# Mandelbrot, Benoit B. 1924-

# MANDELBROT, Benoit B. 1924-

*PERSONAL:* Born November 20, 1924, in Warsaw, Poland; immigrated to France, 1936; immigrated to the United States, 1958; son of Charles (in the clothing business) and Belle (a doctor and dentist; maiden name, Lurie) Mandelbrot; married Aliette Kagan (a biologist), November 5, 1955; children: Laurent, Didier. *Education:* École Polytechnique, Paris, France, diploma, 1947; California Institute of Technology, M.S., 1948; University of Paris, Ph.D., 1952.

*ADDRESSES: Office*—Department of Mathematics, Yale University, P.O. Box 208283, New Haven, CT 06520-8283; International Business Machines, P.O. Box 218, Yorktown Heights, NY 10598-0218.

*CAREER:* Philips Electronics, Paris, France, mathematician, 1950-53; Institute for Advanced Study, Princeton, NJ, junior member and Rockefeller scholar, 1953-54; University of Geneva, Geneva, Switzerland, junior professor of mathematics, 1955-57; University of Lille, Lille, France, junior professor of mathematics, 1957-58; École Polytechnique, Paris, junior professor of mathematics, 1957-58; International Business Machines Co., Thomas J. Watson Research Center, Yorktown Heights, NY, member of research staff, 1958-74, fellow, 1974-93, fellow emeritus, 1993—, member of IBM Academy of Technology, 1989-93. Yale University, Abraham Robinson Professor of Mathematical Sciences, 1987-99, Sterling Professor of Mathematical Sciences, 1999—. Harvard University, visiting professor, 1962-64, 1979-80, professor, 1984-87; Yale University, visiting professor, 1970; Yeshiva University, visiting professor at Albert Einstein College of Medicine, 1970; University of California—Berkeley, Charles M. and Martha Hitchcock Professor, 1992; L'Académie des Sciences, Paris, associate 1995; visiting institute lecturer at Massachusetts Institute of Technology, 1953, University of Paris-Sud, 1966, Collège de France, 1973, Institut des Hautes Études Scientifiques, 1980, Mittag-Leffler Institute, 1984, 2001, and Max Planck Institute for Mathematics, 1988; lecturer at Cambridge University, 1990, Oxford University, 1990, Imperial College of Science and Technology, London, 1991; Sigma Xi, national lecturer, 1980-82; also associate at Institut Henri Poincaré, Paris, 1950s. American Academy of Arts and Sciences, fellow, 1982; National Academy of Sciences, foreign associate, 1987, member, 2001; European Academy of Arts, Sciences, and Humanities, member, 1987; Norwegian academy of Science and Letters, foreign member, 1998. Also worked as apprentice toolmaker and horse caretaker. *Military service:* French Air Force, 1949-50.

*MEMBER:* International Statistics Institute, American Physical Society, Institute for Mathematics and Statistics, Econometric Society, American Geophysicists Union, American Statistical Association, American Mathematical Society, Institute of Electronics and Electrical Engineers, American Association for the Advancement of Science (fellow), French Mathematical Society.

*AWARDS, HONORS:* Guggenheim fellow, 1968; Bernard Medal for Meritorious Service in Science, National Academy of Sciences, 1985; Franklin Medal, Franklin Institute, 1986; Alexander von Humboldt Prize, 1987; Charles Proteus Steinmetz Medal, 1988; Caltech Distinguished Service Award, 1988; Moet-Hennessy Prize, 1988; decorated chevalier, French Legion of Honor, 1989; Harvey Prize, 1989; Nevada Prize, University of Nevada system, 1991; Wolf Prize for physics, 1993; Honda Prize, 1994; Médaille de Vermeil, Ville de Paris, 1996; John Scott Award, John Scott Fund, 1999; Lewis Fry Richardson Medal, European Geophysical Society, 1999; Sven Berggren Priset, Kungliga Fysiografika Sällskapet, Lund, Sweden, 2002; Medal of the President of the Republic of Italy, 2002; William Procter Prize for Scientific Achievement, Sigma Xi, 2002; Japan Prize for Science and Technology, Science and Technology Foundation of Japan, 2003; honorary degrees from Syracuse University, 1986, Laurentian University, 1986, Boston University, 1987, State University of New York, 1988, University of Bremen, 1988, University of Guelph, 1989, Pace University, 1989, University of Dallas, 1992, Union College, Schenectady, NY, 1993, University of Buenos Aires, 1993, University of Tel Aviv, 1995, Open University, Milton Keynes, England, 1998, University of Business and Commerce, Athens, Greece, 1998, University of St. Andrews, 1999, and Emory University, 2002.

## WRITINGS:

*Logique, langage et theorie de l'information,* [France], 1957.

*Les Objets fractals: forme, hasard et dimension,* Flammarion (Paris, France), 1975, 3rd edition, 1989, translation published as *Fractals: Form, Chance, and Dimension,* W. H. Freeman (San Francisco, CA), 1977, expanded edition published as *The Fractal Geometry of Nature,* W. H. Freeman (San Francisco, CA), 1982.

(With Christopher H. Scholz) *Fractals in Geophysics,* Birkauser Verlag (Boston, MA), 1989.

*Fractals in Physics: Essays in Honour of Benoit B. Mandelbrot,* Elsevier Science (New York, NY), 1990.

*Fractals and Scaling in Finance: Discontinuity, Concentration, Risk,* Springer (New York, NY), 1997.

*Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/f Noise and R/S,* Springer (New York, NY), 2003.

(With Michael L. Frame) *Fractals, Graphics, and Mathematical Education,* Mathematical Association of America (Washington, DC), 2002.

(With C. J. G. Evertsz and M. C. Gutzwiller) *Fractals and Chaos: The Mandelbrot Set and Beyond,* Springer (New York, NY), 2004.

Contributor to books, including *Multifractals and 1/f Noise: Wild Self-Affinity in Physics (1963-1976),* Springer (New York, NY), 1999. Contributor of articles to professional journals.

*ADAPTATIONS:* Mandelbrot's work on fractals inspired the symphonic work *The Mandelbrot Echoes,* created by Harri Vuori, published by Edition Love (Helsinki, Finland) in 1998.

*SIDELIGHTS:* Benoit B. Mandelbrot is the mathematician who conceived, developed, and named the field of fractal geometry. This field is devoted to the study of roughness in nature and culture. Thus it describes the everyday forms of nature—such as mountains, clouds, and the path traveled by lightning—that do not fit into the world of straight lines, circles, and smooth curves known as Euclidean geometry. It also describes the charts of the variation of financial prices. Therefore, Mandelbrot also recognized the value of fractal geometry as a tool for analyzing a variety of physical, social, and biological phenomena.

Mandelbrot was born in 1924 to a Lithuanian Jewish family in Warsaw, Poland. His father, the descendant of a long line of scholars, was a manufacturer and wholesaler of children's clothing. His mother, trained as a doctor and dentist, feared exposing her children to epidemics, so instead of sending her son to school, she arranged for him to be tutored at home by his Uncle Loterman. Mandelbrot and his uncle played chess and read maps; he learned to read, but not the whole alphabet. He first attended elementary school in Warsaw. When he was eleven years old, his family moved to France, first to Paris and then to Tulle, in south central France. When Mandelbrot entered secondary school, he was thirteen years old instead of the usual eleven, but he gradually caught up with his age group. His uncle Szolem Mandelbrojt, a mathematician, was a university professor, and Mandelbrot became acquainted with his uncle's mathematician colleagues. Mandelbrot's teenage years were disrupted by World War II, which rendered his school attendance irregular. From 1942 to 1944, he and his younger brother wandered from place to place. He found work as an apprentice toolmaker for the railroad, and for a time he took care of horses at a chateau near Lyon. He carried books with him and tried to study on his own.

After the war, at the age of twenty, Mandelbrot took the month-long entrance exams for the leading science schools. Although he had not received the usual two years of preparation, he did very well. He had not received much formal training in algebra or complicated integrals, but he remembered all kinds of geometric shapes. Faced with an analytic problem, he would make a drawing, and this would often lead him to the solution. He enrolled in École Polytechnique. Graduating two years later, he earned a scholarship to study at the California Institute of Technology. After two years there, he returned to France with a master's degree in aeronautics and spent a year in the French Air Force.

Mandelbrot next found himself in Paris, looking for a topic for his doctoral thesis. One day his uncle, rummaging through his wastebasket for something for Mandelbrot to read on the subway, pulled out a book review of *Human Behavior and the Principle of Least Effort* by George Zipf. The reviewer discussed examples of frequency distributions in the social sciences that did not follow the Gaussian "bell-shaped curve," the so-called normal distribution according to which statistical data cluster around the average. Mandelbrot wrote part of his doctoral thesis on Zipf's claims about word frequencies; the second half was on statistical thermodynamics. Much later, Mandelbrot commented that the book review greatly influenced his early thinking; he saw in Zipf's work flashes of genius, projected in many directions yet nearly overwhelmed by wild notions and extravagance, and he cited Zipf's career as an example of the extraordinary difficulties of doing scientific work that is not limited to one field. At the time, Mandelbrot had read Norbert Wiener on cybernetics and John von Neumann on game theory, and he was inspired to follow their example in using mathematical approaches to solve long-standing problems in other fields.

In the 1960s Mandelbrot studied stock market and commodity price variations and the mathematical models used to predict prices. A Harvard professor had observed that the daily changes in the price of cotton over many years did not follow the Gaussian bell-shaped distribution. Existing statistical models assumed that the rise and fall of stock-market prices was continuous, but Mandelbrot noted that prices may jump or drop suddenly. This showed that a model that assumes continuous prices is wrong. Working at the International Business Machines (IBM) Company's Thomas J. Watson Research Center in Yorktown Heights, New York, using IBM computers to analyze the data, he found that the pattern for daily and monthly price changes are matched. Statistically, the choice of time scale made no difference; the patterns were self-similar. Using this concept, he was able to account for a great part of the observed price variations, where earlier statistical techniques had not succeeded.

Shortly thereafter, IBM scientists asked Mandelbrot to help on a practical problem. In sending computer signals along electric wires, they found occasional random mistakes, or "noise." They suspected that some of the noise was being caused by other technicians tinkering with the equipment. Mandelbrot studied the times when the noise occurred. He found long periods of error-free transmission separated by bursts of noise.

When he looked at a burst of noise in detail, he saw that it, in turn, consisted of smaller error-free periods interspersed with smaller noisy chunks. As he continued to examine chunks at smaller and smaller scales, he found that the pattern noise occurring was statistically the same, regardless of the level of detail he was looking at. He described the probability distribution of the noise pattern as self-similar, or scaling—that is, at every time scale the ratio of noisy to clean transmission remained the same. The noise need not be due to technicians tinkering with screwdrivers; it may be spontaneous—that is, due to physics. In understanding the noise phenomenon, Mandelbrot used as a model the Cantor set, an abstract geometric construction of Georg Cantor, a nineteenth-century German mathematician. The model changed the way engineers viewed and addressed the noise problem.

For centuries humankind has tried to predict the water level of rivers like the Nile in order to prevent floods and crop damage. Engineers have relied on such predictions in building dams and hydroelectric projects. In the 1960s Mandelbrot studied the records of the Nile River level and found that existing statistical models did not fit the long periods of drought and that the longer a drought period, the more likely the drought was to continue. The resulting picture looked like random noise superimposed on a background of random noise. Mandelbrot showed unlabeled graphs of the river's actual fluctuations to a noted hydrologist, along with graphs drawn from the existing statistical models and other graphs based on Mandelbrot's statistical theories. The hydrologist dismissed the graphs from the old models as unrealistic, but he could not distinguish Mandelbrot's graphs from the real ones. For Mandelbrot, this experience illustrated the value of using visual representations to gain insight into natural and social phenomena. Other researchers found support for Mandelbrot's statistical model when they showed fake stock charts to a stockbroker; the stockbroker rejected some of the fakes as unrealistic, but not Mandelbrot's.

Early in the twentieth century, mathematicians and geometers created curves that were infinitely wrinkled and solids that were full of holes. Much later, Mandelbrot found their abstract mathematics indispensable in modeling shapes and phenomena found in nature. He had read an article about the length of coastlines in which Lewis Fry Richardson reported that encyclopedias in Spain and Portugal differed on the length of the border between the two countries; Richardson found similar discrepancies—up to twenty percent—for the border between Belgium and the Netherlands. Mandelbrot took up the question in a paper he called "How Long Is the Coast of Britain?" The answer, he said, depends on the length of the ruler you use. Measuring a rocky shoreline with a foot ruler would produce a longer answer than measuring it with a yardstick. As the scale of measurement becomes smaller, the measured length becomes infinitely large. Mandelbrot also investigated ways of measuring the degree of "wiggliness" of a curve. He worked with programmers to develop computer programs to draw fake coastlines. By changing a number in the program, he could produce relatively smooth or rough coastlines that resembled New Zealand or those of the Aegean Sea. The number determined the degree of wiggliness and came to be identified as the curve's fractal dimension.

Fascinated with this approach, Mandelbrot looked at other patterns in nature, such as the shapes of clouds and mountains, the meanderings of rivers, the patterns of moon craters, the frequency of heartbeats, the structure of human lungs, and the patterns of blood vessels. He found that many shapes in nature—even those of ferns and broccoli and the holes in Swiss cheese—could be described and replicated on the computer screen using fractal formulas.

Mandelbrot's reports and research papers during this period made clear that his methods were part of a more general approach to irregularity and chaos that was applicable to physics as well. Editors, however, usually preferred a more narrowly technical discussion. But then he was invited to give a talk at the Collège de France in 1973. Rather than selecting one of his many areas of research, he decided to explain how his many different interests fit together. A name was needed for this new family of geometric shapes, which typically involved statistical irregularities and scaling. Looking through his son's Latin dictionary, he found the adjective *fractus,* meaning "fragmented, irregular," and the verb *frangere,* "to break," and he came up with "fractal." His lecture aroused considerable interest and was published in expanded form in 1975 in French as *Les Objets fractals: forme, hasard et dimension.* Revised and expanded versions were published later in English in the United States as *The Fractal Geometry of Nature.* The book, which Mandelbrot called a manifesto and a casebook, attracted interest from researchers in fields from mathematics and engineering to economics and physiology.

Using fractal formulas, computer programmers could produce artificial landscapes that were remarkably realistic. This technology could be used in movies and computer games. Among the first films to use fractal landscapes were George Lucas's *Return of the Jedi,* for the surface of the Moons of Endor, *Star Trek II: The Wrath of Khan,* and *The Last Starfighter.* Some fractal formulas produced fantastic abstract designs and strange dragon-like shapes. Mathematicians of the early twentieth century had done research in this area, but they did not have the advantage of seeing visual representations. The formulas were studied as abstract mathematical objects and, because of their strange properties, were called "pathological."

In the 1970s Mandelbrot became interested in a topic he had glimpsed around 1950—investigations carried out during World War I by French mathematicians Pierre Fatou and Gaston Julia, the latter having been one of his teachers years before at the Polytechnique. Julia had worked with mathematical expressions involving complex numbers, which have as a component the square root of negative one. Instead of graphing the solutions of equations in the familiar method of Descartes, Julia used a different approach; he fed a number into an equation, calculated the answer, and then fed the answer back into the equation, recycling again and again, noting what was happening to the answer. Mandelbrot used the computer to explore the patterns generated by this approach. For one set, he used a relatively simple calculation in which he took a complex number, squared it, added the original number, squared the result, continuing again and again; he plotted the original number on the graph only if its answers did not run away to infinity. The figure generated by this procedure turned out to contain a strange cardioid shape with circles and filaments attached. As Mandelbrot made more detailed calculations, he discovered that the outline of the figure contained tiny copies of the larger elements, as well as strange new shapes resembling fantastic seahorses, flames, and spirals. The figure came to be known as the Mandelbrot set. Representations of the Mandelbrot set and the related sets studied by Julia, some in psychedelic colors, soon appeared in books and magazines—some even in exhibits of computer art.

Through his work with fractals and computer projections of various equations, Mandelbrot had discovered tools that could be used by scientists and engineers for strengthening steel, creating polymers, locating underground oil deposits, building dams, and understanding protein structure, corrosion, acid rain, earthquakes, and hurricanes. Physicists studying dynamical systems and fractal basin boundaries could use Mandelbrot's model to better understand phenomena such as the breaking of materials or the making of decisions. If images could be reduced to fractal codes, the amount of data necessary to transmit or store images could be greatly reduced.

Fractal geometry showed that highly complex shapes could be generated by repeating simple instructions, and small changes in the instructions could produce very different shapes. For Mandelbrot, the striking resemblance of some fractal shapes to living organisms raised the possibility that only a limited inventory of genetic coding is needed to obtain the diversity and richness of shapes in plants and animals.

## BIOGRAPHICAL AND CRITICAL SOURCES:

### books

Aharony, Amnon, and Jens Feder, editors, *Fractals in Physics: Essays in Honour of Benoit B. Mandelbrot,* North-Holland (New York, NY), 1990.

Albers, Donald J., and G. L. Anderson, editors, *Mathematical People: Profiles and Interviews,* Birkhauser Verlag (Boston, MA), 1985.

Briggs, John, *Fractals, the Patterns of Chaos: A New Aesthetic of Art, Science, and Nature,* Simon & Schuster (New York, NY), 1992.

Gardner, Martin, *Penrose Tiles to Trapdoor Ciphers,* W. H. Freeman (San Francisco, CA), 1989.

Gleick, James, *Chaos: Making a New Science,* Viking Penguin (New York, NY), 1987.

Peitgen, Heinz-Otto, and C. J. G. Evertsz, editors, *Fractal Geometry and Analysis: The Mandelbrot Festschrift, Curaçao 1995,* World Scientific (River Edge, NJ), 1996.

Peitgen, Heinz-Otto, and Dietmar Saupe, editors, *The Science of Fractal Images,* Springer-Verlag (New York, NY), 1988.

Peitgen, Heinz-Otto, and P. H. Richter, *The Beauty of Fractals,* Springer-Verlag (New York, NY), 1986.

### periodicals

*American Scientist,* May, 2002, "Benoit Mandelbrot Receives 2002 William Procter Prize," p. 293.

*Choice,* February, 2003, H. P. Koirala, review of *Fractals, Graphics, and Mathematics Education,* p. 1034.

*Economist,* December 26, 1987, pp. 99-103; December 6, 2003, "The Father of Fractals: Last Word," p. 36US.

*Java Developer's Journal,* February, 2002, Blair Wyman, "Brushes with Greatness," p. 114.

*Journal of Economic Literature,* June, 2001, Philip Mirowski, review of *Fractals and Scaling in Finance: Discontinuity, Concentration, Risk,* p. 585.

*Mathematics Teacher,* November, 2000, Dane R. Camp, "Benoit Mandelbrot: The Euclid of Fractal Geometry," p. 708.

*New York Times Magazine,* December 8, 1985, p. 64.

*Omni,* February, 1984, pp. 65-66, 102-107.

*Physics Today,* April, 1987, pp. 101-102.

*Scientific American,* August, 1990, pp. 60-67.

*Skeptic,* winter, 2001, "Chaos Forum," p. 26.

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**Mandelbrot, Benoit B. 1924-**