Chaos - James Gleick [83]
It was the same number. Incredibly, this trigonometric function was not just displaying a consistent, geometric regularity. It was displaying a regularity that was numerically identical to that of a much simpler function. No mathematical or physical theory existed to explain why two equations so different in form and meaning should lead to the same result.
Feigenbaum called Paul Stein. Stein was not prepared to believe the coincidence on such scanty evidence. The precision was low, after all. Nevertheless, Feigenbaum also called his parents in New Jersey to tell them he had stumbled across something profound. He told his mother it was going to make him famous. Then he started trying other functions, anything he could think of that went through a sequence of bifurcations on the way to disorder. Every one produced the same number.
Feigenbaum had played with numbers all his life. When he was a teen-ager he knew how to calculate logarithms and sines that most people would look up in tables. But he had never learned to use any computer bigger than his hand calculator—and in this he was typical of physicists and mathematicians, who tended to disdain the mechanistic thinking that computer work implied. Now, though, it was time. He asked a colleague to teach him Fortran, and, by the end of the day, for a variety of functions, he had calculated his constant to five decimal places, 4.66920. That night he read about double precision in the manual, and the next day he got as far as 4.6692016090—enough precision to convince Stein. Feigenbaum wasn’t quite sure he had convinced himself, though. He had set out to look for regularity—that was what understanding mathematics meant—but he had also set out knowing that particular kinds of equations, just like particular physical systems, behave in special, characteristic ways. These equations were simple, after all. Feigenbaum understood the quadratic equation, he understood the sine equation—the mathematics was trivial. Yet something in the heart of these very different equations, repeating over and over again, created a single number. He had stumbled upon something: perhaps just a curiosity; perhaps a new law of nature.
Imagine that a prehistoric zoologist decides that some things are heavier than other things—they have some abstract quality he calls weight—and he wants to investigate this idea scientifically. He has never actually measured weight, but he thinks he has some understanding of the idea. He looks at big snakes and little snakes, big bears and little bears, and he guesses that the weight of these animals might have some relationship to their size. He builds a scale and starts weighing snakes. To his astonishment, every snake weighs the same. To his consternation, every bear weighs the same, too. And to his further amazement, bears weigh the same as snakes. They all weigh 4.6692016090. Clearly weight is not what he supposed. The whole concept requires rethinking.
Rolling streams, swinging pendulums, electronic oscillators—many physical systems went through a transition on the way to chaos, and those transitions had remained too complicated for analysis. These were all systems whose mechanics seemed perfectly well understood. Physicists knew all the right equations; yet moving from the equations to an understanding of global, long-term behavior