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Working 'before the limit'


Measuring roughness
Benoît Mandelbrot Mathematician
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Well, this work on coastlines following Richardson's empirical observation and using this illustration of the notion of the Hausdorff dimension became extremely important. The first application of course is to give a very clear meaning to what dimension is intuitively. Those coastlines that had a high dimension were viewed as rough by the eye, and those coastlines with a very low dimension - very close to one - as smooth. And I became aware increasingly of a very peculiar and marvellous circumstance - that fate had brought me into a position to tackle one of.. in a sense there all along and, which science had not previously tackled. That is, the eye gives us the impression of brightness and colour; the ear of loudness and pitch; the hand of weight, the hand of hotness and coldness, and for all these sensations, pace Plato, physics had created a field which mastered them with proper measurements, with proper simulations, constructions, which maintained the main aspects of it, to a very high degree of precision. What about the idea of roughness? There was nothing about it. It is later when I became very serious about measuring roughness, it was confirmed to me, that there was no agreed-upon measurement of roughness. There was no such thing. Or too many. Or too many which is the same thing. People had used various quantities borrowed from statistics that were not designed on purpose for this goal but were designed for other goals. Now, with fractal dimension therefore I was measuring was roughness in a certain idealised context, and that was marvellous because a number could be given which was, say.. But have you coined the word 'fractal'? Not yet. You just said Did I say 'fractal dimension'? Yes. I went too fast. I was thinking of dimension without the word fractal, because at this point I had not coined the words. Let me say, I had been doing fractal geometry a very long time before the word became necessary, for approximately ten years. I am telling you about the period which came to my book of '75 where the word 'fractal' appeared for the first time. I was using dimensions, sometimes the Hausdorff Besikovitch dimension, though sometimes other words, but the idea was indeed to measure roughness.

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: 3 minutes, 3 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008