Chaos: Making a New Science

by James Gleick

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.

It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all. “If you could write down the solution to a differential equation,” Yorke said, “then necessarily it’s not chaotic, because to write it down, you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things, and that lets you write down a solution. But this is exactly the way to eliminate the possibility of chaos.”

Textbooks showed students only the rare nonlinear systems that would give way to such techniques. They did not display sensitive dependence on initial conditions. Nonlinear systems with real chaos were rarely taught and rarely learned. When people stumbled across such things—and people did—all their training argued for dismissing them as aberrations. Only a few were able to remember that the solvable, orderly, linear systems were the aberrations. Only a few, that is, understood how nonlinear nature is in its soul. Enrico Fermi once exclaimed, “It does not say in the Bible that all laws of nature are expressible linearly!” The mathematician Stanislaw Ulam remarked that to call the study of chaos “nonlinear science” was like calling zoology “the study of non elephant animals.”

James Yorke analyzed this behavior with mathematical rigor in his “Period Three Implies Chaos” paper. He proved that in any one-dimensional system, if a regular cycle of period three ever appears, then the same system will also display regular cycles of every other length, as well as completely chaotic cycles. This was the discovery that came as an “electric shock” to physicists like Freeman Dyson. It was so contrary to intuition. You would think it would be trivial to set up a system that would repeat itself in a period-three oscillation without ever producing chaos. Yorke showed that it was impossible.

“Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties.”

“Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. The rare scholars who are nomads-by–choice are essential to the intellectual welfare of the settled disciplines.”

What is the essence of a coastline, for example? Mandelbrot asked this question in a paper that became a turning point for his thinking: “How Long Is the Coast of Britain?”

In fact, he argued, any coastline is—in a sense—infinitely long. In another sense, the answer depends on the length of your ruler. Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but the surveyor writes the number down anyway. Then he sets the dividers to a smaller length—say, one foot—and repeats the process. He arrives at a somewhat greater length, because the dividers will capture more of the detail and it will take more than three one-foot steps to cover the distance previously covered by a one-yard step. He writes this new number down, sets the dividers at four inches, and starts again. This mental experiment, using imaginary dividers, is a way of quantifying the effect of observing an object from different distances, at different scales. An observer trying to estimate the length of England’s coastline from a satellite will make a smaller guess than an observer trying to walk its coves and beaches, who will make a smaller guess in turn than a snail negotiating every pebble.

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.

THE KOCH SNOWFLAKE. “A rough but vigorous model of a coastline,” in Mandelbrot’s words. To construct a Koch curve, begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and so on. The length of the boundary is 3 × 4/3 × 4/3 × 4/3…—infinity. Yet the area remains less than the area of a circle drawn around the original triangle. Thus an infinitely long line surrounds a finite area.

On reflection, it becomes apparent that the Koch curve has some interesting features. For one thing, it is a continuous loop, never intersecting itself, because the new triangles on each side are always small enough to avoid bumping into each other. Each transformation adds a little area to the inside of the curve, but the total area remains finite, not much bigger than the original triangle, in fact. If you drew a circle around the original triangle, the Koch curve would never extend beyond it. 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.

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.

THE NOTION OF SELF-SIMILARITY strikes ancient chords in our culture. An old strain in Western thought honors the idea. Leibniz imagined that a drop of water contained a whole teeming universe, containing, in turn, water drops and new universes within. “To see the world in a grain of sand,” Blake wrote, and often scientists were predisposed to see it. When sperm were first discovered, each was thought to be a homunculus, a human, tiny but fully formed.

Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world. In the words of Gert Eilenberger, a German physicist who took up nonlinear science after specializing in superconductivity: “Why is it that the silhouette of a storm-bent leafless tree against an evening sky in winter is perceived as beautiful, but the corresponding silhouette of any multi-purpose university building is not, in spite of all efforts of the architect? The answer seems to me, even if somewhat speculative, to follow from the new insights into dynamical systems. Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects—in clouds, trees, mountain ranges, or snow crystals. The shapes of all these are dynamical processes jelled into physical forms, and particular combinations of order and disorder are typical for them.”

In theory the World War II atomic bomb project was a problem in nuclear physics. In reality the nuclear physics had been mostly solved before the project began, and the business that occupied the scientists assembled at Los Alamos was a problem in fluid dynamics.

To produce every rhythm, the orbit would have to be an infinitely long line in a finite area. In other words—but the word had not been invented—it would have to be fractal.

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.

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.

“There’s a fundamental presumption in physics that the way you understand the world is that you keep isolating its ingredients until you understand the stuff that you think is truly fundamental. Then you presume that the other things you don’t understand are details. The assumption is that there are a small number of principles that you can discern by looking at things in their pure state—this is the true analytic notion—and then somehow you put these together in more complicated ways when you want to solve more dirty problems. If you can.

Final cause is cause based on purpose or design: a wheel is round because that shape makes transportation possible. Physical cause is mechanical: the earth is round because gravity pulls a spinning fluid into a spheroid. The distinction is not always so obvious. A drinking glass is round because that is the most comfortable shape to hold or drink from. A drinking glass is round because that is the shape naturally assumed by spun pottery or blown glass.

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.

There is no randomness in the Mandelbrot set,” Hubbard said. “There is no randomness in anything that I do. Neither do I think that the possibility of randomness has any direct relevance to biology. In biology randomness is death, chaos is death. Everything is highly structured. When you clone plants, the order in which the branches come out is exactly the same. The Mandelbrot set obeys an extraordinarily precise scheme leaving nothing to chance whatsoever. I strongly suspect that the day somebody actually figures out how the brain is organized they will discover to their amazement that there is a coding scheme for building the brain which is of extraordinary precision. The idea of randomness in biology is just reflex.”

Farmer said, “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.

And Shaw, settling on an experimental project that would occupy him for years to come, adopted as homely a dynamical system as any physicist could imagine: a dripping faucet. Most people imagine the canonical dripping faucet as relentlessly periodic, but it is not necessarily so, as a moment of experimentation reveals. “It’s a simple example of a system that goes from predictable behavior to unpredictable behavior,” Shaw said. “If you turn it up a little bit, you can see a regime where the pitter-patter is irregular. As it turns out, it’s not a predictable pattern beyond a short time. So even something as simple as a faucet can generate a pattern that is eternally creative.”

Psychiatrists have struggled for generations to define schizophrenia and classify schizophrenics, but the disease has been almost as difficult to describe as to cure. Most of its symptoms appear in mind and behavior. Since 1908, however, scientists have known of a physical manifestation of the disease that seems to afflict not only schizophrenics but also their relatives. When patients try to watch a slowly swinging pendulum, their eyes cannot track the smooth motion. Ordinarily the eye is a remarkably smart instrument. A healthy person’s eyes stay locked on moving targets without the least conscious thought; moving images stay frozen in place on the retina. But a schizophrenic’s eyes jump about disruptively in small increments, overshooting or undershooting the target and creating a constant haze of extraneous movements. No one knows why.

Then, instead of trying to guess at the biochemistry involved, he looked at the problem topologically—that is, he looked at the qualitative shape of the data, instead of the quantitative details.

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 species of firefly that congregates in trees during mating periods, thousands at one time, blinking in a fantastic spectral harmony.

As he saw it, the problem was conceptual. Traditional methods for treating this “most unstable, dynamic, infinite-dimensional machine” were linear and reductionist. “The underlying paradigm remains: one gene → one peptide → one enzyme → one neurotransmitter → one receptor → one animal behavior → one clinical syndrome → one drug → one clinical rating scale. It dominates almost all research and treatment in psychopharmacology. More than 50 transmitters, thousands of cell types, complex electromagnetic phenomenology, and continuous instability based autonomous activity at all levels, from proteins to the electroencephalogram—and still the brain is thought of as a chemical point-to–point switchboard.” To someone exposed to the world of nonlinear dynamics the response could only be: How naïve. Mandell urged his colleagues to understand the flowing geometries that sustain complex systems like the mind.

For them, chaos was the end of the reductionist program in science.

To Ford, the presence of chaos in the Duffing equation was a curious fact—just one of those things he knew to be true, although several years passed before it was published in Physical Review Letters. But he might as well have told a gathering of paleontologists that dinosaurs had feathers. They knew better. “When I said that? Jee-sus Christ, the audience began to bounce up and down. It was, ‘My daddy played with the Duffing equation, and my granddaddy played with the Duffing equation, and nobody seen anything like what you’re talking about.’ You would really run across resistance to the notion that nature is complicated. What I didn’t understand was the hostility.”

“Evolution is chaos with feedback,” Joseph Ford said. The universe is randomness and dissipation, yes. But randomness with direction can produce surprising complexity. And as Lorenz discovered so long ago, dissipation is an agent of order.

“God plays dice with the universe,” is Ford’s answer to Einstein’s famous question. “But they’re loaded dice. And the main objective of physics now is to find out by what rules were they loaded and how can we use them for our own ends.”

Chaos is a creator of information—another apparent paradox.