Chaos: Making a New Science

by James Gleick

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 the brain. The irregular side of nature, the discontinuous and erratic side—these have been puzzles to science, or worse, monstrosities. But in the 1970s a few scientists in the United States and Europe began to find a way through disorder. They were mathematicians, physicists, biologists, chemists, all seeking connections between different kinds of irregularity.

When Mitchell Feigenbaum began thinking about chaos at Los Alamos, he was one of a handful of scattered scientists, mostly unknown to one another. A mathematician in Berkeley, California, had formed a small group dedicated to creating a new study of “dynamical systems.” A population biologist at Princeton University was about to publish an impassioned plea that all scientists should look at the surprisingly complex behavior lurking in some simple models. A geometer working for IBM was looking for a new word to describe a family of shapes—jagged, tangled, splintered, twisted, fractured—that he considered an organizing principle in nature. A French mathematical physicist had just made the disputatious claim that turbulence in fluids might have something to do with a bizarre, infinitely tangled abstraction that he called a strange attractor.

Now that science is looking, chaos seems to be everywhere. A rising column of cigarette smoke breaks into wild swirls. A flag snaps back and forth in the wind. A dripping faucet goes from a steady pattern to a random one. Chaos appears in the behavior of the weather, the behavior of an airplane in flight, the behavior of cars clustering on an expressway, the behavior of oil flowing in underground pipes. No matter what the medium, the behavior obeys the same newly discovered laws. That realization has begun to change the way business executives make decisions about insurance, the way astronomers look at the solar system, the way political theorists talk about the stresses leading to armed conflict.

The most passionate advocates of the new science go so far as to say that twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. Chaos, they contend, has become the century’s third great revolution in the physical sciences. Like the first two revolutions, chaos cuts away at the tenets of Newton’s physics. 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.” Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale.

can always try to solve a problem by proving that no solution exists. Lorenz liked that, as he always liked the purity of mathematics, and when he graduated from Dartmouth College, in 1938, he thought that mathematics was his calling. Circumstance interfered, however, in the form of World War II, which put him to work as a weather forecaster for the Army Air Corps. After the war Lorenz decided to stay with meteorology, investigating the theory of it, pushing the mathematics a little further forward. He made a name for himself by publishing work on orthodox problems, such as the general circulation of the atmosphere. And in the meantime he continued to think about forecasting.

Not only did meteorologists scorn forecasting, but in the 1960s virtually all serious scientists mistrusted computers. These souped-up calculators hardly seemed like tools for theoretical science. So numerical weather modeling was something of a bastard problem. Yet the time was right for it. Weather forecasting had been waiting two centuries for a machine that could repeat thousands of calculations over and over again by brute force. Only a computer could cash in the Newtonian promise that the world unfolded along a deterministic path, rule-bound like the planets, predictable like eclipses and tides. In theory a computer could let meteorologists do what astronomers had been able to do with pencil and slide rule: reckon the future of their universe from its initial conditions and the physical laws that guide its evolution.

Weather was vastly more complicated, but it was governed by the same laws. Perhaps a powerful enough computer could be the supreme intelligence imagined by Laplace, the eighteenth-century philosopher-mathematician who caught the Newtonian fever like no one else: “Such an intelligence,” Laplace wrote, “would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.” In these days of Einstein’s relativity and Heisenberg’s uncertainty, Laplace seems almost buffoon-like in his optimism, but much of modern science has pursued his dream. Implicitly, the mission of many twentieth-century scientists—biologists, neurologists, economists—has been to break their universes down into the simplest atoms that will obey scientific rules. In all these sciences, a kind of Newtonian determinism has been brought to bear. The fathers of modern computing always had Laplace in mind, and the history of computing and the history of forecasting were intermingled ever since John von Neumann designed his first machines at the Institute for Advanced Study in Princeton, New Jersey, in the 1950s. Von Neumann recognized that weather modeling could be an ideal task for a computer. 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. Measurements coul...

With his primitive computer, Lorenz had boiled weather down to the barest skeleton. Yet, line by line, the winds and temperatures in Lorenz’s printouts seemed to behave in a recognizable earthly way. 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. He discovered that when a line went from high to low without a bump, a double bump would come next, and he said, “That’s the kind of rule a forecaster could use.” 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. When he returned an hour later, he saw something unexpected, something that planted a seed for a new science. THIS NEW RUN should have exactly duplicated the old. Lorenz had copied the numbers into the machine himself. The program had not changed. 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 months, all resemblance had disappeared. He looked at one set of numbers, then back at the other. He might as well have chosen two random weathers out of a hat. His first thought was that another vacuum tube had gone bad. Suddenly he realized the truth. There had been no malfunction. The problem lay in the numbers he had typed. In the computer’s memory, six decimal places were stored: .506127. On the printout, to save space, just three appeared: .506. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference—one part in a thousand—was inconsequential. It was a reasonable assumption. If a weather satellite can read ocean-surface temperature to within one part in a thousand, its operators consider themselves lucky. Lorenz...

The Butterfly Effect was no accident; it was necessary. Suppose small perturbations remained small, he reasoned, instead of cascading upward through the system. Then when the weather came arbitrarily close to a state it had passed through before, it would stay arbitrarily close to the patterns that followed. For practical purposes, the cycles would be predictable—and eventually uninteresting. To produce the rich repertoire of real earthly weather, the beautiful multiplicity of it, you could hardly wish for anything better than a Butterfly Effect. The Butterfly Effect acquired a technical name: sensitive dependence on initial conditions. And sensitive dependence on initial conditions was not an altogether new notion. It had a place in folklore: “For want of a nail, the shoe was lost; For want of a shoe, the horse was lost; For want of a horse, the rider was lost; For want of a rider, the battle was lost; For want of a battle, the kingdom was lost!” In science as in life, it is well known that a chain of events can have a point of crisis that could magnify small changes. But chaos meant that such points were everywhere. They were pervasive. In systems like the weather, sensitive dependence on initial conditions was an inescapable consequence of the way small scales intertwined with large.

Every scientist who turned to chaos early had a story to tell of discouragement or open hostility. Graduate students were warned that their careers could be jeopardized if they wrote theses in an untested discipline, in which their advisors had no expertise. A particle physicist, hearing about this new mathematics, might begin playing with it on his own, thinking it was a beautiful thing, both beautiful and hard—but would feel that he could never tell his colleagues about it. Older professors felt they were suffering a kind of midlife crisis, gambling on a line of research that many colleagues were likely to misunderstand or resent. But they also felt an intellectual excitement that comes with the truly new. Even outsiders felt it, those who were attuned to it. To Freeman Dyson at the Institute for Advanced Study, the news of chaos came “like an electric shock” in the 1970s. Others felt that for the first time in their professional lives they were witnessing a true paradigm shift, a transformation in a way of thinking.

As the chaos specialists spread, some departments frowned on these somewhat deviant scholars; others advertised for more. Some journals established unwritten rules against submissions on chaos; other journals came forth to handle chaos exclusively. The chaoticists or chaologists (such coinages could be heard) turned up with disproportionate frequency on the yearly lists of important fellowships and prizes. By the middle of the eighties a process of academic diffusion had brought chaos specialists into influential positions within university bureaucracies. Centers and institutes were founded to specialize in “nonlinear dynamics” and “complex systems.”

Traditionally, a dynamicist would believe that to write down a system’s equations is to understand the system. How better to capture the essential features? For a playground swing or a toy, the equations tie together the pendulum’s angle, its velocity, its friction, and the force driving it. But because of the little bits of nonlinearity in these equations, a dynamicist would find himself helpless to answer the easiest practical questions about the future of the system. A computer can address the problem by simulating it, rapidly calculating each cycle. But simulation brings its own problem: the tiny imprecision built into each calculation rapidly takes over, because this is a system with sensitive dependence on initial conditions. Before long, the signal disappears and all that remains is noise.

Smale’s Fields Medal honored a famous piece of work in topology, a branch of mathematics that flourished in the twentieth century and had a particular heyday in the fifties. Topology studies the properties that remain unchanged when shapes are deformed by twisting or stretching or squeezing. Whether a shape is square or round, large or small, is irrelevant in topology, because stretching can change those properties. Topologists ask whether a shape is connected, whether it has holes, whether it is knotted. They imagine surfaces not just in the one–, two–, and three-dimensional universes of Euclid, but in spaces of many dimensions, impossible to visualize.

RAVENOUS FISH AND TASTY plankton. Rain forests dripping with nameless reptiles, birds gliding under canopies of leaves, insects buzzing like electrons in an accelerator. Frost belts where voles and lemmings flourish and diminish with tidy four-year periodicity in the face of nature’s bloody combat. The world makes a messy laboratory for ecologists, a cauldron of five million interacting species. Or is it fifty million? Ecologists do not actually know. Mathematically inclined biologists of the twentieth century built a discipline, ecology, that stripped away the noise and color of real life and treated populations as dynamical systems. Ecologists used the elementary tools of mathematical physics to describe life’s ebbs and flows. Single species multiplying in a place where food is limited, several species competing for existence, epidemics spreading through host populations—all could be isolated, if not in laboratories then certainly in the minds of biological theorists.

In the Malthusian scenario of unrestrained growth, the linear growth function rises forever upward. For a more realistic scenario, an ecologist needs an equation with some extra term that restrains growth when the population becomes large. The most natural function to choose would rise steeply when the population is small, reduce growth to near zero at intermediate values, and crash downward when the population is very large. By repeating the process, an ecologist can watch a population settle into its long-term behavior—presumably reaching some steady state. A successful foray into mathematics for an ecologist would let him say something like this: Here’s an equation; here’s a variable representing reproductive rate; here’s a variable representing the natural death rate; here’s a variable representing the additional death rate from starvation or predation; and look—the population will rise at this speed until it reaches that level of equilibrium.

Oddly, the flow of numbers begins to misbehave, quite a nuisance for anyone calculating with a hand crank. The numbers still do not grow without limit, of course, but they do not converge to a steady level, either. Apparently, though, none of these early ecologists had the inclination or the strength to keep churning out numbers that refused to settle down. Anyway, if the population kept bouncing back and forth, ecologists assumed that it was oscillating around some underlying equilibrium. The equilibrium was the important thing. It did not occur to the ecologists that there might be no equilibrium.

LATER, PEOPLE WOULD SAY that James Yorke had discovered Lorenz and given the science of chaos its name. The second part was actually true. Yorke was a mathematician who liked to think of himself as a philosopher, though this was professionally dangerous to admit. He was brilliant and soft-spoken, a mildly disheveled admirer of the mildly disheveled Steve Smale. Like everyone else, he found Smale hard to fathom. But unlike most people, he understood why Smale was hard to fathom. When he was just twenty-two years old, Yorke joined an interdisciplinary institute at the University of Maryland called the Institute for Physical Science and Technology, which he later headed.

As Lorenz had discovered a decade before, the only way to make sense of such numbers and preserve one’s eyesight is to create a graph. May drew a sketchy outline meant to sum up all the knowledge about the behavior of such a system at different parameters. The level of the parameter was plotted horizontally, increasing from left to right. The population was represented vertically. For each parameter, May plotted a point representing the final outcome, after the system reached equilibrium. At the left, where the parameter was low, this outcome would just be a point, so different parameters produced a line rising slightly from left to right. When the parameter passed the first critical point, May would have to plot two populations: the line would split in two, making a sideways Y or a pitchfork. This split corresponded to a population going from a one-year cycle to a two-year cycle. As the parameter rose further, the number of points doubled again, then again, then again. It was dumbfounding—such complex behavior, and yet so tantalizingly regular. “The snake in the mathematical grass” was how May put it. The doublings themselves were bifurcations, and each bifurcation meant that the pattern of repetition was breaking down a step further. A population that had been stable would alternate between different levels every other year. A population that had been alternating on a two-year cycle would now vary on the third and fourth years, thus switching to period four. These bifurc...

As May looked at more and more biological systems through the prism of simple chaotic models, he continued to see results that violated the standard intuition of practitioners. In epidemiology, for example, it was well known that epidemics tend to come in cycles, regular or irregular. Measles, polio, rubella—all rise and fall in frequency. May realized that the oscillations could be reproduced by a nonlinear model and he wondered what would happen if such a system received a sudden kick—a perturbation of the kind that might correspond to a program of inoculation. Naïve intuition suggests that the system will change smoothly in the desired direction. But actually, May found, huge oscillations are likely to begin. Even if the long-term trend was turned solidly downward, the path to a new equilibrium would be interrupted by surprising peaks. In fact, in data from real programs, such as a campaign to wipe out rubella in Britain, doctors had seen oscillations just like those predicted by May’s model. Yet any health official, seeing a sharp short-term rise in rubella or gonorrhea, would assume that the inoculation program had failed.

chaos was found in records of New York City measles epidemics and in two hundred years of fluctuations of the Canadian lynx population, as recorded by the trappers of the Hudson’s Bay Company.

The description would not have seemed apt to anyone who knew him in his later years, with his high imposing brow and his list of titles and honors, but Benoit Mandelbrot is best understood as a refugee. He was born in Warsaw in 1924 to a Lithuanian Jewish family, his father a clothing wholesaler, his mother a dentist. Alert to geopolitical reality, the family moved to Paris in 1936, drawn in part by the presence of Mandelbrot’s uncle, Szolem Mandelbrojt, a mathematician. When the war came, the family stayed just ahead of the Nazis once again, abandoning everything but a few suitcases and joining the stream of refugees who clogged the roads south from Paris. They finally reached the town of Tulle. For a while Benoit went around as an apprentice toolmaker, dangerously conspicuous by his height and his educated background. It was a time of unforgettable sights and fears, yet later he recalled little personal hardship, remembering instead the times he was befriended in Tulle and elsewhere by schoolteachers, some of them distinguished scholars, themselves stranded by the war. In all, his schooling was irregular and discontinuous. He claimed never to have learned the alphabet or, more significantly, multiplication tables past the fives. Still, he had a gift.

“How Long Is the Coast of Britain?” Mandelbrot had come across the coastline question in an obscure posthumous article by an English scientist, Lewis F. Richardson, who groped with a surprising number of the issues that later became part of chaos. He wrote about numerical weather prediction in the 1920s, studied fluid turbulence by throwing a sack of white parsnips into the Cape Cod Canal, and asked in a 1926 paper, “Does the Wind Possess a Velocity?” (“The question, at first sight foolish, improves on acquaintance,” he wrote.) Wondering about coastlines and wiggly national borders, Richardson checked encyclopedias in Spain and Portugal, Belgium and the Netherlands and discovered discrepancies of twenty percent in the estimated lengths of their common frontiers.

Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object. A twisting coastline, for example, despite its immeasurability in terms of length, nevertheless has a certain characteristic degree of roughness. Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data, and he allowed his geometry to make a claim about the irregular patterns he had studied in nature. The claim was that the degree of irregularity remains constant over different scales. Surprisingly often, the claim turns out to be true. Over and over again, the world displays a regular irregularity.

One wintry afternoon in 1975, aware of the parallel currents emerging in physics, preparing his first major work for publication in book form, Mandelbrot decided he needed a name for his shapes, his dimensions, and his geometry. His son was home from school, and Mandelbrot found himself thumbing through the boy’s Latin dictionary. He came across the adjective fractus, from the verb frangere, to break. The resonance of the main English cognates—fracture and fraction—seemed appropriate. Mandelbrot created the word (noun and adjective, English and French) fractal.

Yet the curve itself is infinitely long, as long as a Euclidean straight line extending to the edges of an unbounded universe. Just as the first transformation replaces a one-foot segment with four four-inch segments, every transformation multiplies the total length by four-thirds. This paradoxical result, infinite length in a finite space, disturbed many of the turn-of–the-century mathematicians who thought about it. The Koch curve was monstrous, disrespectful to all reasonable intuition about shapes and—it almost went without saying—pathologically unlike anything to be found in nature.

Mandelbrot’s other advantage was the picture of reality he had begun forming in his encounters with cotton prices, with electronic transmission noise, and with river floods. The picture was beginning to come into focus now. His studies of irregular patterns in natural processes and his exploration of infinitely complex shapes had an intellectual intersection: a quality of self-similarity. Above all, fractal meant self-similar. Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern. Mandelbrot’s price charts and river charts displayed self-similarity, because not only did they produce detail at finer and finer scales, they also produced detail with certain constant measurements. Monstrous shapes like the Koch curve display self-similarity because they look exactly the same even under high magnification.

Imagine tracing the surface of the earth as it would look from a distance of one hundred kilometers out in space. The line goes up and down over trees and hillocks, buildings and—in a parking lot somewhere—a Volkswagen. On that scale, the surface is just a bump among many other bumps, a bit of randomness. Or imagine looking at the Volkswagen from closer and closer, zooming in with magnifying glass and microscope. At first the surface seems to get smoother, as the roundness of bumpers and hood passes out of view. But then the microscopic surface of the steel turns out to be bumpy itself, in an apparently random way. It seems chaotic.

Blood vessels, from aorta to capillaries, form another kind of continuum. They branch and divide and branch again until they become so narrow that blood cells are forced to slide through single file. The nature of their branching is fractal. Their structure resembles one of the monstrous imaginary objects conceived by Mandelbrot’s turn-of–the-century mathematicians. As a matter of physiological necessity, blood vessels must perform a bit of dimensional magic. Just as the Koch curve, for example, squeezes a line of infinite length into a small area, the circulatory system must squeeze a huge surface area into a limited volume. In terms of the body’s resources, blood is expensive and space is at a premium. The fractal structure nature has devised works so efficiently that, in most tissue, no cell is ever more than three or four cells away from a blood vessel. Yet the vessels and blood take up little space, no more than about five percent of the body. It is, as Mandelbrot put it, the Merchant of Venice Syndrome—not only can’t you take a pound of flesh without spilling blood, you can’t take a milligram. This exquisite structure—actually, two intertwining trees of veins and arteries—is far from exceptional. The body is filled with such complexity. In the digestive tract, tissue reveals undulations within undulations. The lungs, too, need to pack the greatest possible surface into the smallest space. An animal’s ability to absorb oxygen is roughly proportional to the surface are...

In the end, the word fractal came to stand for a way of describing, calculating, and thinking about shapes that are irregular and fragmented, jagged and broken-up—shapes from the crystalline curves of snowflakes to the discontinuous dusts of galaxies.

Dynamically speaking, a globular cluster is a big many-body problem. The two-body problem is easy. Newton solved it completely. Each body—the earth and the moon, for example—travels in a perfect ellipse around the system’s joint center of gravity. Add just one more gravitational object, however, and everything changes. The three-body problem is hard, and worse than hard. As Poincaré discovered, it is most often impossible. The orbits can be calculated numerically for a while, and with powerful computers they can be tracked for a long while before uncertainties begin to take over. But the equations cannot be solved analytically, which means that long-term questions about a three-body system cannot be answered. Is the solar system stable? It certainly appears to be, in the short term, but even today no one knows for sure that some planetary orbits could not become more and more eccentric until the planets fly off from the system forever.

Most important, no one knew whether strange attractors would say anything about the deepest problem with nonlinear systems. 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. Understanding one seemed to offer no help in understanding the next. An attractor like Lorenz’s illustrated the stability and the hidden structure of a system that otherwise seemed patternless, but how did this peculiar double spiral help researchers exploring unrelated systems? No one knew.

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.

applied the mathematics of renormalization group theory, with its use of scaling to collapse infinities into manageable quantities. In the spring of 1976 he entered a mode of existence more intense than any he had lived through. He would concentrate as if in a trance, programming furiously, scribbling with his pencil, programming again. He could not call C division for help, because that would mean signing off the computer to use the telephone, and reconnection was chancy. He could not stop for more than five minutes’ thought, because the computer would automatically disconnect his line. Every so often the computer would go down anyway, leaving him shaking with adrenalin. He worked for two months without pause. His functional day was twenty-two hours. He would try to go to sleep in a kind of buzz, and awaken two hours later with his thoughts exactly where he had left them. His diet was strictly coffee. (Even when healthy and at peace, Feigenbaum subsisted exclusively on the reddest possible meat, coffee, and red wine. His friends speculated that he must be getting his vitamins from cigarettes.) In the end, a doctor called it off. He prescribed a modest regimen of Valium and an enforced vacation. But by then Feigenbaum had created a universal theory.

That view of science works best when a well-defined discipline awaits the resolution of a well-defined problem. No one misunderstood the discovery of the molecular structure of DNA, for example. But the history of ideas is not always so neat. As nonlinear science arose in odd corners of different disciplines, the flow of ideas failed to follow the standard logic of historians. The emergence of chaos as an entity unto itself was a story not only of new theories and new discoveries, but also of the belated understanding of old ideas. Many pieces of the puzzle had been seen long before—by Poincaré, by Maxwell, even by Einstein—and then forgotten. Many new pieces were understood at first only by a few insiders. A mathematical discovery was understood by mathematicians, a physics discovery by physicists, a meteorological discovery by no one. The way ideas spread became as important as the way they originated. Each scientist had a private constellation of intellectual parents. Each had his own picture of the landscape of ideas, and each picture was limited in its own way. Knowledge was imperfect. Scientists were biased by the customs of their disciplines or by the accidental paths of their own educations. The scientific world can be surprisingly finite. No committee of scientists pushed history into a new channel—a handful of individuals did it, with individual perceptions and individual goals. Afterwards, a consensus began to take shape about which innovations and which contrib...

THE MANDELBROT SET IS the most complex object in mathematics, its admirers like to say. An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God’s personal vine. Examined in color through the adjustable window of a computer screen, the Mandelbrot set seems more fractal than fractals, so rich is its complication across scales. A cataloguing of the different images within it or a numerical description of the set’s outline would require an infinity of information. But here is a paradox: to send a full description of the set over a transmission line requires just a few dozen characters of code. A terse computer program contains enough information to reproduce the entire set. Those who were first to understand the way the set commingles complexity and simplicity were caught unprepared—even Mandelbrot. The Mandelbrot set became a kind of public emblem for chaos, appearing on the glossy covers of conference brochures and engineering quarterlies, forming the centerpiece of an exhibit of computer art that traveled internationally in 1985 and 1986.

Barnsley’s central insight was this: Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process, had a second, equally valid existence as the limit of a random process. By analogy, he suggested, one could imagine a map of Great Britain drawn in chalk on the floor of a room. A surveyor with standard tools would find it complicated to measure the area of these awkward shapes, with fractal coastlines, after all. But suppose you throw grains of rice into the air one by one, allowing them to fall randomly to the floor and counting the grains that land inside the map. As time goes on, the result begins to approach the area of the shapes—as the limit of a random process. In dynamical terms, Barnsley’s shapes proved to be attractors.

In some sense, Barnsley contended, nature must be playing its own version of the chaos game. “There’s only so much information in the spore that encodes one fern,” he said. “So there’s a limit to the elaborateness with which a fern could grow. It’s not surprising that we can find equivalent succinct information to describe ferns. It would be surprising if it were otherwise.”

It was spring in 1978 before the department quite believed that Shaw was abandoning his superconductivity thesis. He was so close to finishing. No matter how bored he was, the faculty reasoned that he could rush through the formalities, get his doctorate and move on to the real world. As for chaos, there were questions of academic suitability. No one at Santa Cruz was qualified to supervise a course of study in this field-without–a-name. No one had ever received a doctorate in it.

“You can’t appreciate the kind of revelation that is unless you’ve been brainwashed by six or seven years of a typical physics curriculum. You’re taught that there are classical models where everything is determined by initial conditions, and then there are quantum mechanical models where things are determined but you have to contend with a limit on how much initial information you can gather. Nonlinear was a word that you only encountered in the back of the book. A physics student would take a math course and the last chapter would be on nonlinear equations. You would usually skip that, and, if you didn’t, all they would do is take these nonlinear equations and reduce them to linear equations, so you just get approximate solutions anyway. It was just an exercise in frustration.

“On a philosophical level, it struck me as an operational way to define free will, in a way that allowed you to reconcile free will with determinism. The system is deterministic, but you can’t say what it’s going to do next. At the same time, I’d always felt that the important problems out there in the world had to do with the creation of organization, in life or intelligence.

THE MOST CHARACTERISTICALLY Santa Cruzian imprint on chaos research involved a piece of mathematics cum philosophy known as information theory, invented in the late 1940s by a researcher at the Bell Telephone Laboratories, Claude Shannon. Shannon called his work “The Mathematical Theory of Communication,” but it concerned a rather special quantity called information, and the name information theory stuck.

THE CHOICE IS ALWAYS the same. You can make your model more complex and more faithful to reality, or you can make it simpler and easier to handle. Only the most naïve scientist believes that the perfect model is the one that perfectly represents reality. Such a model would have the same drawbacks as a map as large and detailed as the city it represents, a map depicting every park, every street, every building, every tree, every pothole, every inhabitant, and every map. Were such a map possible, its specificity would defeat its purpose: to generalize and abstract. Mapmakers highlight such features as their clients choose. Whatever their purpose, maps and models must simplify as much as they mimic the world.

Christian Huygens, the seventeenth-century Dutch physicist who helped invent both the pendulum clock and the classical science of dynamics, stumbled upon one of the great examples of this form of regulation, or so the standard story goes. Huygens noticed one day that a set of pendulum clocks placed against a wall happened to be swinging in perfect chorus-line synchronization. He knew that the clocks could not be that accurate. Nothing in the mathematical description then available for a pendulum could explain this mysterious propagation of order from one pendulum to another. Huygens surmised, correctly, that the clocks were coordinated by vibrations transmitted through the wood. This phenomenon, in which one regular cycle locks into another, is now called entrainment, or mode locking. Mode locking explains why the moon always faces the earth, or more generally why satellites tend to spin in some whole-number ratio of their orbital period: 1 to 1, or 2 to 1, or 3 to 2. When the ratio is close to a whole number, nonlinearity in the tidal attraction of the satellite tends to lock it in. Mode locking occurs throughout electronics, making it possible, for example, for a radio receiver to lock in on signals even when there are small fluctuations in their frequency. Mode locking accounts for the ability of groups of oscillators, including biological oscillators, like heart cells and nerve cells, to work in synchronization. A spectacular example in nature is a Southeast Asian spec...

Now all that has changed. In the intervening twenty years, physicists, mathematicians, biologists, and astronomers have created an alternative set of ideas. Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system’s constituent atoms.

Still, no one could quite agree on the word itself. Philip Holmes, a white-bearded mathematician and poet from Cornell by way of Oxford: The complicated, aperiodic, attracting orbits of certain (usually low-dimensional) dynamical systems. Hao Bai-Lin, a physicist in China who assembled many of the historical papers of chaos into a single reference volume: A kind of order without periodicity. And: A rapidly expanding field of research to which mathematicians, physicists, hydrodynamicists, ecologists and many others have all made important contributions. And: A newly recognized and ubiquitous class of natural phenomena. H. Bruce Stewart, an applied mathematician at Brookhaven National Laboratory on Long Island: Apparently random recurrent behavior in a simple deterministic (clockwork-like) system. Roderick V. Jensen of Yale University, a theoretical physicist exploring the possibility of quantum chaos: The irregular, unpredictable behavior of deterministic, nonlinear dynamical systems.

The ordinary-sized stuff which is our lives, the things people write poetry about—clouds—daffodils—waterfalls—and what happens in a cup of coffee when the cream goes in—these things are full of mystery, as mysterious to us as the heavens were to the Greeks…. The future is disorder. A door like this has cracked open five or six times since we got up on our hind legs. It’s the best possible time to be alive, when almost everything you thought you knew was wrong.

Astronomers had already found the fingerprints of chaos in violence on the sun’s surface, gaps in the asteroid belt, and the distribution of galaxies. Levin and her colleagues have found them in the exit from the big bang and in black holes. They predict that light trapped by a black hole can enter unstable chaotic orbits and be reemitted—making the black hole visible, if only briefly. Yes, chaos can light up black holes. “There are rational numbers to mine, fractal sets, and all kinds of truly beautiful consequences,” she says. “So on the one hand, people are horrified, on the other they’re mesmerized.” She does chaos in curved space-time. Einstein would be proud.

AS FOR ME, I never returned to chaos, but readers might spot seeds of all my later books in this one. I knew hardly anything about Richard Feynman, but he has a cameo here (see here). Isaac Newton has more than a cameo: he seems to be the antihero of chaos, or the god to be overthrown. I discovered only later, reading his notebooks and letters, how wrong I’d been about him. And for twenty years I’ve been pursuing a thread that began with something Rob Shaw told me, about chaos and information theory, as invented by Claude Shannon. Chaos is a creator of information—another apparent paradox. This thread connects with something Bernardo Hubemian said: that he was seeing complex behaviors emerge unexpectedly in information networks.