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The future for fractals


Fractals in education (Part 2)
Benoît Mandelbrot Mathematician
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I am very struck by the use of one of my conjectures, the one about dimension four thirds for Brownian clusters, because its such an easy thing to explain to a child - and it is in a certain sense - a child adolescent rather, it is reassuring for that person to be told that this very simple observation is not known to be either wrong or true. People think it is true, but they can't be sure. They could not give a specific complete argument. I think that the advantage of that is very clear for the mathematicians. The fact that mathematics is well and alive, active and moving amazingly is very difficult to communicate to most people. I don't understand at all most of what's happening in mathematics because it can be understood only after an extraordinarily large number of layers of successive instruction. Where it is possible to show that certain regularities are in a certain sense overwhelmingly likely because, as it turns out in this case, of the eye and of measurements, which is a very much short of a proof, one still establishes that mathematics as a field is open. But I think that beyond these parochial needs of, or uses for, the mathematics community, it is very important to realise once again that the way in which the mind proceeds are not necessarily, in fact seldom are, the same as the ways in which the codification of mathematic observations proceed. To codify observation of nature it had been necessary to begin by Euclid and to go very far in Euclid, and fractals came first not as real things but just as a limitation of Euclid. When they became real things they turned out to be, in a certain sense, more real for many individuals than the simplifications of Euclid. Now why is it so? Why is it that the same kind of techniques, very elementary ones of self-similarity, scaling, self-affinity, and more learned terms like renormalisation and fixed points, why do these techniques apply so generally? I don't think that there is any way of saying that this is the law of structure of the world, but it is a very workable procedure, one which once again very often moves from being very simple to being impossibly difficult; which makes it both challenging and beautiful, but which embodies a certain element of unity among very varied kinds of human behaviour. Which personally for me is the greatest charm and the greatest attraction of this whole enterprise, which has been taking up my life for so many years.

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, 19 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010