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The Hausdorff Dimension


Robert Stewart and a return to an interest in turbulence
Benoît Mandelbrot Mathematician
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The next amazing thing that happened wasn't that- at that point what happened was that Harvard invited me to do a course on the prices that I will discuss later perhaps, and then after this was over, asked me to discuss what came to be this work on telephone errors came to be, and while I was there a speaker, Robert Stewart, who became a very good friend, was speaking about turbulence. Well, Stewart had been the person who had actually found the K-5/3, which is why I went to see his lecture, because he found it in Puget Sound, off the coast of British Columbia, in a very old decommissioned electric submarine. By having a very long snout in front they made measurements of turbulence well under the sea and they found a very substantial range, a scaling range, K-5/3, which in 1948 Caltech was a total pipe dream and everybody said it is impossible, Kolmogorov is wrong. Suddenly Kolmogorov becomes real. But at the same time what they found was that the whole idea of what is turbulent was becoming, shall we say, - was disappearing, because previously with very bad instruments one had a notion that fluid can be either turbulent or laminar. It is laminar and then something happened, it was turbulent, and then something happened, it was laminar. Or you make a cut through a fluid and you find this portion was laminar and this was turbulent. But Stewart and his cohorts who were superb experimentalists, looked carefully at the so-called turbulent region, and they found that most of the turbulent region was in fact laminar. They looked further again and it was laminar. Now, that's what Stewart presented in his lecture, by saying, "This is something impossible. We say it is a property of intervals up or down, up or down - turbulent or laminar. It is not. The turbulent thing is full of holes, laminar." Well, in general it was said that all that was totally beyond any analysis, and I was sitting there in the front row smiling to hear this, because after it was over I told them, "I can tell you how this thing has happened. Turbulence is not a property of intervals; turbulence is a property of Cantor sets and the like," which in terms of understanding this terrifying phenomena I think was a very significant breakthrough. That is, this turbulence had been boxed into this two state dichotomy which was going back to Reynolds or much before, maybe to medieval engineers, and I had given a much more realistic picture of what may be. That explained immediately why turbulence would be very, very difficult, because turbulence was not a mild randomness; it was a very, very wild one. I didn't use the words then, but I became extremely interested in turbulence at that moment.

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, 16 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008