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The haunting beauty in both the Julia set and Mandelbrot set


The first conjecture of the Mandelbrot set
Benoît Mandelbrot Mathematician
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The relation between my original problem, which was the existence of limit cycles, and the problem we were studying, that is the connectedness of the Julia set, became the main issue in front of me. I had been right to follow from this impossible problem to this one that was very easy to program, but right in a sense that visualy it was what I expected, but could I prove it? Well, I spent, I think, twenty minutes on it, a very intense twenty minutes, to convince myself that I shouldn't even try to prove it. It was beyond my capacities. I had stumbled into a field in which very difficult things were happening. I could not do it. I put as a conjecture, and the conjecture is that the first set which I was considering - call it m0 - if you add it to all its limit points, make it the limit points plus the set, you obtain the set m which is obtained by connectedness of the Julia set. This conjecture looked extremely simple. It can be explained to a good high school student, because there was no complicated concept involved. The concept of a limit cycle is straightforward. The concept of connectedness is straightforward and intuitive. The relation between the two was postulated by a very simple identity, that m =m0 plus its limit points. Well, believe it or not, this conjecture is still open and what is so striking about the study of iteration of z2+c is that the first serious difficulty I encountered in its study has remained, after now eighteen, nineteen years of study, totally baffling. First people told me, "I'm going to prove it to you next week," then next month, then next year, then next decade. Now of course they do it next millennium! It's a problem of extraordinary difficulty. Let me comment on this problem. Without computers, it would have been totally unthinkable to even raise it. In 1947 when my uncle spoke to me about that, no-one could have asked seriously the question of the structure of a set for which the limit set is connected, because after a very, very few small pieces that can be done analytically, the rest is just too complicated. With a computer many people have been looking at iteration, outside of my little group. They had computers; they could do iterations, in fact they made a few pictures of Julia sets, but did nothing with them. In fact, they didn't look at them very seriously. In fact, they didn't expect a picture to bring anything new. The idea that a picture could bring anything new in 1970 was simply unthinkable. It was the case in 1830, 1840 perhaps, but not in 1900 and certainly not in 1970. The proof that how should I say, respectful examination of pictures with all the looking again and again, looking closer, looking next to it, experimenting to the hilt, that idea was simply not in anybody's mind. And so my feeling was during that period was of simply walking into a foreign territory which literally nobody had visited, not just wild and there were savages - there was just nobody there. And I was not even working very hard, if you will. I had the courage of looking at it, the courage of taking this very difficult question in my hands, literally in my hands, and in my eyes, and I was leaning by, leaning and collecting diamonds, emeralds and rubies, just lying on the floor. And some of these diamonds, emeralds and rubies gave rise to mathematical conjectures that I could prove myself. Others were proven within a year, others within five years and still more within ten years. And again, the first one, the most obvious, most natural, is still open. It was translated into a variety of other forms. It has now many equivalent expressions, but has become stabilised into an open question that nobody thinks can be solved by a nice simple trick, because almost all the solutions require an enormous amount of effort.

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: 5 minutes, 32 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008