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Geometry; coming home to pictures


Self-affining and self-similar fractals
Benoît Mandelbrot Mathematician
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Well, that lesson was not lost on me, and I am talking now of the late '60s. Shortly afterwards I started working on something different, which gave rise later on to the fractal model of distribution of galaxies. We still had only Calcomp plotters, but to show people some patterns of dots and ask them: do you think that is theory or that's sort of real, became a very important test. And also something else entirely which became very apparent with graphics, that very simple rules that didn't seem to have anything particularly important in them were in fact capable of producing very complicated phenomena. So this distribution of galaxies, the first model - it's not a model even, it's called the scenario, which was a Levy Dust in space. Very easy to construct. Very easy to imitate, to simulate; half a line of formula, fifteen lines of code, we got all this clustering of galaxies, super-clustering, automatically out of it. That is, automatically this model was in a way simplifying, reducing this overwhelming complication of reality to a possible collection of consequences of one very simple principle, which of course is a principle at the core of science. Science tries to duplicate this complication of reality by very simple rules. Now, to come back again to those rivers, the parameter they have is again dimension; more precisely the Holder exponent, again a very esoteric notion of mathematics used in harmonic analysis with which we used to make measurement of roughness. And in each case it became clear that one could gather roughness in at least two different environments: the coastlines which are self-similar, and those other shapes which are self-affining, which means that each is part of the whole but reduced. In one case reduction is isotropic, the same in all directions; in the other reduction it is one in one direction, one in the other direction. Therefore at that point this split occurred between two kinds of fractals, two principal kinds of fractals: self-affining, and self-similar, which have become increasingly important from there on.

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, 44 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008