Benoît Mandelbrot(1), fraktale

Benoît Mandelbrot

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All my life, I have enjoyed the reputation of being someone who disrupted prevailing ideas. Now that I'm in my 80th year, I can play on my age and provoke people even more.

( – ) was a -born French-American mathematician known as the "father of geometry".

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Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless."

· How Long Is the Coast of Britain?

o Part of the title of his paper "" published in Science (1967)

· Being a language, mathematics may be used not only to inform but also, among other things, to seduce.

o Fractals : Form, chance and dimension (1977)

· A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales...

o As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

· I claim that many patterns of Nature are so irregular and fragmented, that, compared with — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."

o As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

· Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. The rare scholars who are nomads by choice are essential to the intellectual welfare of the subtle disciplines

o Benoit Mandelbrot cited in (1987) Chaos: Making a New Science p.70

· I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit.

o As quoted in Encyclopedia of World Biography (1997) edited by

· For most of my life, one of the persons most baffled by my own work was myself.

o Lecture at the University of Maryland (March 2005)

If you have a hammer, use it everywhere you can, but I do not claim that everything is fractal.

· Engineering is too important to wait for science.

o As quoted in

· If you have a hammer, use it everywhere you can, but I do not claim that everything is fractal.

o As quoted in "Fractal Finance" by Greg Phelan in Yale Economic Review (Fall 2005)

The Fractal Geometry of Nature (1982)

Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.

· 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.

· A fractal is by definition a set for which the strictly exceeds the topological dimension.

· Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.

New Scientist interview (2004)

Quotations from

The first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it.

· I always felt that science as the preserve of people from Oxbridge or Ivy League universities — and not for the common mortal — was a very bad idea.

There is no single rule that governs the use of geometry. I don't think that one exists.

· There is nothing more to this than a simple iterative formula. It is so simple that most children can program their home computers to produce the . ... Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it. It was as if somehow I had seen it before. Of course I hadn't. No one had seen it. No one had described it. The fact that a certain aspect of its mathematical nature remains mysterious, despite hundreds of brilliant people working on it, is the icing on the cake to me.

· There is no single rule that governs the use of geometry. I don't think that one exists.

· The Mandelbrot set is the modern development of a theory developed independently in 1918 by and . Julia wrote an enormous book — several hundred pages long — and was very hostile to his rival Fatou. That killed the subject for 60 years because nobody had a clue how to go beyond them. My uncle didn't know either, but he said it was the most beautiful problem imaginable and that it was a shame to neglect it. He insisted that it was important to learn Julia's work and he pushed me hard to understand how equations behave when you iterate them rather than solve them. At first, I couldn't find anything to say. But later, I decided a computer could take over where Julia had stopped 60 years previously.

· My life seemed to be a series of events and accidents. Yet when I look back I see a pattern.

· What motivates me now are ideas I developed 10, 20 or 30 years ago, and the feeling that these ideas may be lost if I don't push them a little bit further.

The Mandelbrot set covers a small space yet carries a large number of different implications...

· The most important thing I have done is to combine something esoteric with a practical issue that affects many people. In this spirit, the stock market is one of the most attractive things imaginable. Stock-market data is abundant so I can check everything. Financial markets are very influential and I want to be part of this field now that it is maturing.

· There is a problem that is specific to financial markets. In most fields of research, when someone makes an important finding, they publish it. In the case of prices, they set up a firm and sell advice about their discovery. If they can make money from it, they will. So the research into market dynamics is a closed field.

· In a different era, I would have called myself a natural philosopher. All my life, I have enjoyed the reputation of being someone who disrupted prevailing ideas. Now that I'm in my 80th year, I can play on my age and provoke people even more.

· My work is more varied than at any other point in my life. I am still carrying out research in pure mathematics. And I am working on an idea that I had several years ago on negative dimensions. ... Negative dimensions are a way of measuring how empty something is. In mathematics, only one set is called empty. It contains nothing whatsoever. But I argued that some sets are emptier than others in a certain useful way. It is an idea that almost everyone greets with great suspicion, thinking I've gone soft in the brain in my old age. Then I explain it and people realise it is obvious. Now I'm developing the idea fully with a colleague. I have high hopes that once we write it down properly and give a few lectures about it at suitable places that negative dimensions will become standard in mathematics.

· The Mandelbrot set covers a small space yet carries a large number of different implications. Is it a fitting epitaph? Absolutely.

A Theory of Roughness (2004)

The notion that these conjectures might have been reached by pure thought — with no picture — is simply inconceivable.

· There is a saying that every nice piece of work needs the right person in the right place at the right time. For much of my life, however, there was no place where the things I wanted to investigate were of interest to anyone. So I spent much of my life as an outsider, moving from field to field, and back again, according to circumstances. Now that I near 80, write my memoirs, and look back, I realize with wistful pleasure that on many occasions I was 10, 20, 40, even 50 years "ahead of my time.

· My ambition was not to create a new field, but I would have welcomed a permanent group of people having interests close to mine and therefore breaking the disastrous tendency towards increasingly well-defined fields. Unfortunately, I failed on this essential point, very badly. Order doesn't come by itself.

· My efforts over the years had been successful to the extent, to take an example, that fractals made many mathematicians learn a lot about physics, biology, and economics. Unfortunately, most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, had been changed by considering the new problems I raised, but largely went their own way.

· For many years I had been hearing the comment that fractals make beautiful pictures, but are pretty useless. I was irritated because important applications always take some time to be revealed. For fractals, it turned out that we didn't have to wait very long. In pure science, fads come and go. To influence basic big-budget industry takes longer, but hopefully also lasts longer.

· One of my conjectures was solved in six months, a second in five years, a third in ten. But the basic conjecture, despite heroic efforts rewarded by two Fields Medals, remains a conjecture, now called MLC: the Mandelbrot Set is locally connected. The notion that these conjectures might have been reached by pure thought — with no picture — is simply inconceivable.

Today, after the fact, the boundary of Brownian motion might be billed as a "natural" concept. But yesterday this concept had not occurred to anyone. And even if it had been reached by pure thought, how could anyone have proceeded to the dimension 4/3?

· I always saw a close kinship between the needs of "pure" mathematics and a certain hero of Greek mythology, . The son of Earth, he had to touch the ground every so often in order to reestablish contact with his Mother; otherwise his strength waned. To strangle him, Hercules simply held him off the ground. Back to mathematics. Separation from any down-to-earth input could safely be complete for long periods — but not forever. In particular, the mathematical study of deserved a fresh contact with reality.

· When you seek some unspecified and hidden property, you don't want extraneous complexity to interfere. In order to achieve homogeneity, I decided to make the motion end where it had started. The resulting motion biting its own tail created a distinctive new shape I call Brownian cluster. ... Today, after the fact, the boundary of Brownian motion might be billed as a "natural" concept. But yesterday this concept had not occurred to anyone. And even if it had been reached by pure thought, how could anyone have proceeded to the dimension 4/3? To bring this topic to life it was necessary for the Antaeus of Mathematics to be compelled to touch his Mother Earth, if only for one fleeting moment.

The infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales.

· How could it be that the same technique applies to the Internet, the weather and the stock market? Why, without particularly trying, am I touching so many different aspects of many different things?
A recent, important turn in my life occurred when I realized that something that I have long been stating in footnotes should be put on the marquee. I have engaged myself, without realizing it, in undertaking a theory of roughness. Think of color, pitch, heaviness, and hotness. Each is the topic of a branch of physics. Chemistry is filled with acids, sugars, and alcohols; all are concepts derived from sensory perceptions. Roughness is just as important as all those other raw sensations, but was not studied for its own sake. ... I was not particularly precocious, but I'm particularly long-lived and continue to evolve even today. Above a multitude of specialized considerations, I see the bulk of my work as having been directed towards a single overarching goal: to develop a rigorous analysis for roughness. At long last, this theme has given powerful cohesion to my life ... my fate has been that what I undertook was fully understood only after the fact, very late in my life.

· To appreciate the nature of fractals, recall 's splendid manifesto that "Philosophy is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which one wanders about in a dark labyrinth." Observe that circles, ellipses, and parabolas are very smooth shapes and that a triangle has a small number of points of irregularity. Galileo was absolutely right to assert that in science those shapes are necessary. But they have turned out not to be sufficient, "merely" because most of the world is of infinitely great roughness and complexity. However, the infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales.

· A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Many plants are like that. A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don't get something smooth but irregularities at a smaller scale.

· Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory, and besides were my love when I was a young man. Cauliflowers exemplify a second area of great simplicity, that of shapes which appear more or less the same as you look at them up close or from far away, as you zoom in and zoom out.
Before my work, those shapes had no use, hence no word was needed to denote them. My work created such a need and I coined "fractals."

An extraordinary amount of arrogance is present in any claim of having been the first in "inventing" something. It's an arrogance that some enjoy, and others do not. Now I reach beyond arrogance when I proclaim that fractals had been pictured forever but their true role remained unrecognized and waited for me to be uncovered.

· Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I'm asserting very strongly is that, when some real thing is found to be un-smooth, the next mathematical model to try is fractal or multi-fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is "not even fractal" is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fracta...

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Benoît Mandelbrot(1), fraktale

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