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A new geometry of nature

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1/f noise, rivers and turbulence

Benoît Mandelbrot
Mathematician

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

91. Meeting at Courchevel (Part 1) | 134 | 02:46 | |

92. Meeting at Courchevel (Part 2) | 124 | 03:36 | |

93. Background to chaos and wild randomness: Galileo, Newton, Laplace | 221 | 03:13 | |

94. The twentieth century - predictability | 187 | 01:25 | |

95. Background to work in mathematics, physics, economics and finance | 205 | 01:10 | |

96. The butterfly effect | 321 | 02:15 | |

97. Fractals as a tool to represent nature | 231 | 04:51 | |

98. 1/f noise, rivers and turbulence | 353 | 06:01 | |

99. A new geometry of nature | 1 | 193 | 01:50 |

100. Lewis Fry Richardson and Leonardo da Vinci | 249 | 04:08 |

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In the early '60s, I became faced with a phenomenon that I discussed in many places, which is distribution of errors communication in telephone channels. This was something very, very powerful, which had surprising success, but which obviously was an over simplification of reality. At this time I was only considering some phenomenon through a very crude reduction; either an error occurs or an error doesn't occur. Almost immediately the task became to understand the underlying phenomena. And what were those underlying phenomena? Essentially there are two of them, or three perhaps. One of them was so-called ^{1}/_{f} noise. Another one was turbulence, and by the very names they're given one realises that they were faced, in a certain sense, very differently. ^{1}/_{f} noise; the very term expresses that if you do spectral analysis of those phenomena you obtain a spectrum that is inversely proportional to frequency. Very, very strange behaviours that were baffling every one of those who discovered them, in totally disconnected places - as a matter of fact in the early '60s, this was esoterica scattered among different disciplines, acoustics, electronics etc., etc. The second phenomenon, which was linked to it, is the behaviour of rivers over the long run. That was not presented as ^{1}/_{f} noise in any certain form, but in a certain sense like a story of the Bible, the story of Joseph, the son of Jacob, and the seven fat years, seven lean years. The Bible, like the expression of any culture which lives in a dry climate, had to take account of something fundamental about Egypt, which is that some periods are very wet, other periods are very dry; some periods of seven or seventeen or seventy years are either wet or dry. The very successful Egyptian dynasties were ones that were blessed with lots of water in the Nile, and those that had no water in the Nile had no resources to either build the great buildings or wage great wars. Therefore, certain cultures are very much dominated by this behaviour of very great irregularity. If I was to define the very notion of what the average level of the Nile is, it almost goes through the fingers. There is no average level around which it fluctuates. Do you mean this dynasty, or that dynasty? This period, or that period? Therefore the way of thinking, which we have been used to, which Laplace, around 1800, taught us very strongly, of randomness as being fluctuations around a middle value, doesn't apply easily to the Nile. As a matter of fact, the people who were studying the Nile were very emphatic in saying that one needs art, experience, memory to study it because mathematics was too blunt a tool for that. What about turbulence? Turbulence is a phenomenon, which ordinarily enough has been named after a phenomenon of sociology: a crowd, an unruly crowd, that's what turbulent meant. But Osborne Reynolds and others in the late nineteenth century classified the behaviour of fluids as being turbulent or non-turbulent. Turbulence is one of the greatest, the most difficult problems that physics faces us with. The story is told - it may be apocryphal but is a very profound one - of Enrico Fermi close to death. He was dying of radiation exposure and, very conscious to cheer him up, his friends asked him, "When you meet our Maker, what will you ask our Maker?" And he said, "There are two questions that now bother me. One is quantum electrodynamics, the other is turbulence. Quantum electrodynamics, well, if I could live longer it will be solved s there are enough bright people to do it. Turbulence is harder. I'll ask our Maker how turbulence functions." A feeling of mystery has always enclosed it. It is not defined by a formula, ^{1}/_{f}, but by phenomena. Somehow an airplane is in what was smooth air, then being shaken, then again the air becomes smooth. It turns out that techniques that are used for the ^{1}/_{f} noises, for the rivers, and for turbulence form a whole. As a matter of fact, I started with one, a very simplified view of the ^{1}/_{f} noise, which was lucky, because had I seen the whole monster I would have been totally overwhelmed. So chance served me once again. But very soon I realised that the ^{1}/_{f} noise was not a well-defined notion, that in fact you could have a spectrum, ^{1}/_{f} , while being of very many different kinds, and that is a discovery that at the same time is, how should I say, a shallow observation and at the same time also very profound, because ^{1}/_{f} is a formula. The same formula can be used as caption to all kinds of different phenomena. This proves the formula does not have much precision to it. Phenomena are very complicated. Again, these phenomena are very rough, irregular. They are not fluctuations around a straight line, certainly not straight lines. They are very, very difficult, yet ^{1}/_{f} finds some commonality among them but doesn't go any deeper.

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: **1/f noise, rivers and turbulence

**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:**
6 minutes, 1 second

**Date story recorded:**
May 1998

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