# Benoît Mandelbrot

**Benoît B. Mandelbrot** (born November 20, 1924) is a Polish-born French mathematician and leading proponent of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University and IBM Fellow Emeritus at the Thomas J. Watson Research Center.

## Contents

## Early years

Born in Warsaw, Mandelbrot lived in France from the age of 12 to the end of his college studies. He was born into a family with a strong academic tradition - his mother was a medical doctor and his uncle, Szolem Mandelbrojt, was a famous Parisian mathematician. His father, however, made his living trading clothing. His family left Poland for Paris in 1936. There, Benoît was introduced to mathematics by his two uncles.

Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle. In 1944 he returned to Paris to attend the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. He graduated from the École Polytechnique in 1947, and spent two years at the California Institute of Technology where he studied aeronautics. Back in France, he studied for a Ph.D. in Mathematical Sciences at the University of Paris, graduating in 1952.

From 1949 to 1957 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva.

In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York. He remained at IBM for the rest of his working life, becoming an IBM Fellow, and later Fellow Emeritus.

## Later years

From 1955 onwards Mandelbrot worked on problems and published papers in fields as diverse as information theory, economics and fluid dynamics. He became convinced that a common theme of self-similar structures ran through all of these real-world problems. In 1975 Mandelbrot coined the term *fractal* to describe these structures, and published his ideas in *Les objets fractals, forme, hasard et dimension* (translated into English as *Fractals: form, chance and dimension*) in 1977.

In 1979, while on secondment as Visiting Professor of Mathematics at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets of the formula *z*^{2} - μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he discovered the Mandelbrot set fractal that is now named after him (note that the Mandelbrot set is now usually defined in terms of the formula *z*^{2} + *c*, so Mandelbrot's early plots in terms of his parameter μ are left-right mirror images of more recent plots in terms of the parameter *c*) .

In 1982 Mandelbrot published an expanded and updated version of his ideas in *The Fractal Geometry of Nature*. This influential work brought fractals into the mainstream of both professional and popular mathematics.

On his retirement from IBM in 1987, Mandelbrot joined the faculty of Yale as Sterling Professor of Mathematical Sciences. Mandelbrot was awarded the prestigious Japan Prize in 2003. The asteroid 27500 Mandelbrot was named in his honour.

## Mandelbrot and fractals

Although Mandelbrot invented the word *fractal*, many of the objects featured in *The Fractal Geometry of Nature* had been previously described by other mathematicians (the Mandelbrot set being a notable exception). However, they had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and highlighted their common properties, such as self-similarity (sometimes partial or statistical), scale invariance and (usually) non-integer Hausdorff dimension.

He also emphasized the use of fractals as realistic and useful models of many natural phenomena, including the shape of coastlines and river basins; the structure of plants, blood vessels and lungs; the clustering of galaxies; Brownian motion; and stock market prices. Far from being unnatural, Mandelbrot held the view that fractals were, in many ways, more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry. As he says in the Introduction to *The Fractal Geometry of Nature*:

*Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.*

Mandelbrot's informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made *The Fractal Geometry of Nature* accessible to non-specialists. It sparked a widespread popular interest in fractals as well as contributing to new fields of science such as chaos theory.

## Pronunciation

In English, "Mandelbrot" is pronounced [ˈmændəlbrɒt] (rhymes with *cot*). In Yiddish and German it is pronounced as mand'l*brot* and the name means "almond bread".

## See also

## External links

- Mandelbrot's page at Yale
- Interview of the École Polytechnique site
- Biography of Mandelbrot
- Video stream of Mandelbrot lecturing at MIT

## Suggested reading

*The Fractal Geometry of Nature*, by Benoît Mandelbrot; W H Freeman & Co, 1982; ISBN 0716711869*The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward*, by Benoit Mandelbrot and Richard L. Hudson; Basic Books, 2004; ISBN 0-465-04355-0

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