Real Beauty

Last weekend found us in Cambridge, Mass., on the first warm days of spring. The Charles was jammed with sailboats and sculls, and a large MIT lecture hall nearby was jammed with several hundred students and other interested persons who had come to hear a day-long lecture series on the latest in fractal geometry. We followed the crowd into the lecture hall and took our place, raising the swinging desk beside the seat into position for taking notes.

Fractal geometry is based on the idea that the natural world is not made up of the familiar objects of geometry--circles and triangles and the like. The natural world of clouds, coastlines and mountains cannot be fully described by the geometry of circles and squares. When looking for signs of humanity in aerial photographs, look for straight lines. There are very few of them in nature.

Pythagoras discovered that the real world cannot be expressed completely by mathematics, but he tried to keep it a secret. Mathematicians are still striving to make closer the fit between their tools and the world around us. Fractal geometry is an effort to get a mathematical handle on nature’s infinite complexity. It tries to deal with the irregularities of nature in an orderly way.

How long is the coastline of California? It depends on how closely you want to measure it. If you smooth out the roughness, you can come up with a number, but if you try to measure every nook and cranny you will have quite a task ahead of you, and the length will keep getting larger.

Coastlines, perhaps, but when it comes to clouds you might ask what business it is of mathematicians, anyway. Shouldn’t clouds be left to poets and artists? It turns out, said Benoit (pronounced Ben-WAH) Mandelbrot of Harvard and IBM Research, that “great painters are very bad at painting clouds,” just as mathematicians cannot describe them in a formula.

Mandelbrot himself invented fractal geometry a decade ago, and touched off a booming field of mathematical inquiry in the bargain. His contribution was lauded by Michael Barnsley of the Georgia Institute of Technology, who began his talk by noting that Mandelbrot continued a long line of distinguished scientists whose contribution consisted of seeing the obvious, which everyone else had missed. Of course, it becomes obvious to the rest of us only after the genius points it out.

“How to deal with this space?” Barnsley went on to ask. “There are wonderful spatial intuitions that are not provided to us by the mathematics. Good mathematics, good geometry, should allow us to handle spatial intuitions and bring them under our control.”

All day long the speakers showed beautiful pictures of mathematical shapes and graphs created by the computer and fractal geometry. Robert L. Devaney of Boston University showed mathematical movies made on a Cray X-MP supercomputer at Digital Productions in Hollywood, makers of very high-tech special-effects theatrical films. Richard Voss of IBM Research among other things played fractal music, which tries to mimic the unpredictability of music composed by humans. Heinz-Otto Peitgen of UC Santa Cruz and the University of Bremen showed slides of fractal shapes from his recently published, stunning book, “The Beauty of Fractals” (Springer-Verlag), written with Peter H. Richter of the University of Bremen.

All this makes for pretty pictures, but it also has great value as well. The mathematics of fractals has helped physics, chemistry and biology unify previously unrelated areas of research. Mandelbrot noted, for example, that the functioning of DNA, which carries the genetic code in all living things, is so complicated that it is still not fully understood. “A small change of DNA gives rise to a great wealth of shape in simple organisms,” he said. The mathematics of fractals might be useful in explaining these things and others like the weather or the economy.

But it is the beauty of fractal pictures that is so captivating. “Why are fractals beautiful?” Mandelbrot asked. Because they show that even the most complicated organic structures can yield to concise mathematical analysis. “There is a degree of simplicity of nature that is astonishing.”