly found that he could generate all the now-classic fractals from Mandelbrot’s book. Mandelbrot’s technique had been an infinite succession of construction and refinement. For the Koch snowflake or the Sierpiński gasket, one would remove line segments and replace them with specified figures. By using the chaos game instead, Barnsley made pictures that began as fuzzy parodies and grew progressively sharper. No refinement process was necessary: just a single set of rules that somehow embodied the final shape.
Barnsley and his co-workers now embarked on an out-of–control program of producing pictures, cabbages and molds and mud. The key question was how to reverse the process: given a particular shape, how to choose a set of rules. The answer, which he called “collage theorem,” was so inanely simple to describe that listeners sometimes thought there must be some trick. You would begin with a drawing of the shape you wanted to reproduce. Barnsley chose a black spleenwort fern for one of his first experiments, having long been a fern buff. Then using a computer terminal and a mouse as pointing device, you would lay small copies over the original shape, letting them overlap sloppily if need be. A highly fractal shape could easily be tiled with copies of itself, a less fractal shape less easily, and at some level of approximation every shape could be tiled.
THE CHAOS GAME. Each new point falls randomly, but gradually the image of a fern emerges. All the necessary information is encoded in a few simple rules.
“If the image is complicated, the rules will be complicated,” Barnsley said. “On the other hand, if the object has a hidden fractal order to it—and it’s a central observation of Benoit’s that much of nature does have this hidden order—then it will be possible with a few rules to decode it. The model, then, is more interesting than a model made with Euclidean geometry, because we know that when you look at the edge of a leaf you don’t see straight lines.” His first fern, produced with a small desktop computer, perfectly matched the image in the fern book he had since he was a child. “It was a staggering image, correct in every aspect. No biologist would have any trouble identifying it.”
In some sense, Barnsley contended, nature must be playing its own version of the chaos game. “There’s only so much information in the spore that encodes one fern,” he said. “So there’s a limit to the elaborateness with which a fern could grow. It’s not surprising that we can find equivalent succinct information to describe ferns. It would be surprising if it were otherwise.”
But was chance necessary? Hubbard, too, thought about the parallels between the Mandelbrot set and the biological encoding of information, but he bristled at any suggestion that such processes might depend on probability. “There is no randomness in the Mandelbrot set,” Hubbard said. “There is no randomness in anything that I do. Neither do I think that the possibility of randomness has any direct relevance to biology. In biology randomness is death, chaos is death. Everything is highly structured. When you clone plants,