Benoit B. Mandelbrot

Polish-born French-American mathematician

• Born: November 20, 1924
• Birthplace: Warsaw, Poland
• Died: October 14, 2010
• Place of death: Cambridge, Massachusetts

Mandelbrot’s research in applied mathematics led him to discover a pattern appearing across the disciplines: the duplication of a pattern or shape at any scale (scale invariance), which he called “self-similarity.” He then identified the mathematics underlying that pattern and gave it the name fractal, thereby creating a new branch of research in pure and applied mathematics.

Early Life

Benoit B. Mandelbrot (behn-WAH MAHN-dehl-broht) was born in Poland shortly after that country regained its independence after more than a century of Russian, Prussian, and Austro-Hungarian rule. His father was in the clothing business but descended from a line of Jewish scholars, while his mother had medical and dental training. The name Mandelbrot was distinctively Jewish, if only because of the popularity of the pastry bearing that name in the Jewish communities of Eastern Europe (Mandelbrot means “almond bread”). Mandelbrot attributed some of his later intellectual independence to having been schooled at home for his first couple of years, after which he attended elementary school in Warsaw. When he was eleven years old, his family moved to France, in part because of the increasingly anti-Semitic atmosphere in Poland and because of his uncle’s established position in the French mathematical community.

After the Nazi conquest of France in 1940, life became uncertain for the Jewish population of the Nazi-occupied and the Vichy parts of the country. Mandelbrot wandered between 1942 and 1944, taking up various jobs and keeping out of sight. After the end of the Nazi regime, he was able to return to Paris for college, and he started off at the École Normale, the leading university for mathematics in the country. He left after a few days and went to the École Polytechnique, which did not have quite the standing of the École Normale. His reason for making the change was that he found the mathematics at the École Normale too abstract, while the mathematics at the École Polytechnique was more applied.

Abstraction in mathematics was part of a general development in the field, spearheaded by a contingent of French mathematicians known collectively under the pseudonym Nicolas Bourbaki. They pushed French mathematics in the direction of abstraction, claiming that German mathematics had taken a lead in research in this area. The change was anathema to the young Mandelbrot, who found that his imagination and predilection for geometry was much stronger than his fondness for axiomatic deductions. After getting a degree from the École Polytechnique, he proceeded to the California Institute of Technology (Caltech) in Pasadena, California, and earned a master’s degree in 1948. However, because the person with whom he wanted to work was no longer at Caltech, he returned to Paris for his doctoral study.

The combination of topics in Mandelbrot’s dissertation (from word frequencies to statistical thermodynamics) was indicative of his reluctance to confine his research within too narrow a discipline. After a number of positions of short duration, he took a position as a member of the research staff at International Business Machines in the United States in 1958 and stayed with that company, with only a short break, until 1993. He married Aliette Kagan in 1955, and the couple had two children.

Life’s Work

One of the problems Mandelbrot addressed early in his career was the daily and long-term fluctuation in stock prices. Economists tried to predict the motion of stock prices, but price fluctuations always seemed to get in the way of these predictions. One solution to the problem was to see how much time had to be taken into account before the effects of the fluctuations could be disregarded as insignificant. What Mandelbrot discovered was that a pattern of fluctuations remained, regardless of the time period he chose to examine. This indicated that fluctuations could not be ignored. Indeed, any model attempting to predict the motion of prices would have to take into account fluctuations in stock prices.

Mandelbrot also considered issues connected with what is called statistical noise in the transmission of information. The problem is that the transmission of a signal (information) is always accompanied by some distortion, and the receiver has to distinguish the signal from the distortion. The receiver needs to have a guarantee that there exists enough signal to eliminate the distortion. Mandelbrot discovered that, just as with stock prices, there is no way to be sure of telling distortion from signal, no matter how large a sample is taken. This duplication of a pattern at any scale (scale invariance) he called “self-similarity,” and its appearance in widely different areas of study suggested that it was worth considering in its own right.

Mathematical constructions with the property of self-similarity existed prior to Mandelbrot’s time. These constructions included, for example, the Cantor set. Named for nineteenth century German mathematicianGeorg Cantor, the Cantor set involved taking the set of points on the number line between 0 and 1 and deleting the middle third (leaving two disconnected pieces). The middle third of each of those pieces was then deleted, and the process continued. The result served as an important example for pure mathematics, but before Mandelbrot, no one had suspected that phenomena in the real world could be modeled by such constructions.

Perhaps the best-known example of this sort of self-similarity arose from considering the issue of measuring the length of a coastline. Mandelbrot himself devoted an article to the coastline of Great Britain. If one imagined measuring the coastline with a ruler of length one mile, he wrote, the coastline could be given a certain length. When one used a smaller ruler, however, then more of the little curves and inlets would have to be counted and the coast would end up with a slightly longer length. The smaller the size of the ruler, the longer the coastline. It was impossible to tell from looking at a drawing of the coastline what scale was used to measure it.

During the nineteenth century and into the twentieth century, scientists speculated that the atoms of the physical universe had a structure similar to those of the solar system, with the nucleus corresponding to the sun and the electrons to the planets. With the advent of quantum mechanics, however, the similarity between solar system and atomic structure waned. One of the reasons for the popularity of Mandelbrot’s notion of self-similarity was that it revived the pre-quantum theory picture of identical structure at different scales. In searching for a name to describe the kind of curve that had this property of self-similarity.

It was important to establish that fractals were not just a curiosity, and Mandelbrot proceeded to do so through lectures and writings. In 1973 he lectured at the Collège de France, tying together his thoughts on the various applications of the idea of a fractal. In particular, he devised a notion of fractal dimension corresponding to the idea of dimension for ordinary geometrical shapes. What was new for mathematics was that the dimension of a fractal could end up as a fraction rather than a whole number. The dimension could be calculated by seeing what happened to the length of a fractal curve when a ruler of different size was used to measure it. The exponent that resulted was the fractal dimension. Mandelbrot wrote up his lecture in an published it as Les Objets fractals (1975; Fractals, 1977). “Fractal” derives from the Latin adjectival root fractus, the same root as in fraction, fracture, and fragment. Mandelbrot claims to have consulted a Latin textbook used by his son to coin the term “fractal.” The beauty of the pictures in Fractals helped guarantee an audience beyond that of professional mathematicians.

Another discipline that was coming into its own during the 1970’s was chaos theory , which dealt with the complicated patterns that arose in the solution of problems in nonlinear dynamics. What made chaos theory appealing was that problems could be expressed in terms of the iteration of simple processes. As with fractals, a variety of simple processes could still lead to the same kind of result. What was even more striking was that the curves that resulted from such iterations looked like fractals. Chaos theory could be used to explain where fractals came from, a problem Mandelbrot was not able to solve. The presence of fractals in chaos theory ensured that Mandelbrot’s ideas would survive. With the success of fractal research programs, Mandelbrot received a chair in mathematics at Yale University. A volume of papers in his honor was published by the American Mathematical Society in 2004.

Significance

Mathematics has its share of objects that lack any obvious connection to the real world. There are also problems in the real world that seem to have solutions only through extensive calculation and are not easily generalized. What Mandelbrot did was to find fractals across many areas of application, and not just within pure mathematics. He considered concrete examples of scaling, which led to his discovery of fractals and their wider use.

The conjunction of fractals research with chaos theory provided an explanation for how these self-similar curves arose and predicted where else they might be applicable. Many without mathematical training, or even interest, have been fascinated by pictures of fractals. A colored fractal can be the most accurate way of representing a repeated process, and it is certainly the most beautiful. Mandelbrot’s work helped bring the geometrical back into the world of abstract mathematics.

Bibliography

Albers, Donald J., and G. L. Alexanderson, eds. Mathematical People. Boston: Birkhäuser, 1985. Includes an interview with Mandelbrot, who discusses and explains the origins of some of his theories.

Gleick, James. Chaos: Making a New Science. New York: Viking, 1987. A popular account of chaos theory, with an important chapter on Mandelbrot.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983. Beautiful illustrations accompany an enumeration of those instances in which fractals appear.

‗‗‗‗‗‗‗. Fractals: Form, Chance, and Dimension. San Francisco, Calif.: W. H. Freeman, 1977. The first English-language general-audience monograph on the history and concepts of fractional dimension.

‗‗‗‗‗‗‗. Fractals and Multifractals: Noise, Turbulence, and Galaxies. New York: Springer, 1990. Mandelbrot’s most comprehensive and rigorous treatment of fractals.

Peitgen, Heinz-Otto, and Peter H. Richter. The Beauty of Fractals. New York: Springer, 1986. Places Mandelbrot in perspective rather than in the foreground, allowing the book’s illustrations to speak for themselves.

Smith, Peter. Explaining Chaos. New York: Cambridge University Press, 1998. An excellent exploration of the philosophical background for fractals.

Stewart, Ian. Does God Play Dice? The New Mathematics of Chaos. Malden, Mass.: Blackwell, 2002. Extensive tribute to Mandelbrot that also explains how the mathematics of chaos developed.

‗‗‗‗‗‗‗, ed. The Colours of Infinity: The Beauty and Power of Fractals. Bath, England: Clear Books, 2004. Perhaps the most straightforward, nontechnical outline and discussion of what Mandelbrot came to call fractals. Highly recommended for all readers at all levels. Includes a stunning DVD of the British television documentary that helped to popularize the subject. DVD introduced by writer Arthur C. Clarke.