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Mandelbrot on Fractals, Academia, and Industry

By Akshay Patil

staff writer

The Tech had an opportunity to talk to math and physics legend Benoit B. Mandelbrot during his short visit to MIT. One of the fathers of fractal science, Mandelbrot discovered a mathematical set of numbers whose graphical representation is so stunning that it is often considered the face of fractals and chaos today.

The Tech: Do you have any personal heroes and inspirations that have driven you over the years?

Benoit Mandelbrot: For a long time my hero was John von Neumann, who was, among other things, one of the pioneers of computers. I was a post-doc with von Neumann when Dr. von Neumann died and he was my hero because he succeeded during his life in doing work in mathematics and application based technologies; all without compromising his perfectly rigorous manner of doing things.

In time, more heroes appeared. One that is not so widely known I think, a pity, is a Spaniard who lived a hundred years ago, his name was Santiago Ramon y Cajal. Do you know his name? Ramon y Cajal was a doctor in Spain who described the structures of the nervous system, which is made of molecules, if you wish, which are the neurons, and atoms, which are parts of neurons, and how they interact.

He started in a world where nothing was known, a world in which the eye was paramount. When I first looked at pictures of the Mandelbrot set which are sort of lighter gray, darker gray, on a horrible electronic screen which was worn by excessive use with a graphic system which again gave dark gray over light gray, I was thinking of Ramon y Cajal because he looked at structures that could not be photographed. There wasn’t enough contrast but, by playing with depths of field, by his extraordinary visual skills, he was marvelous.

He then drew pictures of all these neurons. It was so perfect, so early, that in the early 1950s when neuron anatomy awoke again, because of new progress here at MIT, my friends at MIT were using as the reference for the nervous system, a book, first published in Spanish 60 years before. They were using the French translation from 1903 .

Now I think that Cajal had this combination of working on the very border of what was known, tools were not available, but combining every trick, skill, eye, he did provide this picture of what the brain is which has not been changed. So that is an extreme, I can also name heroes that are exemplified more than others.

Henri Poincare is a hero for reasons more general. Poincare is such a basic name in pure mathematics that one could think that he was an unquestioned person in his time, but actually in his time he was extraordinarily controversial. He defied categories, because on the one hand he was, by quite a long stretch, the most amazing man of his time in many different areas but he never proved anything rigorously, so his community disliked him for his desire to leave difficulties to others because they enjoyed it and he didn’t.

TT: What was your first thought when you first saw your Mandelbrot Set?

BM: Well it was the middle of the night and I thought I was dreaming a nightmare or something. Truly I thought some machine had gone haywire or something. So the cure to that is to change some place and look at it again.

The next day, we, my programmer and myself, came back to it and we checked again differently and again zooming in and out. And then the third day, came back to it and the thing was totally familiar.

It was an uncanny situation -- something moved from being totally wild to something I felt as if I had known forever. I’ve been telling that to people, and whenever I tell that story among people my age, who are of the age to see it when it was still totally new, they say they had the same experience. I just attended a meeting on this topic with several known physiologists, the question is whether it is part of the wiring of the human brain which is trained into these shapes, it is totally hypothetical, that possibility.

TT: With the discovery of self similarity everywhere, how has this effected or reinforced your belief in God or the supernatural?

BM: No comment.

TT: It seems that science now is obsessed with fractals and chaos. It’s almost buzzword. Do you think the theories are being overused in other fields of science?

BM: Well I think that what you say “buzzword” and so on, was much more true 20 years ago. I remember the first few years after my book as being a combination of great pleasure and great pain because I simply couldn’t stand to be in the middle of a fad. My impression is that the fad aspect has very much decreased. If you say what you say, I believe you.

You feel it, but it’s nothing compared to 20 years ago. And the fad was in place by just the completely natural use of these techniques in many areas. Now it’s different if seen by me or by people who are actually professionals working in the field and by the outside world.

Let me elaborate. The outside world knows almost nothing of science. Most of science, even in the very beginning, is difficult. To say what the problem is, is very difficult. Once you pose open questions it becomes shut off. Fractals are different insofar as the results are so striking so soon. You start with very childish assumptions you turn the thing once or twice and you get questions nobody can answer. So the absence of competition in that world means that people who are interested in science of this sort have nowhere to go except at this point: chaos and fractals.

So it is not something that is good or bad, it’s just lack of competition. Yesterday I gave this lecture at MIT, and students run after me saying they heard me or my friends speak a few years ago and that’s what brought them to science and to MIT. It’s actually a wonderful feeling for me because it’s much better to influence a nature so young then to influence some of one’s peers. But it doesn’t affect the professional development of the field.

In many areas people just take these things for granted, the tools are provided, and they go on. Where the tool came from, the motivation, my personality, the eye matters very little. And that is how it should be; I mean there is no way of it. So perception depends very much on where you stand.

TT: I know you did a lot of your work at IBM, what are you feelings about the intersection between industry and academia?

BM: Well, IBM. I was at IBM during the 35 years of history which were quite extraordinary. IBM research began, as it turned out, for internal reasons not because of choice, around the time of Sputnik. In other words, in 1957-58 IBM was a beggar who couldn’t be a chooser. Anyone with a clean record, a good degree, and belonging to a well-established group with a good number of recommendations could get any number of jobs at MIT, at Bell Labs, you name it. IBM was not even in the running.

It was a mechanical engineering outfit which was suddenly transforming itself into an electronics outfit and facing questions which the old IBM could not even begin to face. It was run by people who had very long-view and who therefore accepted that it was necessary to follow criteria which were totally independent of academic criteria.

That is again, the top of the crop in academia was unavailable and so the question of academic publications was irrelevant and the reason why after coming to IBM by some accident, as it turned out, I stayed on and on and finally stayed half of my life is that it was for that period an extraordinary place.

And a statement was made, which I don’t think was made anywhere nearly so strongly about what happens if the academic criteria are abandoned because there is no choice. There is no question that a large part of my life has been spent fighting academic criteria. At one time people tell me I’m not really a mathematician. Well, I know what they mean, but I’m in for the math more than many real mathematicians. People say I’m not a really a physicist, I know what they mean.

I received the Wolf Prize in Physics. Some people believe otherwise. There is a way of handling phenomena which academia does not do well. Today it is particularly severe, because all the institutions of research in the United States were set up in the late 40’s under conditions which are long gone. Those institutions are, well, not necessarily fitted for plain reality; in fact they are not fitted for reality.

Academia, as I know it from years at Harvard, before I went to Yale -- academia simply does not know how to deal with administration. It doesn’t have structures for it.

We do our best at Yale, in fact the reason why I stay at Yale and I’m happy there is because it turns out that the math department is a particularly open-minded department which has a great deal of variety of activities. Everybody can interpret mathematics very broadly.

I think that’s how it should be, that is, in the near future. Now, academic specifications must be modified. I mean some activities simply must be replaced by other activities. Change is indispensable. I see very well how the lessons are influenced by my work, not because some horrible committee decided for it, because actually even with what you said about the popularity of chaos and fractals, many committees are rather hostile to it because they say we should not let that be alone.

So they are hostile, but people vote with their feet. They come to these courses, they enjoy it. We have, at Yale, courses for non-mathematicians, little people who take fractals.

They love it and they learn more mathematics than they could ever learned any of the standard ways of teaching mathematics to a laymen.

So I think the world will change and in this sense, what happened at IBM not by design, but because of historical accident with the ’57 situation and then the presence of several very hostile Russians who were directors of research and had a very broad view of science. That example should be instructive because science had become extraordinarily over organized, it had been specialized in different fields, and I think to its own detriment.

TT: One final question: what’s your favorite math or physics constant?

BM: Oh! No comment ...

TT: No comment? Alright ...

BM: One!