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1/f noise, rivers and turbulence


Fractals as a tool to represent nature
Benoît Mandelbrot Mathematician
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My whole scientific life was spent wandering between disciplines. I have often had the feeling of them as being cities, somewhat fortified, not completely fortified, and of myself coming and asking to be welcomed and trying to contribute to their welfare. But to explain this work I am obliged in a way to disentangle different stories that occurred together and now would like to proceed to story of fractals as tools to represent nature. At the beginning of my book, The Fractal Geometry of Nature, I observe that mountains are not cones, clouds are not spheres, islands are not circles, rivers do not run straight, lightening is not a straight line; and more generally that the shapes that nature offers us for contemplation and understanding are only extraordinarily, exceptionally represented by shapes of the geometry of Euclid. It's a miracle, in a certain sense, a central one but still totally surprising, that shapes like circles which we see in nature only when the moon is full, or when a tree is particularly straight and you cut it - those shapes give an idea of a circle, but there are very few circles in nature. There are no straight lines in nature. Painters, who are very careful of these matters, particularly so Eugene Delacroix who wrote about these topics very eloquently, says that there's no straight line in nature, there is no circle, there is no triangle of course. Yet these shapes have enabled man to obtain an extraordinary degree of understanding, control and explanation about many aspects of nature. However, for everyone, the ordinary man and woman and the scientist, it was always a source of surprise that, in a certain sense, the limitations of science should be so obvious. The poets, who often do not like the scientists, are quite generous with comments in which they say that geometry is hard and dull and dry and that nature has a richness of shape, a richness of expression that goes beyond anything that intellectual minds, the mind of a scientist could imagine. Well, this thought is very, very deeply embodied in our culture, but the fractals prove that it is without foundation. One of the most striking features of fractals is that they enable us to imitate nature and thereby to, I think, gather immediately a great deal of understanding about nature by creating shapes that are completely understood, totally grasped by thought, which are completely described by very simple formulas that can be combined, divided, altered, modified, and which in different forms provide us with very straightforward forgeries of nature. In most cases 'forgery' is a bad word because one wants to repeat, in a certain sense, the thinking of Vermeer or some other painter who painted only a little, and it has become very valuable; in this case 'forgery' is a very positive word. We do not claim to represent nature in all its richness. The claim is not to obtain pictures that are indistinguishable from nature, because had that been our goal we'd be led to enormous levels of complication and the simplicity, the straightforwardness of fractal geometry would be lost or at least diminished. The goal is to find a rather reasonable imitation; I wouldn't say caricature, I wouldn't say cartoon, but a good imitation of what nature is about. The most widely known of these pictures, are, of course, pictures of mountains, of clouds, of trees, which fractal geometry produces so easily. Instead of starting on this topic I would like to go straight to more fundamental issues of physics because my involvement with mountains did not occur because of a particular love for mountains but as a rhetoric, to explain how other things worked.

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: 4 minutes, 52 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008