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The River Nile and Infinite Systems

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The birth of Fractals (Part 2)
Benoît Mandelbrot Mathematician
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So, Richardson was for me a very important point, because first of all it came back to simple shapes: coastlines. Everybody knows coastlines. There was not one geographer of my acquaintance; therefore I had no vested interest in geography. Besides Richardson had provided all his examples, to which I added, of coastlines or geographical lines measured by different people giving entirely different results, like the border between Spain and Portugal which Portugal has claimed is thirty percent longer than what Spain claims. So, the indeterminacy of these quantities, the fact I'd been drumming in the context of economics that the quantities underlying their most important theories were in fact ill-defined, that indeterminancy had moved all over. It had not been eliminated by physical cleanliness, but it had moved into the domain of physics. In the domain of geophysics or geology, which is physical sciences if not physics proper, one had quantities which were totally ill-defined and for which one could not give a number like length; one had to deal with them differently. And so I published this paper first in Science, called How long is the coast of Britain? which became well known. It was published in 1967. It took very long time publishing, and so could I say that there was much actual graphics in it? No. The graphics were very primitive, I just drew by the hand some classical constructions by Koch and others, and variants, and argued that any sensible person would consider a coastline as being made of capes and bays and then smaller capes and bays and so on, and unless something drastic is true, for which there's no evidence, the length is going to increase more and more as you go into details. I thought that this was a great insight; of course I was totally wrong; it was absolutely well-known to everybody, but it was, how to say, suppressed. I'm told by scholars of Greek history that Greek sailors sailing from Athens towards the western Mediterranean were reporting a very different length for Sardinia's coast, saying if you go on our ship it's so much, but if you go on, say, the little lifeboat, then it's much longer, and if you walk along it, it's much longer. They knew it. "So which length do you want?" they asked the Admiralty. The Admiralty did not know. The idea of area, of length and so on, became separate at that time, but certainly by the '60s, the power of school books, the power of school teachers, the power of people who barely understood the bare rudiments of mathematics was such that length was something which was very entrenched, and also very fragile. Infinite variance was not as real as infinite length. Infinite dependence, as I was supposing for these errors, was absolutely not real. Nobody can think simultaneously of these things. One can see the coastline. And so this paper and several papers that followed began fractals, by introducing specifically the eye into scientific research. I would say that '67, '68 were the critical years for that.

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, 58 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008