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Fractals as a tool to represent nature


The butterfly effect
Benoît Mandelbrot Mathematician
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Perhaps before I proceed to other fields in mathematics and physics and elsewhere, I should give the classic example. I have a certain resistance to examples that have become cliches, even when the cliche really embodies very strongly the true situation. In the case of the weather, the great contribution of Edward Lorenz has been to observe that even if one assumes that the weather follows extremely precise laws which are differential equations of fluid motion, and if everything is under perfect control, even so a very tiny change in initial conditions? the classical butterfly effect. A butterfly moving up or moving sideways somewhere in the Caribbean would affect the process of a storm moving towards the US coast; because the butterfly went up, the storm much later moved up the coast. The storm involves a gigantic amount of energy, the butterfly a very tiny one. If one knew exactly the initial conditions, if one knew exactly how the butterfly moved, one could predict in that model where the hurricane would move. Since actual prediction of this butterfly or other butterflies is impossible, despite the existence, presumed in this thinking, of very strict laws, it's impossible to know where the storm will move. That is, a system can be both completely predictable in principle, and completely unpredictable in practice. Needless to say, in actual situations the complete predictability in principle is seldom satisfied, but as a background for understanding many phenomena, the butterfly effect is absolutely essential.

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, 16 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008