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Preparation for exams - Monsieur Croissal


Return to education - thinking in pictures
Benoît Mandelbrot Mathematician
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So in early '44 I found myself in Lyon, in what is called commonly in France 'le top', the official name is post-graduate high school, Preparatoire, and the purpose of this group was primarily to just cram for an exam, in fact for several exams the École Polytechnique, the École Normale, etc., etc. The fact was that there was and there is a gap between high school, on the one hand, and the elite universities, which are called Grandes Écoles. This gap was filled by classes given by some high schools for students who were deemed to have a chance of being accepted to these elite schools. Now the reason I was put in that class was not that anybody felt that I could, after all these years of not studying, catch up or do anything like that. In fact it was very difficult - this planning to keep me out of harm's way for a few weeks, or a few months, so I could recover and perhaps participate more actively again in the events of the day. But things turned out very differently. First of all, in the first week I understood nothing, which was not incomprehensible after having missed a year and a half of studying. I was with students who had studied for a year and a half, again under conditions that were disturbed every so often, very roughly, but on the whole were comparatively undisturbed. So I understood nothing. In the second week, I understood nothing. And then towards the third or fourth week of this 'forcible' stay in such an environment I suddenly realised that there was a game being played. That is, the professor would give us exams or problems in algebra of a very abstract kind, but I didn't perceive them as being abstract; I perceived them as being concrete, and not with anything conscious about it. I didn't just sit down and say, "Can I think of a geometric construction, which would be exemplified by these questions of analysis?" I just simply, spontaneously, thought of such shapes. It was a freakish gift which I must have had for a long time but which events did not have an opportunity of bringing out as they did in that winter of early '44. And I would raise my hand and say, "But isn't your question equivalent to asking how these two surfaces relate to each other, and whether intersection has this and that property?" The professor would say, "Well, yes, but after all, the question is not to know what surfaces, but the question is to learn about how to do algebra very rapidly." One was learning algebra, and this was a pretext for learning algebra, and not for doing geometry. It's very strange that in high school I never knew, I never felt that I had this very particular gift, but in that year in that special cramming school it became more and more pronounced, and in fact in many ways saved me. In the fourth week again I understood nothing, but after five or six weeks of this game it became established that I could spontaneously just listen to the problem and do one geometric solution, then a second and a third. Whilst the professor was checking whether they were the same, I would provide other problems having the same structure. It went on. I didn't learn much algebra. I just learned how better to think in pictures because I knew how to do it. I would see them in my mind's eye, intersecting, moving around, or not intersecting, having this and that property, and could describe what I saw in my eye. Having described it, I could write two or three lines of algebra, which is much easier if you know the results than if you don't.

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: 4 minutes, 24 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008