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Solving the problem of limit sets (Part 2)

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Solving the problem of limit sets (Part 1)
Benoît Mandelbrot Mathematician
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Now the development of these two topics, from my viewpoint, took entirely different aspects. In the theory of Kleinian groups I did not contribute to any new problem. As a matter of fact, it might have been just playing by me over a few years, if it were not that I solved a problem, which Poincaré and Felix Klein at the same time had seen about a hundred years before and couldn't solve, and this was something in which the eye plays a role which is completely amazing. Poincaré and Klein had to deal with limit sets of these groups; that is, after you repeat these operation many times there are some sets which are defined as limit sets. They can be called attractors in a certain sense, which is a better known word now in the theory of chaos, but they called them limit sets. And they had an algorithm for certain limit sets. This algorithm was converging with excruciating slowness, that is, after many, many operations it gave a very, very crude idea of what the limit is going to be. And there was a very widespread feeling that somehow this was a gap; that there should exist a simple way of constructing those limit sets without going through very slow and unsatisfactory algorithm. Well, what I did first was not to think of the algorithm, but simply to see whether those people in 1870s, 1880s or 1890s had been right in their evaluation of the limit sets. With a computer it was very easy: where they had great trouble we could just take the pictures, measure all the circles they had and then redo it and take orbits and just see the orbit- and the limit set grow in front of our eyes. The limit sets were very different from what the books of the time presented. On one hand, very much more complicated, but on the other hand extraordinarily orderly. In fact, in the limit sets that were obtained by pictures, the self-similar aspect of these constructions was very apparent. A small piece of the limit set looked like the whole limit set but smaller, and deformed - not linearly, but some were deformed - but definitely the same structure in different scale. In the pictures of the 1980s, self-similarity was not visible. I don't even remember whether they were aware of self-similarity, because it is not something which one can think of so easily in these very complicated shapes. And by and by, looking at the variety of these constructions, I had a very nice and very good assistant at the time, the programmer who was working for me all the time, but he was not mathematically sophisticated. But I had for a month a mathematically very sophisticated programmer, a student at Princeton, and so I was having him redo these things with slightly different initial conditions, so we had limits set for this set; well, actually, we were looking at inversions and circles, which is almost like Kleinian groups, not quite, but very, very close. So you select a collection of circles and then you invert. It's a very easy thing to do. And I would say, "Well move these circles, move these circles, move those circles." He'd say, "Why? I mean it's all the same." I 'd say, "No, no. At this point it doesn't talk to me. I don't see anything. I would like to establish a connection with this alien in front of me." And then at one point the solution became obvious. I didn't do any kind of highfaluting mathematical investigations; I just experimented. Experimentally the solution to this problem became totally obvious. That is, there was a way of constructing a limit set, which after two or three stages gave a very, clear idea of what the limit set was. Moreover, this way had nothing, how to say, bizarre about it because a curve which is one of the major monsters of mathematics round 1900, namely the Koch curve, the snowflake curve, can be constructed by having the inside and the outside made of little triangles that become smaller and smaller, and each new iteration gives more detail. You get an overall shape after one stage; a very good approximation after two: an excellent one after three or four - the eye sees how to continue. Well the construction I found for the limit sets of groups based upon inversions, which are almost like Kleinian groups, consisted in replacing the triangles with circles, and those circles I called 'oscillatory', because they kiss the limit set in a certain sense.

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, 12 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008