Benoît Mandelbrot

Benoît Mandelbrot
Benoit Mandelbrot mg 1804.jpg
Mandelbrot under et foredrag på EPFL, 2007.
Personlig information
Født20. november 1924
Warszawa, Polen
Død14. oktober 2010
Cambridge, Storbritannien Rediger på Wikidata
NationalitetFrankrig Fransk
USA Amerikansk
BopælFrankrig, USA
Uddannelse og virke
Uddannelses­stedÉcole Polytechnique
California Institute of Technology
Université de Paris
Forsknings­områdeMatematiker
ArbejdsstedYale University
International Business Machines (IBM)
Pacific Northwest National Laboratory
Betydningsfulde eleverF. Kenton Musgrave
Eugene F. Fama
Murad S. Taqqu
Nassim Nicholas Taleb
Daniel Zajdenweber
Yuval Gefen
Charles V. Meneveau
Adlai J. Fisher
Laurent E. Calvet
Kendt forFraktalgeometri
Mandelbrotmængden
Nomineringer og priser
UdmærkelserWolf Prize (1993)
Japanprisen (2003)
Information med symbolet Billede af blyant hentes fra Wikidata. Kildehenvisninger foreligger sammesteds.

Benoît Mandelbrot (20. november 192414. oktober 2010) var en fransk matematiker, født i Polen; ansat ved IBM's Thomas J. Watson Research Center i New York 195893 (som fellow fra 1974) og professor ved forskellige amerikanske universiteter, senest Yale fra 1987. I 1975 indførte han betegnelsen fraktalgeometri som en fælles ramme for strukturer, der ikke falder ind under den klassiske geometri, men kan bruges som modeller for objekter i naturen som fx kystlinjer og snefnug. Hans bog The Fractal Geometry of Nature (1982) har været medvirkende til emnets opblomstring og popularitet.

Liv

Han blev født i Warszawa, men familien flyttede til Frankrig, da han var 12 år gammel. Hans familie havde stærke akademiske traditioner, hans mor var doktor, og en af hans farbrødre, Szolem Mandelbrojt, var professor i matematik i Paris.

Mandelbrot studerede i Paris indtil 2. verdenskrig brød ud, hvorefter familien flyttede til Tulle i det sydlige Frankrig. I 1944 vendte han tilbage til Paris og påbegyndte studier i matematik ved École polytechnique med blandt andet Gaston Julia og Paul Pierre Lévy som lærere. Han tog sin eksamen i 1947 og tilbragte derefter to år ved California Institute of Technology, hvor han studerede aerodynamik. Tilbage i Frankrig fortsatte han sine matematiske studier ved Universitetet i Paris og fik der en doktorgrad 1952.

Mellem 1949 og 1957 var Mandelbrot medlem af Centre national de la recherche scientifique, og han tilbragte også et år ved Institute for Advanced Study i Princeton i USA, med støtte fra John von Neumann. I 1955 giftede han sig med Aliette Kagan, og de flyttede til Genève.

I 1958 flyttede familien til USA, hvor Mandelbrot blev ansat ved IBM's J. Watson Research Centre i delstaten New York. Han forblev hos IBM under resten af sit virksomme liv og udnævntes til IBM Fellow, en hæderstitel for fremstående forskningsarbejde.

Efter at have forladt IBM i 1987 blev han Sterling Professor of Mathematical Sciences ved Yale University. Han tildeltes i 2003 den prestigefyldte Japanpris for sin forskning.

Forskning

Fra 1955 arbejdede Mandelbrot med problemer og publicerede artikler inden for så vidt forskellige områder som informationsteori, økonomi og strømningsmekanik.

I 1967 publicerede han en artikel i tidskriftet Science med titlen "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". I denne artikel diskuterer Mandelbrot selvsimulerende kurver, dvs. kurver med mønstre og strukturer, der gentager sig selv regelbundet i stadig mindre skala, så at alt ser ens ud uafhængigt af i hvilken skala, man betragter dem. Sådanne kurver har en fraktal dimension mellem 1 og 2 og er eksempler på "fraktaler", et begreb Mandelbrot siden formulerede; ordet anvendte han første gang i 1975 i publikationen Les objets fractals, forme, hasard et dimension.

I 1979, mens han var gæsteprofessor ved Harvard University, begyndte Mandelbrot at studere den fraktal, som kaldes Juliamængden. Han startede fre tidligere arbejder af Gaston Julia og Pierre Fatou, og ved hjælp af datamaskiner fremstillede Mandelbrot billeder af Juliamængder til ligningen z2c. Ved undersøgelser af, hvordan disse Juliamængder afhang af den komplekse parameter c, opdagede han den efter ham opkaldte Mandelbrotmængde. I 1982 publicerede han sine opdagelser i den meget kendte bog The Fractal Geometry of Nature. Bogen beskriver ikke kun Mandelbrot- og Juliamængderne, men desuden et stort antal fraktaler, som bygger på det såkaldte L-system. Mest kendt af disse er antagelig "Abetræet" (engelsk: monkeys' tree), som i lighed med mange andre fraktaler i bogen bygger på von Kochs snefnug, en kurve, som Mandelbrot ofte har anvendt som udgangspunkt for sine overvejelser og slutninger.


Litteratur

  • James Gleick: Chaos: Making a New Science, Viking Penguin 1987; ISBN 0-14-009250-1

Eksterne henvisninger

Medier brugt på denne side

Mandel zoom 04 seehorse tail.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 4 of a zoom sequence: The central endpoint of the "seahorse tail" is also a Misiurewicz point.
  • Coordinates of the center: Re(c) = -.743,566,9, Im(c) = . 131,402,3
  • Horizontal diameter of the image: .002,287,8
  • Magnification relative to the initial image: 1,344.9
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 14 satellite julia island.jpg
Forfatter/Opretter: Wolfgang Beyer, Licens: CC BY-SA 3.0
* Partial view of the Mandelbrot set. Step 14 of a zoom sequence: On the first sight these islands seem to consist on infinitely many parts like Cantor sets, as it is actually the case for the corresponding Julia set Jc. Here they are connected by tiny structures so that the whole represents a simply connected set. These tiny structures meet each other at a satellite in the center which is too small to be recognized at this magnification. The value of c for the corresponding Jc is not that of the image center but has relative to the main body of the Mandelbrot set the same position as the center of this image relative to the satellite shown in zoom step 7.
  • Coordinates of the center: Re(c) = -.743,643,887,037,151, Im(c) = .131,825,904,205,330
  • Horizontal diameter of the image: .000,000,000,051,299
  • Magnification relative to the initial image: 59,979,000,000
Mandel zoom 09 satellite head and shoulder.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 9 of a zoom sequence: The "seahorse valley" of the satellite. All the structures from the image of zoom step 1 reappear.
  • Coordinates of the center: Re(c) = -.743,644,099,61, Im(c) = .131,826,046,88
  • Horizontal diameter of the image: .000,000,662,08
  • Magnification relative to the initial image: 4,647,300
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 10 satellite seehorse valley.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 10 of a zoom sequence: Double-spirals and "seahorses". Unlike the image of zoom step 2 they have appendices consisting of structures like "seahorse tails". This demonstrates the typical linking of n+1 different structures in the environment of satellites of the order n, here for the simplest case n=1.
  • Coordinates of the center: Re(c) = -.743,643,862,69, Im(c) = .131,825,902,71
  • Horizontal diameter of the image: .000,000,135,26
  • Magnification relative to the initial image: 22,748,000
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 02 seehorse valley.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 2 of a zoom sequence: On the left double-spirals, on the right "seahorses".
  • Coordinates of the center: Re(c) = -.759,856, Im(c) = .125,547
  • Horizontal diameter of the image: .051,579
  • Magnification relative to the initial image: 59,654
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 00 mandelbrot set.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Mandelbrot set. Initial image of a zoom sequence: Mandelbrot set with continuously colored environment.
  • Coordinates of the center: Re(c) = -.7, Im(c) = 0
  • Horizontal diameter of the image: 3.076,9
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 05 tail part.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 5 of a zoom sequence: Part of the "tail". There is only one path consisting of the thin structures which leads through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" on the inner and outer sides of the "tail". It makes sure, that the Mandelbrot set is a so called simply connected set. That means there are no islands and no loop roads around a hole.
  • Coordinates of the center: Re(c) = -.743,649,90, Im(c) = . 131,882,04
  • Horizontal diameter of the image: .000,738,01
  • Magnification relative to the initial image: 4,169.2
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 08 satellite antenna.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 8 of a zoom sequence: "Antenna" of the satellite. Several satellites of second order can be recognized.
  • Coordinates of the center: Re(c) = -.743,644,786,0, Im(c) = .131,825,253,6
  • Horizontal diameter of the image: .000,002,933,6
  • Magnification relative to the initial image: 1,048,800
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 06 double hook.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 6 of a zoom sequence: Satellite. The two "seahorse tails" are the beginning of a series of concentrical crowns with the satellite in the center.
  • Coordinates of the center: Re(c) = -.743,640,85, Im(c) = .131,827,33
  • Horizontal diameter of the image: .000,120,68
  • Magnification relative to the initial image: 25,497
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 11 satellite double spiral.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 4 of a zoom sequence: The central endpoint of the "seahorse tail" is also a Misiurewicz point.
  • Coordinates of the center: Re(c) = -.743,643,900,055, Im(c) = .131,825,890,901
  • Horizontal diameter of the image: .000,000,049,304
  • Magnification relative to the initial image: 62,407,000
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 13 satellite seehorse tail with julia island.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 13 of a zoom sequence: Part of the "double-hook".
  • Coordinates of the center: Re(c) = -.743,643,887,173,42, Im(c) = .131,825,904,251,82
  • Horizontal diameter of the image: .000,000,000,598,49
  • Magnification relative to the initial image: 5,141,100,000
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 12 satellite spirally wheel with julia islands.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 12 of a zoom sequence: In the outer part of the appendices islands of structures can be recognized. They have a shape like Julia sets Jc. The largest of them can be found in the center of the "double-hook" on the right side.
  • Coordinates of the center: Re(c) = -.743,643,888,570,6, Im(c) = .131,825,904,312,4
  • Horizontal diameter of the image: .000,000,004,149,3
  • Magnification relative to the initial image: 741,550,000
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 01 head and shoulder.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 1 of a zoom sequence: Gap between the "head" and the "body" also called the "seahorse valley".
  • Coordinates of the center: Re(c) = -.875,91, Im(c) = .204,64
  • Horizontal diameter of the image: .531,84
  • Magnification relative to the initial image: 5.785,4
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Mandel zoom 07 satellite.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 7 of a zoom sequence: Each of these crowns consists of similar "seahorse tails". Their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center mentioned in zoom step 5 passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head". Also observe that the starting view is located in the center.
  • Coordinates of the center: Re(c) = -.743,643,135, Im(c) = .131,825,963
  • Horizontal diameter of the image: .000,014,628
  • Magnification relative to the initial image: 210,350
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.
Benoit Mandelbrot mg 1804.jpg
Forfatter/Opretter: Rama, Licens: CC BY-SA 2.0 fr
Benoît Mandelbrot at the EPFL, on the 14h of March 2007
Mandel zoom 03 seehorse.jpg
Forfatter/Opretter: Created by Wolfgang Beyer with the program Ultra Fractal 3., Licens: CC BY-SA 3.0
Partial view of the Mandelbrot set. Step 3 of a zoom sequence: "Seahorse" upside down. Its "body" is composed by 25 "spokes" consisting of 2 groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These 2 groups can be attributed by some kind of metamorphosis to the 2 "fingers" of the "upper hand" of the Mandelbrot set. Therefore the number of "spokes" increases from one "seahorse" to the next by 2. The "hub" is a so called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted satellite can be recognized.
  • Coordinates of the center: Re(c) = -.743,030, Im(c) = .126,433
  • Horizontal diameter of the image: .016,110
  • Magnification relative to the initial image: 190.99
  • Created by Wolfgang Beyer with the program Ultra Fractal 3.
  • Uploaded by the creator.