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kind of behavior over a very long time, yet a completely different kind of behavior could be just as natural for the system. Such a system is called intransitive. It can stay in one equilibrium or the other, but not both. Only a kick from outside can force it to change states. In a trivial way, a standard pendulum clock is an intransitive system. A steady flow of energy comes in from a wind-up spring or a battery through an escapement mechanism. A steady flow of energy is drained out by friction. The obvious equilibrium state is a regular swinging motion. If a passerby bumps the clock, the pendulum might speed up or slow down from the momentary jolt but will quickly return to its equilibrium. But the clock has a second equilibrium as well—a second valid solution to its equations of motion—and that is the state in which the pendulum is hanging straight down and not moving. A less trivial intransitive system—perhaps with several distinct regions of utterly different behavior—could be climate itself.

Climatologists who use global computer models to simulate the long-term behavior of the earth’s atmosphere and oceans have known for several years that their models allow at least one dramatically different equilibrium. During the entire geological past, this alternative climate has never existed, but it could be an equally valid solution to the system of equations governing the earth. It is what some climatologists call the White Earth climate: an earth whose continents are covered by snow and whose oceans are covered by ice. A glaciated earth would reflect seventy percent of the incoming solar radiation and so would stay extremely cold. The lowest layer of the atmosphere, the troposphere, would be much thinner. The storms that would blow across the frozen surface would be much smaller than the storms we know. In general, the climate would be less hospitable to life as we know it. Computer models have such a strong tendency to fall into the White Earth equilibrium that climatologists find themselves wondering why it has never come about. It may simply be a matter of chance.

To push the earth’s climate into the glaciated state would require a huge kick from some external source. But Lorenz described yet another plausible kind of behavior called “almost-intransitivity.” An almost-intransitive system displays one sort of average behavior for a very long time, fluctuating within certain bounds. Then, for no reason whatsoever, it shifts into a different sort of behavior, still fluctuating but producing a different average. The people who design computer models are aware of Lorenz’s discovery, but they try at all costs to avoid almost-intransitivity. It is too unpredictable. Their natural bias is to make models with a strong tendency to return to the equilibrium we measure every day on the real planet. Then, to explain large changes in climate, they look for external causes—changes in the earth’s orbit around the sun, for example. Yet it takes no great imagination for a climatologist to see that almost-intransitivity might well explain why the earth’s climate has drifted in and out of long Ice Ages at mysterious, irregular intervals. If so, no physical cause need be found for the timing. The Ice Ages may simply be a byproduct of chaos.

LIKE A GUN COLLECTOR wistfully recalling the Colt .45 in the era of automatic weaponry, the modern scientist nurses a certain nostalgia for the HP–65 hand-held calculator. In the few years of its supremacy, this machine changed many scientists’ working habits forever. For Feigenbaum, it was the bridge between pencil-and–paper and a style of working with computers that had not yet been conceived.

He knew nothing of Lorenz, but in the summer of 1975, at a gathering in Aspen, Colorado, he heard Steve Smale talk about some of the mathematical qualities of the same quadratic difference equation. Smale seemed to think that there were some interesting open questions about the exact point at which the mapping changes from periodic to chaotic. As always, Smale had a sharp instinct for questions worth exploring.