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Self-affining and self-similar fractals


Writing and publishing work on rivers
Benoît Mandelbrot Mathematician
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But to come back to '67, '68, in addition to this work on coastlines, I became very much engaged in the River Nile and such phenomena, and decided very deliberately and tactically to do it right. I had felt that in all these contexts, when one needs to deal with a very advanced technique, applicable to a very practical problem, one must break it down so that everybody agrees, everybody understands, everybody sees what's happening. And so with a collaborator who was a hydrologist named Wallis, we wrote half a dozen papers in this light. Now I may add that Wallis was also the programmer of this work, and that I would like to mention because from there on I became increasingly active in using computers and graphics, but almost totally inactive in terms of programming. I'm a very bad programmer for reasons that are quite understandable. I'm not detail conscious; I am more conscious of the big framework, and in programming, every small detail is make or break. Therefore, to me, debugging was an impossible task. Besides, I would say that the programming of this work became so time-consuming and one or sometimes several programmers were engaged in it; if I'd been doing programming myself I would just have competed with them on unequal terms. It was not worthwhile. But I became very skilled because it was so gradual, in explaining to my collaborators or my assistants what kind of program they would write, what things one should beware of, etc., etc. And it goes on very effectively to this day. Now, with Wallis we wrote these papers, and then the question of publication was raised, and Wallace said, "Well, let's publish that in a statistics journal." I said, "Never! Never. I'm not competing in statistics. I would like these ideas to become standard knowledge in hydrology." But he said, "But they won't accept these papers. They're too complicated." So the following scheme was hatched. There was going to be a meeting of the American Geophysical Union, I think in Baltimore, or maybe Washington, and we prepared on the same machine, which was a Calcomp tracer, we prepared charts of a large number of records of rivers, the Nile, Colorado, Amazon, St. Lawrence, etc., etc. and of all the existing models, and of my model, which was based upon what I called fractional Gaussian noise and all the captions were at the back; everything was done in the same style. We went to see the editor of the main journal in hydrology, a man named Langbine, a very brilliant old-fashioned engineer, but an extraordinarily honest straightforward person, and told him the following story. "We have a crazy way of representing river behaviour. If we present you with this thing you will not understand because it's just mathematical esoterica at this point, and certainly you wouldn't see how it applies to your problem. But perhaps you should accept to play a game with us. Please tell us which of these graphics are real rivers and which are theories?" Well, the man started shuffling (the papers) and said, "Oh, this one's ridiculous." Well, that was his theory! Well, as a witness of his honesty and competence he said, "I knew it was bad; it didn't think it was that bad, but it was very short term dependence." He put it aside. And then put aside others and yet others; then he said, "Well, I don't know. I don't know. They all look like the things which I deal with every day." Then he turns to the Amazon, Nile, St Lawrence models - model one, model two, model three of my model. He said, "I can't believe it. So you have found some part of a secret of the world." And he accepted the papers, how to say, cumulatively; he was saying, "Somebody's going to read them and to help you, you understand, but you will not get any flak because it's incomprehensible" And he delivered. That was to me an amazing situation because, again, given the extraordinary badness of the tools - the Calcomp plotters were high-tech - we couldn't hope to have much better than these tracings, but even those simple tracings, in the eyes of this man, very skilled, very old-fashioned, who was looking at these things all the time and commenting about them, were sufficient to make a difference between inacceptable esoterica, and something that for some reason he had to learn.

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: 5 minutes, 21 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008