"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." So writes acclaimed mathematician Benoit Mandelbrot in his path-breaking book The Fractal Geometry of Nature. Instead, such natural forms, and many man-made creations as well, are "rough," he says. To study and learn from such roughness, for which he invented the term fractal, Mandelbrot devised a new kind of visual mathematics based on such irregular shapes. Fractal geometry, as he called this new math, is worlds apart from the Euclidean variety we all learn in school, and it has sparked discoveries in myriad fields, from finance to metallurgy, cosmology to medicine. In this interview, hear from the father of fractals about why he disdains rules, why he considers himself a philosopher, and why he abandons work on any given advance in fractals as soon as it becomes popular.
NOVA: You've said, "My whole career is an ardent pursuit of the concept of roughness." What exactly do you mean by that?
Benoit Mandelbrot: Actually, this word roughness has different meanings according to context. Until I turned 20 and World War II ended, my life was becoming increasingly rough because of historical events over which no one I knew had any control. But after I turned 20, things changed. Without any clear plan or conscious decision, I became fascinated by, and then almost exclusively devoted to, all kinds of phenomena in which irregularity and variability dominate but are so great that they don't average out. This led me first to disbelieve and then to contradict in a radical fashion what everybody else was saying about those phenomena.
In what way?
The predominant view of irregularity continues to follow Galileo's famous saying that the Great Book of Nature is written in the language of mathematics, the characters being circles, triangles, and other such shapes. A circle is perfectly regular. A triangle has three corners and is otherwise very smooth. And the great bulk of science studies smooth behavior, in particular using equations that assume that everything evolves in a very regular fashion.
And you were studying the unsmooth.
That's right. I soon came to devote my life to phenomena that may belong to very different organized sciences but have the common characteristic of being irregular and fragmented at many scales. Like the weather, for instance. I could not possibly anticipate the level of complication this youthful choice would bring to my life trajectory. Each year of my life brought change in my interests, my tools, and my self-confidence, and I greatly learned from both the kind and the unkind comments I continually received.
But I was so alone that the direction I was following was not described by any existing word. In 1975, my work forced me to coin one: fractal. The Latin adjective, fractus, can denote anything that is like a broken-up stone—irregular and fragmented. The sudden realization that "fractal" deserved to be put in a book's title changed nothing in the substance but brought considerable change in the perception of my work. The word is now found in many dictionaries.
"With two hands, you can count all the simple shapes of nature. Everything else is rough."
And did that one word change your own perception of your work?
Yes. And it provided a degree of order to a myriad of other objects I went on to study. In a way, I became attracted to a host of phenomena that for me smelled the same. My friends praised my vision, but I rather felt that I was good at sniffing [laughs]. The word vision only became appropriate when I had reached the end and could look back.
Fractals are beautiful, but they're much more than that. The uses for fractal geometry just keep growing, don't they?
And how! I heard fractals described endless times as "pretty pictures but pretty useless." Ridiculous! To give only one example, my study of fractals began with the stock market, which certainly deserves close attention.
And today there are numerous other applications.
So numerous and diverse you wouldn't believe it. Many came together within just the last few years. Some of them may seem obvious, but they're extremely important. For example, take wireless antennas. They used to be sticks. Then sticks with bends and crosses. Such more complicated antennas were not good ideas, because one could not make calculations to see how they perform. Then, at one point, enough young people active in that field read my books. Why not make an antenna fractal? Fractal antennas are now almost routine.
And concrete. Concrete has gone fractal, right?
Yes. Concrete was known to the Romans, then was forgotten and had to be reinvented. The result was not impermeable. Water that came through dissolved it, and chunks of it were falling off.
Now concrete has come into its own. Many former physicists moved into studying it, because it's a very well-supported field. At a conference on fractal applications that took place around my birthday in 2004, a speaker introduced an altogether new kind of concrete that is enormously stronger and more durable, a form entirely based upon his deep understanding of fractals.
Take something else that wasn't anticipated but is very simple: People living along highways scream about noise, but the flat walls put in place to placate them were very ineffective, because the noise that hit them simply bounced off. Responding to some political pressure, a friend of mine had the brilliant idea that a wall having a fractal surface would be far better because it would absorb the noise. That insight works and underlies the current technology.
These walls are rough, kind of like nature would have made them.
Precisely. In raw nature, very few shapes are simple: the pupil, the iris, the moon—with two hands, you can count all the simple shapes of nature. Everything else is rough. But if you look around us, almost everything industrial is very smooth, round, flat, corrugated, and so on. Now that is changing. Engineers everywhere know how to use fractals.
Even medicine is embracing fractals, such as in interpreting illness, right?
Yes. Anybody who has had a mild operation may realize how uneven a healthy heartbeat is. I knew this only theoretically, until a special dental surgeon put a little thing on my finger to see how my heart was reacting. So I could listen to my heartbeat. I'm a healthy old man, but my heartbeat is very irregular, in a way very well described by fractals. Many people hope that this will help enormously in following the progression of illness.
Do you keep up on advances that use fractals?
Read everything written about fractals? I don't even try. People very often report to me, "Here's something you may have missed that is very interesting." I do go to meetings and hear new things, but there is no way to follow everything. I abandon problems when a constituency gets created around them.
Well, for example, the Mandelbrot set took off like a rocket. Within a year, millions of people had become involved. I felt overgunned by their number and tenacity. Besides, I can stand loneliness. In fact, I'm rarely comfortable in a big crowd, because big crowds automatically are very specifically organized by dates, by tradition, by training. And I don't sound like a mathematician. I don't sound like a physicist either. Nor do I sound like an art critic. There's very great strength in being a stranger, if one brings something new.
You call yourself a maverick. Why did you choose a lifestyle that's "dangerous"?
Earlier European politics handed me an extremely complicated youth. My parents were from Lithuania, but I was born in Warsaw, Poland. I was at least twice a foreigner, which made an enormous difference. I was forced very early to take very big risks—often big gambles.
Is that why your family moved to France in 1936, when you were 11?
My mother was a highly educated woman, one of the first few Russian doctors in medicine. She had enormous ambitions for her sons. But we lived in a world where everything was unstable. We didn't read newspapers by curiosity but because, within a week or a month, what we read could affect our whole life.
"I don't only study books; I study nature."
We moved to France when my mother, at age 50, made a decision I still find admirable and incomprehensible. She abandoned her profession, her roots, her friends, and her place in a well-defined society to become a lonely housewife in a slum in Paris. My parents figured this gave us a better chance to survive.
And what was the war like for you personally?
The critical year was 1944, when I was 19. I turned 20 just too late to be in the military during the war. But certainly not too late to watch what was happening. A constant feeling of anxiety made me develop what many people call a survival instinct. In a dangerous situation I was very careful to abstain from intervening, or even to skip away.
What kind of education did you get early on?
I don't remember about learning to read and write, but I do remember very well learning to play chess very early. This is a very geometric game, and geometric memory is essential. In Poland chess is very popular. I was a local champion and soon would have been pushed to compete seriously. Also, my father was a passionate collector of maps, so I could read them as far back as I can remember. In fact, I learned to crawl on a Caucasian rug that my parents had received as a wedding present. Caucasian rugs are very geometric.
You've said you played "intuitive" chess. What do you mean by that?
Well, I had an uncle who, like most people in Poland at that time, seemed to be chronically unemployed. He was kept alive by the family. In particular, he was my tutor—today this would be called home-schooling. A very cultured person and a truly nice man, yet very ill-prepared. But he taught me the rules of chess and played with me—not according to a textbook but in a very, very informal fashion. He didn't explain to me the famous games with all the old champions' names but instead kept asking, "What do you think is the best move?" It was my good luck to leave Poland and abandon chess about the time when I would have started reading the books about the famous games, the championship of 1870 when so-and-so beat so-and-so, and so on. I never did that.
Free of constraints
You've said there is a danger of making rules absolute.
In France, one must belong to one well-defined and very stable sub-community. You do your work and don't look "over the fence" or at what your neighbor does. This is an extraordinary burden largely established in Napoleonic times. By the late 19th century, it was deeply rooted. The French school of mathematics was very strong, and gifted young people were virtually pushed into joining. To the contrary, France had no counterpart to James Maxwell in Britain or Max Planck in Germany, who were creating theoretical physics.
But you didn't let such constraints hold you back.
I didn't. Perhaps my early rootlessness gave me an awareness that one can live without being so completely specified.
You've been interested in the revolution in thinking that took place during the Renaissance. I love the term "natural philosophy" from that period.
It is lovely indeed. Too bad it hasn't been used since the 18th century.
What does that term mean to you?
Before Galileo, a philosopher was somebody who studied the great books. Many of those people were extraordinarily brilliant, but their absolute obedience to books was destructive. What Galileo did was to say that natural philosophy is written in the Great Book of Nature and that one must move from reading the books in the library to reading the books around us—that is, use the experimental method and believe in the power of the eye. That was the big thing. Newton was called a natural philosopher. And in the 18th century, the professions of mathematics and physics were not deeply distinguished, but now they are.
I'm certainly a philosopher—how do you say?—entranced with unifying ideas. However, I don't only study books; I study nature. Also art of the past, for the purpose of finding artifacts that I could embrace.
Doesn't such a stance have dangers of its own, like being too much of a generalist, perhaps?
This is a fundamental question that deserves a very careful answer. Close and competent authority has continually reminded me that every generalist is a gambler and faces very serious professional threats. Perhaps so, but I was lucky. I am glad to be able to say that many distinct fields forgave me the time I spent on other projects and gave me a very nice share of very nice awards. Had I been nothing but a gambler, I would have vanished absolutely. But I have always been extremely disciplined and conservative.
Let me elaborate. I did start as a gambler but soon enough realized that all my successful "plays" had a strong common feel, a common flavor. As a result, I gradually made myself into a red-blooded true innovator/ specialist—the organizer of a fractal theory of roughness as a new field following its own ways.
"What I did was totally despised by my peers, who felt that I'd completely destroyed my promise."
Back to your question, which I understand perfectly. Many people I have known were blessed with comparable intelligence, memory, and independence of mind but just abandoned themselves to the pleasure of commenting about everything that comes out, on new discoveries made in, say, biology or cosmology. They are the ones who live a very "dangerous" life. One very close friend, whom I admired endlessly and rated as superior to me in many of the basic tools of our trade, never knew how to discipline himself. By the end of his life, he had already been forgotten.
Is this point of view what led you in 1958 to consider working as a scientist at IBM's Thomas J. Watson Research Center?
No. Going to IBM was initially pure accident, and it took me many years to realize how lucky I'd been. In fact, my story provides an interesting reflection on what it is to be lucky. I had the chance of enjoying a very long and varied apprenticeship, which included working with people nobody would expect me to work with. For example, I spent two years in Geneva as an associate of Jean Piaget, the child psychologist.
However, in the mood that prevailed in France, I belonged nowhere. What I did was totally despised by my peers, who felt that I'd completely destroyed my promise, that I was just playing around with things of no particular interest. A good older friend commented that my Ph.D. was one half in a field that didn't yet exist, and one half in a field that no longer existed. But that didn't disturb me.
I was trying to tell my colleagues and everybody that, assuming there is a choice, one should not force people to move as fast as possible and decide as early as possible where to go. One should not force people to classify themselves before they are ready. But it was talking to the deaf. In France, if you tried to change from field to field, you had to start from scratch.
You've said you were forced out of France.
I was referring to an intense feeling of not belonging. Also, I was a junior professor of mathematics, but my senior professor, whom I liked very much, was about to retire, and the idea of working for his replacement was totally unbearable. I could have stuck around, I'm sure, but I would not have liked it.
How did you get to IBM then?
Pure luck. Some people I knew were joining IBM Research, and they invited me for a summer in 1958, just after I had started teaching in France and was very dubious about my prospects there. IBM was changing all the time, so I thought I might, perhaps for a couple of years, do enough to please the establishment, yet save enough time to do some gambles of mine.
Wasn't IBM the image of a highly structured environment, though?
This image was strong and pervasive, but the reality was different. The goal was to create a freewheeling academic atmosphere. While they lasted, IBM's laboratories were highly renowned.
That summer was very nice, so after a substantial discussion my wife and I decided to ask France for a leave and stay at IBM for a year or two. During the first two years at IBM I hit a very conspicuous jackpot and was invited by Harvard as Visiting Professor of Economics [laughs]. So we postponed our return to France.
"It dawned on him that I was unpredictable and might do things about which he didn't give a hoot."
Which you eventually postponed indefinitely.
Yes. After a year as Visiting Professor of Economics, I hit a second jackpot: Harvard asked me to return as a Visiting Professor of Applied Physics. So while I had left France temporarily, I eventually faced the reality of the situation.
I was helped by Jerry Weisner, a very remarkable fellow and good friend, who later became President of M.I.T. I asked him for advice about what to do next. It's a long story, but his conclusion was that IBM was no longer a chancy situation. There was no other place in any country where I could do my kind of work.
Because of that freewheeling academic atmosphere.
Yes. IBM was very keen to achieve intellectual renown and technological success. So they wanted me back, while everywhere else I had this problem of being unclassifiable. People would tell me, "If we knew that you were going to stop running around and, from today on, become an honest economist or an honest electrical engineer or an honest this or that, we would offer you a job instantly." At one point a university offered me a very high position. But the next day the Dean called to withdraw the offer, because it dawned on him that I was unpredictable and might do things about which he didn't give a hoot.
[laughs] You've had a lot of opposition over the years, haven't you?
I have. For example, a proposal I made to the National Science Foundation was turned down. I was very surprised when a man telephoned and told me, "In order to fund this thing, we need six Outstanding reviews, and you have five Outstanding and one Excellent. So you can't be funded."
I was so shocked that I raised my voice. In Europe, to raise one's voice is okay, but not in America. I told this man, "Please consider the fact that what I propose is completely at variance with everything else you're supporting. In a certain sense, this proposal is criticizing my peers, because I think they do things in a narrow fashion, and I'm offering a new way. So five Outstanding and one Excellent should be viewed not as a failure but as something of a success." To his credit, he agreed—sort of. "Yes, okay," he said. "I'll fund you. But I'm going to cut your grant in one-half."
Now, in mathematics, your main contribution has not been proofs but new questions, correct?
That's been the case in pure mathematics, in which the overwhelming bulk of mathematical work consists of proving or extending existing statements. But my work in other fields has had a very, very different aspect. In economics there are no proofs. Science is "proven" by its applicability.
Now, my uncle, who was a mathematician given to strong opinions, was very scornful of some of his peers. He said that they were very, very good, but they were just theorem-provers. They have an extraordinary arsenal of techniques, remember many previous results, and put them together in new ways. But they don't have the creativity to ask new questions. So in mathematics there has been historically this more or less sharp distinction between those who are best known for asking questions and those who are best known for proving theorems that others have conjectured.
Who among mathematicians do you most admire for asking questions?
The greatest mathematician in my private pantheon has been Henri Poincaré. Altogether a very great man, he started many branches of mathematics from scratch, but he acknowledged himself that he didn't prove any difficult theorem and cared about proofs less than about concepts. I'm nowhere near Poincaré [laughs]—don't misunderstand me. My point is that a large number of truths that I discovered did not result from purely mathematical deduction but from skilled examination of mathematical pictures.
And do you see the world differently now because of those mathematical pictures, because of fractals?
I certainly see the world today differently from the way I saw it early on. And friends of mine who are mountain climbers tell me they see mountains differently now than before. People who just like to look out the window when they fly—they tell me that they see mountains differently now than before. They see an orderliness to mountains, piles upon piles of pyramids that before they did not see.