More (#3) from Chaos:

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 showed complicated underlying organization that could never have been seen by earlier mathematicians, able only to calculate a point here and a point there. While some starting guesses converged quickly to a root, others bounced around seemingly at random before finally converging to a solution. Sometimes it seemed that a point could fall into a cycle that would repeat itself forever—a periodic cycle—without ever reaching one of the three solutions.

As Hubbard pushed his computer to explore the space in finer and finer detail, he and his students were bewildered by the picture that began to emerge. Instead of a neat ridge between the blue and red valleys, for example, he saw blotches of green, strung together like jewels. It was as if a marble, caught between the conflicting tugs of two nearby valleys, would end up in the third and most distant valley instead. A boundary between two colors never quite forms. On even closer inspection, the line between a green blotch and the blue valley proved to have patches of red. And so on—the boundary finally revealed to Hubbard a peculiar property that would seem bewildering even to someone familiar with Mandelbrot’s monstrous fractals: no point serves as a boundary between just two colors. Wherever two colors try to come together, the third always inserts itself, with a series of new, self-similar intrusions. Impossibly, every boundary point borders a region of each of the three colors.

And:

For... Peitgen the study of complexity provided a chance to create new traditions in science instead of just solving problems. “In a brand new area like this one, you can start thinking today and if you are a good scientist you might be able to come up with interesting solutions in a few days or a week or a month,” Peitgen said. The subject is unstructured.

“In a structured subject, it is known what is known, what is unknown, what people have already tried and doesn’t lead anywhere. There you have to work on a problem which is known to be a problem, otherwise you get lost. But a problem which is known to be a problem must be hard, otherwise it would already have been solved.”

Peitgen shared little of the mathematicians’ unease with the use of computers to conduct experiments. Granted, every result must eventually be made rigorous by the standard methods of proof, or it would not be mathematics. To see an image on a graphics screen does not guarantee its existence in the language of theorem and proof. But the very availability of that image was enough to change the evolution of mathematics. Computer exploration was giving mathematicians the freedom to take a more natural path, Peitgen believed. Temporarily, for the moment, a mathematician could suspend the requirement of rigorous proof. He could go wherever experiments might lead him, just as a physicist could. The numerical power of computation and the visual cues to intuition would suggest promising avenues and spare the mathematician blind alleys. Then, new paths having been found and new objects isolated, a mathematician could return to standard proofs. “Rigor is the strength of mathematics,” Peitgen said. “That we can continue a line of thought which is absolutely guaranteed — mathematicians never want to give that up. But you can look at situations that can be understood partially now and with rigor perhaps in future generations. Rigor, yes, but not to the extent that I drop something just because I can’t do it now.”