Editor's note: This article originally appeared in the August 1985 issue of Scientific American under the title "Computer Recreations: A computer microscope zooms in for a look at the most complex object in mathematics." The article helped make famous the Mandelbrot set, named for mathematician Benoit Mandelbrot; it is being posted now following the October 14 death of Mandelbrot at age 85.
The Mandelbrot set broods in silent complexity at the center of a vast two-dimensional sheet of numbers called the complex plane. When a certain operation is applied repeatedly to the numbers, the ones outside the set flee to infinity. The numbers inside remain to drift or dance about. Close to the boundary minutely choreographed wanderings mark the onset of the instability. Here is an infinite regress of detail that astonishes us with its variety, its complexity and its strange beauty.
The set is named for Benoit B. Mandelbrot, a research fellow at the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y. From his work with geometric forms Mandelbrot has developed the field he calls fractal geometry, the mathematical study of forms having a fractional dimension. In particular the boundary of the Mandelbrot set is a fractal, but it is also much more.
With the aid of a relatively simple program a computer can be converted into a kind of microscope for viewing the boundary of the Mandelbrot set. In principle one can zoom in for a closer look at any part of the set at any magnification. From a distant vantage the set resembles a squat, wart-covered figure eight lying on its side. The inside of the figure is ominously black. Surrounding it is a halo colored electric white, which gives way to deep blues and blacks in the outer reaches of the plane.
Approaching the Mandelbrot set, one finds that each wart is a tiny figure shaped much like the parent set. Zooming in for a close look at one of the tiny figures, however, opens up an entirely different pattern: a riot of organic-looking tendrils and curlicues sweeps out in whorls and rows. Magnifying a curlicue reveals yet another scene: it is made up of pairs of whorls joined by bridges of filigree. A magnified bridge turns out to have two curlicues sprouting from its center. In the center of this center, so to speak, is a four-way bridge with four more curlicues, and in the center of these curlicues another version of the Mandelbrot set is found.
The magnified version is not quite the same Mandelbrot set. As the zoom continues, such objects seem to reappear, but a closer look always turns up differences. Things go on this way forever, infinitely various and frighteningly lovely.
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