Chaos: Making a New Science

by James Gleick

So is contact in machine joints, or electrical contact. Contacts between surfaces have properties quite independent of the materials involved. They are properties that turn out to depend on the fractal quality of the bumps upon bumps upon bumps. One simple but powerful consequence of the fractal geometry of surfaces is that surfaces in contact do not touch everywhere. The bumpiness at all scales prevents that. Even in rock under enormous pressure, at some sufficiently small scale it becomes clear that gaps remain, allowing fluid to flow. To Scholz, it is the Humpty-Dumpty Effect. It is why two pieces of a broken teacup can never be rejoined, even though they appear to fit together at some gross scale. At a smaller scale, irregular bumps are failing to coincide.

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. Categories mislead. The ends of the continuum are of a piece with the middle. 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. And, with some limitations, small storms act like large storms. 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.

Typical human lungs pack in a surface bigger than a tennis court.

Lorenz Albatractor

Koch Curve

THE COMPLEX BOUNDARIES OF NEWTON’S METHOD.

FRACTAL CLUSTERS. A random clustering of particles generated by a computer produces a “percolation network,” one of many visual models inspired by fractal geometry. Applied physicists discovered that such models imitate a variety of real-world processes, such as the formation of polymers and the diffusion of oil through factured rock. Each color in the percolation network represents a grouping that is connected throughout. THE GREAT RED SPOT: REAL AND SIMULATED. The Voyager satellite revealed Jupiter’s surface is a seething, turbulent fluid, with horizontal bands of east-west flow. The Great Red Spot is seen from above the planet’s equator and also in a view looking down on the South Pole. Computer graphics from Phillip Marcus’s simulation present the South Pole view. The color shows the direction of spin for particular pieces of fluid: pieces turning counterclockwise are red, and pieces turning clock-wise are blue. No matter what the starting configuration, clumps of blue tend to break up, while the red tends to merge into a single spot, stable and coherent amid the surrounding tumult.

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. A fractal curve implies an organizing structure that lies hidden among the hideous complication of such shapes. High school students could understand fractals and play with them; they were as primary as the elements of Euclid. Simple computer programs to draw fractal pictures made the rounds of personal computer hobbyists.

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. Every new accelerator, with its increase in energy and speed, extends science’s field of view to tinier particles and briefer time scales, and every extension seems to bring new information. At first blush, the idea of consistency on new scales seems to provide less information. In part, that is because a parallel trend in science has been toward reductionism. Scientists break things apart and look at them one at a time. If they want to examine the interaction of subatomic particles, they put two or three together. There is complication enough. The power of self-similarity, though, begins at much greater levels of complexity. It is a matter of looking at the whole.

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.”