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Brownian motion and the four-thirds conjecture


The branching structure of Mandelbrot sets
Benoît Mandelbrot Mathematician
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After the story of z2+c proved to be so fruitful, it was natural for many people to go to study of z3+c or other such transformations. It turned out to be extraordinarily more complicated, not just a little more complicated, and what was so striking in the study of these simple transformations is that you had a stage that was totally elementary; a stage which is, how should I say, was very rich, but you could grasp it; and then - a total unknown. There is an image that I use for that purpose, which I think makes the point. In most sciences there's a core that is very simple, at least very simple to write. Anybody can understand gravitation, F = Gm1m2/r2, it's a formula that is very simple. As you go out and out and out it becomes more and more complicated. As you continue to go out it becomes more complicated again, more technical. But there's a big domain around the formula in which one doesn't encounter impossible difficulties. The study of fractals, of chaos in general - and I will say in a minute how it relates to chaos - is more like the situation of a very branching structure. There is a core that is very simple, which children can understand, which anybody can copy and implement, but in order to continue doing something doable you must follow a finger. If you go in a natural direction looking at natural questions, most likely you'll get into an abyss, which means some things that are extraordinarily difficult. Difficulty does not increase gradually as you go away from the simple core. And that is true of fractals in general. It's illegitimate as you look at the parts of fractal geometry that were written down for children to believe it's an easy subject, because the main constructions are like playing, the main early results are like play, and in general there's almost no effort needed to obtain them, or so it seems. Then people say it's an easy subject; it's far from true. If you follow very peculiar lines you again find easy considerations, but if you follow a natural line, which was a matter, for example, of the structure of the boundary of the Mandelbrot set, you very rapidly find things, which are beyond the most refined techniques we have.

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: Daniel Zajdenweber Bernard Sapoval

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

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.

Duration: 2 minutes, 54 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008