Chaos - James Gleick [102]
One might ask exactly what sort of route Newton’s method traces as it winds toward a root of a degree-two polynomial on the complex plane. One might answer, thinking geometrically, that the method simply seeks out whichever of the two roots is closer to the initial guess. That is what Hubbard told his students at Orsay when the question arose one day.
“Now, for equations of, say, degree three, the situation seems more complicated,” Hubbard said confidently. “I will think of it and tell you next week.”
He still presumed that the hard thing would be to teach his students how to calculate the iteration and that making the initial guess would be easy. But the more he thought about it, the less he knew—about what constituted an intelligent guess or, for that matter, about what Newton’s method really did. The obvious geometric guess would be to divide the plane into three equal pie wedges, with one root inside each wedge, but Hubbard discovered that that would not work. Strange things happened near the boundaries. Furthermore, Hubbard discovered that he was not the first mathematician to stumble on this surprisingly difficult question. Arthur Cayley had tried in 1879 to move from the manageable second-degree case to the frighteningly intractable third-degree case. But Hubbard, a century later, had a tool at hand that Cayley lacked.
Hubbard was the kind of rigorous mathematician who despised guesses, approximations, half-truths based on intuition rather than proof. He was the kind of mathematician who would continue to insist, twenty years after Edward Lorenz’s attractor entered the literature, that no one really knew whether those equations gave rise to a strange attractor. It was unproved conjecture. The familiar double spiral, he said, was not proof but mere evidence, something computers drew.
Now, in spite of himself, Hubbard began using a computer to do what the orthodox techniques had not done. The computer would prove nothing. But at least it might unveil the truth so that a mathematician could know what it was he should try to prove. So Hubbard began to experiment. He treated Newton’s method not as a way of solving problems but as a problem in itself. Hubbard considered the simplest example of a degree-three polynomial, the equation x3– 1 =0. That is, find the cube root of 1. In real numbers, of course, there is just the trivial solution: 1. But the polynomial also has two complex solutions: –½ + i√3/2, and –½ – i√3/2. Plotted in the complex plane, these three roots mark an equilateral triangle, with one point at three o’clock, one at seven o’clock, and one at eleven o’clock. Given any complex number as a starting point, the question was to see which of the three solutions Newton’s method would lead to. It was as if Newton’s method were a dynamical system and the three solutions were three attractors. Or it was as if the complex plane were a smooth surface sloping down toward three deep valleys. A marble starting from anywhere on the plane should roll into one of the valleys—but which?
Hubbard set about sampling the infinitude of points that make up the plane. He had his computer sweep from point to point, calculating the flow of Newton’s method for each one, and color-coding the results. Starting points that led to one solution were all colored blue. Points that led to the second solution were red, and points that led to the third were green. In the crudest approximation, he found, the dynamics of Newton’s method did indeed divide the plane into three pie wedges. Generally the points near a particular solution led quickly into that solution. But systematic computer exploration