Mandelbrot , Benoit

(1924–) Polish-born American mathematician

The son of a Lithuanian Jewish merchant, Mandelbrot was born in the Polish capital Warsaw but moved with his parents to Paris in 1936. In 1939 they found it necessary to flee once more and lived in Tulle in southern France for the duration of World War II. Despite an interrupted and irregular education, Mandelbrot gained acceptance at the École Polytechnique after the war even though, he later claimed, he had never learned the alphabet, nor progressed beyond the five-times table. He gained his PhD from the University of Paris in 1952 and spent several years in short-term appointments at the Institute of Advanced Studies, Princeton, and at the University of Geneva and Lille University. In 1958 he moved to the IBM Research Center, Yorktown Heights, New York, where he remained until 1987, when he was appointed professor of mathematics at Yale.

Mandelbrot studied a number of such seemingly unrelated topics as fluctuations in commodity prices, noise in telephone lines, and linguistics. He also considered the seemingly innocent question, “How long is the coast of Britain?” Encyclopedias gave lengths differing by as much as 20%. Mandelbrot pointed out that it depended on how the measurement was done. From a distant space craft, many inlets would reveal their own inlets. Mandelbrot dealt with this and other matters in his The Fractal Geometry of Nature (1982). “Clouds are not spheres,” he declared, “mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” To understand this structured irregularity of nature Mandelbrot introduced the term ‘fractal’ based on the idea of fractional dimension.

An example of a fractal is the snowflake curve first described by Helge von Koch in 1904. It begins with an equilateral triangle. Each side is divided into three equal parts and the middle section is used as the base of a smaller equilateral triangle, resulting in a six-pointed star. The process can be continued indefinitely and has an infinite perimeter bounding a finite area. It is fractal in the sense that it is self-similar, and also in the sense that it has fractional dimension.

Mandelbrot saw that shrinking the unit that a side is measured in by a factor of P, increases the number of units along that side by a factor of Q. In the case of the Koch curve, shrinking the side by a factor of 3 increases the units by a factor of 4. The fractal dimension A can be defined as:

A = logQ/logP = log4/log3 = 1.2618

Mandelbrot went on to determine the fractal dimensions of other similar objects.

Mandelbrot is equally well known for his discovery of theMandelbrot set. The set is constructed from the simple mapping z-> z² + c, where z and c are complex numbers, with z arbitrarily chosen and c fixed. If a fixed value is assigned to c and z = 0, the answer is calculated and fed back into the mapping as a new value for z. The process is repeated, substituting each new output for z. Some values for c when plugged back into the mapping rapidly approach infinity; other values remain within a certain boundary. For example, when c = 1 + 0i, the sequence begins 0, 1, 2, 5, 26, 677, 458, 330… and is unbounded. But when c = –1 + 0i, the sequence is 0, –1, 0, –1, 0, –1… and is clearly bounded.

The set is constructed by marking a black dot on the complex plane for those points c where the sequence is unbounded, and leaving all other values white. The result, best displayed in color on a computer screen, takes on the distinctive shape described as a warty figure of eight on its side. Yet at higher magnifications borders reveal endless detail and startling images, apparently copies of the original but also displaying small differences.