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Question: What is fractal geometry?
Benoit Mandelbrot: Well, regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don’t become any less complicated. Closer and closer, or you go farther or farther, they remain equally complicated. There is never a plane, never a straight line, never anything smooth and ordinary. The idea is very, very vague, is expressed – it’s an expression of reality.
Fractal geometry is a new subject and each definition I try to give for it has turned out to be inappropriate. So I’m now being cagey and saying there are very complex shapes which would be the same from close by and far away.
Question: What does it mean to say that fractal shapes are self-similar?
Benoit Mandelbrot: Well, if you look at a shape like a straight line, what’s remarkable is that if you look at a straight line from close by, from far away, it is the same; it is a straight line. That is, the straight line has a property of self-similarity. Each piece of the straight line is the same as the whole line when used to a big or small extent. The plane again has the same property. For a long time, it was widely believed that the only shapes having these extraordinary properties are the straight line, the whole plane, the whole space. Now in a certain sense, self-similarity is a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar again, the same seen from close by and far away, and which are far from being straight or plane or solid. And those shapes, which I studied and collected and put together and applied in many, many domains, I called fractals.
Question: How can complex natural shapes be represented mathematically?
Benoit Mandelbrot: Well, historically, a mountain could not be represented, except for a few mountains which are almost like cones. Mountains are very complicated. If you look closer and closer, you find greater and greater details. If you look away until you find that bigger details become visible, and in a certain sense this same structure appears at those scales. If you look at coastlines, if you look at that them from far away, from an airplane, well, you don’t see details, you see a certain complication. When you come closer, the complication becomes more local, but again continues. And come closer and closer and closer, the coastline becomes longer and longer and longer because it has more detail entering in. However, these details amazingly enough enters this certain this certain regular fashion. Therefore, one can study a coastline **** object because the geometry for that existed for a long time, and then I put it together and applied it to many domains.
Question: What was the discovery process behind the Mandelbrot set?
Benoit Mandelbrot: The Mandelbrot set in a certain sense is a **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but not happy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, who was a very well-known pure mathematician, wanted me to study a certain theory which was then many years old, 30 years old or something, but had in a way stopped developing. When he was young he had tried to get this theory out of a rut and he didn’t succeed, nobody succeeded. So, there was a case of two men, Julia, a teacher of mine, and Fatu, who had died, had a very good year in 1910 and then nothing was happening. My uncle was telling me, if you look at that, if you find something new, it would be a wonderful thing because I couldn’t – nobody could.
I looked at it and found it too difficult. I just could see nothing I could do. Then over the years, I put that a bit in the back of my mind until one day I read an obituary. It is an interesting story that I was motivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this question was raised again. At that time, I had a computer and I had become quite an expert in the use of the computer for mathematics, for physics, and for many sciences. So, I decided, perhaps the time has come to please my uncle; 35 years later, or something. To please my uncle and do what my uncle had been pushing me to do so strongly.
But I approached this topic in a very different fashion than my uncle. My uncle was trying to think of something, a new idea, a new problem, a new way of developing the theory of Fatu and Julia. I did something else. I went to the computer and tried to experiment. I introduced a very high level of experiment in very pure mathematics. I was at IBM, I had the run of computers which then were called very big and powerful, but in fact were less powerful than a handheld machine today. But I had them and I could make the experiments. The conditions were very, very difficult, but I knew how to look at pictures. In fact, the reason I did not go into pure mathematics earlier was that I was dominated by visual. I tried to combine the visual beauty and the mathematics.
So, I looked at the picture for a long time in a very unsystematic fashion just to become acquainted in a kind of physical fashion with those extraordinary difficult and complicated shapes. Two were extraordinarily difficult. Computer graphics did not exist back then, but to have a machine which was – made it seem doable. And I started finding extraordinary complications, extraordinary structure, extraordinary beauty of both a theoretical kind, mathematical, and a visual kind. And collected observations of my trip in this new territory. When I presented that work to my colleagues, it was an explosion of interest. Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract. Pictures were completely eliminated from mathematics; in particular when I was young this happened in a very strong fashion.
Some mathematicians didn’t even perceive of the possibility of a picture being helpful. To the contrary, I went into an orgy of looking at pictures by the hundreds; the machines became a little bit better. We had friends who improved them, who wrote better software to help me, which was wonderful. That was the good thing about being at IBM. And I had this collection of observations, which I gave to my friends in mathematics for their pleasure and for simulation. The extraordinary fact is that the first idea I had which motivated me, that worked, is conjecture, a mathematical idea which may or may not be true. And that idea is still unproven. It is the foundation, what started me and what everybody failed to **** prove has so far defeated the greatest efforts by experts to be proven. In a certain sense it’s a very, very strange because the object itself is understandable, even for a child. If the object can be drawn by a child with new computers, with new graphic devices, and still the basic idea has not been proven.
But the development of it has been extraordinary, then it was slowed down a bit, and now again it is going up. New people are coming in and they prove extraordinary results which nobody was hoping to prove, and I am astonished and of course, very pleased by this development.
Question: What is the unproven conjecture that drove you?
Benoit Mandelbrot: The conjecture itself consists in two different issues in Mandelbrot set – two alternative definitions which are too technical to describe without a blackboard, but which are both very simple and which I assumed naively to be equivalent. Why did I assume so? Because on the pictures I could not see any difference. Obtaining pictures in one way or another way, I couldn’t tell them apart. Therefore, I assumed they were identical and I went on studying this piece. I found that, again, many interesting observations of which most were very preferred by many other very, very skilled mathematicians. But the idea that these two conditions, definitions, are identical is still open. So there are two definitions in Mandelbrot set, the usual one and another one, and they may theoretically be different. People are getting closer, but have not proven it completely.
Question: Why do people find fractals beautiful?
Benoit Mandelbrot: Well, first of all, one explanation of that is that the feeling for fractality is not new. It is one very surprising and extraordinary discovery I made gradually, very slowly by looking again at the paintings of the past. Many painters had a clear idea of what fractals are. Take a French classic painter named Poussin. Now, he painted beautiful landscapes, completely artificial ones, imaginary landscapes. And how did he choose them? Well, he had the balance of trees, of lawns, of houses in the distance. He had a balance of small objects, big objects, big trees in front and his balance of objects at every scale is what gives to Poussin a special feeling.
Take Hokusai, a famous Chinese painter of 1800. He did not have any mathematical training; he left no followers because his way of painting or drawing was too special to him. But it was quite clear by looking at how Hokusai, the eye, which had been trained from the fractals, that Hokusai understood fractal structure. And again, had this balance of big, small, and intermediate details, and you come close to these marvelous drawings, you find that he understood perfectly fractality. But he never expressed it. Nobody ever expressed it, and then the next stage of Japanese image experts did some other things.
So humanity has known for a long time what fractals are. It is a very strange situation in which an idea which each time I look at all documents have deeper and deeper roots, never (how to say it), jelled. Never got together until I started playing with the computer and playing with topics which nobody was touching because they were just desperate and hopeless.
Question: How has computer technology impacted your work?
Benoit Mandelbrot: Well, the computer had been sort of spoken about since the early 19th century, even before. But until the electronic computers came, which was in reality during World War II, or shortly afterwards. They could not be used for any purpose in science. They were just too slow, too limited in their capacity. My chance was that I was myself a very visual person. Again, a mathematician who had started a very unconventional career because my interest was both mathematics and in the eye. And with IBM very primitive picture-making machines became available and we had to program everything. It was heroic. And my friends at IBM who helped me deserve a great thank you. With these two, I could begin to do things which before had been impossible. I could begin to implement an idea of how a mountain looked like. To reduce a mountain, which is something most complicated to a very simple idea – how do you do it? Well you make a conjecture, have positives about shapes of mountains, and you don’t think about the mathematics of it, you must make a picture of it.
If the picture is – everybody to be a mountain, then there’s something true about it. Or a cloud. It was astonishing when at one point, I got the idea of how to make artifical clouds with a collaborator, we had pictures made which were theoretically completely artificial pictures based upon that one very simple idea. And this picture everybody views as being clouds. People don’t believe that they aren’t photographs. So, we have certainly found something true about nature.
And on the other side, the completely artificial shapes, the shapes that don’t exist in nature, which, for example, the Mandelbrot set, which was completely came out of the blue out of a very simple formula which is about one inch long and which gives us an endless, endless stream of questions and results. There what happened is that to everybody’s surprise there’s a very strong inner resemblance between those shapes and the shapes of nature, which I have been studying. And again, I spent half my life, roughly speaking, doing the study of nature in many aspects and half of my life studying completely artificial shapes. And the two are extraordinarily close; in one way both are fractal.
Question: What does the word “chaos” mean to mathematicians?
Benoit Mandelbrot: The theory of chaos and theory of fractals are separate, but have very strong intersections. That is one part of chaos theory is geometrically expressed by fractal shapes. Another part of chaos theory is not expressed by fractal shapes. And other part of fractals does not belong to chaos theory so that two theories which overlap very strongly and do not coincide. One of them, chaos theory, is based on behavior of systems defined by equations. Equations of motion, for example, and classical mathematics, and around 1900, Poincaré and ****, two great mathematicians at the time, have realized that sometimes the solution of very simple looking equations can be extremely complicated. But in 1900, it was too early to develop that idea. It was very well expressed and very much discussed, but did not – could not grow.
Much later, of course, with computers this idea came to life again and became the very important part of science. So both chaos theory and fractal have had contacts in the past when they are both impossible to develop and in a certain sense not ready to be developed. And again, they intersect very strongly but they are very distinct.
Question: Do mathematical descriptions of chaos define some order within chaos?
Benoit Mandelbrot: Well a very strong distinction was made between chaos and fractals. For example, the rules which generate most of natural fractals, models of mountains, of clouds, and many other phenomena involve change. And therefore they are not at all chaotic in the ordinary sense of the word, in an ordinary, current, modern sense of the word. Not chaotic in the old sense of the word, which doesn’t have any specific meaning. But I don’t like to discuss the question of terms. The term chaos came, but you know something which was very confused, it helped it jell, but the use of a biblical name in a certain sense forces us to find the implications which were not important in mathematics. That’s why when the time came to give a name to my work, I chose the word fractal which was new. Before that, there was no need of a word at all because again there were only a few undeveloped ideas in the very many great minds. But when a word became necessary, I preferred not to use an old word, but to create a new one.
Question: How did you come up with the word “fractal”?
Benoit Mandelbrot: Well, it was a very, very interesting story. At one point, a friend of mine, an older person, told me that he saw a paper of mine on a new topic. And he said, “Look Benoit, I tell you, you must stop writing all of these papers in that field, that field, that field, that field. Nobody knows where you are, what you are doing. You just sit down and write a book. A short book, a clear 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 object of any theory whatsoever. And there are many words which mean nothing, but many fields which have no name because they don’t exist. So, the publisher didn’t like this very ponderous title, said, “Look.” And a friend of mine, another friend, told me, “Look, you create a new field, you are entitled to give it a name.” So, I had Latin in high school and it turned out that one of my son’s was taking Latin in the United States, and so there was a Latin dictionary in our house, which was an exception.
I went in there and tried to look for a word which fitted what I had been working on. And when I was playing with the word fraction, and looked in the dictionary for a word where fraction came from. It came from a Latin word which meant, how to say disconnect – rough and disconnected, it was a very general – the idea of roughness originally in Latin. So, I started playing withfractus, which I named it that and coined the word fractal. First of all, I put it in this book, Objets Fractals, in French as it turned out, and then the English translation of the book, and then the word took off. First of all, people applied it in ways in which I didn’t find sensible, but there was nothing I could say about it. So, then the dictionary started defining it, each a little bit differently. And in a certain sense the word became alive and independent of me. I could scream and say, I don’t like it, but it made no difference.
I had once a curiosity of looking on the Web in different countries having different languages, what is a fractal, and found that in one country, I will not mention, it’s a word that has become applied to some nightclubs. A fractal nightclub is a kind of nightclub. I don’t know which, because I haven’t been there, but, and I don’t know the language, but I guess, from what I could guess, what it was. It’s a word which has its own life. I gave it a definition, but that definition became too narrow because some objects I want to go fractal did not fit the old definition.
So some people asked me would I still believe the definition of whatever – 40 years ago. I don’t. But I have no control. It’s something which works by itself. The fact that very many adults I know never heard of it, but the children have, is what gives me particular pleasure because high school students, even the bright ones, are very resistant to, how to say, imposed terms. And the combination of pictures and of deep theory, you can look at the picture and find something, some idea about this picture is sensible, and then be told that very great scientists either can’t prove it, or has taken 40 years to prove it, or had to be several of them together to prove it because it was so difficult. And it can be seen by a child, understood by a child. That aspect is one which very many people find particularly attractive in the field.
In mathematics and science definition are simple, but bare-bones. Until you get to a problem which you understand it takes hundreds and hundreds of pages and years and years of learning. In this case, you have this formula, you track in a computer and from a simple formula, in a very short time amazingly beautiful things come out, which sometimes people can prove instantly and sometimes great scientists take forever to prove. Or don’t even succeed in proving it.
Question: How can we understand financial market fluctuations in fractal terms?
Benoit Mandelbrot: Well, what I discovered quite a while ago in fact, that was my first major piece of work is that a model of price variation which everybody was adopting was very far from being applicable. It’s a very curious story.
In 1900, a Frenchman named Bachelier, who was a student of mathematics, wrote a thesis on the theory of speculation. It was not at all an acceptable topic in 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 motion, which is in physics. So, Bachelier was a pioneer of a very marvelous essential theory in physics. But to economics, it didn’t apply at all, it was very ingenious, but Bachelier had no data, in fact no data was available at that time in 1900, so he imagined an artificial market in which certain rules may apply. Unfortunately, the theory which was developed by economists when computers came up was Bachelier’s theory. It does not account for any of the major effects in economics. For example, it assumes prices are continuous when everybody knows the prices are not continuous. Some people say, well, all right, there are discontinuities but they are a different kind of economics that we are doing, not because certain discontinuities become too complicated and only will the **** look more or less continuous. But it turns out that discontinuities are as important, or more important than the rest.
Bachelier assumed that each price changes in compared of the preceding price change. It’s a very beautiful assumption, but it’s completely incorrect because we know very well, especially today that for a long time prices may vary moderately and then suddenly they begin to vary a great deal. So, even we’re saying that the theory changes or you say the theory which exists is not appropriate. What I found that Bachelier’s theory was defective on both grounds. That was in 1961, 1962, I forgot the exact dates and when the development of Bachelier became very, very rapid. Since nobody wanted to listen to me, I did other things. Many other things, but I was waiting because it was quite clear that my time would have to come. And unfortunately, it has come, that is, the fluctuation of the economy, the stock market, and commodity markets today are about as they were in historical times. There was no change which made the stock market different today than it was long ago. And the lessons which are drawn from **** peers do represent today’s events very accurately. But the situation is much more complicated than Bachelier had assumed. Bachelier, again, was a genius, Bachelier had an excellent idea which happened to be very useful in physics, but economics, he just lacked data. He did not have awareness of discontinuity which is essential in this context. Not having an awareness of dependence, which is also essential in this context. So, his theory is very, very different from what you observe in reality.
Question: As you write your memoirs, which memories are the most fun and the most difficult to look back on?
Benoit Mandelbrot: Well, my life has been extremely complicated. Not by choice at the beginning at all, but later on, I had become used to complication and went on accepting things that other people would have found too difficult to accept. I was born in Poland and moved to France as a child shortly before World War II. During World War II, I was lucky to live in the French equivalent of Appalachia, a region which is sort of not very high mountains, but very, very poor, and Appalachia we are poorer even, so poorer 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 hills and had mostly a private intellectual life. I read many books; there were many books, a very good library. I had many books and I had dreams of all kinds. Dreams in which were in a certain sense, how to say, easy to make because the near future was always extremely threatening. It was a very dangerous period. But since I had nothing to lose, I was dreaming of what I could do.
Then the war ended. I had very, very little training in taking an exam to determine a scientist’s life in France. There were two schools, both very small. One tiny, and one small, which in a certain sense was the place that I was sure I wanted to go. I had only a few months of finding out how the exam proceeded, but I took the exam and perhaps because of inherited gifts, I did very well. In fact, I barely missed being number one in France in both schools. In particular I did very well in mathematical problems. The physics I could not guess, other things I could not guess. But then I had a big choice, should I go into mathematics in a small and **** school. Or should I go to a bigger school in which, in a certain sense would give me time to decide what I wanted to do?
First I entered the small school where I was, as a matter of fact, number one of the students who entered then. But immediately, I left because that school, again, was going to teach me something which I did not fully believe, namely mathematics separate from everything else. It was excellent mathematics, French mathematics was very high level, but in everything else it was not even present. And I didn’t want to become a pure mathematician, as a matter of fact, my uncle was one, so I knew what the pure mathematician was and I did not want to 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, with the real world of which I was very, very curious.
And so, I did not go to École Polytechnique. It was a very rough decision, and the year when I took this decision remembers my memory very, very strongly. Then for several years, I just was lost a bit. I was looking for a good place. I spent my time very nicely in many ways, but not fully satisfactory. Then I became Professor in France, but realized that I was not – for the job that I should spend my life in. Fortunately, IBM was building a research center, I went there for a summer thing, for a summer only. I knew this summer, decided to stay. It was a very big gamble. I lost my job in France, I received a job in which was extremely uncertain, how long would IBM be interested in research, but the gamble was taken and very shortly afterwards, I had this extraordinary fortune of stopping at Harvard to do a lecture and learning about the price variation in just the right way. At a time when nobody was looking, was realizing that either one needed, or one could make a theory of price variation other than the theory of 1900 at which Bachelier had proposed, which was very, very far from being representative of the actual thing.
So, I went to IBM and I was fortunate in being allowed – to be successful as to go from field to field, which in a way was what I had been hoping for. I didn’t feel comfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of everything and explore the world. And IBM let me do so. I touched on far more topics than anybody would have found reasonable. I was often told, “Settle down, stay in one field, don’t go all the time to another field.” But I was just compelled to move from one thing to another.
And fractal geometry was not an idea which I had early on, for something was developed progressively. I didn’t choose to go into the topic because of any compelling reason, but because the problems there seemed to be somehow similar to the ones I knew how to handle. I had experienced this kind of problem and gradually realized that I was truly putting together a new theory. A theory of roughness. What is roughness? Everybody knows what is roughness. When was roughness discovered? Well, prehistory. Everything is roughness, except for the circles. How many circles are there in nature? Very, very few. The straight lines. Very shapes are very, very smooth. But geometry had laid them aside because they were too complicated. And physics had laid them aside because they were too complicated. One couldn’t even measure roughness. So, by luck, and by reward for persistence, I did found the theory of roughness, which certainly I didn’t expect and expecting to found one would have been pure madness.
So, one of the high points of my life was when I suddenly realized that this dream I had in my late adolescence of combining pure mathematics, very pure mathematics with very hard things which had been long a nuisance to scientists and to engineers, that this combination was possible and I put together this new geometry of nature, the fractal geometry of nature.
Question: Which honor means more to you: your Légion d’Honneur medal or the “Mandelbrot Set” rock song?
Benoit Mandelbrot: Well, I happen to know this song, it was sent to me and I was very impressed by it and by its popularity. In a certain sense, it is not which one, but the combination. When people ask me what’s my field? I say, on one hand, a fractalist. Perhaps the only one, the only full-time one. On the other hand, I’ve been a professor of mathematics at Harvard and at Yale. At Yale for a long time. But I’m not a mathematician only. I’m a professor of physics, of economics, a long list. Each element of this list is normal. The combination of these elements is very rare at best. And so in a certain sense, it is not the fact that I was a professor of mathematics at these great universities, or professor of physics at other great universities, or that I received, among other doctorates, one in medicine, believe it or not. And one in civil engineering. It is the coexistence of these various aspects that in one lifetime it is possible, if one takes the kinds of risks which I took, which are colossal, but taking risks, I was rewarded by being able to contribute in a very 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 certain sense, they were no longer being pursued. And nobody – I didn’t know anybody who was trying to define roughness of ****. It was a hopeless subject. But I did it and there’s a whole field by which has been created by that.
In a certain sense the beauty of what I happened by extraordinary chance to put together is that nobody would have believed that this 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 held together, and very lately I see that in each field very old problems could be if not solved, at least advanced or reawakened, and therefore gradually very much improved in your understanding.
Recorded on February 17, 2010
Interviewed by Austin Allen
Binary Heap 