WHERE CHAOS begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder: in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and brain. The irregular side of nature, the discontinuous and erratic side -- these have been puzzles to science, or worse, monstrosities.

As recently as two decades ago, most practicing scientists shared these beliefs:

Simple systems behave in simple ways. A mechanical contraption like a pendulum, a small electrical circuit, an idealized population of fish in a pond -- as long as these systems could be reduced to a few perfectly understood, perfectly deterministic laws, their long-term behavior would be stable and predictable.

Complex behavior implies complex causes. A mechanical device, an electrical circuit, a wildlife population, a fluid flow, a biological organ, a particle beam, an atmospheric storm, a national economy -- a system that was visibly unstable, unpredictable, or out of control must either be governed by a multitude of independent components or subject to random external influences.

Now all that has changed. In the intervening 20 years, scientists have created an alternative set of ideas: Simple systems give rise to complex behavior. {See box.} Complex systems give rise to simple behavior. More important, the laws of complexity hold universally, caring not at all for the details of a system's constituent atoms. And "chaos" -- the obstinate element of disorder within order, of variation where predicability was expected -- has become a shorthand name for a fast-growing movement that is reshaping the fabric of the scientific establishment.

Its advocates contend that it has become the century's third great revolution in the physical sciences. As one physicist put it: "Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability."

Clouds of Knowing The modern study of chaos began with the creeping realization in the 1960s that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output -- a phenomenon given the name "sensitive dependence on initial conditions." In weather, for example, this translates into what is known as the Butterfly Effect -- the notion that a butterfly stirring the air today in Beijing can transform storm systems next month in New York.

In 1960, MIT research meteorologist Edward Lorenz had developed a simulated weather model in his new electronic computer, based on 12 numerical rules -- equations that expressed the relationships between temperature and pressure, pressure and wind speed, and so forth. Lorenz understood that he was putting into practice the laws of Newton. Thanks to the determinism of physical law, futher intervention would then be unnecessary. Those who made such models took for granted that, from present to future, the laws of motion provide a bridge of mathematical certainty. Understand the laws and you understand the universe.

But there was always one small compromise, so small that working scientists usually forgot it was there, lurking in a corner of their philosophies like an unpaid bill: Measurement could never be perfect. Scientists marching under Newton's banner actually waved another flag that said something like this: Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the heart of science. As one theoretician liked to tell his students: "There's a convergence in the way things work, and arbitrarily small influences don't blow up to have arbitrarily large effects."

At first, Lorenz's printouts seemed to behave in those recognizable ways. They matched his cherished intuition about the weather, his sense that it repeated itself, displaying familiar patterns over time, pressure rising and falling, the airstream swinging north and south. But the repetitions were never quite exact. There was pattern, with disturbances. An orderly disorder.

One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee.

The new run should have exactly duplicated the old. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few simulated "months," all resemblance had disappeared. At first he suspected a malfunction. But suddenly he realized the truth.In the computer's memory, six decimal places were stored. On the printout, to save space, just three appeared. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference -- one part in a thousand -- was inconsequential.

A small numerical error was like a small puff of wind -- surely the small puffs faded or canceled each other out before they could change important, large-scale features of the weather. Yet in Lorenz's system of equations, small error proved catastrophic. For reasons of mathematical intuition that his colleagues would understand only later, Lorenz felt a jolt. The practical import could be staggering.

But Lorenz saw beyond the randomness embedded in his weather model. He saw a fine geometrical structure, order masquerading as randomness. He turned his attention more and more to the mathematics of systems that never found a steady state, that almost repeated themselves but never quite succeeded, trying to find simple equations that would produce the aperiodicity he was seeking. At first his computer tended to lock into repetitive cycles. But he finally succeeded when he put an equation that varied the amount of heating from east to west, corresponding to the real-world variation between the way the sun warms the east coast of North America and the way it warms the Atlantic Ocean. The repetition disappeared.

He continued to seek simpler and simpler models and to examine other dynamic fluid systems. His models turned out to have exact analogues in real systems. For example, his equations precisely describe an old-fashioned electrical dynamo, where current flows through a disc that rotates through a magnetic field. Under certain conditions, the dynamo can reverse itself. Such behavior, scientists later suggested, might provide an explanation for another peculiar reversing phenomenon: the earthly magnetic field, or "geodynamo," that is known to have flopped many times during the earth's history, at intervals that seem erratic and inexplicable. Another system precisely described by Lorenz' equations is a certain kind of water wheel. {See box.}

The Capricious Swing

As Lorenz' ideas spread and others arrived at similar kinds of conclusions, many scientists felt an intellectual excitement that comes with the truly new. To physicist Freeman Dyson at Princeton's Institute for Advanced Study, the news of chaos came "like an electric shock" in the '70s. Physicists began to face up to what many believed was a deficiency in their education about even such simple systems as the pendulum.

The pendulum was the classical model of measurable regularity. Galileo contended that a pendulum of a given length not only keeps precise time but keeps the same time no matter how wide or narrow the angle of its swing. He made his claim in terms of experimentation, but the theory made it convincing -- so much so that it is still taught as gospel in most high school physics classes. But it is wrong. The regularity Galileo saw is only an approximation. The changing angle of the bob's motion creates a slight nonlinearity in the equations.

Consider a playground swing. It accelerates on its way down, decelerates on its way up, all the while losing a bit of speed to friction. It gets a regular push -- say, from some clockwork machine. All our intuition tells us that, no matter where the swing might start, the motion will eventually settle down to a regular back-and-forth pattern, with the swing coming to the same height each time. That can happen. Yet odd as it seems, the motion can also turn erratic, first high, then low, never settling down to a steady state and never exactly repeating a pattern of swings that came before.

The surprising behavior comes from a nonlinear twist in the flow of energy in and out of this simple oscillator. The swing is damped and it is driven: Damped because friction is trying to bring it to a halt; driven because it is getting a periodic push. Even when a damped, driven system is at equilibrium, it is not at equilibrium -- and the world is full of such systems, beginning with the weather, damped by the friction of moving air and water and by the dissipation of heat to outer space, and driven by the constant push of the sun's energy.

As chaos began to unite the study of different systems, pendulum dynamics broadened to cover high technologies from lasers to superconductors. Some chemical reactions displayed pendulum-like behavior, as did the beating heart. There has long been a feeling that theoretical physics -- with its emphasis on subatomic particles -- has strayed far from human intuition about the world. But the revolution in chaos applies to the universe we see and touch, to objects at human scale.

Random Regularity

Nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal (i.e, composed of "fractional dimensions"), exhibiting structures self-similar in scale. {Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern, like an aerial picture of a coastline or the "Koch curve" in the illustration above.} Others give rise to steady states or oscillating ones. Pattern formation has now become a branch of physics and of materials science, allowing scientists to model the aggregation of particles into clusters, the fractured spread of electrical discharges, and the growth of crystals in ice and metal alloys.

The dynamics seem so basic -- shapes changing in space and time -- yet only now are the tools available to understand them. It is a fair question now to ask a physicist, "Why are all snowflakes different?"

Ice crystals form in the turbulent air with a famous blending of symmetry and chance, the special beauty of six-fold indeterminancy, obeying mathematical laws of surprising subtlety, making it formerly impossible to predict how fast a flake tip would grow, how narrow it would be, or how often it would branch. When a crystal solidifies outward from an initial seed -- as a snowflake does, grabbing water molecules as it falls -- the process becomes unstable.

The physics of heat diffusion cannot completely explain the patterns. But recently scientists worked out a way to incorporate another process -- surface tension. The heart of the new snowflake model is the essence of chaos: a delicate balance between forces of stability and forces of instability, a powerful interplay of forces on atomic scales and forces on everyday scales.

Because the laws of growth are purely deterministic, snowflakes maintain a near-perfect symmetry. But the nature of the turbulent air is such that any pair of snowflakes will experience very different paths. The final flake records the history of all the changing weather conditions it has experienced, and the combinations may as well be infinite.

Classical physics could complete its mission without answering some of the most fundamental questions about nature. How does life begin? What is turbulence? And above all, in an universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise? Only a new kind of science could begin to address those questions, begin to cross the great gulf between knowledge of what one thing does -- one water molecule, one neuron -- and what millions of them do.

"God plays dice with the universe," says the physicist Joseph Ford in answer to Einstein's famous question. "But they're loaded dice. And the main object of physics now is to find out by what rules were they loaded and how we can we use them for our own ends."