Big Think Interview With Benoit Mandelbrot
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A conversation with the mathematician and Professor Emeritus at Yale University.
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Benoît B. Mandelbrot:
Benoît B. Mandelbrot is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was born in Poland and educated in France, and is now a dual French and American citizen. His books include the classic "The Fractal Geometry of Nature" (1982).
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TRANSCRIPT:
Benoit Mandelbrot: Benoit Mandelbrot, Sterling Professor, Emeritus at Yale University, IBM Fellow Emeritus at IBM Research Center.
Question: What is fractal geometry?
Benoit Mandelbrot: Well, regular geometry, the geometry ofrnEuclid, is concerned with shapes which are smooth, except perhaps for cornersrnand lines, special lines which are singularities, but some shapes in nature arernso complicated that they are equally complicated at the big scale and comerncloser and closer and they don’t become any less complicated. Closer and closer, or you go farther orrnfarther, they remain equally complicated. rnThere is never a plane, never a straight line, never anything smooth andrnordinary. The idea is very, veryrnvague, is expressed - it’s an expression of reality.
Fractal geometry is a new subject and each definition I tryrnto give for it has turned out to be inappropriate. So I’m now being cagey and saying there are very complexrnshapes which would be the same from close by and far away.
Question: What does it mean to say that fractal shapes arernself-similar?
Benoit Mandelbrot: Well, if you look at a shape like arnstraight line, what’s remarkable is that if you look at a straight line fromrnclose by, from far away, it is the same; it is a straight line. That is, the straight line has arnproperty of self-similarity. Eachrnpiece of the straight line is the same as the whole line when used to a big orrnsmall extent. The plane again hasrnthe same property. For a longrntime, it was widely believed that the only shapes having these extraordinaryrnproperties are the straight line, the whole plane, the whole space. Now in a certain sense, self-similarityrnis a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similarrnagain, the same seen from close by and far away, and which are far from beingrnstraight or plane or solid. Andrnthose shapes, which I studied and collected and put together and applied inrnmany, many domains, I called fractals.
Question: How can complex natural shapes be representedrnmathematically?
Benoit Mandelbrot: Well, historically, a mountain could notrnbe represented, except for a few mountains which are almost like cones. Mountains are very complicated. Ifrnyou look closer and closer, you find greater and greater details. If you look away until you find thatrnbigger details become visible, and in a certain sense this same structurernappears at those scales. If yournlook at coastlines, if you look at that them from far away, from an airplane,rnwell, you don’t see details, you see a certain complication. When you come closer, the complicationrnbecomes more local, but again continues. rnAnd come closer and closer and closer, the coastline becomes longer andrnlonger and longer because it has more detail entering in. However, these details amazingly enoughrnenters this certain this certain regular fashion. Therefore, one can study a coastline **** object because therngeometry for that existed for a long time, and then I put it together and appliedrnit to many domains.
Question: What was the discovery process behind thernMandelbrot set?
Benoit Mandelbrot: The Mandelbrot set in a certain sense isrna **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but notrnhappy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, whornwas a very well-known pure mathematician, wanted me to study a certain theoryrnwhich was then many years old, 30 years old or something, but had in a wayrnstopped developing. When he wasrnyoung he had tried to get this theory out of a rut and he didn’t succeed,rnnobody succeeded.
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