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Iteration; background to the work of Fatou and Julia


Pathological shapes
Benoît Mandelbrot Mathematician
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The people who around 1900 introduced the shapes called them mathematical monsters, pathological shapes, because their only intent was to show that certain ideas that had become intuitive were incorrect. Now, intuition was certainly believed at the time to be quite invariant in time, quite inbred in humans, and it didn't occur to them that what they viewed as intuitive had any relation with their time and space. Fifty years later- well for some, seventy-five for others - I thought that these constructions could be models of reality and increasingly proved them to be so, and then built upon an enormous variety of variants which very many people continue to do. Then we went to the next stage which was to construct those random novelties and random shapes obtained by iteration, of which I will speak later - the theory of iteration, the theory of Julia sets and Mandelbrot sets and the like. Those shapes are found by many people to be very attractive visually, in fact artistically. And in due time the readers of my books pointed out to me that the shapes which I was creating around 1900 in the work of like Peano, Sierpinski, like Cantor even; in fact familiar in decoration in many, many cultures. So we find here a circle, a very virtuous circle I must say immediately, which goes from art to mathematics to physics and to art again, with the help of the computer in the passage from the second to third stage.

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

Date story recorded: May 1998

Date story went live: 24 January 2008