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The physics of earthquake behavior is mostly independent of scale. A large earthquake is just a scaled-up version of a small earthquake.
Clouds, on the other hand, are scaling phenomena like earthquakes. Their characteristic irregularity—describable in terms of fractal dimension—changes not at all as they are observed on different scales. That is why air travelers lose all perspective on how far away a cloud is.
But the claim of fractal geometry is that, for some elements of nature, looking for a characteristic scale becomes a distraction. Hurricane. By definition, it is a storm of a certain size. But the definition is imposed by people on nature. In reality, atmospheric scientists are realizing that tumult in the air forms a continuum, from the gusty swirling of litter on a city street corner to the vast cyclonic systems visible from space.
It happens that the equations of fluid flow are in many contexts dimensionless, meaning that they apply without regard to scale. Scaled-down airplane wings and ship propellers can be tested in wind tunnels and laboratory basins.
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.
An animal’s ability to absorb oxygen is roughly proportional to the surface area of its lungs. Typical human lungs pack in a surface bigger than a tennis court. As an added complication, the labyrinth of windpipes must merge efficiently with the arteries and veins.
The language of anatomy tends to obscure the unity across scales. The fractal approach, by contrast, embraces the whole structure in terms of the branching that produces it, branching that behaves consistently from large scales to small.
The standard “exponential” description of bronchial branching proved to be quite wrong; a fractal description turned out to fit the data. The urinary collecting system proved fractal. The biliary duct in the liver. The network of special fibers in the heart that carry pulses of electric current to the contracting muscles.
Considerable work on healthy and abnormal hearts turned out to hinge on the details of how the muscle cells of the left and right pumping chambers all manage to coordinate their timing. Several chaos-minded cardiologists found that the frequency spectrum of heartbeat timing, like earthquakes and economic phenomena, followed fractal laws, and they argued that one key to understanding heartbeat timing was the fractal organization of the His-Purkinje network, a labyrinth of branching pathways organized to be self-similar on smaller and smaller scales.
Mandelbrot’s point is that the complications exist only in the context of traditional Euclidean geometry. As fractals, branching structures can be described with transparent simplicity, with just a few bits of information.
DNA surely cannot specify the vast number of bronchi, bronchioles, and alveoli or the particular spatial structure of the resulting tree, but it can specify a repeating process of bifurcation and development.
Cohen had scoured the annals of discovery for years, looking for scientists who had declared their own work to be “revolutions.” All told, he found just sixteen. Robert Symmer, a Scots contemporary of Benjamin Franklin whose ideas about electricity were indeed radical, but wrong. Jean-Paul Marat, known today only for his bloody contribution to the French Revolution. Von Liebig. Hamilton. Charles Darwin, of course. Virchow. Cantor. Einstein. Minkowski. Von Laue. Alfred Wegener—continental drift. Compton. Just. James Watson—the structure of DNA. And Benoit Mandelbrot.
Nor were they mollified that Mandelbrot was equally copious with his references to predecessors, some thoroughly obscure. (And all, as his detractors noted, quite safely deceased.) They thought it was just his way of trying to position himself squarely in the center, setting himself up like the Pope, casting his benedictions from one side of the field to the other.
The attracting pull of four points—in the four dark holes—creates “basins of attraction,” each a different color, with a complicated fractal boundary. The image represents the way Newton’s method for solving equations leads from different starting points to one of four possible solutions (in this case the equation is x4 - 1 = 0).
A random clustering of particles generated by a computer produces a “percolation network,” one of many visual models inspired by fractal geometry.
A branch of physics, once it becomes obsolete or unproductive, tends to be forever part of the past. It may be a historical curiosity, perhaps the source of some inspiration to a modern scientist, but dead physics is usually dead for good reason. Mathematics, by contrast, is full of channels and byways that seem to lead nowhere in one era and become major areas of study in another.
Mandelbrot found his most enthusiastic acceptance among applied scientists working with oil or rock or metals, particularly in corporate research centers. By the middle of the 1980s, vast numbers of scientists at Exxon’s huge research facility, for example, worked on fractal problems. At General Electric, fractals became an organizing principle in the study of polymers and also—though this work was conducted secretly—in problems of nuclear reactor safety.
But self-similarity withered as a scientific principle, for a good reason. It did not fit the facts. Sperm are not merely scaled-down humans—they are far more interesting than that—and the process of ontogenetic development is far more interesting than mere enlargement.
The first discoveries were realizations that each change of scale brought new phenomena and new kinds of behavior. For modern particle physicists, the process has never ended.
Although Mandelbrot made the most comprehensive geometric use of it, the return of scaling ideas to science in the 1960s and 1970s became an intellectual current that made itself felt simultaneously in many places. Self-similarity was implicit in Edward Lorenz’s work. It was part of his intuitive understanding of the fine structure of the maps made by his system of equations, a structure he could sense but not see on the computers available in 1963.
By the late twentieth century, in ways never before conceivable, images of the incomprehensibly small and the unimaginably large became part of everyone’s experience. The culture saw photographs of galaxies and of atoms.
To Mandelbrot the epitome of the Euclidean sensibility outside mathematics was the architecture of the Bauhaus.
Mathematicians like Cantor and Koch had delighted in their originality. They thought they were outsmarting nature—when actually they had not yet caught up with nature’s creation.
Only later, after Steve Smale brought mathematicians back to dynamical systems, could a physicist say, “We have the astronomers and mathematicians to thank for passing the field on to us, physicists, in a much better shape than we left it to them, 70 years ago.”
TURBULENCE WAS A PROBLEM with pedigree. The great physicists all thought about it, formally or informally. A smooth flow breaks up into whorls and eddies.
There was a story about the quantum theorist Werner Heisenberg, on his deathbed, declaring that he will have two questions for God: why relativity, and why turbulence. Heisenberg says, “I really think He may have an answer to the first question.”
Fortunately, a smooth-flowing fluid does not act as though it has a nearly infinite number of independent molecules, each capable of independent motion.
Engineers have workable techniques for calculating flow, as long as it remains calm. They use a body of knowledge dating back to the nineteenth century, when understanding the motions of liquids and gases was a problem on the front lines of physics.
A practical interest in turbulence has always been in the foreground, and the practical interest is usually one-sided: make the turbulence go away. In some applications, turbulence is desirable—inside a jet engine, for example, where efficient burning depends on rapid mixing. But in most, turbulence means disaster.
Researchers must worry about flow in blood vessels and heart valves. They worry about the shape and evolution of explosions. They worry about vortices and eddies, flames and shock waves. 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.
Suppose you have a perfectly smooth pipe, with a perfectly even source of water, perfectly shielded from vibrations—how can such a flow create something random?
When flow is smooth, or laminar, small disturbances die out. But past the onset of turbulence, disturbances grow catastrophically. This onset—this transition—became a critical mystery in science.
Traditionally, knowledge gained has always been special, not universal. Research by trial and error on the wing of a Boeing 707 aircraft contributes nothing to research by trial and error on the wing of an F–16 fighter.
Something shakes a fluid, exciting it. The fluid is viscous—sticky, so that energy drains out of it, and if you stopped shaking, the fluid would naturally come to rest. When you shake it, you add energy at low frequencies, or large wavelengths, and the first thing to notice is that the large wavelengths decompose into small ones.
In the 1930s A. N. Kolmogorov put forward a mathematical description that gave some feeling for how these eddies work. He imagined the whole cascade of energy down through smaller and smaller scales until finally a limit is reached, when the eddies become so tiny that the relatively larger effects of viscosity take over.
The vorticity is localized. Energy actually dissipates only in part of the space. At each scale, as you look closer at a turbulent eddy, new regions of calm come into view. Thus the assumption of homogeneity gives way to the assumption of intermittency.
Closely related, but quite distinct, was the question of what happens when turbulence begins. How does a flow cross the boundary from smooth to turbulent? Before turbulence becomes fully developed, what intermediate stages might exist?
When more energy comes into a system, he conjectured, new frequencies begin one at a time, each incompatible with the last, as if a violin string responds to harder bowing by vibrating with a second, dissonant tone, and then a third, and a fourth, until the sound becomes an incomprehensible cacophony.
In Landau’s view, these unstable new motions simply accumulated, one on top of another, creating rhythms with overlapping speeds and sizes. Conceptually, this orthodox idea of turbulence seemed to fit the facts, and if the theory was mathematically useless—which it was—well, so be it.
Physicists accepted this picture, but no one had any idea how to predict when an increase in energy would create a new frequency, or what the new frequency would be. No one had seen these mysteriously arriving frequencies in an experiment because, in fact, no one had ever tested Landau’s theory for the onset of turbulence.
They need each other, but theorists and experimenters have allowed certain inequities to enter their relationships since the ancient days when every scientist was both. Though the best experimenters still have some of the theorist in them, the converse does not hold.
In condensed matter physics, the machinery was simpler. The gap between theorist and experimenter remained narrower. Theorists expressed a little less snobbery, experimenters a little less defensiveness.
“That’s true if you have an infinite amount of noise-free data.” And wheel dismissively back toward the blackboard, adding, “In reality, of course, you have a limited amount of noisy data.”
He designed an apparatus to measure how well carbon dioxide conducted heat around the critical point where it turned from vapor to liquid. Most people thought that the thermal conductivity would change slightly. Swinney found that it changed by a factor of 1,000. That was exciting—alone in a tiny room, discovering something that no one else knew.
Like so much of chaos itself, phase transitions involve a kind of macroscopic behavior that seems hard to predict by looking at the microscopic details. When a solid is heated, its molecules vibrate with the added energy. They push outward against their bonds and force the substance to expand. The more heat, the more expansion. Yet at a certain temperature and pressure, the change becomes sudden and discontinuous.
The average atomic energy has barely changed, but the material—now a liquid, or a magnet, or a superconductor—has entered a new realm.
The march of phase transition research had proceeded along stepping stones of analogy: a nonmagnet-magnet phase transition proved to be like a liquid-vapor phase transition. The fluid-superfluid phase transition proved to be like the conductor-superconductor phase transition. The mathematics of one experiment applied to many other experiments.
Typically, the inner cylinder spins inside a stationary shell, as a matter of convenience. As the rotation begins and picks up speed, the first instability occurs: the liquid forms an elegant pattern resembling a stack of inner tubes at a service station. Doughnut-shaped bands appear around the cylinder, stacked one atop another. A speck in the fluid rotates not just east to west but also up and in and down and out around the doughnuts.
As the rate of spin is increased, the structure grows more complex. First the water forms a characteristic pattern of flow resembling stacked doughnuts. Then the doughnuts begin to ripple. The physicists used a laser to measure the water’s changing velocity as each new instability appeared.
The two had in mind a legitimate scientific task that would have brought them a standard bit of recognition for their work and would then have been forgotten. Swinney and Gollub intended to confirm Landau’s idea for the onset of turbulence.
