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recorded the time. Meanwhile Shaw had his three arbitrary equations up and running on the analog computer, producing a stream of imaginary data. One day he performed some show-and–tell for the faculty—a “pseudocolloquium,” as Crutch-field said, because graduate students were not permitted to give formal colloquiums. Shaw played a tape of a faucet making its drumbeat on a piece of tin. And he had his computer going click-click–click in a crisp syncopation, revealing patterns to the ear. He had solved the problem simultaneously from front and back, and his listeners could hear the deep structure in this seemingly disorderly system. But to go further, the group needed a way of taking raw data from any experiment and working backward to equations and strange attractors that characterized chaos.

With a more complicated system, one could imagine plotting one variable against another, relating changes in temperature or velocity to the passage of time. But the dripping faucet provided only a series of times. So Shaw tried a technique that may have been the Santa Cruz group’s cleverest and most enduring practical contribution to the progress of chaos. It was a method of reconstructing a phase space for an unseen strange attractor, and it could be applied to any series of data at all. For the dripping faucet data, Shaw made a two-dimensional graph in which the x axis represented a time interval between a pair of drops and the y axis represented the next time interval. If 150 milliseconds passed between drop one and drop two, and then 150 milliseconds passed between drop two and drop three, he would plot a point at the position 150–150.

That was all there was to it. If the dripping was regular, as it tended to be when the water flowed slowly and the system was in its “water clock regime,” the graph would be suitably dull. Every point would land at the same place. The graph would be a single dot. Or almost. Actually, the first difference between the computer dripping faucet and the real dripping faucet was that the real version was subject to noise, and exceedingly sensitive. “It turns out that the thing is an excellent seismometer,” Shaw said ironically, “very efficient in bringing noise up from the little-league scales to the big-league scales.” Shaw ended up doing most of his work at night, when foot traffic in the physics corridors was lightest. Noise meant that, instead of the single dot predicted by theory, he would see a slightly fuzzy blob.

As the flow rate was increased, the system would go through a period-doubling bifurcation. Drops would fall in pairs. One interval might be 150 milliseconds, and the next might be 80. So the graph would show two fuzzy blobs, one centered at 150–80 and the other at 80–150. The real test came when the pattern became chaotic. If it were truly random, points would be scattered all over the graph. There would be no relation to be found between one interval and the next. But if a strange attractor were hidden in the data, it might reveal itself as a coalescence of fuzziness into distinguishable structures.

Often three dimensions were necessary to see the structure, but that was no problem. The technique could easily be generalized to higher-dimensional graph-making. Instead of plotting interval n against interval n +1, one could plot interval n against interval n + 1 against interval n + 2. It was a trick—a gimmick. Ordinarily a three-dimensional graph required knowledge of three independent variables in a system. The trick gave three variables for the price of one. It reflected the faith of these scientists that order was so deeply ingrained in apparent disorder that it would find a way of expressing itself even to experimenters who did not know which physical variables to measure or who were not able to measure such variables directly. As Farmer said, “When you think about a variable, the evolution of it must be influenced by whatever other variables it’s interacting with. Their values must somehow be contained in the history of that thing. Somehow their mark must be there.” In the case of