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The branching structure of Mandelbrot sets


The Mandelbrot set and fractals
Benoît Mandelbrot Mathematician
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Now I took the short path to tell you the story, but let me recount some more episodes. The Julia sets again are self-similar; each part is like any other part, but one of the observations made about the Mandelbrot set was the following: that a small corner of the Mandelbrot set looks in many ways like the Julia sets corresponding to it; therefore the Mandelbrot set included as parts of it reduced scale images of an immense variety of Julia sets; therefore it is not self-similar in that respect. It is self-similar in the sense that the islands are like the continent but slightly deformed; but not in a sense that the way in which the islands are arranged, the kind of strings that link them together are the same; in fact, they're different in every point. Therefore the complication of a set goes beyond fractals. Julia sets are self-similar, they are fractals by every definition and by intuition, except when they are just straight intervals which happens for some cases, in which case they are too simple to be fractal. The Mandelbrot set is a complication which includes a huge number of different fractals in its structure and is therefore beyond any fractal. It is a paradox of sorts that this has become the icon of fractality, whereas it does not fit the definition of the concept at all.

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: 1 minute, 43 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008