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The inevitability of mathematical development


Benoît Mandelbrot Mathematician
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Perhaps I will say a few words again about dimension. Dimension ordinarily is a number which is like zero for points, one for lines, two for squares, three for cubes, etc., if you can expand in more than three dimensional spaces, which is comparatively immidiate mathematically. What fractal geometry does is to create a continuity dimension. You can have a set, dimension one half, or one and a half or two and a half, whatever. Any number can be a dimension of a set. This dimension, as I said before, represents in a very concrete sense, roughness, and irregularity, both for a coastline and for a curve that represents a function. But there's something more to it. The notion of dimension is very central to physics; that is, physics in one dimension is different from physics in two dimensions or in three dimensions. Now, take magnets. There are no magnets in one dimension; there are magnets in two and three dimensions, because of various very complicated fundamental reasons, and my friends and I have been preoccupied with the question of understanding more precisely in a more demanding fashion, all of dimension in these phenomena. We have written about it. I think that it is a wide-open topic. Perhaps it is mathematics, perhaps it is physics: I'm not sure. Anyway, it deserves to be taken up again and looked at much more closely. But the idea of dimension is something, which has become truly fundamental out of being just a very abstract idea. And there again, dimension is substantial progress, but that is not the end. To represent a shape by numbers one must go beyond dimension. There is a whole enormous chapter that I opened many years ago and that moves very, very slowly because it is so difficult. I call this notion lacunarity, that is, you can have shapes having the same dimension and looking completely different. And that touches a very basic point of interaction between the analytic way of doing science, which is everybody's way and that has to be the way when you go to equations, and the geometric component of it, because a shape, a very complicated shape, cannot be totally specified by a few numbers, and the very complicated shapes we have around us very often have to be approximated by one number then another and then a third. When I said earlier on that dimension is for roughness what temperature is for hotness, it is part of the truth but roughness requires more numbers than just one - not very many but still more than one. And this particularly is so if one wants to examine closely various phenomena that the world faces us with.

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, 7 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008