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By Root 789 0

to infinity—all the real numbers, that is—lie on a line stretching from the far west to the far east, with zero at the center. But this line is only the equator of a world that also stretches to infinity in the north and the south. Each number is composed of two parts, a real part, corresponding to east-west longitude, and an imaginary part, corresponding to north-south latitude. By convention, these complex numbers are written this way: 2 + 3i, the i signifying the imaginary part. The two parts give each number a unique address in this two-dimensional plane. The original line of real numbers, then, is just a special case, the set of numbers whose imaginary part equals zero. In the complex plane, to look only at the real numbers—only at points on the equator—would be to limit one’s vision to occasional intersections of shapes that might reveal other secrets when viewed in two dimensions. So Barnsley suspected.

The names real and imaginary originated when ordinary numbers did seem more real than this new hybrid, but by now the names were recognized as quite arbitrary, both sorts of numbers being just as real and just as imaginary as any other sort. Historically, imaginary numbers were invented to fill the conceptual vacuum produced by the question: What is the square root of a negative number? By convention, the square root of –1 is i, the square root of –4 is 2i, and so on. It was only a short step to the realization that combinations of real and imaginary numbers allowed new kinds of calculations with polynomial equations. Complex numbers can be added, multiplied, averaged, factored, integrated. Just about any calculation on real numbers can be tried on complex numbers as well. Barnsley, when he began translating Feigenbaum functions into the complex plane, saw outlines emerging of a fantastical family of shapes, seemingly related to the dynamical ideas intriguing experimental physicists, but also startling as mathematical constructs.

These cycles do not appear out of thin air after all, he realized. They fall into the real line off the complex plane, where, if you look, there is a constellation of cycles, of all orders. There always was a two-cycle, a three-cycle, a four-cycle, floating just out of sight until they arrived on the real line. Barnsley hurried back from Corsica to his office at the Georgia Institute of Technology and produced a paper. He shipped it off to Communications in Mathematical Physics for publication. The editor, as it happened, was David Ruelle, and Ruelle had some bad news. Barnsley had unwittingly rediscovered a buried fifty-year–old piece of work by a French mathematician. “Ruelle shunted it back to me like a hot potato and said, ‘Michael, you’re talking about Julia sets,’” Barnsley recalled.

Ruelle added one piece of advice: “Get in touch with Mandelbrot.”

JOHN HUBBARD, AN AMERICAN MATHEMATICIAN with a taste for fashionable bold shirts, had been teaching elementary calculus to first-year university students in Orsay, France, three years before. Among the standard topics that he covered was Newton’s method, the classic scheme for solving equations by making successively better approximations. Hubbard was a little bored with standard topics, however, and for once he decided to teach Newton’s method in a way that would force his students to think.

Newton’s method is old, and it was already old when Newton invented it. The ancient Greeks used a version of it to find square roots. The method begins with a guess. The guess leads to a better guess, and the process of iteration zooms in on an answer like a dynamical system seeking its steady state. The process is fast, the number of accurate decimal digits generally doubling with each step. Nowadays, of course, square roots succumb to more analytic methods, as do all roots of degree-two polynomial equations—those in which variables are raised only to the second power. But Newton’s method works for higher-degree polynomial equations that cannot be solved directly. The method also works beautifully in a variety of computer algorithms, iteration being,