A fractal life

Few people would recognise Benoit Mandelbrot in the street, but the intricate pattern of blobs, swirls and spikes that bears his name - the Mandelbrot set - is an icon of science. It has come to symbolise the geometry of fractals, patterns whose shape stays the same whatever scale you view them on. His life has followed a path as jagged as any fractal. Next week he turns 80. He tells Valerie Jamieson that he still has plenty of work to do

<>How did you feel when you discovered it?

Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it. It was as if somehow I had seen it before. Of course I hadn't. No one had seen it. No one had described it. The fact that a certain aspect of its mathematical nature remains mysterious, despite hundreds of brilliant people working on it, is the icing on the cake to me.

<>What's the mystery?
It relates to a rather subtle mathematical property. In simple terms, there are two ways to define the Mandelbrot set. It is rather like proving that 3+1 and 2+2 give the same result. I have always thought that the two definitions were equivalent. But one is easy to study whereas the other is extremely difficult. So far, the proof has defeated many people. The fact that my conjecture is so simple to state, yet baffles everybody, makes it attractive to mathematicians. The conjecture is the mathematical face of the Mandelbrot set, and the T-shirts are the popular face. <>Fractals seem to appear all over nature and in economics. Even the internet is fractal. What does that say about the underlying nature of these phenomena?
Well, it depends on the field. Circles and straight lines also appear everywhere. Does this mean that all those phenomena have something in common? Of course not. The roughly circular trajectory of a planet around the sun is due to gravitational interactions. Berries are round because a sphere has a smaller skin. The beauty of geometry is that it is a language of extraordinary subtlety that serves many purposes.

<>So fractals don't point to a single rule underlying reality?
There is no single rule that governs the use of geometry. I don't think that one exists.

<>Your uncle Szolem Mandelbrojt was also a mathematician. How has he influenced your life and work?
In every way imaginable. I was 13 when my uncle became professor at Collège de France in Paris. I learned early on that mathematics is an honourable profession from which you can make a living. My uncle was a pure mathematician on weekdays and a painter and fanatic museum visitor on Sundays. He had a gifted eye. But he felt that beauty and mathematics were completely separate. These two gifts probably existed in the family and I put them together, which made a big difference to my work. I could relate mathematical formulae to pictures.

Is that where the Mandelbrot set came from?

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

<>What are you working on now?
My work is more varied than at any other point in my life. I am still carrying out research in pure mathematics. And I am working on an idea that I had several years ago on negative dimensions. <>

What are they?
Negative dimensions are a way of measuring how empty something is. In mathematics, only one set is called empty. It contains nothing whatsoever. But I argued that some sets are emptier than others in a certain useful way. It is an idea that almost everyone greets with great suspicion, thinking I've gone soft in the brain in my old age. Then I explain it and people realise it is obvious. Now I'm developing the idea fully with a colleague. I have high hopes that once we write it down properly and give a few lectures about it at suitable places that negative dimensions will become standard in mathematics.

<>When you were 20, you said that you wanted to be the Johannes Kepler of a new branch of science. What did you mean?
What Kepler did was to make sense of the motion of planets around the sun. He replaced an earlier accumulation of fixes with a beautiful collection of three laws that truly explained the behaviour of planets. Kepler used the mathematics of ellipses, a great achievement of Greek mathematics, for something practical. My childish ambition was to find a field that nobody had studied, then study it using sophisticated mathematical tools which I would create and manipulate if necessary. <>

And have you succeeded in that ambition?
Yes. Before my first paper on cotton prices in 1963, the model in circulation was pretty bad. I proposed a different model. People came proposing cycles, epicycles and so on which would mimic my model to a point. But they were much more complicated and less complex.