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Fractals and beauty


The uses of fractals
Benoît Mandelbrot Mathematician
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I think that the last part of this interview should cover, how should I say, the applications of fractals. The applications - 'applications' actually is a word I do not like because it expresses a kind of superiority of a theory over the uses to which it is put, so let me speak of the uses of fractals. I wouldn't like to resume what was said before but just sketch a few scattered ones to give an idea of the extraordinary variety. To begin with, the notion of roughness that was undefined or measured by too many irrelevant quantities can be measured now by one, two or a few numbers. That is, one of our principle senses has been tamed, to a large extent, whereas it was completely untamed before. In things like, for example, oil exploration. Oil is found in rocks, which are porous under very complicated conditions. The structure of these porous rocks is essential; to define it accurately is fundamental. Fractals have allowed scientists to describe those rocks much better and to go quite far in improving procedures in various fields of that domain. In an entirely different context there is the problem of image compression. I'm jumping from one to one because I don't want to give the impression of any exhaustive presentation. Images must often be compressed to be transmitted, with a little bit of distortion and a great deal of economy. There are two major competing ways of image compression at this point. One is based on fractals and the other on the wavelets. They are very different from each other, hybrids exist, but altogether the combination of fractal thinking and wavelet thinking allows one to replace a picture that takes an enormous number of bytes to transmit by a picture which will be much more economical and which cannot be distinguished very much from the original one. I emphasis techniques using all these applications are very, very different from each other.

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: 2 minutes, 38 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010