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would drastically undermine a system’s stability. But folding turned out to be necessary, and folding allowed sharp changes in dynamical behavior. Smale’s horseshoe stood as the first of many new geometrical shapes that gave mathematicians and physicists a new intuition about the possibilities of motion. In some ways it was too artificial to be useful, still too much a creature of mathematical topology to appeal to physicists. But it served as a starting point. As the sixties went on, Smale assembled around him at Berkeley a group of young mathematicians who shared his excitement about this new work in dynamical systems. Another decade would pass before their work fully engaged the attention of less pure sciences, but when it did, physicists would realize that Smale had turned a whole branch of mathematics back toward the real world. It was a golden age, they said.

“It’s the paradigm shift of paradigm shifts,” said Ralph Abraham, a Smale colleague who became a professor of mathematics at the University of California at Santa Cruz.

“When I started my professional work in mathematics in 1960, which is not so long ago, modern mathematics in its entirety—in its entirety—was rejected by physicists, including the most avant-garde mathematical physicists. So differentiable dynamics, global analysis, manifolds of mappings, differential geometry—everything just a year or two beyond what Einstein had used—was all rejected. The romance between mathematicians and physicists had ended in divorce in the 1930s. These people were no longer speaking. They simply despised each other. Mathematical physicists refused their graduate students permission to take math courses from mathematicians: Take mathematics from us. We will teach you what you need to know. The mathematicians are on some kind of terrible ego trip and they will destroy your mind. That was 1960. By 1968 this had completely turned around.” Eventually physicists, astronomers, and biologists all knew they had to have the news.

A MODEST COSMIC MYSTERY: the Great Red Spot of Jupiter, a vast, swirling oval, like a giant storm that never moves and never runs down. Anyone who saw the pictures beamed across space from Voyager 2 in 1978 recognized the familiar look of turbulence on a hugely unfamiliar scale. It was one of the solar system’s most venerable landmarks—“the red spot roaring like an anguished eye/ amid a turbulence of boiling eyebrows,” as John Updike described it. But what was it? Twenty years after Lorenz, Smale, and other scientists set in motion a new way of understanding nature’s flows, the other-worldly weather of Jupiter proved to be one of the many problems awaiting the altered sense of nature’s possibilities that came with the science of chaos.

For three centuries it had been a case of the more you know, the less you know. Astronomers noticed a blemish on the great planet not long after Galileo first pointed his telescopes at Jupiter. Robert Hooke saw it in the 1600s. Donati Creti painted it in the Vatican’s picture gallery. As a piece of coloration, the spot called for little explaining. But telescopes got better, and knowledge bred ignorance. The last century produced a steady march of theories, one on the heels of another. For example:

The Lava Flow Theory, Scientists in the late nineteenth century imagined a huge oval lake of molten lava flowing out of a volcano. Or perhaps the lava had flowed out of a hole created by a planetoid striking a thin solid crust.

The New Moon Theory. A German scientist suggested, by contrast, that the spot was a new moon on the point of emerging from the planet’s surface.

The Egg Theory. An awkward new fact: the spot was seen to be drifting slightly against the planet’s background. So a notion put forward in 1939 viewed the spot as a more or less solid body floating in the atmosphere the way an egg floats in water. Variations of this theory—including the notion of a drifting bubble of hydrogen or helium—remained current for decades.

The Column-of-Gas Theory. Another new fact: even though the spot drifted, somehow it never drifted