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Beginning to work on the problems of Julia and Fatou


Imitation of nature and creation of shapes
Benoît Mandelbrot Mathematician
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I would like to add a few words about these iterations of inversions and Kleinian groups. First of all the self-similarity is obvious on these pictures; that is one looks at a small piece of it and it is like the whole except smaller, and deformed - that's very essential, that part. The second point is that it is not imitation of nature. In fractals there are two aspects, which I like to keep very much in balance. For example, all my book jackets, the front and the back jacket, one part is imitation of nature, and one part is the creation of these amazing shapes - which, again, chaos theory embodies - which are created from very, very simple formulas in a fractal. The two are very closely related in my mind, in the motivation, but quite separate in, how to say, the daily handling and style. The iteration, the Kleinian group part did not become wildly popular. It had a very strong influence on some of my friends, and I think it had a strong influence on research on Kleinian groups, which was very much in the doldrums and became again very active. But to my disappointment people did not start drawing such shapes and to correct for it I decided just a few weeks ago to use such a structure as a cover on a forthcoming book. Today's computers are so much better than what I had in the late '70s, that it could really play much further than before; and what you obtain - in particular, one structure that is in my book of 1982, which I called 'the pharaoh's breastplate' was re-rendered not as a kind of shadow of jewellery found in the Pyramid, but as a jewel. It is astounding how rich those shapes can be. That is, you take a collection of six circles, not chosen at random, but chosen according to my experience with what is going to be nice, and do this - apply my algorithm from my book of '82, and lo and behold, you get objects which are certainly not an imitation of nature, but which are, in a certain very strong sense, powerfully decorative. And frankly, I am going to have them made up as jewels as soon as I can manage to corner a high class jeweller to do it!

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: 3 minutes, 4 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008