Big Think Interview With Benoit Mandelbrot
Question: What is fractal geometry?\r\n\r\n
Benoit Mandelbrot: Well, regular geometry, the geometry of\r\nEuclid, is concerned with shapes which are smooth, except perhaps for corners\r\nand lines, special lines which are singularities, but some shapes in nature are\r\nso complicated that they are equally complicated at the big scale and come\r\ncloser and closer and they don’t become any less complicated. Closer and closer, or you go farther or\r\nfarther, they remain equally complicated. \r\nThere is never a plane, never a straight line, never anything smooth and\r\nordinary. The idea is very, very\r\nvague, is expressed – it’s an expression of reality.\r\n\r\n
Fractal geometry is a new subject and each definition I try\r\nto give for it has turned out to be inappropriate. So I’m now being cagey and saying there are very complex\r\nshapes which would be the same from close by and far away.\r\n\r\n
Question: What does it mean to say that fractal shapes are\r\nself-similar?\r\n\r\n
Benoit Mandelbrot: Well, if you look at a shape like a\r\nstraight line, what’s remarkable is that if you look at a straight line from\r\nclose by, from far away, it is the same; it is a straight line. That is, the straight line has a\r\nproperty of self-similarity. Each\r\npiece of the straight line is the same as the whole line when used to a big or\r\nsmall extent. The plane again has\r\nthe same property. For a long\r\ntime, it was widely believed that the only shapes having these extraordinary\r\nproperties are the straight line, the whole plane, the whole space. Now in a certain sense, self-similarity\r\nis a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar\r\nagain, the same seen from close by and far away, and which are far from being\r\nstraight or plane or solid. And\r\nthose shapes, which I studied and collected and put together and applied in\r\nmany, many domains, I called fractals.\r\n\r\n
Question: How can complex natural shapes be represented\r\nmathematically?\r\n\r\n
Benoit Mandelbrot: Well, historically, a mountain could not\r\nbe represented, except for a few mountains which are almost like cones. Mountains are very complicated. If\r\nyou look closer and closer, you find greater and greater details. If you look away until you find that\r\nbigger details become visible, and in a certain sense this same structure\r\nappears at those scales. If you\r\nlook at coastlines, if you look at that them from far away, from an airplane,\r\nwell, you don’t see details, you see a certain complication. When you come closer, the complication\r\nbecomes more local, but again continues. \r\nAnd come closer and closer and closer, the coastline becomes longer and\r\nlonger and longer because it has more detail entering in. However, these details amazingly enough\r\nenters this certain this certain regular fashion. Therefore, one can study a coastline **** object because the\r\ngeometry for that existed for a long time, and then I put it together and applied\r\nit to many domains.\r\n\r\n
Question: What was the discovery process behind the\r\nMandelbrot set?\r\n\r\n
Benoit Mandelbrot: The Mandelbrot set in a certain sense is\r\na **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but not\r\nhappy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, who\r\nwas a very well-known pure mathematician, wanted me to study a certain theory\r\nwhich was then many years old, 30 years old or something, but had in a way\r\nstopped developing. When he was\r\nyoung he had tried to get this theory out of a rut and he didn’t succeed,\r\nnobody succeeded. So, there was a\r\ncase of two men, Julia, a teacher of mine, and Fatu, who had died, had a very\r\ngood year in 1910 and then nothing was happening. My uncle was telling me, if you look at that, if you find\r\nsomething new, it would be a wonderful thing because I couldn’t – nobody\r\ncould.\r\n\r\n
I looked at it and found it too difficult. I just could see nothing I could\r\ndo. Then over the years, I put\r\nthat a bit in the back of my mind until one day I read an obituary. It is an interesting story that I was\r\nmotivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this\r\nquestion was raised again. At that\r\ntime, I had a computer and I had become quite an expert in the use of the\r\ncomputer for mathematics, for physics, and for many sciences. So, I decided, perhaps the time has\r\ncome to please my uncle; 35 years later, or something. To please my uncle and do what my uncle\r\nhad been pushing me to do so strongly.\r\n\r\n
But I approached this topic in a very different fashion than\r\nmy uncle. My uncle was trying to\r\nthink of something, a new idea, a new problem, a new way of developing the\r\ntheory of Fatu and Julia. I did\r\nsomething else. I went to the\r\ncomputer and tried to experiment. \r\nI introduced a very high level of experiment in very pure\r\nmathematics. I was at IBM, I had\r\nthe run of computers which then were called very big and powerful, but in fact\r\nwere less powerful than a handheld machine today. But I had them and I could make the experiments. The conditions were very, very\r\ndifficult, but I knew how to look at pictures. In fact, the reason I did not go into pure mathematics\r\nearlier was that I was dominated by visual. I tried to combine the visual beauty and the\r\nmathematics.\r\n\r\n
So, I looked at the picture for a long time in a very\r\nunsystematic fashion just to become acquainted in a kind of physical fashion\r\nwith those extraordinary difficult and complicated shapes. Two were extraordinarily\r\ndifficult. Computer graphics did\r\nnot exist back then, but to have a machine which was – made it seem\r\ndoable. And I started finding\r\nextraordinary complications, extraordinary structure, extraordinary beauty of\r\nboth a theoretical kind, mathematical, and a visual kind. And collected observations of my trip\r\nin this new territory. When I\r\npresented that work to my colleagues, it was an explosion of interest. Everybody in mathematics had given up\r\nfor 100 years or 200 years the idea that you could from pictures, from looking\r\nat pictures, find new ideas. That\r\nwas the case long ago in the Middle Ages, in the Renaissance, in later periods,\r\nbut then mathematicians had become very abstract. Pictures were completely eliminated from mathematics; in\r\nparticular when I was young this happened in a very strong fashion.\r\n\r\n
Some mathematicians didn’t even perceive of the possibility\r\nof a picture being helpful. To the\r\ncontrary, I went into an orgy of looking at pictures by the hundreds; the\r\nmachines became a little bit better. \r\nWe had friends who improved them, who wrote better software to help me,\r\nwhich was wonderful. That was the good\r\nthing about being at IBM. And I\r\nhad this collection of observations, which I gave to my friends in mathematics\r\nfor their pleasure and for simulation. \r\nThe extraordinary fact is that the first idea I had which motivated me,\r\nthat worked, is conjecture, a mathematical idea which may or may not be\r\ntrue. And that idea is still\r\nunproven. It is the foundation,\r\nwhat started me and what everybody failed to **** prove has so far defeated the\r\ngreatest efforts by experts to be proven. \r\nIn a certain sense it’s a very, very strange because the object itself\r\nis understandable, even for a child. \r\nIf the object can be drawn by a child with new computers, with new\r\ngraphic devices, and still the basic idea has not been proven.\r\n\r\n
But the development of it has been extraordinary, then it\r\nwas slowed down a bit, and now again it is going up. New people are coming in and they prove extraordinary\r\nresults which nobody was hoping to prove, and I am astonished and of course,\r\nvery pleased by this development.\r\n\r\n
Question: What is the unproven conjecture that drove you?\r\n\r\n
Benoit Mandelbrot: The conjecture itself consists in two\r\ndifferent issues in Mandelbrot set – two alternative definitions which are too\r\ntechnical to describe without a blackboard, but which are both very simple and\r\nwhich I assumed naively to be equivalent. \r\nWhy did I assume so? \r\nBecause on the pictures I could not see any difference. Obtaining pictures in one way or\r\nanother way, I couldn’t tell them apart. \r\nTherefore, I assumed they were identical and I went on studying this\r\npiece. I found that, again, many\r\ninteresting observations of which most were very preferred by many other very,\r\nvery skilled mathematicians. But\r\nthe idea that these two conditions, definitions, are identical is still\r\nopen. So there are two definitions\r\nin Mandelbrot set, the usual one and another one, and they may theoretically be\r\ndifferent. People are getting\r\ncloser, but have not proven it completely.\r\n\r\n
Question: Why do people find fractals beautiful?\r\n\r\n
Benoit Mandelbrot: Well, first of all, one explanation of\r\nthat is that the feeling for fractality is not new. It is one very surprising and extraordinary discovery I made\r\ngradually, very slowly by looking again at the paintings of the past. Many painters had a clear idea of what\r\nfractals are. Take a French\r\nclassic painter named Poussin. \r\nNow, he painted beautiful landscapes, completely artificial ones,\r\nimaginary landscapes. And how did\r\nhe choose them? Well, he had the\r\nbalance of trees, of lawns, of houses in the distance. He had a balance of small objects, big\r\nobjects, big trees in front and his balance of objects at every scale is what\r\ngives to Poussin a special feeling.\r\n\r\n
Take Hokusai, a famous Chinese painter of 1800. He did not have any mathematical\r\ntraining; he left no followers because his way of painting or drawing was too\r\nspecial to him. But it was quite\r\nclear by looking at how Hokusai, the eye, which had been trained from the\r\nfractals, that Hokusai understood fractal structure. And again, had this balance of big, small, and intermediate\r\ndetails, and you come close to these marvelous drawings, you find that he\r\nunderstood perfectly fractality. \r\nBut he never expressed it. \r\nNobody ever expressed it, and then the next stage of Japanese image\r\nexperts did some other things.\r\n\r\n
So humanity has known for a long time what fractals\r\nare. It is a very strange\r\nsituation in which an idea which each time I look at all documents have deeper\r\nand deeper roots, never (how to say it), jelled. Never got together until I started playing with the computer\r\nand playing with topics which nobody was touching because they were just\r\ndesperate and hopeless.\r\n\r\n
Question: How has computer technology impacted your work?\r\n\r\n
Benoit Mandelbrot: Well, the computer had been sort of spoken about since the\r\nearly 19th century, even before. \r\nBut until the electronic computers came, which was in reality during\r\nWorld War II, or shortly afterwards. \r\nThey could not be used for any purpose in science. They were just too slow, too limited in\r\ntheir capacity. My chance was that\r\nI was myself a very visual person. \r\nAgain, a mathematician who had started a very unconventional career\r\nbecause my interest was both mathematics and in the eye. And with IBM very primitive\r\npicture-making machines became available and we had to program everything. It was heroic. And my friends at IBM who helped me\r\ndeserve a great thank you. With\r\nthese two, I could begin to do things which before had been impossible. I could begin to implement an idea of\r\nhow a mountain looked like. To\r\nreduce a mountain, which is something most complicated to a very simple idea –\r\nhow do you do it? Well you make a\r\nconjecture, have positives about shapes of mountains, and you don’t think about\r\nthe mathematics of it, you must make a picture of it.\r\n\r\n
If the picture is – everybody to be a mountain, then there’s\r\nsomething true about it. Or a\r\ncloud. It was astonishing when at\r\none point, I got the idea of how to make artifical clouds with a collaborator,\r\nwe had pictures made which were theoretically completely artificial pictures\r\nbased upon that one very simple idea. \r\nAnd this picture everybody views as being clouds. People don’t believe that they aren’t\r\nphotographs. So, we have certainly\r\nfound something true about nature. \r\n\r\n\r\n
\r\n\r\n And on the other side, the completely artificial shapes, the\r\nshapes that don’t exist in nature, which, for example, the Mandelbrot set,\r\nwhich was completely came out of the blue out of a very simple formula which is\r\nabout one inch long and which gives us an endless, endless stream of questions\r\nand results. There what happened is\r\nthat to everybody’s surprise there’s a very strong inner resemblance between\r\nthose shapes and the shapes of nature, which I have been studying. And again, I spent half my life,\r\nroughly speaking, doing the study of nature in many aspects and half of my life\r\nstudying completely artificial shapes. \r\nAnd the two are extraordinarily close; in one way both are fractal.
And on the other side, the completely artificial shapes, the\r\nshapes that don’t exist in nature, which, for example, the Mandelbrot set,\r\nwhich was completely came out of the blue out of a very simple formula which is\r\nabout one inch long and which gives us an endless, endless stream of questions\r\nand results. There what happened is\r\nthat to everybody’s surprise there’s a very strong inner resemblance between\r\nthose shapes and the shapes of nature, which I have been studying. And again, I spent half my life,\r\nroughly speaking, doing the study of nature in many aspects and half of my life\r\nstudying completely artificial shapes. \r\nAnd the two are extraordinarily close; in one way both are fractal.
Question: What does the word “chaos” mean to mathematicians?\r\n\r\n
Benoit Mandelbrot: The theory of chaos and theory of\r\nfractals are separate, but have very strong intersections. That is one part of chaos theory is\r\ngeometrically expressed by fractal shapes. Another part of chaos theory is not expressed by fractal\r\nshapes. And other part of fractals\r\ndoes not belong to chaos theory so that two theories which overlap very\r\nstrongly and do not coincide. One\r\nof them, chaos theory, is based on behavior of systems defined by\r\nequations. Equations of motion,\r\nfor example, and classical mathematics, and around 1900, Poincaré and ****, two great\r\nmathematicians at the time, have realized that sometimes the solution of very\r\nsimple looking equations can be extremely complicated. But in 1900, it was too early to\r\ndevelop that idea. It was very\r\nwell expressed and very much discussed, but did not – could not grow.\r\n\r\n
Much later, of course, with computers this idea came to life\r\nagain and became the very important part of science. So both chaos theory and fractal have had contacts in the\r\npast when they are both impossible to develop and in a certain sense not ready\r\nto be developed. And again, they\r\nintersect very strongly but they are very distinct.\r\n\r\n
Question: Do mathematical descriptions of chaos define some\r\norder within chaos?\r\n\r\n
Benoit Mandelbrot: Well a very strong distinction was made\r\nbetween chaos and fractals. For\r\nexample, the rules which generate most of natural fractals, models of\r\nmountains, of clouds, and many other phenomena involve change. And therefore they are not at all\r\nchaotic in the ordinary sense of the word, in an ordinary, current, modern\r\nsense of the word. Not chaotic in\r\nthe old sense of the word, which doesn’t have any specific meaning. But I don’t\r\nlike to discuss the question of terms. \r\nThe term chaos came, but you know something which was very confused, it\r\nhelped it jell, but the use of a biblical name in a certain sense forces us to\r\nfind the implications which were not important in mathematics. That’s why when the time came to give a\r\nname to my work, I chose the word fractal which was new. Before that, there was no need of a\r\nword at all because again there were only a few undeveloped ideas in the very\r\nmany great minds. But when a word\r\nbecame necessary, I preferred not to use an old word, but to create a new one.\r\n\r\n
Question: How did you come up with the word “fractal”?\r\n\r\n
Benoit Mandelbrot: \r\nWell, it was a very, very interesting story. At one point, a friend of mine, an older person, told me\r\nthat he saw a paper of mine on a new topic. And he said, “Look Benoit, I tell you, you must stop writing\r\nall of these papers in that field, that field, that field, that field. Nobody knows where you are, what you\r\nare doing. You just sit down and\r\nwrite a book. A short book, a\r\nclear book, a book of things which you have done.” So, I sat down and wrote the book. Now, the book had to title, why? Because the topics I had been studying had not been the\r\nobject of any theory whatsoever. \r\nAnd there are many words which mean nothing, but many fields which have\r\nno name because they don’t exist. \r\nSo, the publisher didn’t like this very ponderous title, said, “Look.” And a friend of mine, another friend,\r\ntold me, “Look, you create a new field, you are entitled to give it a\r\nname.” So, I had Latin in high\r\nschool and it turned out that one of my son’s was taking Latin in the United\r\nStates, and so there was a Latin dictionary in our house, which was an\r\nexception.\r\n\r\n
I went in there and tried to look for a word which fitted\r\nwhat I had been working on. And\r\nwhen I was playing with the word fraction, and looked in the dictionary for a\r\nword where fraction came from. It\r\ncame from a Latin word which meant, how to say disconnect – rough and\r\ndisconnected, it was a very general – the idea of roughness originally in Latin. So, I started playing with fractus, which I named it that and coined\r\nthe word fractal. First of all, I\r\nput it in this book, Objets Fractals,\r\nin French as it turned out, and then the English translation of the book, and\r\nthen the word took off. First of\r\nall, people applied it in ways in which I didn’t find sensible, but there was\r\nnothing I could say about it. So,\r\nthen the dictionary started defining it, each a little bit differently. And in a certain sense the word became\r\nalive and independent of me. I\r\ncould scream and say, I don’t like it, but it made no difference.\r\n\r\n
I had once a curiosity of looking on the Web in different\r\ncountries having different languages, what is a fractal, and found that in one\r\ncountry, I will not mention, it’s a word that has become applied to some\r\nnightclubs. A fractal nightclub is\r\na kind of nightclub. I don’t know\r\nwhich, because I haven’t been there, but, and I don’t know the language, but I\r\nguess, from what I could guess, what it was. It’s a word which has its own life. I gave it a definition, but that\r\ndefinition became too narrow because some objects I want to go fractal did not\r\nfit the old definition.\r\n\r\n
So some people asked me would I still believe the definition\r\nof whatever – 40 years ago. I\r\ndon’t. But I have no control. It’s something which works by\r\nitself. The fact that very many\r\nadults I know never heard of it, but the children have, is what gives me\r\nparticular pleasure because high school students, even the bright ones, are\r\nvery resistant to, how to say, imposed terms. And the combination of pictures and of deep theory, you can\r\nlook at the picture and find something, some idea about this picture is\r\nsensible, and then be told that very great scientists either can’t prove it, or\r\nhas taken 40 years to prove it, or had to be several of them together to prove\r\nit because it was so difficult. \r\nAnd it can be seen by a child, understood by a child. That aspect is one which very many\r\npeople find particularly attractive in the field.\r\n\r\n
In mathematics and science definition are simple, but\r\nbare-bones. Until you get to a problem which you understand it takes hundreds\r\nand hundreds of pages and years and years of learning. In this case, you have this formula,\r\nyou track in a computer and from a simple formula, in a very short time\r\namazingly beautiful things come out, which sometimes people can prove instantly\r\nand sometimes great scientists take forever to prove. Or don’t even succeed in proving it.\r\n\r\n
Question: How can we understand financial market\r\nfluctuations in fractal terms?\r\n\r\n
Benoit Mandelbrot: Well, what I discovered quite a while ago\r\nin fact, that was my first major piece of work is that a model of price\r\nvariation which everybody was adopting was very far from being applicable. It’s a very curious story.\r\n\r\n
In 1900, a Frenchman named Bachelier, who was a student of\r\nmathematics, wrote a thesis on the theory of speculation. It was not at all an acceptable topic\r\nin pure mathematics and he had a very miserable life. But his thesis was extraordinary. Extraordinary in a very strange way. It applied very well to Brownian\r\nmotion, which is in physics. So,\r\nBachelier was a pioneer of a very marvelous essential theory in physics. But to economics, it didn’t apply at\r\nall, it was very ingenious, but Bachelier had no data, in fact no data was\r\navailable at that time in 1900, so he imagined an artificial market in which\r\ncertain rules may apply. \r\nUnfortunately, the theory which was developed by economists when\r\ncomputers came up was Bachelier’s theory. \r\nIt does not account for any of the major effects in economics. For example, it assumes prices are\r\ncontinuous when everybody knows the prices are not continuous. Some people say, well, all right, there\r\nare discontinuities but they are a different kind of economics that we are\r\ndoing, not because certain discontinuities become too complicated and only will\r\nthe **** look more or less continuous. \r\nBut it turns out that discontinuities are as important, or more\r\nimportant than the rest.\r\n\r\n
Bachelier assumed that each price changes in compared of the\r\npreceding price change. It’s a\r\nvery beautiful assumption, but it’s completely incorrect because we know very\r\nwell, especially today that for a long time prices may vary moderately and then\r\nsuddenly they begin to vary a great deal. \r\nSo, even we’re saying that the theory changes or you say the theory\r\nwhich exists is not appropriate. \r\nWhat I found that Bachelier’s theory was defective on both grounds. That was in 1961, 1962, I forgot the\r\nexact dates and when the development of Bachelier became very, very rapid. Since nobody wanted to listen to me, I\r\ndid other things. Many other\r\nthings, but I was waiting because it was quite clear that my time would have to\r\ncome. And unfortunately, it has\r\ncome, that is, the fluctuation of the economy, the stock market, and commodity\r\nmarkets today are about as they were in historical times. There was no change which made the\r\nstock market different today than it was long ago. And the lessons which are drawn from **** peers do represent\r\ntoday’s events very accurately. \r\nBut the situation is much more complicated than Bachelier had\r\nassumed. Bachelier, again, was a\r\ngenius, Bachelier had an excellent idea which happened to be very useful in\r\nphysics, but economics, he just lacked data. He did not have awareness of discontinuity which is\r\nessential in this context. Not\r\nhaving an awareness of dependence, which is also essential in this context. So, his theory is very, very different\r\nfrom what you observe in reality.\r\n\r\n
Question: As you write your memoirs, which memories are the\r\nmost fun and the most difficult to look back on?\r\n\r\n
Benoit Mandelbrot: Well, my life has been extremely\r\ncomplicated. Not by choice at the\r\nbeginning at all, but later on, I had become used to complication and went on\r\naccepting things that other people would have found too difficult to\r\naccept. I was born in Poland and\r\nmoved to France as a child shortly before World War II. During World War II, I was lucky to\r\nlive in the French equivalent of Appalachia, a region which is sort of not very\r\nhigh mountains, but very, very poor, and Appalachia we are poorer even, so\r\npoorer than Appalachia of the United States. And for me, I was in high school where things were very easy. It was a small high school way up in the\r\nhills and had mostly a private intellectual life. I read many books; there were many books, a very good\r\nlibrary. I had many books and I\r\nhad dreams of all kinds. Dreams in\r\nwhich were in a certain sense, how to say, easy to make because the near future\r\nwas always extremely threatening. \r\nIt was a very dangerous period. \r\nBut since I had nothing to lose, I was dreaming of what I could do.\r\n\r\n
Then the war ended. \r\nI had very, very little training in taking an exam to determine a\r\nscientist’s life in France. There\r\nwere two schools, both very small. \r\nOne tiny, and one small, which in a certain sense was the place that I\r\nwas sure I wanted to go. I had\r\nonly a few months of finding out how the exam proceeded, but I took the exam\r\nand perhaps because of inherited gifts, I did very well. In fact, I barely\r\nmissed being number one in France in both schools. In particular I did very well in mathematical problems. The physics I could not guess, other\r\nthings I could not guess. But then\r\nI had a big choice, should I go into mathematics in a small and ****\r\nschool. Or should I go to a bigger\r\nschool in which, in a certain sense would give me time to decide what I wanted\r\nto do?\r\n\r\n
First I entered the small school where I was, as a matter of\r\nfact, number one of the students who entered then. But immediately, I left because that school, again, was\r\ngoing to teach me something which I did not fully believe, namely mathematics\r\nseparate from everything else. It\r\nwas excellent mathematics, French mathematics was very high level, but in\r\neverything else it was not even present. \r\nAnd I didn’t want to become a pure mathematician, as a matter of fact,\r\nmy uncle was one, so I knew what the pure mathematician was and I did not want\r\nto be a pure – I wanted to do something different. Not less, not more but different. Namely, combine pure mathematics at which I was very good,\r\nwith the real world of which I was very, very curious.\r\n\r\n
And so, I did not go to École Polytechnique. It was a very rough decision, and the\r\nyear when I took this decision remembers my memory very, very strongly. Then for several years, I just was lost\r\na bit. I was looking for a good\r\nplace. I spent my time very nicely\r\nin many ways, but not fully satisfactory. \r\nThen I became Professor in France, but realized that I was not – for the\r\njob that I should spend my life in. \r\nFortunately, IBM was building a research center, I went there for a\r\nsummer thing, for a summer only. I\r\nknew this summer, decided to stay. \r\nIt was a very big gamble. I\r\nlost my job in France, I received a job in which was extremely uncertain, how\r\nlong would IBM be interested in research, but the gamble was taken and very\r\nshortly afterwards, I had this extraordinary fortune of stopping at Harvard to\r\ndo a lecture and learning about the price variation in just the right way. At a time when nobody was looking, was realizing\r\nthat either one needed, or one could make a theory of price variation other\r\nthan the theory of 1900 at which Bachelier had proposed, which was very, very\r\nfar from being representative of the actual thing.\r\n\r\n
So, I went to IBM and I was fortunate in being allowed – to\r\nbe successful as to go from field to field, which in a way was what I had been\r\nhoping for. I didn’t feel\r\ncomfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of\r\neverything and explore the world. \r\nAnd IBM let me do so. I\r\ntouched on far more topics than anybody would have found reasonable. I was often told, “Settle down, stay in\r\none field, don’t go all the time to another field.” But I was just compelled to move from one thing to\r\nanother.\r\n\r\n
And fractal geometry was not an idea which I had early on,\r\nfor something was developed progressively. I didn’t choose to go into the topic because of any\r\ncompelling reason, but because the problems there seemed to be somehow similar\r\nto the ones I knew how to handle. \r\nI had experienced this kind of problem and gradually realized that I was\r\ntruly putting together a new theory. \r\nA theory of roughness. What\r\nis roughness? Everybody knows what\r\nis roughness. When was roughness\r\ndiscovered? Well, prehistory. Everything is roughness, except for the\r\ncircles. How many circles are\r\nthere in nature? Very, very\r\nfew. The straight lines. Very shapes are very, very smooth. But geometry had laid them aside\r\nbecause they were too complicated. \r\nAnd physics had laid them aside because they were too complicated. One couldn’t even measure roughness. So, by luck, and by reward for\r\npersistence, I did found the theory of roughness, which certainly I didn’t expect\r\nand expecting to found one would have been pure madness.\r\n\r\n
So, one of the high points of my life was when I suddenly\r\nrealized that this dream I had in my late adolescence of combining pure\r\nmathematics, very pure mathematics with very hard things which had been long a\r\nnuisance to scientists and to engineers, that this combination was possible and\r\nI put together this new geometry of nature, the fractal geometry of nature.\r\n\r\n
Question: Which honor means more to you: your Légion\r\nd'Honneur medal or the “Mandelbrot Set” rock song?\r\n\r\n
Benoit Mandelbrot: Well, I happen to know this song, it was\r\nsent to me and I was very impressed by it and by its popularity. In a certain sense, it is not which\r\none, but the combination. When\r\npeople ask me what’s my field? I say,\r\non one hand, a fractalist. Perhaps\r\nthe only one, the only full-time one. \r\nOn the other hand, I’ve been a professor of mathematics at Harvard and\r\nat Yale. At Yale for a long\r\ntime. But I’m not a mathematician\r\nonly. I’m a professor of physics,\r\nof economics, a long list. Each\r\nelement of this list is normal. \r\nThe combination of these elements is very rare at best. And so in a certain sense, it is not\r\nthe fact that I was a professor of mathematics at these great universities, or\r\nprofessor of physics at other great universities, or that I received, among\r\nother doctorates, one in medicine, believe it or not. And one in civil engineering. It is the coexistence of these various aspects that in one\r\nlifetime it is possible, if one takes the kinds of risks which I took, which\r\nare colossal, but taking risks, I was rewarded by being able to contribute in a\r\nvery substantial fashion to a variety of fields. I was able to reawaken and solve some very old problems. The problems are just so old that in a\r\ncertain sense, they were no longer being pursued. And nobody – I didn’t know anybody who was trying to define roughness\r\nof ****. It was a hopeless subject. \r\nBut I did it and there’s a whole field by which has been created by\r\nthat.\r\n\r\n
In a certain sense the beauty of what I happened by\r\nextraordinary chance to put together is that nobody would have believed that\r\nthis is possible, and certainly I didn’t expect that it was possible. I just moved from step to step to step. Lately I realized that all these things\r\nheld together, and very lately I see that in each field very old problems could\r\nbe if not solved, at least advanced or reawakened, and therefore gradually very\r\nmuch improved in your understanding.
Recorded on February 17, 2010
Interviewed by Austin Allen
A conversation with the mathematician and Professor Emeritus at Yale University.
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