Chaos: Making a New Science

by James Gleick

As long as the starting point lies somewhere near the attractor, the next few points will converge to the attractor with great rapidity.

As an element in the world revealed by computer exploration, the strange attractor began as a mere possibility, marking a place where many great imaginations in the twentieth century had failed to go. Soon, when scientists saw what computers had to show, it seemed like a face they had been seeing everywhere, in the music of turbulent flows or in clouds scattered like veils across the sky.

Disorder was channeled, it seemed, into patterns with some common underlying theme.

Strange attractors seemed fractal, implying that their true dimension was fractional, but no one knew how to measure the dimension or how to apply such a measurement in the context of engineering problems.

Unlike linear systems, easily calculated and easily classified, nonlinear systems still seemed, in their essence, beyond classification—each different from every other. Scientists might begin to suspect that they shared common properties, but when it came time to make measurements and perform calculations, each nonlinear system was a world unto itself.

Sometime in his last year of college, it struck him that he had missed his adolescence, and he made a deliberate project out of regaining touch with humanity. He would sit silently in the cafeteria, listening to students chatting about shaving or food, and gradually he relearned much of the science of talking to people.

As singular boundaries between two realms of existence, phase transitions tend to be highly nonlinear in their mathematics. The smooth and predictable behavior of matter in any one phase tends to be little help in understanding the transitions.

As Kadanoff viewed the problem in the 1960s, phase transitions pose an intellectual puzzle. Think of a block of metal being magnetized. As it goes into an ordered state, it must make a decision. The magnet can be oriented one way or the other. It is free to choose. But each tiny piece of the metal must make the same choice. How?

Kadanoff’s insight was that the communication can be most simply described in terms of scaling. In effect, he imagined dividing the metal into boxes. Each box communicates with its immediate neighbors. The way to describe that communication is the same as the way to describe the communication of any atom with its neighbors. Hence the usefulness of scaling: the best way to think of the metal is in terms of a fractal-like model, with boxes of all different sizes.

Renormalization had entered physics in the 1940s as a part of quantum theory that made it possible to calculate interactions of electrons and photons. A problem with such calculations, as with the calculations Kadanoff and Wilson worried about, was that some items seemed to require treatment as infinite quantities, a messy and unpleasant business.

Such quantities seemed to float up or down depending on the scale from which they were viewed. It seemed absurd. Yet it was an exact analogue of what Benoit Mandelbrot was realizing about geometrical shapes and the coastline of England. Their length could not be measured independent of scale.

There was a kind of relativity in which the position of the observer, near or far, on the beach or in a satellite, affected the measurement.

Variability in the standard measures of mass or length meant that a different sort of quantity was remaining fixed. In the case of fractals, it was the fractional dimension—a constant that could be calculated and used as a tool for further calculations. Allowing mass to vary depending ...

In practice the renormalization group was far from foolproof. It required a good deal of ingenuity to choose just the right calculations to capture the self-similarity. However, it worked well enough and often enough to inspire some physicists, Feigenbaum included, to try it on the problem of turbulence.

But what about the onset of turbulence—the mysterious moment when an orderly system turned chaotic. There was no evidence that the renormalization group had anything to say about this transition. There was no evidence, for example, that the transition obeyed laws of scaling.

On second thought the connection between shrinking and loss of meaning was not so obvious. Why should it be that as things become small they also become incomprehensible?

One of the minor skirmishes of science in the first years of the nineteenth century was a difference of opinion between Newton’s followers in England and Goethe in Germany over the nature of color. To Newtonian physics, Goethe’s ideas were just so much pseudoscientific meandering. Goethe refused to view color as a static quantity, to be measured in a spectrometer and pinned down like a butterfly to cardboard. He argued that color is a matter of perception.

He finally did track down a copy, and he found that Goethe had actually performed an extraordinary set of experiments in his investigation of colors. Goethe began as Newton had, with a prism. Newton had held a prism before a light, casting the divided beam onto a white surface. Goethe held the prism to his eye and looked through it. He perceived no color at all, neither a rainbow nor individual hues. Looking at a clear white surface or a clear blue sky through the prism produced the same effect: uniformity. But if a slight spot interrupted the white surface or a cloud appeared in the sky, then he would see a burst of color. It is “the interchange of light and shadow,” Goethe concluded, that causes color.