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Fractals and rules


Fractality and the end of regularity
Benoît Mandelbrot Mathematician
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Same thing for fluid flow; the equation of Navier-Stokes' rules the behaviour where it is smooth, and there are places where the behaviour is not smooth at all. Of course it is not completely rough because of viscosity, but let's forget viscosity for the moment. The idea is that the irregularities that represent turbulence are manifestations of the same phenomenon that gave us the smooth flows. Take DLA. The way I describe it by these particles floating around, as a matter of fact performing a Brownian motion, is the way they came in initially. But this can also be rephrased as a problem in Laplace's equation in which the boundary is variable. That is, in solving a problem with a fixed box as I always did as a student, as millions of books were written to do, you imagine that the solution to the problem changes its own boundaries. It is extraordinary broadening. That is, a block, nobody thought of that because there was, in a certain sense, no need for it, but this experiment of Whitney and Sandor which creates DLA shows us that starting with very, very smooth boundaries, for example a little circle or a straight line, one makes them rougher and rougher and rougher, and the roughening is not haphazard, irregular, incomprehensible, lawless; quite the contrary, it follows very, very strict rules of fractality. So, what I would like to say is that the various examples in physics at very different scales, again skipping many examples, tends to point in that single direction: that fractality is not a separate phenomenon from the smooth phenomena, it is part of the same reality. That in a certain sense fractality is at the end of regularity, the boundary of regularity, the frontiers of regularity, and those frontiers of regularity become destroyed and are replaced by fractality.

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, 6 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008