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Self-similarity

Benoît Mandelbrot
Mathematician

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

61. Writing and publishing work on rivers | 272 | 05:20 | |

62. Self-affining and self-similar fractals | 276 | 02:43 | |

63. Geometry; coming home to pictures | 290 | 04:13 | |

64. Origins and publication of Fractal Objects (Part 1) | 275 | 04:32 | |

65. Origins and publication of Fractal Objects (Part 2) | 232 | 04:12 | |

66. Commonality of structure | 224 | 01:11 | |

67. Fractals and the importance of proper description (Part 1) | 1058 | 03:13 | |

68. Fractals and the importance of proper description (Part 2) | 344 | 05:54 | |

69. Self-similarity | 289 | 03:53 | |

70. Cartoons | 316 | 02:15 |

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I would like to say more about what fractals are and what they are not. First of all, I don't think that fractals should be defined in a very strict fashion. Not because of any kind of a priori dislike for definitions - I am quite neutral on that account, - but because since I've been working with them several would-be definitions were proposed by me and others and then found to be wanting. They did not represent the distinction that we wished to create and the lack of their definition never did any harm I think, at all. So let us speak about fractals that are the most clear cut. If you look at the Cantor set being obtained by taking an interval, taking away the middle third, and then for each remaining third taking away the third and so on, the main fact is that the side thirds are identical to the whole set, except that they are reduced in a ratio of one third. Therefore the notion of self-similarity is overwhelming in that context. I'm sure that Cantor when he defined his set had this self-similarity in mind. However, the self-similarity was viewed by him as being inessential, more a matter of convenience, of ease of writing a formula, than as a matter of anything useful. Mathematicians very rapidly went beyond self-similarity to shapes rather like Cantor but more general. The world 'self-similarity' itself in English dates, I think, to a paper I wrote in 1964. Now the word 'fractal' is one with which I have certain pride because it corresponded to bringing together a variety of shapes and circumstances which were not identified as being special at all, and of underlining their co-ordination, their usefulness in the description of nature. But 'self-similar' was a word, which was not used before because nobody thought it was necessary. The idea was in some cases trivial and transient, in other cases simply happenstance, something of no significance. Therefore when I had to distinguish self-similar shapes from others, I had to coin a word that was self explanatory, which was not needed before and which on purpose is very, very simple. Now in the case of the Cantor set again the parts on the side are, like the whole thing, one third. The case of this shape which I called the Sierpinski gasket or Sierpinski triangle; you take a triangle, divide it into four equal parts and you erase the middle one. Again, quite clearly, the final shape is made of three parts, each of them reduced in the ratio of one half. These shapes are self-similar in a linear stick-fashion, and totally non-random in a lengthless fashion. Those shapes are in a certain sense very dull and very useless because there's not much that one discovers by studying them which is not already present in the definition. Therefore self-similarity has this strength, in those cases, of being expressed in the simplest possible terms. But to get shapes that are of great, great richness and unpredictability and which are of greatest usefulness in terms of physics and economics and also dynamics and so on, one needs shapes which are either random or in which the reduction from the whole to the parts are non-linear. The idea of similarity changes from being strictly deduction of directions to being deduction, with a certain amount of random or non-linear deformation.

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: **Self-similarity

**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:**
3 minutes, 53 seconds

**Date story recorded:**
May 1998

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