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Development of the Mandelbrot set; 'dirt' in the picture


Julia sets
Benoît Mandelbrot Mathematician
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In the case of z2+c one can ask the question, which was asked by Julia and Fatou; under which conditions is it true that the Julia set is connected? Now what is the Julia set? The Julia set is in this case the set you obtain by working back this transformation, not going to z2+c, but working back from one value of zn to one which is the inverse of f. There are two possible values, therefore by working back there's a tree of possibilities that occurs, and one chooses one of these possibilities at random; and one sees how it behaves. I was postulating, I was implying all kinds of theorems I could not prove, some of which were proven by others I didn't know, and others are still open in this process, because I was trying not to do mathematics, but to explore the unknown numerically at that time. And so the second set we could look at was whether the other - for c - for which the inverse backing up on this transformation would lead to a set which is connected or not connected. That question was raised by Fatou and Julia because it was mathematically simple. It had and still has no intuitive meaning to me. I felt it was close to a question that is physically interesting. So after playing for a long time with Julia sets, finding their complications, finding their extraordinary structure, finding their self-similarity, which was so striking, the small piece and the whole thing - but again smaller, not reduced linearly but by squaring but that's alright, it is still a form of self-similarity - I tried to make a map. It was a map of possible behaviours, in a certain sense a dictionary, in certain sence an index. Since I did not know that there was no theory that says if c is in that region, then that thing happens; if c is at other region, then something else happens. I decided to establish this map, first, again, on the basis of existent limit cycles, and then on the basis from Fatou and Julia's criteria of when the Julia set is connected.

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, 33 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008