WHOSE SET, NOW? A graphical representation of the Mandelbrot set. Image: WOLFGANG BEYER VIA WIKIMEDIA COMMONS
Editor's note: This article originally appeared in the April 1990 issue of Scientific American, under the title "Mandelbrot Set-To." We are posting it now to coincide with our reporting on a talk this week by Benoit Mandelbrot at Columbia University on fractals and financial markets. The phrasing of some references to dates has been changed, in brackets, for clarity.
Who discovered the Mandelbrot set? This is not a trick question—or a trivial one. The set has been called (in this magazine) "the most complex object in mathematics." That is debatable, yet it is almost certainly the most famous such object. The infinitely intricate computer-generated image of the set serves as an icon for the burgeoning field of chaos theory and has attracted enormous public attention.
The set is named after Benoit B. Mandelbrot, a mathematician at the IBM Thomas J. Watson Research Center. He is best known for coining the term fractal to describe phenomena (such as coastlines, snowflakes, mountains and trees) whose patterns repeat themselves at smaller and smaller scales. Mandelbrot claims that he and he alone discovered the Mandelbrot set—which has fractal properties—about a decade ago. He refers to its image as his "signature."
Three other mathematicians have challenged his claim. Two maintain that they independently discovered and described the set at about the same time as Mandelbrot did. A third asserts that his work on the set not only predated Mandelbrot's efforts but also helped to guide them. These assertions have long circulated in the mathematics community but have only recently surfaced in print.
Mathematicians are not known for priority battles, but Mandelbrot—a self-described "black sheep "—has often bumped heads with colleagues. "Were it not for his personality," remarks Robert L. Devaney of Boston University, who says he admires Mandelbrot's work, "there would be no controversy."
The scientific stakes are also high. Even those who scorn the set's popularity acknowledge its mathematical significance. Dennis P. Sullivan of the City University of New York calls it a "crucible" for testing ideas about the behavior of dynamical (or nonlinear, or complex, or chaotic) systems. "It is really quite fundamental," he says.
Part of the charm of the set is that it springs from such a simple equation: z2 + c. The terms z and c are complex numbers, which consist of an imaginary number (a multiple of the square root of –1) combined with a real number. One begins by assigning a fixed value to c, letting z = 0 and calculating the output. One then repeatedly recalculates, or iterates, the equation, substituting each new output for z. Some values of c, when plugged into this iterative function, produce outputs that swiftly soar toward infinity. Other values of c produce outputs that eternally skitter about within a certain boundary. This latter group of c's, or complex numbers, constitutes the Mandelbrot set.
When plotted on a graph consisting of all complex numbers, the members of the set cluster into a distinctive shape. From afar, it is not much to look at: it has been likened to a tumor-ridden heart, a beetle, a badly burned chicken and a warty figure eight on its side.
A closer look reveals that the borders of the set do not form crisp lines but seem to shimmer like flames. Repeated magnification of the borders plunges one into a bottomless phantasmagoria of baroque imagery. Some forms, such as the basic heartlike shape, keep recurring but always with subtle differences.
Today virtually anyone with a personal computer can "discover" the set. But [in 1979] computers were much less powerful, and few mathematicians associated computers with serious mathematics.
Even Mandelbrot has described his first tentative steps toward the set in 1979 as "mindless fun." He began using a computer to map out Julia sets, which are generated by plugging complex numbers into iterative functions. The sets' peculiar properties had been described as early as 1906 by the French mathematician Pierre Fatou. They were named later for Gaston Julia, who successfully claimed that his work on the sets some dozen years later had greater significance than Fatou's. Mandelbrot, who was born [in 1924] in Poland, had read the work of both men and studied under Julia in the 1940's.