Benoit B. Mandelbrot

(1924–  ; b. Warsaw, Poland) French mathematician and engineer, famed for his work on fractals. Mandelbrot graduated from the École Polytechnique in Paris in 1947 with a diploma in engineering. He obtained his MSc from Caltech in 1948 and his doctorate from the Faculté des Sciences de Paris in 1952. In 1958 he joined the IBM research centre in New York State. From 1987 to 2005 he was on the faculty of yale U. He was made a fellow of the AAAS in 1982. He was made a Chevalier, L'Ordre National de la Légion d'Honneur in 1989.

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Polish-born American mathematician (1924–)

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 = log Q/log P = log 4/log 3 = 1.2618

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

Mandelbrot is equally well known for his discovery of the Mandelbrot 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.

The Polish-born French-American mathematician Benoit B. Mandelbrot (born 1924) was the inventor of fractals. Fractal geometry has been described as one of the major developments of 20th-century mathematics. He called himself "a physicist also, and an economist, and an artist of sorts, and…."

Benoit Mandelbrot was born in Warsaw, Poland, on November 20, 1924. He described his father (1883-1952) as "a very scholarly person, and the descendant of long lines of scholars. In fact, it often seemed everyone in the family was - or was expected to become - a scholar of some kind, at least part-time. Unfortunately, many were starving scholars, and my father - being a practical man - saw virtues in a good steady job." So Mandelbrot manufactured and sold clothing. He helped raise his youngest (by 16 years) brother, Szolem Mandelbrot, who later became a famous mathematician. His mother was a doctor. Afraid of epidemics, she tried to keep him out of school. His uncle Loterman, unemployed, was his tutor, and from him Mandelbrot mastered chess and maps and learned to read very fast. In 1929, when he was five, his uncle Szolem became professor at the University of Clermont-Ferrand in France, and in 1938 at the Collège de France in Paris.

In 1936 Mandelbrot's family moved to Paris, where he attended the lycée, or secondary school. When World War II broke out, he moved south to Tulle, where he attended the lycée in Clermont-Ferrand. As he later recalled, "poverty and the wish to keep away from big cities to maximize the chances of survival made me skip most of what you might call college, so I am essentially self-taught in many ways."

College and Early Career

When Paris was liberated in 1944, Mandelbrot took the entrance exams of both the Ecole Normale Supérieure and Ecole Polytechnique. He started Ecole Normale (ranking first among an entering class of 15) but after a few days transferred to Polytechnique. Here his hopes "were thoroughly romantic: to be the first to find order where everyone else had only seen chaos." In 1947 Mandelbrot graduated from Polytechnique as Ingénieur diplômé. He obtained French and American scholarships to study in the United States.

Mandelbrot went for two years to Caltech, in Pasadena, California, earning the titles of Master of Science and Professional Engineer in Aeronautics in 1949. Back in France, he spent a year with the Air Force, then developed his doctoral thesis at the University of Paris (Facultédes Sciences). In December 1952 he was awarded a Doctorat d'Etat ès Sciences Mathématiques. His thesis title was Games of Communication, due to the influence of mathematicians John von Neumann and Norbert Wiener. From 1949 to 1957 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique, Paris. From 1950 to 1953 he was ingénieur, Group de Télévision en Couleur: LEP, S.A. (Groupe Philips), Paris.

The last man whom Von Neumann sponsored at the Institute for Advanced Study in Princeton was Mandelbrot, who spent a "marvelous year" there in 1953-1954. From 1953 to 1971 he often visited the Massachusetts Institution of Technology in Cambridge as a research associate, then lecturer in electrical engineering, and then Institute Lecturer.

Mandelbrot returned to France, married Aliette in 1955 (they later had two children), and moved to Geneva. From 1955 to 1975 he was chargéde cours de mathématiques and belonged to the seminar of psychologist Jean Piaget at the University of Geneva.

French universities suddenly started expanding and were looking for applied mathematicians. Mandelbrot became maître de conférences d'analyse mathématique at the University of Lille and, at the request of his former mathematics teacher Paul Lévy, at the Ecole Polytechnique in Paris.

Career at IBM

Mandelbrot went to IBM as a faculty visitor in the summer of 1958 and "decided to take the gamble of staying a bit longer." He was a research staff member at IBM Thomas J. Watson Research Center, Yorktown Heights, New York, from 1958 to 1974. From 1974 to 1997 he was an IBM fellow. As Mandelbrot noted, "A few dozen IBM'ers are designated as IBM Fellows…. Thus, it was stated officially that my work had become widely respected, and that I could proceed in my very own way."

As Mandelbrot put it, "My wild gamble started paying off during 1961-1962. By then, there was no question in my mind that I had identified a new phenomenon present in many aspects of nature, but all the examples were peripheral in their fields, and the phenomenon itself eluded definition." He added: "Many years were to go by before I formulated fractal geometry, and became able to say that I had long been concerned with the fractal aspects of nature, with seeking them out and with building theories around them."

In 1961 he established the new phenomenon as central to economics. Next, he established it was central to vital parts of physical science. And finally, he "was back to geometry after years of analytic wilderness."

In 1967 Mandelbrot raised the question, "How long is the coast of Britain?" The usual answer was, "It all depends." But he was able to show the wiggliness of a coastline can be measured using the notion of fractal dimension: this is a number like 1.15 or 1.21 which can be measured quite accurately. A favorite line of Mandelbrot became, he said, "an instant cliché": "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."

As Mandelbrot summed up: "I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit."

In the Proceedings of the Royal Society in 1989 Mandelbrot summarized fractal geometry as a "workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics." It was based on a form of symmetry that had previously been underused. It can be used in art and pure mathematics, being without practical application.

Many Honors and Awards

At Harvard University Mandelbrot was visiting professor of economics and research fellow in psychology (1962-1963), visiting professor of applied mathematics and staff member of the Joint Committee on Biomedical Computer Science (1963-1964), and visiting professor, later professor of the practice of mathematics, Mathematics Department (1979-1980; 1984-1987). Beginning in 1987 Mandelbrot was Abraham Robinson Adjunct Professor of Mathematical Sciences at Yale University.

He was a chevalier of the Légion d'Honneur, France (1989); a fellow of the American Academy of Arts and Sciences (1982); a foreign associate of the U.S. National Academy of Sciences (1987); a member of the European Academy of Arts, Sciences and Humanities (1987); and a member of the IBM Academy of Technology (1989).

He was made a doctor honoris causa of Syracuse University (1986), Laurentian University (1986), Boston University (1987), SUNY at Albany (1988), Universität Bremen, (then West) Germany (1988), Pace University (1988), and University of Guelph (1989).

He was a scholar, Rockefeller Foundation (1953) and a fellow, John Simon Guggenheim Memorial Foundation (1968, resigned). He received the Research Division outstanding innovation award (1983) and corporate award (1984) from IBM; the 1985 Barnard Medal for meritorious service to science, Magna est Veritas, U.S. National Academy of Sciences and Columbia University; the 1986 Franklin Medal for signal and eminent service in science from the Franklin Institute; the 1988 Charles Proteus Steinmetz Medal, IEEE and Union College; the 1988 alumni distinguished service award for outstanding achievement, Caltech; the 1988 senior award (Humboldt Preis), Alexander von Humboldt-Stiftung, Bonn, West Germany; the 1988 "Science for Art" prize, Fondation Moet-Hennessy-Louis Vuitton, Paris; the 1989 Harvey prize for science and technology, Technion-Israel Institute of Technology, Haifa, Israel; and the 1991 Nevada prize, University of Nevada System. He also received the 1993 Wolf Foundation Prize for Physics from the Wolf Foundation of Israel to Promote Science and Art for the Benefit of Mankind. He shared the 1994 Honda Prize with Abraham Robinson Adjunct Professor of Mathematical Sciences at Yale University. Mandelbrot was cited by the Honda Foundation "for contributing to the establishment of a harmony between mathematics and science and culture and the environment that surrounds human activities, and to a better understanding worldwide of science and for new tools to solve the problems induced by modern progress."

He has been visiting professor of engineering and applied science (Yale University, 1970) and visiting professor of physiology (Albert Einstein College of Medicine, Bronx, 1972; SUNY Downstate Medical Center, Brooklyn, 1974). Other institutions where he lectured included the Collége de France (1973, 1974, 1977) and as Hitchcock professor, University of California, Berkeley (1991-1992). He also belonged to the U.S. National Academy of Sciences, the American Academy of Arts and Sciences, and the European Academy. As of the (mid to late) 1990s, he was still an IBM Fellow at IBM T.J. Watson Research Center, Yorktown Heights, New York. Since 1987 he has been the Abraham Robinson Professor of Mathematical Sciences, Yale University, New Haven, Connecticut.

Further Reading

The most important autobiographical piece was "Benoit Mandelbrot, Interview by Anthony Barcellos," in Mathematical People: Profiles and Interviews, Donald J. Albers and G. L. Alexanderson (1985). See also James Gleick, "The Man Who Reshaped Geometry," The New York Times Magazine (December 8, 1985); John Rockwell, "Review/Music. Fractals: A Mystery Lingers," The New York Times (April 26, 1990); and L. R. Shannon, "Peripherals," The New York Times (October 2, 1990).

Mandelbrot was the author of Logique, language et théorie de l'information (with Leo Apostel and Albert Morf; 1957); Les objets fractals: forme, hasard et dimension (1975, 1984, 1989; translated into Hungarian, Italian, and Spanish); Fractals: Form, Chance and Dimension (1977); The Fractal Geometry of Nature (1982; translated into German and Japanese); La geometria della natura (1987, 1989); and Noise and Multifractals, 1963-1976.

In addition to books Mandelbrot published hundreds of research papers and less technical articles. The latter included "Exiles in Pursuit of Beauty," The Scientist (March 23, 1987); "Towards a Second Stage of Indeterminism in Science," Interdisciplinary Science Reviews (1987); "Fractals and the Re-birth of Iteration Theory," in The Beauty of Fractals, Heinz-Otto Peitgen and Peter H. Richter, editors (1986); "Foreword. People and events behind the 'Science of Fractal Images,"' in The Science of Fractal Images, Heinz-Otto Peitgen and Dictmar Saupe, editors (1988); "Fractal geometry: what is it, and what does it do?" Proc. R. Soc. Lond. A 423 (1989); and " Fractals and the rebirth of experimental mathematics" in Fractals for the Classroom, Heinz-Otto Paetgen et al (1991).

See also Fractals in Physics. Essays in Honor of Benoit M. Mandelbrot, Proceedings of the International Conference (Vence, France, 1-4 October, 1989), Amnon Aharong and Jens Feder, editors (1989).