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The first conjecture of the Mandelbrot set


Development of the Mandelbrot set; 'dirt' in the picture
Benoît Mandelbrot Mathematician
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That is how the Mandelbrot set came to be. We drew this set as a - well, we call it a map; on this map we should see in which parts of a plane different behaviours occurred. The first impression of this map, of course, was extraordinary complication. The first impressions we had were of shapes that were dirty. Now, I must emphasise, what kind of instruments I was using. This work was started when I was at IBM, and continued when I was a visiting professor of mathematics at Harvard. Harvard then had very poor computer facilities, extremely poor ones. We had a small DEK computer, and we had a Tektronix screen, a graphic screen, which was very old and very worn out, and a Versatec printer, with a very old-fashioned processor which gave rise to very dirty pictures under ordinary conditions. So the first drawing of what became known as the Mandelbrot set a few years later, was very, very messy, and I thought it was the fault of our terrible equipment. Well, to make sure, we zoomed; that is, we looked in greater detail at a small part of the field. Some pieces were clearly dirt; other pieces we didn't know. I took advantage of a trip back to IBM and went to beg my previous assistant, and remind him that I was a nice person to work for, that I let him do all kinds of things which I didn't care about because that would help his career after he stopped working for me, and finally convinced him to give me a day. So, he had some difficulty reproducing the program done at Harvard, and then he reproduced it: their machines were far better, much faster; the screen was extremely high quality; and I was expecting a nice clean picture. The picture was worse than ever. It was a very peculiar cardioid, then a circle, then this and that, and then at the boundary- mess beyond imagination. Well I couldn't beg Mark Laff to continue, so I went back to Harvard, then started looking at the boundaries in greater detail and greater detail. At some point - we had picture after picture; we repeated the picture - and the dirt was different. Where it was different in different transparencies, probably it was dirt. And we zoomed in, - it was dirt. In other places, a strange thing was happening. We saw the same piece of dirt on top and it was symmetrical at the bottom. That was very unlikely to happen because of just a bad system. We looked at these pieces in greater detail: total shock. Those pieces were exactly like the whole thing but smaller and a little bit deformed. Well, we looked at some other pieces in greater detail and then we started taking values of the parameter c, in z2+c. Actually, I was using a different formalism, but it was equivalent. When c is in those pieces - I called them islands - and when it was there, inside, we got a limit cycle; when it was outside we didn't get a limit cycle. More precisely, the thing was going on moving around and we couldn't see whether it was a very big cycle or it was no cycle: therefore, chaos.

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, 58 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008