Benoit Mandelbrot

Benoit B.  ^{[n 1]} Mandelbrot  ^{[n 2]} (20 November 1924 to 14 October 2010) was a Polish-born, French and American mathematician with broad interests in the practical-sciences, met name Regarding what he, key as “the art of roughness “of physical phenomena and” the uncontrolled element in life. ” ^{[6] [7]} He referred to himself as a “fractalist”. ^[8] He was honored for his contribution to this club to the field of fractal geometry , welke included coining the word “fractal”, “as well as ontwikkelingslanden a theory of” roughness and self-similarity “in nature. ^[9]

In 1936, while he was a child, Mandelbrot’s family migrated to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and Receiving a master’s degree in Aeronautics from the California Institute of Technology . He spent musts or his career in zowel the United States and France, maintaining dual French and American citizenship. In 1958, he Began a 35-year career at IBM , where he became an IBM Fellow , and Periodically took leaves of Absence to teach at Harvard University . At Harvard, volgende the publication of his study of US commodity markets in relatie to cotton futures, he taught economics and toegepast sciences.

Because of his access to IBM’s computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his Discovering the Mandelbrot set in 1979. He Showed how visual complexity kan be created from simple rules. He zegt dat things Typically Considered to be “rough”, a “mess” or “chaotic”, like clouds or Shorelines, actually had a “degree of order.” ^[10] His math and geometry-centered research career included contributions to industry leaders such areas as statistical physics , meteorology , hydrology , Geomorphology , anatomy , taxonomy , neurology , linguistics , information technology , computer graphics , economics , geology , medicine , physical kosmologie , engineering , chaos theory , econophysics , metallurgy and the social sciences . ^[11]

Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University , where he was the oldest professor in Yale’s history not to receive tenure. ^[12]Mandelbrot ook hero positions at the Pacific Northwest National Laboratory , Université Lille Nord de France , Institute for Advanced Study and Centre National de Recherche Scientifique . During his career, he RECEIVED over 15 honorary doctorates and served on many science journals, Along with winning numerous awards. His Autobiography, The Fractalist: Memoir of a Scientific Maverick , was published in 2012.

Early years

Mandelbrot was born in Warsaw prolongation the Second Polish Republic . His family was Jewish . Hoewel de his Father made his living trading clothing, the family had a strong academic tradition and his mother was a dental surgeon. ^[13] He was first introduced to mathematics at two or his Uncles, one or Whom, Szolem Mandelbrojt was a mathematician who resided in Paris. Volgens to Mandelbrot’s Autobiography, The Fractalist – Memoir of a Scientific Maverick , ^[14] “[t] he love of his [Szolem’s] mind was mathematics”. ^{[8] : 16}

The family emigrated from Poland to France in 1936-when he was 11. “The fact dat my parents, as economic and political refugees, joined Szolem in France saved our lives,” he writes. ^{[8] : 17 [15]} Mandelbrot attended the Lycée Rolin in Paris Until the start of World War II , als his family-then moved to Tulle , France. He was helped by Rabbi David Feuer worker , the Rabbi of Brive-la-Gaillarde , to continuous his studies. ^{[8] : 62-63 [16]} Much of France was Occupied by the Nazis at the time, and Mandelbrot recalls this period:

Our constant fear was dat a sufficiently Determined foe Might report us to an authority and we mention anything be penny to our deaths. This happened to a close friend from Paris, Zina Morhange, a physician in a nearby county seat. Simply to Eliminate the competition, Another physician denounced her … We escaped this fate. Who knows why? ^{[8] : 49}

In 1944, Mandelbrot Returned to Paris, studied at the Lycée du Parc in Lyon , and in 1945 to 1947 attended the Ecole 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 . ^[2] Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952. ^[13]

Research career

From 1949 to 1958, Mandelbrot was a staff member at the Centre National de 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 University Lille Nord de France . ^[17] 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 . ^[17] He remained at IBM for 35 years, Becoming an IBM Fellow , and later Fellow Emeritus . ^[13]

From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in toegepast areas zoals information theory , economics, and fluid dynamics .

States of randomness and financial markets

Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution , but Rather Lévy stable distributions keeping theoretically infinite variance . He found, for example, therein cotton prices Followed a Lévy stable distribution with parameter α equal to 1.7 Rather dan 2 than in a Gaussian distribution. “Stable” distributions harbor the property therein the sum of many instances of a random variable follows the co-distribution but with a larger scale parameter . ^[18]

Developing “fractal geometry” and the Mandelbrot set

As a visiting professor at Harvard University , Mandelbrot Began to study fractals called Julia sets therein ulcers invariant under certainement 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. While investigating the topology or synthesis Julia sets, have studied the Mandelbrot set fractal dat is now named after im. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature . ^[19] This Influential work brought` fractals withinto the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as ” program artifacts .”

In 1975, Mandelbrot coined the term fractal to DESCRIBE synthesis structures and first published his ideas, and later translated, Fractals: Form, Chance and Dimension . ^[20] volgens to mathematics scientist Stephen Wolfram , the book was a “Breakthrough” for Mandelbrot, who Until dan mention anything Typically “apply Fairly straightforward mathematics … to areas dat had barely seen the light or serious mathematics voordat.” ^[10] Wolfram adds therein as a result of this new research, he was no longer a “wandering scientist”, and later called him “the Father of fractals”

Mandelbrot ended up doing a great piece of science and Identifying a much stronger and more fundamentele idea pit simply, dat there are some geometric shapes, welke have called “fractals” therein are equally “rough” at all scales. No matter how close you look, they ‘never get simpler, much as the section or a rocky coastline u can see at your feet looks just as Jagged than the stretch u can see from space. ^[10]

Wolfram briefly describes fractals as a form of geometric Repetition, “in welke narrower and narrower copies or a pattern are successively nested inside eachother, so dat de co intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesco broccoli are two examples from nature. ” ^[10] He points out an unexpected conclusion:

One Might harbor thought dat zoals a simple and fundamentele form or regularity mention anything about port leg studied for hundreds, if not duizendtallen or years. But it was not. In fact, it rose to Prominence only over the Past 30 or so years-almost entirely through the policymaking or one man, the mathematician Benoit Mandelbrot. ^[10]

Mandelbrot-used the term “fractal” as it derived from the Latin word “fractus”, defined as broken or shattered glass. Using the newly developed IBM computers at his Disposal, Mandelbrot was loveable to create fractal images using graphic computer code, images therein an interviewer DESCRIBED as looking like “the Delirious exuberance of the 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body. ” He’ll be saw himself as a “would-be Kepler” after the 17th-century scientist Johannes Kepler , who calculated and DESCRIBED the orbits of the planets. ^[21]

Mandelbrot, however, never have been fact represented Inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke:

Exploring this set I Certainly never had the feeling of invention It. I never had the feeling dat my imagination was rich enough to invent all Those extraordinary things on Discovering them. Way Down were there, even though nobody had seen Them voordat. It’s marvelous, a very simple formula wordt uitgelegd all synthesis very complicated things. So the goal of science is starting with a mess, and explanatory it with a simple formula, a child or dream of science. ^[22]

Volgens to Clarke, “the Mandelbrot set is indeed one of the musts Astonishing Discoveries in the entire history of mathematics. Who Could port Dreamed therein zoals an incredibly simple equation Could port generated images or In English one infinite complexity?” Clarke ook notes an “odd Coincidence,” “the names Mandelbrot, and the word” mandala “-for a religious symbol-welke I’m sure is a pure Coincidence, but indeed the Mandelbrot set does seem to contain ‘an enormous number of mandalas . ” ^[22]

Mandelbrot left IBM in 1987, after 35 years and 12 days, als IBM decided to end pure research in his division. ^[23] He joined the Department of Mathematics at Yale , and obtained his first tenured post in 1999 at the age of 75. ^[24] At the time of his Retirement in 2005, he was Sterling Professor of Mathematical Sciences.

Fractals and the “theory of roughness”

Mandelbrot created the first-ever “theory of roughness,” and he saw “roughness” in the shapes of mountains, coastlines and river basins ; the structures of plants, blood vessels and lungs ; the clustering of galaxies . His personal quest was to create some mathematical formula to measure the overall “roughness” or zoals objects in nature. ^{[8] : xi} He Began at Asking himself verschillende childhood or questions related to nature:

Can geometry deliver what the Greek root of zijn names [geo-] seemed to promise-truthful measurement, not only of cultivated fields Along the Nile River but ook or Untamed Earth? ^{[8] : xii}

In his paper entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension published in Science in 1967 Mandelbrot discusses self-similar curves dat port Hausdorff dimension therein are examples of fractals , hoewel de Mandelbrot does not use this term in the paper, as he did not coin it Until 1975. The paper is one of Mandelbrot’s first publications on the topic or fractals. ^{[25] [26]}

Mandelbrot emphasized the use of fractals as realistic and useful models for Describing many “rough” phenomena in the real world. He concluded that ‘real roughness is of or in fractal and kan be Measured. ” ^{[8] : 296} hoewel de Mandelbrot coined the term ” fractal “, some of the mathematical objects have Presented in The Fractal Geometry of Nature had leg to post DESCRIBED by other mathematicians. Before Mandelbrot, however, they ‘ulcers regarded as isolated Curiosities with Unnatural and non-INTUITIVE properties. Mandelbrot brought` synthesis objects together for the first time and turned Them JSON essential tools for the long-stalled effort to extendwatchlist the scope of science to explanatory non-smooth, “rough” objects in the real world. His methods of research ulcers zowel old and new:

The form or geometry I are getting favored is the oldest, must concrete, and must inclusive, specifiek Empowered by the eye and helped by the hand and, today, transformed by the computer … Bringing an element of unity to the worlds of knowing and feeling … and, unwittingly, as a bonus, for the purpose of customizing beauty. ^{[8] : 292}

Fractals are’ll be found in human pursuits, zoals music, painting, architecture, and stock market prices. Mandelbrot fractals believed therein, far from being Unnatural, ulcers in many ways more INTUITIVE and natural dan the artificially smooth objects or traditional Euclidean geometry :

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 bone called a visionary ^[27] and a maverick. ^[28] 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 areas of science and mathematics.

Mandelbrot ook pit his ideas to work in kosmologie. 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 therein if the stars in the universe ulcers fractally distributed (for example, like Cantor dust ), it mention anything not be Necessary to Rely on the Big Bang theory to explain the paradox. His model mention anything not rule out a big bang, but mention anything allow directive for a dark sky as if the Big Bang had not occurred. ^[29]

Awards and receptacles

Mandelbrot’s 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, ^[30] 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 . ^[31] Mandelbrot was promoted to Officer of the Legion of Honour in January 2006. ^[32] An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 COMMENCEMENT exercises. ^[33]

A partial list of awards RECEIVED by Mandelbrot: ^[34]

Death and legacy

Mandelbrot mayest from pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts on 14 October 2010. ^[1][37] -reacting to news or his death, mathematician Heinz-Otto Peitgen zegt: “[I] f we talk about impact inside mathematics, and applications in the sciences, he is one of the must important figures of the last fifty years. ” ^[1]

Chris Anderson , TED conference curator, DESCRIBED Mandelbrot as “an icon who changed how we see the world”. ^[38] Nicolas Sarkozy , President of France at the time of Mandelbrot’s death, zegt Mandelbrot had “a powerful, original mind dat never shied away from Innova and shattering preconceived notions [… h] is work, developed entirely outside mainstream research, led to modern information theory. ” ^[39]Mandelbrot’s Obituary in The Economist points out his fame as “celebrity beyond the academy” and lauds im as the “Father of fractal geometry”. ^[40]

Best-selling essayist-author Nassim Nicholas Taleb , a Mandelbrot protege and a scientific adviser at Universa Investments , has remarked dat Mandelbrot’s book The (Mis) Behavior of Markets in his opinion “The DEEPEST and musts realistic finance book ever published.” ^[9]

References in popular culture

In 2004, the American singer-songwriter Jonathan Coulton wrote “The Mandelbrot Set”, a song dedicated to Mandelbrot and his famous fractal.

Notes

References