Benoît Mandelbrot
Encyclopedia
Benoît B. Mandelbrot was a French American
mathematician
. Born in Poland
, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and working in the United States
, and he acquired dual
French
and American citizenship.
Mandelbrot worked on a wide range of mathematical problems, including mathematical physics
and quantitative finance, but is best known as the father of fractal geometry
. He coined the term fractal
and described the Mandelbrot set
. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.
Mandelbrot spent most of his career at IBM's Thomas J. Watson Research Center
, and was appointed as an IBM Fellow
. He later became a Sterling Professor
of Mathematical Sciences at Yale University
, where he was the oldest professor in Yale's history to receive tenure. Mandelbrot also held positions at the Pacific Northwest National Laboratory
, Université Lille Nord de France
, Institute for Advanced Study
and Centre National de la Recherche Scientifique
.
into a Jewish
family from Lithuania
. Mandelbrot was born into a family with a strong academic tradition - his mother was a dentist
and he was introduced to mathematics
by two of his uncles, one of whom, Szolem Mandelbrojt, was a mathematician who resided in Paris
. However, his father made his living trading clothing. Anticipating the threat posed by Nazi Germany
, the family fled from Poland to France
in 1936 when he was 11. Mandelbrot attended the Lycée Rolin in Paris until the start of World War II
, when his family moved to Tulle
, France. He was helped by Rabbi
David Feuerwerker
, the Rabbi of Brive-la-Gaillarde
, to continue his studies. In 1944 he returned to Paris. He studied at the Lycée du Parc
in Lyon
and in 1945 - 47 attended the École Polytechnique
, where he studied under Gaston Julia
and Paul Lévy
. From 1947 to 1949 he studied at California Institute of Technology
, where he earned a master's degree in aeronautics
. Returning to France, he obtained his Ph.D. degree in Mathematical Sciences at the University of Paris
in 1952.
From 1949 to 1958 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique
. During this time he spent a year at the Institute for Advanced Study
in Princeton, New Jersey
, where he was sponsored by John von Neumann
. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland, and later to the Université Lille Nord de France
. In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center
in Yorktown Heights, New York
. He remained at IBM for 35 years, becoming an IBM Fellow
, and later Fellow Emeritus
.
, economics
, and fluid dynamics
. He became convinced that two key themes, fat tail
s and self-similar structure, ran through a multitude of problems encountered in those fields.
Mandelbrot found that price changes in financial market
s did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite variance
. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter
.
Mandelbrot also put his ideas to work in cosmology
. He offered in 1974 a new explanation of Olbers' paradox
(the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the star
s in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang
theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.
In 1975, Mandelbrot coined the term fractal
to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer
, demographer and statistician
Jaromír Korčák.
While on secondment as Visiting Professor at Harvard University
in 1979, Mandelbrot began to study fractals called Julia set
s that were invariant under certain transformations of the complex plane
. Building on previous work by Gaston Julia
and Pierre Fatou
, Mandelbrot used a computer to plot images of the Julia sets of the formula z2 − μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set
fractal that is now named after him. (Note that the Mandelbrot set is now usually defined in terms of the formula z2 + c, so Mandelbrot's early plots in terms of the earlier parameter μ are left–right mirror images of more recent plots in terms of the parameter c.)
In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature
. This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".
Mandelbrot left IBM
in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale
, and obtained his first tenure
d post in 1999, at the age of 75. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson
Prize of the European Geophysical Society in 2000, the Japan Prize
in 2003, and the Einstein Lectureship of the American Mathematical Society
in 2006.
The small asteroid 27500 Mandelbrot
was named in his honor. In November 1990, he was made a Knight in the French Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory
. Mandelbrot was promoted to Officer of the Legion of Honour in January 2006. An honorary degree from Johns Hopkins University
was bestowed on Mandelbrot in the May 2010 commencement exercises.
, some of the mathematical objects he presented in The Fractal Geometry of Nature
had been described by other mathematicians. Before Mandelbrot, they had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to non-smooth objects in the real world. He highlighted their common properties, such as self-similarity
(linear, non-linear, or statistical), scale invariance
, and a (usually) non-integer Hausdorff dimension
.
He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountain
s, coastlines and river basins; the structures of plants, blood vessel
s and lung
s; the clustering of galaxies
; and Brownian motion
. Fractals are found in human pursuits, such as music
, painting
, architecture
, and stock market
prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry
:
Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory
and other fields of science and mathematics.
When visiting the Museu de la Ciència de Barcelona in 1988, he told its director that the painting The Face of War
had given him "the intuition about the transcendence of the fractal geometry when making intelligible the omnipresent similitude in the forms of nature". He also said that, fractally, Gaudí was superior to Van der Rohe.
in Cambridge, Massachusetts
, on 14 October 2010 from pancreatic cancer
, at the age of 85. Reacting to news of his death, mathematician Heinz-Otto Peitgen
said "if we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last 50 years." Chris Anderson described Mandelbrot as "an icon who changed how we see the world." French President Nicolas Sarkozy
said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions". Sarkozy also added, "His work, developed entirely outside mainstream research, led to modern information theory." Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry."
Member of the Norwegian Academy of Science and Letters
.
French American
French Americans or Franco-Americans are Americans of French or French Canadian descent. About 11.8 million U.S. residents are of this descent, and about 1.6 million speak French at home.An additional 450,000 U.S...
mathematician
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. Born in Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and working in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, and he acquired dual
Multiple citizenship
Multiple citizenship is a status in which a person is concurrently regarded as a citizen under the laws of more than one state. Multiple citizenships exist because different countries use different, and not necessarily mutually exclusive, citizenship requirements...
French
French nationality law
French nationality law is historically based on the principles of jus soli , according to Ernest Renan's definition, in opposition to the German's definition of nationality, Jus sanguinis , formalized by Fichte.The 1993 Méhaignerie Law required children born in France of foreign parents to request...
and American citizenship.
Mandelbrot worked on a wide range of mathematical problems, including mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and quantitative finance, but is best known as the father of fractal geometry
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
. He coined the term fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
and described the Mandelbrot set
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.
Mandelbrot spent most of his career at IBM's Thomas J. Watson Research Center
Thomas J. Watson Research Center
The Thomas J. Watson Research Center is the headquarters for the IBM Research Division.The center is on three sites, with the main laboratory in Yorktown Heights, New York, 38 miles north of New York City, a building in Hawthorne, New York, and offices in Cambridge, Massachusetts.- Overview :The...
, and was appointed as an IBM Fellow
IBM Fellow
An IBM Fellow is an appointed position at IBM made by IBM’s CEO. Typically only 4 to 9 IBM Fellows are appointed each year, at the annual Corporate Technical Recognition Event in May or June. It is the highest honor a scientist, engineer, or programmer at IBM can achieve.The IBM Fellows program...
. He later became a Sterling Professor
Sterling Professor
A Sterling Professorship is the highest academic rank at Yale University, awarded to a tenured faculty member considered one of the best in his or her field...
of Mathematical Sciences at Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, where he was the oldest professor in Yale's history to receive tenure. Mandelbrot also held positions at the Pacific Northwest National Laboratory
Pacific Northwest National Laboratory
Pacific Northwest National Laboratory is one of the United States Department of Energy National Laboratories, managed by the Department of Energy's Office of Science. The main campus of the laboratory is in Richland, Washington....
, Université Lille Nord de France
Université Lille Nord de France
The University of Lille -Nord de France , located in Lille, France, is a center for higher education, academic research and doctoral studies located over multiple campuses in the Academie de Lille....
, Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
and Centre National de la Recherche Scientifique
Centre national de la recherche scientifique
The National Center of Scientific Research is the largest governmental research organization in France and the largest fundamental science agency in Europe....
.
Early years
Mandelbrot was born in WarsawWarsaw
Warsaw is the capital and largest city of Poland. It is located on the Vistula River, roughly from the Baltic Sea and from the Carpathian Mountains. Its population in 2010 was estimated at 1,716,855 residents with a greater metropolitan area of 2,631,902 residents, making Warsaw the 10th most...
into a Jewish
Lithuanian Jews
Lithuanian Jews or Litvaks are Jews with roots in the Grand Duchy of Lithuania:...
family from Lithuania
Lithuania
Lithuania , officially the Republic of Lithuania is a country in Northern Europe, the biggest of the three Baltic states. It is situated along the southeastern shore of the Baltic Sea, whereby to the west lie Sweden and Denmark...
. Mandelbrot was born into a family with a strong academic tradition - his mother was a dentist
Dentist
A dentist, also known as a 'dental surgeon', is a doctor that specializes in the diagnosis, prevention, and treatment of diseases and conditions of the oral cavity. The dentist's supporting team aides in providing oral health services...
and he was introduced to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
by two of his uncles, one of whom, Szolem Mandelbrojt, was a mathematician who resided in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. However, his father made his living trading clothing. Anticipating the threat posed by Nazi Germany
Nazi Germany
Nazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...
, the family fled from Poland to France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
in 1936 when he was 11. Mandelbrot attended the Lycée Rolin in Paris until the start of World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, when his family moved to Tulle
Tulle
Tulle is a commune and capital of the Corrèze department in the Limousin region in central France. It is also the episcopal see of the Roman Catholic Diocese of Tulle...
, France. He was helped by Rabbi
Rabbi
In Judaism, a rabbi is a teacher of Torah. This title derives from the Hebrew word רבי , meaning "My Master" , which is the way a student would address a master of Torah...
David Feuerwerker
David Feuerwerker
- Born in Geneva :He was born on October 2, 1912, at 11 Rue du Mont-Blanc, in Geneva, Switzerland. He was the seventh of eleven children. His father Jacob Feuerwerker was born in Sighet, now Sighetu Marmatiei, Maramureş, then Hungary, now Rumania...
, the Rabbi of Brive-la-Gaillarde
Brive-la-Gaillarde
Brive-la-Gaillarde is a commune of France. It is a sub-prefecture of the Corrèze department. The population of the urban area was 89,260 as of 1999. Although it is by far the biggest commune in Corrèze, the capital is Tulle.-History:...
, to continue his studies. In 1944 he returned to Paris. He studied at the Lycée du Parc
Lycée du Parc
The Lycée du Parc is a public secondary school located in the sixth arrondissement of Lyon, France. Its name comes from the Parc de la Tête d'Or, one of Europe's largest urban parks, which is situated nearby....
in Lyon
Lyon
Lyon , is a city in east-central France in the Rhône-Alpes region, situated between Paris and Marseille. Lyon is located at from Paris, from Marseille, from Geneva, from Turin, and from Barcelona. The residents of the city are called Lyonnais....
and in 1945 - 47 attended the École Polytechnique
École Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
, where he studied under Gaston Julia
Gaston Julia
Gaston Maurice Julia was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoît Mandelbrot; the Julia and Mandelbrot fractals are closely related....
and Paul Lévy
Paul Pierre Lévy
Paul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...
. From 1947 to 1949 he studied at California Institute of Technology
California Institute of Technology
The California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
, where he earned a master's degree in aeronautics
Aeronautics
Aeronautics is the science involved with the study, design, and manufacturing of airflight-capable machines, or the techniques of operating aircraft and rocketry within the atmosphere...
. Returning to France, he obtained his Ph.D. degree in Mathematical Sciences at the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
in 1952.
From 1949 to 1958 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique
Centre national de la recherche scientifique
The National Center of Scientific Research is the largest governmental research organization in France and the largest fundamental science agency in Europe....
. During this time he spent a year at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, where he was sponsored by John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland, and later to the Université Lille Nord de France
Université Lille Nord de France
The University of Lille -Nord de France , located in Lille, France, is a center for higher education, academic research and doctoral studies located over multiple campuses in the Academie de Lille....
. In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center
Thomas J. Watson Research Center
The Thomas J. Watson Research Center is the headquarters for the IBM Research Division.The center is on three sites, with the main laboratory in Yorktown Heights, New York, 38 miles north of New York City, a building in Hawthorne, New York, and offices in Cambridge, Massachusetts.- Overview :The...
in Yorktown Heights, New York
Yorktown Heights, New York
Yorktown Heights is a census-designated place in the town of Yorktown in Westchester County, New York, United States. The population was 1,781 at the 2010 census.-Geography:Yorktown Heights is located at ....
. He remained at IBM for 35 years, becoming an IBM Fellow
IBM Fellow
An IBM Fellow is an appointed position at IBM made by IBM’s CEO. Typically only 4 to 9 IBM Fellows are appointed each year, at the annual Corporate Technical Recognition Event in May or June. It is the highest honor a scientist, engineer, or programmer at IBM can achieve.The IBM Fellows program...
, and later Fellow Emeritus
Emeritus
Emeritus is a post-positive adjective that is used to designate a retired professor, bishop, or other professional or as a title. The female equivalent emerita is also sometimes used.-History:...
.
Research career
From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theoryInformation theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
. He became convinced that two key themes, fat tail
Fat tail
A fat-tailed distribution is a probability distribution that has the property, along with the heavy-tailed distributions, that they exhibit extremely large skewness or kurtosis. This comparison is often made relative to the ubiquitous normal distribution, which itself is an example of an...
s and self-similar structure, ran through a multitude of problems encountered in those fields.
Mandelbrot found that price changes in financial market
Financial market
In economics, a financial market is a mechanism that allows people and entities to buy and sell financial securities , commodities , and other fungible items of value at low transaction costs and at prices that reflect supply and demand.Both general markets and...
s did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter
Scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
.
Mandelbrot also put his ideas to work in cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
. He offered in 1974 a new explanation of Olbers' paradox
Olbers' paradox
In astrophysics and physical cosmology, Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. The argument is also...
(the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.
In 1975, Mandelbrot coined the term fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer
Geographer
A geographer is a scholar whose area of study is geography, the study of Earth's natural environment and human society.Although geographers are historically known as people who make maps, map making is actually the field of study of cartography, a subset of geography...
, demographer and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...
Jaromír Korčák.
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1979, Mandelbrot began to study fractals called Julia set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
s that were invariant under certain transformations of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Building on previous work by Gaston Julia
Gaston Julia
Gaston Maurice Julia was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoît Mandelbrot; the Julia and Mandelbrot fractals are closely related....
and Pierre Fatou
Pierre Fatou
Pierre Joseph Louis Fatou was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory...
, Mandelbrot used a computer to plot images of the Julia sets of the formula z2 − μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
fractal that is now named after him. (Note that the Mandelbrot set is now usually defined in terms of the formula z2 + c, so Mandelbrot's early plots in terms of the earlier parameter μ are left–right mirror images of more recent plots in terms of the parameter c.)
In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature
The Fractal Geometry of Nature
The Fractal Geometry of Nature is an influential book written in 1982 by the Franco-American mathematician Benoît Mandelbrot. It is a revised and enlarged version of his 1977 book entitled Fractals: Form, Chance and Dimension.- See also :* Fractal...
. This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, and obtained his first tenure
Tenure
Tenure commonly refers to life tenure in a job and specifically to a senior academic's contractual right not to have his or her position terminated without just cause.-19th century:...
d post in 1999, at the age of 75. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...
Prize of the European Geophysical Society in 2000, the Japan Prize
Japan Prize
is awarded to people from all parts of the world whose "original and outstanding achievements in science and technology are recognized as having advanced the frontiers of knowledge and served the cause of peace and prosperity for mankind."- Explanation :...
in 2003, and the Einstein Lectureship of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
in 2006.
The small asteroid 27500 Mandelbrot
27500 Mandelbrot
27500 Mandelbrot is a main-belt asteroid discovered on April 12, 2000 by P. G. Comba at Prescott. It is named for mathematician Benoit Mandelbrot.- External links :...
was named in his honor. In November 1990, he was made a Knight in the French Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory
Pacific Northwest National Laboratory
Pacific Northwest National Laboratory is one of the United States Department of Energy National Laboratories, managed by the Department of Energy's Office of Science. The main campus of the laboratory is in Richland, Washington....
. Mandelbrot was promoted to Officer of the Legion of Honour in January 2006. An honorary degree from Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
was bestowed on Mandelbrot in the May 2010 commencement exercises.
Fractals and regular roughness
Although Mandelbrot coined the term fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
, some of the mathematical objects he presented in The Fractal Geometry of Nature
The Fractal Geometry of Nature
The Fractal Geometry of Nature is an influential book written in 1982 by the Franco-American mathematician Benoît Mandelbrot. It is a revised and enlarged version of his 1977 book entitled Fractals: Form, Chance and Dimension.- See also :* Fractal...
had been described by other mathematicians. Before Mandelbrot, they had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to non-smooth objects in the real world. He highlighted their common properties, such as self-similarity
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
(linear, non-linear, or statistical), scale invariance
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
, and a (usually) non-integer Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
.
He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountain
Mountain
Image:Himalaya_annotated.jpg|thumb|right|The Himalayan mountain range with Mount Everestrect 58 14 160 49 Chomo Lonzorect 200 28 335 52 Makalurect 378 24 566 45 Mount Everestrect 188 581 920 656 Tibetan Plateaurect 250 406 340 427 Rong River...
s, coastlines and river basins; the structures of plants, blood vessel
Blood vessel
The blood vessels are the part of the circulatory system that transports blood throughout the body. There are three major types of blood vessels: the arteries, which carry the blood away from the heart; the capillaries, which enable the actual exchange of water and chemicals between the blood and...
s and lung
Lung
The lung is the essential respiration organ in many air-breathing animals, including most tetrapods, a few fish and a few snails. In mammals and the more complex life forms, the two lungs are located near the backbone on either side of the heart...
s; the clustering of galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
; and Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
. Fractals are found in human pursuits, such as music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, painting
Painting
Painting is the practice of applying paint, pigment, color or other medium to a surface . The application of the medium is commonly applied to the base with a brush but other objects can be used. In art, the term painting describes both the act and the result of the action. However, painting is...
, architecture
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
, and stock market
Stock market
A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
:
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.
—Mandelbrot, in his introduction to The Fractal Geometry of NatureThe Fractal Geometry of NatureThe Fractal Geometry of Nature is an influential book written in 1982 by the Franco-American mathematician Benoît Mandelbrot. It is a revised and enlarged version of his 1977 book entitled Fractals: Form, Chance and Dimension.- See also :* Fractal...
Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
and other fields of science and mathematics.
When visiting the Museu de la Ciència de Barcelona in 1988, he told its director that the painting The Face of War
The Face of War
The Face of War is a painting by the Spanish surrealist Salvador Dalí. It was painted during a brief period when the artist lived in California....
had given him "the intuition about the transcendence of the fractal geometry when making intelligible the omnipresent similitude in the forms of nature". He also said that, fractally, Gaudí was superior to Van der Rohe.
Death
Mandelbrot died in a hospiceHospice
Hospice is a type of care and a philosophy of care which focuses on the palliation of a terminally ill patient's symptoms.In the United States and Canada:*Gentiva Health Services, national provider of hospice and home health services...
in Cambridge, Massachusetts
Cambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...
, on 14 October 2010 from pancreatic cancer
Pancreatic cancer
Pancreatic cancer refers to a malignant neoplasm of the pancreas. The most common type of pancreatic cancer, accounting for 95% of these tumors is adenocarcinoma, which arises within the exocrine component of the pancreas. A minority arises from the islet cells and is classified as a...
, at the age of 85. Reacting to news of his death, mathematician Heinz-Otto Peitgen
Heinz-Otto Peitgen
thumb|180px|right|Heinz-Otto PeitgenHeinz-Otto Peitgen is a German mathematician. Peitgen is one of the most prominent researchers in the study of fractals.- Life :...
said "if we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last 50 years." Chris Anderson described Mandelbrot as "an icon who changed how we see the world." French President Nicolas Sarkozy
Nicolas Sarkozy
Nicolas Sarkozy is the 23rd and current President of the French Republic and ex officio Co-Prince of Andorra. He assumed the office on 16 May 2007 after defeating the Socialist Party candidate Ségolène Royal 10 days earlier....
said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions". Sarkozy also added, "His work, developed entirely outside mainstream research, led to modern information theory." Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry."
Honors and awards
A partial list of awards received by Mandelbrot:
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Harvey Prize The Harvey Prize is awarded by the Technion in Haifa, Israel. It is awarded in different disciplines of Science, Technology, Human Health, and Contributions to Peace in the Middle East. Two awards - each of $75,000 - are given away annually... IBM Fellow An IBM Fellow is an appointed position at IBM made by IBM’s CEO. Typically only 4 to 9 IBM Fellows are appointed each year, at the annual Corporate Technical Recognition Event in May or June. It is the highest honor a scientist, engineer, or programmer at IBM can achieve.The IBM Fellows program... ship Japan Prize is awarded to people from all parts of the world whose "original and outstanding achievements in science and technology are recognized as having advanced the frontiers of knowledge and served the cause of peace and prosperity for mankind."- Explanation :... John Scott Award The John Scott Legacy Medal and Premium, created in 1816, is a medal presented to men and women whose inventions improved the "comfort, welfare, and happiness of human kind" in a significant way... |
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Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
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See also
- "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional DimensionHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2...
", a 1967 paper by Mandelbrot - Louis BachelierLouis Bachelier-External links:** Louis Bachelier webpage at the Université de Franche-Comté, Besançon / France. Text in French.** also from Index Funds Advisors, this discussion of...
- Mandelbrot CompetitionMandelbrot CompetitionNamed in honor of the Mandelbrot set, the Mandelbrot Competition is a mathematics competition founded by Sam Vandervelde, Richard Rusczyk and Sandor Lehoczky that allows high school students to compete individually and in four-person teams.-Competition:...
- Skewness riskSkewness riskSkewness risk in financial modeling denotes that observations are not spread symmetrically around an average value. As a result, the average and the median can be different...
and Kurtosis riskKurtosis riskKurtosis risk in statistics and decision theory denotes the fact that observations are spread in a wider fashion than the normal distribution entails...
Further reading
- Heinz-Otto PeitgenHeinz-Otto Peitgenthumb|180px|right|Heinz-Otto PeitgenHeinz-Otto Peitgen is a German mathematician. Peitgen is one of the most prominent researchers in the study of fractals.- Life :...
, Hartmut JürgensHartmut JürgensHartmut Jürgens is a German mathematician, born in 1955 in Bremen, Germany. He received his doctorate in 1983 from the University of Bremen. He has worked in the computer industry, and was the Director of the Dynamical Systems Graphics Laboratory at the University of Bremen...
, Dietmar SaupeDietmar SaupeDietmar Saupe is a fractal researcher and professor of computer science, Department of Computer and Information Science, University of Konstanz, Germany....
and Cornelia Zahlten: Fractals: An Animated Discussion (63 min video film, interviews with Benoît Mandelbrot and Edward Lorenz, computer animations), W.H. Freeman and Company, 1990. ISBN 0-716-72213-5 (re-published by Films for the Humanities & Sciences, ISBN 978-0-7365-0520-8) - "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature", NOVANOVA (TV series)Nova is a popular science television series from the U.S. produced by WGBH Boston. It can be seen on the Public Broadcasting Service in the United States, and in more than 100 other countries...
, WGBH TV, PBS, October 28, 2008.
External links
- Mandelbrot's page at Yale
- Ted talk: "Benoît Mandelbrot: Fractals and the art of roughness"
- Interview at FT.com, in which Mandelbrot discusses his early work with stock market behavior, 2009-09-30
- Benoît Mandelbrot tells his life story (video)
- Interview with Benoît Mandelbrot (January 1, 1981, Ithaca, NY), Eugene DynkinEugene DynkinEugene Borisovich Dynkin is a Soviet and American mathematician. He has made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes...
Collection of Mathematics Interviews, Cornell University LibraryCornell University LibraryThe Cornell University Library is the library system of Cornell University. In 2010 it held 8 million printed volumes in open stacks, 8.5 million microfilms and microfiches, more than of manuscripts, and close to 500,000 other materials, including motion pictures, DVDs, sound recordings, and...
.