## Benoit B Mandelbrot

**-**

## Benoit B. Mandelbrot

# Benoit B. Mandelbrot

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). □

## Mandelbrot, Benoit B.

# Mandelbrot, Benoit B.

*American Geometer1924–*

Benoit B. Mandelbrot, called the father of **fractal geometry** , was born November 20, 1924, in Warsaw, Poland, into a well-educated Jewish family. In 1936 the family moved to France where Benoit spent time with his uncle, Szolem Mandelbrojt, who was a professor of mathematics at the prestigious Collège de France in Paris, and who took an interest in Benoit's education.

Szolem Mandelbrojt recommended that Benoit study the work of Gaston Julia, whose 1918 paper was considered a mathematical masterpiece and a source of good problems. At the time, Benoit expressed little interest for the kind of mathematics that he found in Julia's paper, much to the dismay of his uncle, but instead showed an interest in geometry. In 1944 Benoit was accepted into the Ecole Polytechnique and studied under the direction of Paul Lévy, who embraced Mandelbrot's interest in geometry.

In 1952 Mandelbrot received his Ph.D. (Docteur ès Sciences Mathématiques) from the University of Paris. After completing his degree, Mandelbrot went to the United States, where he held a position at the School of Mathematics at the Institute for the Advanced Studies (under J. von Neumann) at Princeton University. He returned to France in 1955, at which time he married Ailette Kagan.

Discontented with the style of mathematics work in France at the time, Mandelbrot returned to the United States in 1958 to accept a position as a fellow and professor in the research department at the world-famous laboratories of International Business Machines (IBM) in New York. IBM was beginning to lead the computer industry, and the company provided Mandelbrot with the freedom and resources to pursue his research interests.

## Chaos and Fractals

During the 1970s, Mandelbrot's research examined unusual or chaotic patterns of behavior in geometric shapes. In 1975 Mandelbrot coined the term "fractal," from the Latin *fractus* (meaning fragmented, irregular), as a way to describe the self-similar geometric patterns he had discovered. In addition, Mandelbrot revisited Julia's earlier work and, with the aid of computer graphics, illustrated Julia's work as a source of some of the most beautiful fractal images known today. His work was first published in English in his book *Fractals: Form, Chance, and Dimension* (1977).

The connection between chaos and **geometry** was further established with Mandelbrot's discovery in 1980 of what we have come to call the Mandelbrot Set. Named in his honor, it is certainly the most popular fractal and is often noted as the most popular object of contemporary mathematics. In addition, Mandelbrot had discovered fractal geometry and chaotic behavior in many aspects of nature. In his most recognized book, *The Fractal Geometry of Nature* (1982), Mandelbrot demonstrated that mathematical fractals have many features in common with shapes found in nature, such as snowflakes, mountains, ferns, and coastlines.

## Honors and Achievements

Throughout his career, Mandelbrot has held many academic positions. In addition to being a Fellow of the IBM Thomas J. Watson Research Center and of the American Academy of Arts and Sciences, he held appointments as professor of the practice of mathematics and economics at Harvard University; professor of engineering at Yale University; and professor of physiology at the Einstein College of Medicine. He is recognized for his many remarkable achievements, prizes, and honors in the fields of mathematics, physics, engineering, and medicine, including the Barnard Medal for Meritorious Service to Science (1985), the Franklin Medal (1986), the Alexander von Humboldt Prize (1987), the Steinmetz Medal (1988), the Nevada Medal (1991), and the Wolf prize for physics (1993). On June 23, 1999, Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews.

see also Chaos; Fractals.

*Gay A. Ragan and*

*Óscar Chávez*

## Bibliography

Hall, Nina. *Exploring Chaos: A Guide to the New Science of Disorder.* New York: W. W. Norton & Company, Inc., 1993.

Henderson, Harry. "Benoit Mandelbrot." In *Modern Mathematicians.* New York: Facts on File, Inc., 1996.

Mandelbrot, Benoit B. *The Fractal Geometry of Nature.* New York: W. H. Freeman and Company, 1982.

———. *Fractals: Form, Chance, and Dimension.* San Francisco: W. H. Freeman and Company, 1977.

### Internet Resources

"Benoit Mandelbrot." *School of Mathematical and Computational Sciences.* University of St Andrews. <http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html>.

"Gaston Maurice Julia." *School of Mathematical and Computational Sciences.* University of St Andrews. <http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Julia.html>.

## Benoit B. Mandelbrot

# Benoit B. Mandelbrot

**1924-**

**Polish-born French Mathematician**

Polish-born French mathematician Benoit Mandelbrot is widely acclaimed as one of the founding fathers of fractal geometry. Educated in France, Mandelbrot shunned the prevailing French emphasis on pure mathematics to take and early and strong interest in applied mathematics.

Due to the Second World War, Mandelbrot's education was, at times, sporadic. In many academic areas, including some mathematical subjects, he was self-taught. This reliance on his own ability to investigate and prove mathematical concepts lead him to construct geometrical proofs. Mandelbrot often credits these formative years as a source of stimulus and necessity that helped him develop his capacity for geometric thought and his geometrical approach to mathematics.

Mandelbrot eventually studied at the Ecole Polytechnique. After obtaining his doctorate, he took on academic posts at the California Institute of Technology and at Princeton's Institute for Advanced Study. In 1955, for a brief interval, Mandelbrot returned to France to accept a post at the National Center for Scientific Research (Centre National de la Récherche Scientific) before returning to the United States to accept a position with IBM laboratories. Mandelbrot has also held teaching and research posts in mathematics, engineering, physiology, and economics at various institutions, including Yale, the Einstein College of Medicine, and the French Ecole Polytechnique.

Using the emerging tools of computer graphics, Mandelbrot developed many of the well known concepts associated with fractal geometry. In 1963 Mandelbrot published what was termed the fractal concept. Fractals are geometric shapes that maintain their similar properties and relationships at all levels of magnification.

In contrast to ordinary geometry, a fractal geometric object may have fractional dimensions or use infinities to represent dimensions. Ordinary geometric objects have integer representations of their dimensions. Planes are two dimensional, and lines are one dimensional objects. Fractal objects exhibit a property described as self-similarity. The degree of self-similarity between fractals may vary, but, in general, within a self-similar object the component parts resemble the whole regardless of scale (this property is also termed scaling symmetry). In practical terms, this means that if any fractal component of an object is magnified it resembles the structure as a whole.

This fractal scaling is not the same type of scaling found in the objects familiar to classical geometry involving translational symmetry (that is, objects with translational, rotational, or reflective symmetry.) Natural fractals can be seemingly chaotic or random, yet, when this is the case, they retain the overall structure in only a statistical sense. What always remains invariant with fractals is their factual dimension.

Mandelbrot's work in fractal geometry created a mathematical school with broad scope and application. Fractals seemed to be everywhere—a universality in nature. Mandelbrot's vision was, however, inconsistent and at odds with the mathematical descriptions of natural events routinely used by physicists. In essence, while physicists tried to smooth data to explain seemingly chaotic phenomena such as turbulence, Mandelbrot saw the profound differences in behavior that could be characterized using the methodologies of factual geometry.

Fractal concepts are now used by astrophysicists to construct computer simulations depicting the collapse of systems in a gravitational field. As such, they may help formulate an understanding of the dynamics involved in the highly complex formation of cosmic structures (for example, galaxies, galactic clusters, and planetary systems).

During his career, Mandelbrot moved fractal concept from the realm of geometry into a vast array of scientific disciplines. Fractals were used to describe, unite, and relate seemingly far-flung and separate phenomena. The compartmentalized fractal concept was even used to describe cellular processes and behaviors.

In addition to his mathematical work, Mandelbrot worked on the development of computer graphics programs that could be used to represent his concepts. Fractal concepts have found widespread use in computer animation.

Mandelbrot's long and productive career has garnered significant honors, including the 1993 Wolf Prize for Physics. His published works include the 1982 book *The Factual Geometry Of Nature.*

**ADRIENNE WILMOTH LERNER**

## Mandelbrot, Benoit B.

# MANDELBROT, Benoit B.

**MANDELBROT, Benoit B.** American (born Poland), b. 1924. **Genres:** Economics, Mathematics/Statistics, Physics, Sciences, Autobiography/ Memoirs. **Career:** Mathematician. Philips Electronics, Paris, France, mathematician, 1950-53; Institute for Advanced Study, Princeton, NJ, junior member and Rockefeller scholar, 1953-54; University of Geneva, Switzerland, junior professor of math, 1955-57; University of Lille, France, junior professor of math, 1957-58; Ecole Polytechnique, Paris, junior professor of math, 1957-58; IBM Thomas J. Watson Research Center, Yorktown Heights, NY, member of research staff, 1958-74, IBM fellow, 1974-93, IBM fellow emeritus, 1993-; Yale University, New Haven, CT, visiting professor of engineering, 1970, Abraham Robinson Professor of Mathematical Sciences, 1987-99, Sterling Professor of Mathematical Sciences, 1999-; Academie des Sciences, Paris, professor, 1995. Harvard University, visiting professor, 1962-64, 1979-80, professor, 1984-87; Albert Einstein College of Medicine, visiting professor, 1970; University of California at Berkeley, Charles M. and Martha Hitchcock Professor, 1992. Visitor and institute lecturer at universities in the US and France. Lecturer at universities worldwide. **Publications:** Logique, langage et theorie de l'information, 1957; Les objets fractals: forme, hasard et dimension, c.1976, 4th ed., 1995, in US as Fractals: Form, Chance, and Dimension, 1977; The Fractal Geometry of Nature, 1982; (with C.H. Scholz) Fractals in Geophysics, 1989; Fractals and Scaling in Finance, 1997; Multifractals and 1/f Noise, 1999; Gaussian Self-Affinity and Fractals, 2002; (with M.L. Frame) Fractals, Graphics, and Mathematics Education, 2002; Fractals and Chance, 2004. Contributor to professional journals. **Address:** Dept of Math, Yale University, New Haven, CT 06520, U.S.A. **Online address:** [email protected]

## Mandelbrot, Benoît B.

Benoît B. Mandelbrot: (bənwä´ măn´dəlbrō´, Fr. mäNdĕlbrô´), 1924–2010, French-American mathematician, b. Warsaw, Poland, Ph.D. Univ. of Paris, 1952. Largely self-taught and considered a maverick in the field of mathematics, he was uncomfortable with the rigorously pure logical analysis prescribed by Nicolas Bourbaki and relied instead on his talent for visualizing natural phenomena. A pioneer of chaos theory, he conceived, developed, and applied fractal geometry, which is used to find order in complex, apparently erratic shapes and processes. His work has had implications for and influence in such diverse fields as geometry, medicine, cosmology, economics, and computer graphics. Mandelbrot worked at IBM from 1958, holding an emeritus position there from 1993, and taught at Yale from 1987, where he was Sterling Professor of Mathematics (1999–2004). He wrote *The Fractal Geometry of Nature* (1982) and other works.

See his *The Fractalist: Memoir of a Scientific Maverick* (2012).

## Mandelbrot, Benoit B.

**Mandelbrot, Benoit B.** (1942– ) US mathematician, b. Poland. He made major contributions to chaos theory, and is best known for coining the term fractal. His book The Fractal Geometry of Nature (1982) contains many examples of natural fractals, such as ferns, trees and rivers. The Mandelbrot set, a well-known fractal object, is named after him.