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to the physical origins of the system and concentrate only on the geometrical essence he wanted to explore. Where Lorenz and others had stuck to differential equations—flows, with continuous changes in space and time—he turned to difference equations, discrete in time. The key, he believed, was the repeated stretching and folding of phase space in the manner of a pastry chef who rolls the dough, folds it, rolls it out again, folds it, creating a structure that will eventually be a sheaf of thin layers. Hénon drew a flat oval on a piece of paper. To stretch it, he picked a short numerical function that would move any point in the oval to a new point in a shape that was stretched upward in the center, an arch. This was a mapping—point by point, the entire oval was “mapped” onto the arch. Then he chose a second mapping, this time a contraction that would shrink the arch inward to make it narrower. And then a third mapping turned the narrow arch on its side, so that it would line up neatly with the original oval. The three mappings could be combined into a single function for purposes of calculation.

In spirit he was following Smale’s horseshoe idea. Numerically, the whole process was so simple that it could easily be tracked on a calculator. Any point has an x coordinate and a y coordinate to fix its horizontal and vertical position. To find the new x, the rule was to take the old y, add 1 and subtract 1.4 times the old x squared. To find the new y, multiply 0.3 by the old x. That is: xnew = y +1 – 1.4x2 and ynew = 0.3x. Hénon picked a starting point more or less at random, took his calculator and started plotting new points, one after another, until he had plotted thousands. Then he used a real computer, an IBM 7040, and quickly plotted five million. Anyone with a personal computer and a graphics display could easily do the same.

At first the points appear to jump randomly around the screen. The effect is that of a Poincaré section of a three-dimensional attractor, weaving erratically back and forth across the display. But quickly a shape begins to emerge, an outline curved like a banana. The longer the program runs, the more detail appears. Parts of the outline seem to have some thickness, but then the thickness resolves itself into two distinct lines, then the two into four, one pair close together and one pair farther apart. On greater magnification, each of the four lines turns out to be composed of two more lines—and so on, ad infinitum. Like Lorenz’s attractor, Hénon’s displays infinite regress, like an unending sequence of Russian dolls one inside the other.

The nested detail, lines within lines, can be seen in final form in a series of pictures with progressively greater magnification. But the eerie effect of the strange attractor can be appreciated another way when the shape emerges in time, point by point. It appears like a ghost out of the mist. New points scatter so randomly across the screen that it seems incredible that any structure is there, let alone a structure so intricate and fine. Any two consecutive points are arbitrarily far apart, just like any two points initially nearby in a turbulent flow. Given any number of points, it is impossible to guess where the next will appear—except, of course, that it will be somewhere on the attractor.

The points wander so randomly, the pattern appears so ethereally, that it is hard to remember that the shape is an attractor. It is not just any trajectory of a dynamical system. It is the trajectory toward which all other trajectories converge. That is why the choice of starting conditions does not matter. As long as the starting point lies somewhere near the attractor, the next few points will converge to the attractor with great rapidity.

YEARS BEFORE, WHEN DAVID RUELLE arrived at the City College laboratory of Gollub and Swinney in 1974, the three physicists found themselves with a slender link between theory and experiment. One piece of mathematics, philosophically bold but technically uncertain. One cylinder of turbulent fluid, not much to look at, but clearly