Editing Benoit Mandelbrot (section)

===Developing "fractal geometry" and the Mandelbrot set===
As a visiting professor at [[Harvard University]], Mandelbrot began to study mathematical objects called [[Julia set]]s that were [[Invariant (mathematics)|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. While investigating the topology of these Julia sets, he studied the [[Mandelbrot set]] which was introduced by him in 1979.

[[File:Mandelbrot p1130876.jpg|thumb|right|Mandelbrot speaking about the [[Mandelbrot set]], during his acceptance speech for the [[Légion d'honneur]] in 2006]]

In 1975, Mandelbrot coined the term ''[[fractal]]'' to describe these structures and first published his ideas in the French book ''Les Objets Fractals: Forme, Hasard et Dimension'', later translated in 1977 as ''Fractals: Form, Chance and Dimension''.<ref>''Fractals: Form, Chance and Dimension'', by Benoît Mandelbrot; W H Freeman and Co, 1977; {{isbn|0-7167-0473-0}}</ref> According to computer scientist and physicist [[Stephen Wolfram]], the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before".<ref name=Wolfram>{{cite news| last= Wolfram| first= Stephen| url= https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 |title= The Father of Fractals| archiveurl= https://web.archive.org/web/20170825102714/https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 |archivedate=25 August 2017 | work= [[The Wall Street Journal]]| date= 22 November 2012| access-date= }}</ref> Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":

{{blockquote|Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.<ref name=Wolfram />}}

Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. [[Fern|Fern leaves]] and [[Romanesco broccoli|Romanesque broccoli]] are two examples from nature."<ref name=Wolfram /> He points out an unexpected conclusion:

{{blockquote|One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot.<ref name=Wolfram />}}

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 able to create fractal images using graphics computer code, images that 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 also saw himself as a "would-be Kepler", after the 17th-century scientist [[Johannes Kepler]], who calculated and described the orbits of the planets.<ref>{{cite web| last= Ivry| first= Benjamin| url= http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all |title= Benoit Mandelbrot Influenced Art and Mathematics| archiveurl= https://web.archive.org/web/20130602171300/http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all |archivedate= 2 June 2013 | website=[[The Jewish Daily Forward]] | date= 17 November 2012| access-date= }}</ref>

[[File:Newton-lplane-Mandelbrot.jpg|thumb|A Mandelbrot set]]

Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke:

{{blockquote|Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.<ref name=Clarke>{{citation| url= https://www.youtube.com/watch?v=Lk6QU94xAb8 |title= Arthur C Clarke – Fractals – The Colors Of Infinity|date= 25 December 2010| archiveurl= https://web.archive.org/web/20170531193057/https://www.youtube.com/watch?v=Lk6QU94xAb8 |archivedate= 31 May 2017 | access-date= | via= YouTube}}</ref>}}

According to Clarke, "the [[Mandelbrot set]] is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally ''infinite'' complexity?" Clarke also notes an "odd coincidence":

<blockquote>the name Mandelbrot, and the word "[[mandala]]"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas.<ref name=Clarke /></blockquote>

In 1982, Mandelbrot expanded and updated his ideas in ''[[The Fractal Geometry of Nature]]''.<ref>{{cite book| url= https://books.google.com/books?id=xJ4qiEBNP4gC |title= The Fractal Geometry of Nature |archiveurl= https://web.archive.org/web/20151130231048/https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover |archivedate=30 November 2015 | first= Benoît| last= Mandelbrot| publisher= W H Freeman & Co| year= 1982 |isbn= 0-7167-1186-9}}</ref> This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "[[Artifact (observational)|program artifacts]]".

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.<ref name="wos44">{{cite web|url=http://www.webofstories.com/play/10483|title=Benoît Mandelbrot • IBM: background and policies |last=Mandelbrot |first=Benoît |author2=Bernard Sapoval |author3=Daniel Zajdenweber|date=May 1998|publisher=[[Web of Stories]]|access-date=17 October 2010|archive-date=8 September 2011|archive-url=https://web.archive.org/web/20110908162215/http://www.webofstories.com/play/10483|url-status=live}}</ref> He joined the Department of Mathematics at [[Yale]], and obtained his first [[tenure]]d post in 1999, at the age of 75.<ref name="Tenner">{{cite news|url=https://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|title=Benoît Mandelbrot the Maverick, 1924–2010|last=Tenner|first=Edward|date=16 October 2010|work=[[The Atlantic]]|access-date=16 October 2010|archive-date=18 October 2010|archive-url= https://web.archive.org/web/20101018132145/http://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|url-status=live}}</ref> At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences.