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Benoit Mandelbrot and Fractals

Abstract

Benoît Mandelbrot (1924–2010) spent 35 years at IBM Research studying irregular things that classical mathematics ignored: cotton prices, coastlines, turbulence. In 1975 he named such self-similar shapes fractals, and in 1980 he printed the first picture of the set that now carries his name. Coloured by mathematicians in Bremen, the “Apfelmännchen” became the emblem of chaos theory and a favourite program for home-computer owners, while fractal algorithms generated landscapes for film and, for a time, compressed images.

A Maverick at IBM

Mandelbrot was born in Warsaw on 20 November 1924; his uncle was the mathematician Szolem Mandelbrojt. The family moved to Paris in 1936 and survived the war and the German occupation in the French provinces. From 1944 he studied in Lyon and Paris and aeronautics at Caltech in Pasadena, and took his doctorate at the University of Paris in 1952 under Paul Lévy. After posts in France, the United States and Switzerland he joined IBM’s research centre in 1958, stayed 35 years and became an IBM Fellow (see IBM: The Company).

IBM gave him freedom and computers. In 1963 he showed that cotton prices did not follow the bell curve of classical economics but heavy-tailed Lévy distributions, a theme he returned to in The (Mis)behavior of Markets (2004). In 1967 he asked in Science “How Long Is the Coast of Britain?”: the answer depends on the length of the ruler, because a coastline shows new detail at every scale. In 1975 he coined the word fractal in Les objets fractals, and in 1982 The Fractal Geometry of Nature argued that clouds, mountains, trees and blood vessels were better described by fractals than by smooth curves.

The Set

The set grows out of a simple iteration in the plane of complex numbers. Start with z = 0 and a point c, and repeat z → z² + c. For some values of c the numbers stay bounded; for others they run off to infinity. The points that stay bounded form the Mandelbrot set. The French mathematician Gaston Julia (1893–1978) had studied related iterations, the Julia sets, early in the twentieth century, but without computers nobody could see what the sets looked like.

Mandelbrot saw it as a visiting professor at Harvard. On 27 March 1980, at night, he printed the first image on an electrostatic printer at the university’s computing centre: a blurry figure with a round head and round limbs, standing upright, which he described as discs, sprouts and islands. He published it in a paper at the end of 1980.

The Apfelmännchen

The set became famous in Germany first. At the University of Bremen, where a research group on dynamical systems had existed since 1977, the mathematician Heinz-Otto Peitgen and the physicist Peter Richter founded a computer-graphics laboratory in 1983. Peitgen devised the colouring that made the images popular: points outside the set are coloured by how many iterations they take to escape, which produces bands of colour, while the set itself is left black. His wife Karin gave the shape its German name, Apfelmännchen, “little apple man.” GEO put one on its cover in June 1984, the Goethe-Institut sent an exhibition, Schönheit im Chaos (The Beauty of Chaos), designed by the computer artist Michael Weisser, from Bremen through West Germany, America and Japan from 1985, and in August 1985 the set appeared on the cover of Scientific American with the Bremen images inside. Peitgen and Richter’s The Beauty of Fractals (1986) became a scientific bestseller.

Between about 1985 and 1995 the set conquered home computers (see Computer Art and Generative Art). Programs to draw it circulated for every machine, and a single picture could take hours; fractals, chaos theory and “self-similarity” became part of popular culture. Not everyone was charmed: in 1993 Der Spiegel warned that the science of chaos was a breeding ground for “fairy-tale false expectations and pseudo-religious speculation.”

Fractals at Work

In 1980 Loren Carpenter showed Vol Libre at SIGGRAPH, a flight over fractally generated mountains, and in 1982 his algorithms produced the planet-forming “Genesis Effect” in Star Trek II, the first entirely computer-generated sequence in a major film (see Pixar and Computer Animation). Fractal terrain became standard in film and games.

Fractal image compression was a shorter-lived application. Michael Barnsley began research on it at Georgia Tech in 1985 with DARPA support and in 1987 founded Iterated Systems with Alan Sloan; in 1992 his student Arnaud Jacquin implemented the first automatic encoder, and Microsoft licensed the technique for its Encarta encyclopedia. Decoding was fast, but encoding required an expensive search for self-similar regions, and JPEG and wavelet methods won.

Late Recognition

Mandelbrot received the Wolf Prize in Physics in 1993 and the Japan Prize in 2003, and at 75 became Sterling Professor of Mathematical Sciences at Yale. He died of pancreatic cancer in Cambridge, Massachusetts, on 14 October 2010; his memoir, The Fractalist, appeared in 2012. In 1998 he recorded his life story for Web of Stories.

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