Benoît B. Mandelbrot (born 20 November 1924) is a
French AmericanFrench Americans or Franco-Americans are Americans of French descent. About 11.8 million U.S. residents are of French descent, and about 1.6 million speak French at home.An additional 450,000 U.S...
mathematicianMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, best known as the father of
fractal geometryA fractal is "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 is
Sterling ProfessorA 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,
EmeritusEmeritus is an adjective that is used in the title of a retired professor, bishop, or other professional. Emerita is often used as the female equivalent, although avoided by purists, since phrases such as professor emerita are ungrammatical in Latin...
at
Yale UniversityYale University is a private research university in New Haven, Connecticut, and a member of the Ivy League. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States. Yale has produced many notable alumni, including five...
;
IBM FellowAn 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...
Emeritus at the
Thomas J. Watson Research CenterThe 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 Battelle Fellow at the
Pacific Northwest National LaboratoryPacific Northwest National Laboratory is one of the U.S. Department of Energy’s national laboratories, managed by DOE’s Office of Science. PNNL scientists conduct basic and applied research and development to strengthen U.S...
. He was born in
PolandPoland , officially the Republic of Poland , is a country in Central Europe . Poland is 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...
. His family moved to France when he was a child, and he was educated in France. He is a
dual French and AmericanMultiple 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...
citizen. Mandelbrot now lives and works in the
United StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
.
Early years
Mandelbrot was born in
WarsawWarsaw is the capital and largest city of Poland. It is located on the Vistula River roughly from both the Baltic Sea coast and the Carpathian Mountains. Its population as of 2009 was estimated at 1,709,781, and the Warsaw metropolitan area at approximately 2,785,000...
in a
JewishLithuanian Jews are Ashkenazi Jews with roots in the Grand Duchy of Lithuania ....
family from
LithuaniaLithuania , officially the Republic of Lithuania is a country in Northern Europe, the southernmost of the three Baltic states. Situated along the southeastern shore of the Baltic Sea, it shares borders with Latvia to the north, Belarus to the southeast, Poland, and the Russian exclave of...
. Anticipating the threat posed by Nazi Germany, the family fled from Poland to
FranceFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
in 1936 when he was 11. He remained in France through the war to near the end of his college studies. He was born into a family with a strong academic tradition—his mother was a medical doctor and he was introduced to mathematics by two uncles. His uncle,
Szolem MandelbrojtSzolem Mandelbrojt was a Jewish-Polish mathematician. He worked mainly in classical analysis; he was a student of Jacques Hadamard, and became Hadamard's successor as Professor at the Collège de France....
, was a Parisian mathematician. His father, however, made his living trading clothing.
Mandelbrot attended the Lycée Rolin in Paris until the start of
World War IIWorld War II, or the Second World War , was a global military conflict which involved a majority of the world's nations, including all great powers, organized into two opposing military alliances: the Allies and the Axis...
, when his family moved to
TulleTulle is a commune of France, capital of the Corrèze department in the Limousin region in central France and the episcopal see of the eponymous Roman Catholic diocese, the Bishopric of Tulle...
. He was helped by Rabbi
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, Maramures, then Hungary, now Rumania...
, the Rabbi of Brive-la-Gaillarde, to continue his studies. In 1944 he returned to Paris. He studied at the
Lycée du ParcThe 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 , often Anglicized as Lyons, is a city in east-central France in the region Rhône-Alpes, situated between Paris and Marseille. Its name is pronounced in French and Arpitan, and or in English...
and in 1945-47 attended the
École PolytechniqueThe École Polytechnique is the foremost French engineering school. Known for its extremely competitive entrance exam, it produces graduates that occupy outstanding positions in industry and research...
, where he studied under
Gaston JuliaGaston Maurice Julia was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoît Mandelbrot, and the Julia and Mandelbrot fractals are closely related.-Military service:Julia was born in the Algerian town of Sidi Bel Abbes, at...
and
Paul LévyPaul Levy is a US/British author and journalist. He lives with his wife, Penelope Marcus, and children in Oxfordshire and London, UK....
. From 1947 to 1949 he studied at
California Institute of TechnologyThe California Institute of Technology is a private research university located in Pasadena, California, United States. The Institute maintains a strong emphasis on the natural sciences and engineering, and operates and manages NASA's neighboring Jet Propulsion Laboratory...
where he studied aeronautics. Back in France, he obtained a Ph.D. in Mathematical Sciences at the
University of ParisThe historic University of Paris was founded in the mid 12th century, likely between 1160 and 1170 , In 1970 it was reorganized as 13 autonomous universities...
in 1952.
From 1949 to 1957 Mandelbrot was a staff member at the
Centre National de la Recherche ScientifiqueThe 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 StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is a center for theoretical research and intellectual inquiry. The Institute is perhaps best known as the academic home of Albert Einstein, John von Neumann, and Kurt Gödel, after their immigration to the United...
in
Princeton, New JerseyPrinceton, New Jersey is located in Mercer County, New Jersey, United States. Princeton University has been sited in the town since 1756. Although Princeton is a "college town", there are other important institutions in the area, including the Institute for Advanced Study, Educational Testing...
where he was sponsored by
John von NeumannJohn von Neumann was a Hungarian American mathematician who made major contributions to a vast range of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics John...
. In 1955 he married Aliette Kagan and moved to
GenevaGeneva, is the second-most-populous city in Switzerland and is the most populous city of Romandie...
,
SwitzerlandSwitzerland , officially the Swiss Confederation , is a federal republic consisting of 26 states named cantons, with Bern as the seat of the federal authorities...
then
LilleThe 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....
,
FranceFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
.
In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM
Thomas J. Watson Research CenterThe 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 HeightsYorktown Heights is a census-designated place in the town of Yorktown in Westchester County, New York, United States. The population was 7,972 people at the 2000 census.-Geography:Yorktown Heights is located at ....
,
New YorkNew York is a state in the Mid-Atlantic and Northeastern regions of the United States and is the nation's third most populous. The state is bordered by New Jersey and Pennsylvania to the south, and Connecticut, Massachusetts and Vermont to the east...
. He remained at IBM for thirty-two years, becoming an
IBM FellowAn 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
EmeritusEmeritus is an adjective that is used in the title of a retired professor, bishop, or other professional. Emerita is often used as the female equivalent, although avoided by purists, since phrases such as professor emerita are ungrammatical in Latin...
.
Later years
From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on compressing and reliably storing and communicating data...
,
economicsEconomics is the social science that studies 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 dynamicsIn 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 tailA fat tail is a property of some probability distributions exhibiting extremely large kurtosis particularly relative to the ubiquitous normal which itself is an example of an exceptionally thin tail distribution. Fat tail distributions have power law decay...
s and self-similar structure, ran through a multitude of problems encountered in those fields.
Mandelbrot found that price changes in
financial marketIn economics, a financial market is a mechanism that allows people to easily buy and sell financial securities , commodities , and other fungible items of value at low transaction costs and at prices that reflect the efficient-market hypothesis.Financial markets have evolved significantly over...
s did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite
varianceIn probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
. 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 parameterIn 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
cosmologyCosmology is the study of the Universe in its totality, and by extension, humanity's place in it...
. He offered in 1974 a new explanation of
Olbers' ParadoxIn 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 "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
starA star is a massive, luminous ball of plasma that is held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun...
s in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the
Big BangThe Big Bang is the cosmological model of the initial conditions and subsequent development of the Universe that is supported by the most comprehensive and accurate explanations from current scientific evidence and observation...
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
fractalA fractal is "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
geographerA geographer is a scientist whose area of study is geography, the study of Earth's physical environment and human habitat.Though 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
statisticianStatisticians work with theoretical and applied statistics 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.
While on
secondmentSecondment is the transfer of a person from their regular organization for temporary assignment elsewhere, to transfer an employee, official, or soldier temporarily to other duties....
as Visiting Professor of Mathematics at
Harvard UniversityHarvard University is a private university located in Cambridge, Massachusetts and a member of the Ivy League. Founded in 1636 by the colonial Massachusetts legislature, Harvard is the oldest institution of higher learning in the United States and currently comprises ten separate academic units...
in 1979, Mandelbrot began to study fractals called
Julia setIn 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 planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis...
. Building on previous work by
Gaston JuliaGaston Maurice Julia was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoît Mandelbrot, and the Julia and Mandelbrot fractals are closely related.-Military service:Julia was born in the Algerian town of Sidi Bel Abbes, at...
and
Pierre FatouPierre 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 setIn mathematics the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal...
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".
Upon his retirement from
IBMInternational Business Machines Corporation, abbreviated IBM, is a multinational computer technology and IT consulting corporation headquartered in Armonk, Town of North Castle, New York, United States. The company is one of the few information technology companies with a continuous history dating...
in 1987, Mandelbrot joined the
YaleYale University is a private research university in New Haven, Connecticut, and a member of the Ivy League. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States. Yale has produced many notable alumni, including five...
Department of Mathematics. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His awards include the
Wolf PrizeThe Wolf Prize is an international award, that has been presented annually since 1978 to living scientists and artists for "achievements in the interest of mankind and friendly relations among peoples ... irrespective of nationality, race, colour, religion, sex or political views."The prize is...
for Physics in 1993, the
Lewis Fry RichardsonLewis 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 PrizeThe 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 :It is presented by The...
in 2003, and the Einstein Lectureship of the
American Mathematical SocietyThe 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 Mandelbrot27500 Mandelbrot is a Main-belt Asteroid discovered on April 12, 2000 by P. G. Comba at Prescott. It is named for mathematician Benoit Mandelbrot....
was named in his honor. In November 1990, he was made a Knight in the French
Legion of HonourThe Légion d'honneur or Ordre national de la Légion d'honneur is a French order established by Napoleon Bonaparte, First Consul of the First Republic, on 19 May 1802...
. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the
Pacific Northwest National LaboratoryPacific Northwest National Laboratory is one of the U.S. Department of Energy’s national laboratories, managed by DOE’s Office of Science. PNNL scientists conduct basic and applied research and development to strengthen U.S...
. Mandelbrot was promoted to Officer of the French Legion of Honour
in January 2006.
Fractals and regular roughness
Although Mandelbrot coined the term
fractal, 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-similarityIn 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 invarianceIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales are multiplied by a common factor...
, and a (usually) non-integer
Hausdorff dimensionIn mathematics, the Hausdorff dimension is an extended non-negative real number associated to 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
mountainA mountain is a large landform that stretches above the surrounding land in a limited area usually in the form of a peak. A mountain is generally steeper than a hill. The adjective montane is used to describe mountainous areas and things associated with them...
s, coastlines and river basins; the structures of plants,
blood vesselThe blood vessels are the part of the circulatory system that transport 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
lungThe lung or pulmonary system is the essential respiration organ in 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 in the chest on either side of the heart...
s; the clustering of
galaxiesA 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 name is from the Greek root galaxias [γαλαξίας], meaning "milky," a reference...
; and
Brownian motionBrownian motion is the seemingly random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a particle theory....
. Fractals are found in human pursuits, such as
musicMusic is an art form whose medium is sound. Common elements of music are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
,
paintingPainting is the practice of applying paint, pigment, color or other medium to a surface . In art, the term describes both the act and the result, which is called a painting. Paintings may have for their support such surfaces as walls, paper, canvas, wood, glass, lacquer, clay or concrete...
,
architectureFor a topical guide to this subject, see Outline of architecture. Architecture is the art and science of designing and constructing buildings and other physical structures for human shelter or use....
, and
stock marketA stock market is a public market 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....
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 geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
:
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 Nature
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 theoryChaos theory is a branch of mathematics which studies the behavior of certain dynamical systems that may be highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. As a result of this sensitivity, which manifests itself as an exponential growth of...
and other fields of science and mathematics.
Honors and Awards
A partial list of awards received by Mandelbrot:
- 2004 Best Business Book of the Year Award
- AMS Einstein Lectureship
- Barnard Medal
- Caltech Service
- Casimir Frank Natural Sciences Award
- Charles Proteus Steinmetz Medal
- Franklin Medal
|
Harvey Prize
Honda Prize
Humboldt Preis
IBM Fellowship
Japan Prize
John Scott Award
Lewis Fry Richardson Medal |
Medaglia della Presidenza della Repubblica Italiana
Médaille de Vermeil de la Ville de Paris
Nevada Prize
Science for Art
Sven Berggren-Priset
Władysław Orlicz Prize
Wolf Foundation Prize for Physics |
See also
- " How 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
- Mandelbrot Competition
Named 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...
- skewness risk
Skewness 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...
, kurtosis riskIn statistics and decision theory, kurtosis risk denotes that observations are spread in a wider fashion than the normal distribution entails. In other words, fewer observations cluster near the average and more observations populate the extremes either far above or far below the average compared...
Further reading
- The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward, by Benoît Mandelbrot and Richard L. Hudson; Basic Books, 2004; ISBN 0-465-04355-0
- A focus on the exceptions that prove the rule, by Benoît Mandelbrot and Nassim Taleb; Financial Times, 23 March 2006.
- 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 :...
, Hartmut 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 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)
- A Multifractal Walk down Wall Steet by Benoit Mandelbrot, Scientific American, February 1999 http://www.scribd.com/doc/7130201/UnprotectedBenoit-Mandelbrot-A-Multi-Fractal-Walk-Down-at-Wall-Street
- "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature", NOVA
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