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Meeting at Courchevel (Part 1)

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Multifractals

Benoît Mandelbrot
Mathematician

Views | Duration | ||
---|---|---|---|

81. Beginning to work on the problems of Julia and Fatou | 166 | 03:42 | |

82. Julia sets | 180 | 02:33 | |

83. Development of the Mandelbrot set; 'dirt' in the picture | 251 | 03:57 | |

84. The first conjecture of the Mandelbrot set | 233 | 05:31 | |

85. The haunting beauty in both the Julia set and Mandelbrot set | 253 | 01:16 | |

86. The Mandelbrot set and fractals | 1 | 583 | 01:42 |

87. The branching structure of Mandelbrot sets | 211 | 02:53 | |

88. Brownian motion and the four-thirds conjecture | 323 | 06:06 | |

89. The four-thirds conjecture and proof that mathematics is still... | 219 | 03:26 | |

90. Multifractals | 210 | 05:35 |

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I would like to add a few words to these examples about mathematics, the limit set of Kleinian groups, iteration of z^{2}+c and the four thirds. One of the fields which my work opened is that of multifractals. Now, multifractals are measures which mean the following: most fractals are sets, which means that you are either in a set or outside; you have a straight line, you are either on the line or outside; a circle, either on the circle or outside, either in the disc which is inside the circle or outside. Therefore it's a matter of black and white. If you are black and if you are inside, they are white on the outside, if you think of a printed picture of a set infinitely fine. But most of physics is concerned with shades of grey. Things are not black and white, don't have nice weather and stormy weather- more or less stormy. And one of the most important structures, which I introduced over the years is that of multifractals. Now I would like to emphasise again its roots in preceding mathematics that I did before very strongly. Some multifractals were known in the literature, again in this collection of esoterica, pathology etc. Very few - just a hint of some complication was known. I began to study them in the context of 'intermittency' of turbulence. Turbulence is not something that comes as no turbulence, then turbulence; but no turbulence ,then turbulence at a more or less higher levels of intensity. And a concept that is very important, and that I hope to discuss in the part concerned with physics, is that of intermittency. I constructed a model for it and encountered some contradictions, some paradoxes. The formalism was becoming ridiculous if the parameter was above a certain value. Now, if you encounter ridiculous phenomena in mathematics there are many possibilities. One is that you made a mistake and it is very often the case. Sometimes, often in my experience, it means that if you look carefully there is something very interesting hidden in the paradox. And in this case - this was the case. Now how did I approach it? First, not by analysis. Not because I couldn't do it; I could do it and I did it shortly afterwards, at least up to a point. Because I wanted to understand in my fingers what was happening. Therefore it was in the early '70s I would say, around 1970 or perhaps even slightly before that with heroic calculations - because then computers were very slow, and these were very big calculations - I tried to make constructions both below and above this threshold where things were getting funny, and acquired a very concrete feeling of what becoming funny meant. It is not a feeling in mathematics that says, in this case, that the measure of the generator is almost surely zero, in the other case it's a well-defined limit. That is something very important, but it is not the whole thing. I looked at those shapes, I looked at what becoming degenerate meant. And only after having this very strong feeling did I collect my thoughts and show by heuristic, meaning incomplete and not completely rigorous arguments, that indeed where the parameter, when things start becoming funny, a certain result that was quite unexpected and that again informed the degeneracy. I would not like to emphasis the details because it's a delicate issue, though I believe that a few questions were later to give much work to mathematician friends because it became very difficult to prove my loose arguments to be correct. All those difficulties were not of an artificial kind, not of a hypothetical kind, not of a kind that refers only in infinity and are in the limit where nobody can have any feeling for what's happening. Look, one sees them occurring. In mathematics the idea that refined results, difficult results, are beyond intuition, are beyond the eye and almost the hand that idea I think was simply, simply wrong. The advent of the computer and the use of computer as we did to create computer graphics for our purposes, to expand its uses, have shown it again and again and again.

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.

**Title: **Multifractals

**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, 36 seconds

**Date story recorded:**
May 1998

**Date story went live:**
24 January 2008