Chaos - James Gleick [103]
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.
Hubbard embarked on a study of these complicated shapes and their implications for mathematics. His work and the work of his colleagues soon became a new line of attack on the problem of dynamical systems. He realized that the mapping of Newton’s method was just one of a whole unexplored family of pictures that reflected the behavior of forces in the real world. Michael Barnsley was looking at other members of the family. Benoit Mandelbrot, as both men soon learned, was discovering the granddaddy of all these shapes.
BOUNDARIES OF INFINITE COMPLEXITY. When a pie is cut into three slices, they meet at a single point, and the boundaries between any two slices are simple. But many processes of abstract mathematics and real-world physics turn out to create boundaries that are almost unimaginably complex.
Above, Newton’s method applied to finding the cube root of –1 divides the plane into three identical regions, one of which is shown in white. All white points are “attracted” to the root lying in the largest white area; all black points are attracted to one of the other two roots. The boundary has the peculiar property that every point on it borders all three regions. And, as the insets show, magnified segments reveal a fractal structure, repeating the basic pattern on smaller and smaller scales.
THE MANDELBROT SET IS the most complex object in mathematics, its admirers like to say. An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God’s personal vine. Examined in color through the adjustable window of a computer screen, the Mandelbrot set seems more fractal than fractals, so rich is its complication across scales. A cataloguing of the different images within it or a numerical description of the set’s outline would require an infinity of information. But here is a paradox: to send a full description of the set over a transmission line requires just a few dozen characters of code. A terse computer program contains enough information to reproduce the entire set. Those who were first to understand the way the set commingles complexity and simplicity were caught unprepared—even Mandelbrot. The Mandelbrot set became a kind of public emblem for chaos, appearing on the glossy covers of conference brochures and engineering quarterlies, forming the centerpiece of an exhibit of computer art that traveled internationally in 1985 and 1986. Its beauty was easy to feel from these pictures; harder to grasp was the meaning it had for the mathematicians