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Imitation of nature and creation of shapes


Solving the problem of limit sets by using computers
Benoît Mandelbrot Mathematician
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Well, the result of it was very strange. On the one hand neither I nor anybody else could view that as being a breakthrough when we saw it. It's as if a gap which was an illusion had vanished, because it was so simple. It is a well-known construction; I don't think that it is attributed to anybody in literature; I'm not even angry about it because I felt it didn't require any initiative, but I think it's worth recording because of the way it was obtained. For a hundred years people far more specialised in this field than myself, and very brilliant people had been looking for this solution, but were just looking at the sky, dreaming, "How can I make it?" I was looking down on pictures. They were looking for an algorithm; I was looking at actual constructions that the computer by then could give very easily, because iteration is very, very easy to implement on the computer. And was I the first to use the computer to look at these solutions? Of course not. As a matter of fact, when I found this solution I was convinced that it must be well known. I went to see a very well known person in the field, a man named Wilhelm Magnus, a professor at New York University and asked him - he had written a book on the subject - whether anybody had told him about the solution. He said, "No." He was amazed: "It's so simple, and I never saw it. But," he said, "after my book came out and computers came out, I received enormous correspondence from many people who've sent me drawings of these limit sets." So he give me a big file. I looked at the file and was totally amazed. No-one had experimented. That is, those people wanted to use the computer to make better pictures for a book by Klein, from 1885 or 1890, but they didn't take the initiative of thinking of the picture as being something that is creative. A picture is something that visualises results obtained, otherwise it is not a tool for invention, a tool for creation, a tool for helping the mind obtain results which pure thought cannot allow us to obtain. It would not have been possible without the advent of computers? I think that the computer by itself was necessary, not sufficient. If one did not break from the mould of mathematics in the '70s as I did, one could not imagine using computer for this purpose. I was not the first. People who knew mathematics far better than I did were there, but they could not have obtained this result, I think, at all, because it required a leap of faith.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Bernard Sapoval Daniel Zajdenweber

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Duration: 3 minutes, 23 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008