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By Root 784 0

the closer it came to unraveling. One day Bernardo Huberman called. He asked for Rob Shaw, but he happened to get Crutchfield. Huberman needed a collaborator for a tight, simple paper about chaos. Crutchfield, the youngest member of the collective, concerned about being thought of as merely its “hacker,” was beginning to realize that in one respect the Santa Cruz faculty had been right all along: each of the students was someday going to have to be judged as an individual. Huberman, furthermore, had all the sophistication about the profession of physics that the students lacked, and in particular he knew how to get the most mileage from a given piece of work. He had his doubts, having seen their laboratory—“It was all very vague, you know, sofas and bean bags, like stepping into a time machine, flower children and the 1960s again.” But he needed an analog computer, and in fact Crutchfield managed to get his research program running in hours. The collective was a problem, though. “All the guys want in,” Crutchfield said at one point, and Huberman said absolutely not. “It’s not just the credit, it’s the blame. Suppose the paper is wrong—you’re going to blame a collective? I’m not part of a collective.” He wanted one partner for a clean job.

The result was just what Huberman had hoped for: the first paper about chaos to be published in the premier American journal for reporting breakthroughs in physics, Physical Review Letters. In terms of scientific politics this was a nontrivial achievement. “To us it was fairly obvious stuff,” Crutchfield said, “but what Bernardo understood was that it would have a huge impact.” It was also one beginning of the group’s assimilation into the real world. Farmer was angered, seeing in Crutchfielďs defection an undermining of the collective spirit.

Crutchfield was not alone in stepping outside the group. Soon Farmer himself, and Packard, too, were collaborating with established physicists and mathematicians: Huberman, Swinney, Yorke. The ideas formed in the cauldron at Santa Cruz became a firm part of the framework of the modern study of dynamical systems. When a physicist with a mass of data wanted to investigate its dimension or its entropy, the appropriate definitions and working techniques might well be those created in the years of patching plugs in the Systron-Donner analog computer and staring at the oscilloscope. Climate specialists would argue about whether the chaos of the world’s atmosphere and oceans had infinite dimensions, as traditional dynamicists would assume, or somehow followed a low-dimensional strange attractor. Economists analyzing stock market data would try to find attractors of dimension 3.7 or 5.3. The lower the dimension, the simpler the system. Many mathematical peculiarities had to be sorted and understood. Fractal dimension, Hausdorff dimension, Lyapunov dimension, information dimension—the subtleties of these measures of a chaotic system were best explained by Farmer and Yorke. An attractor’s dimension was “the first level of knowledge necessary to characterize its properties.” It was the feature that gave “the amount of information necessary to specify the position of a point on the attractor to within a given accuracy.” The methods of the Santa Cruz students and their older collaborators tied these ideas to the other important measures of systems: the rate of decay of predictability, the rate of information flow, the tendency to create mixing. Sometimes scientists using these methods would find themselves plotting data, drawing little boxes, and counting the number of data points in each box. Yet even such seemingly crude techniques brought chaotic systems for the first time within the reach of scientific understanding.

Meanwhile, having learned to look for strange attractors in flapping flags and rattling speedometers, the scientists made a point of finding the symptoms of deterministic chaos all through the current literature of physics. Unexplained noise, surprising fluctuations, regularity mixing with irregularity—these effects popped up in papers from