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A new alphabet


Fractals and chaos theory
Benoît Mandelbrot Mathematician
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I spoke of iteration without using the word 'chaos', or perhaps only in passing. But iteration has become one of the principle tools of chaos theory, which again was sketched, seen, but not followed up, long ago, and became a very major aspect of science later on. Let me say two words about the relation between chaos theory and fractals. Much of chaos theory is made of mechanical considerations that are totally analytic. They don't concern shapes, they concern something else. That is not fractal. When chaos generates shapes, the shapes it generates are invariably fractal, that is, they have the property of self-similarity or self-affinity because of the way they were generated, because one of the reasons for fractality is indeed chaos - differential equations. I spoke of fractality in the visible space, in turbulence, in stars and so on; there is also fractality in phase space, in invisible representational spaces, and that is generated by chaos in dynamical systems. In both cases one has fractality. Fractals themselves have at least two aspects. One of them is random, as I mentioned; the other is non-linear. There are also, how to say, basic fractals that are neither random nor non-linear; they are very intuitive, very useful, but they're only cartoons of reality because they lack flexibility in a very strong sense. The flexible fractals are either random or non-linear. If you put together all my work on iteration, and I could say on other aspects of chaos but I will not enter into it because it would become too disconnected, then my work on these phenomena in physics, which can be described as the theory of irregularity- the word escapes me- in general the phenomena that I described as being a matter of complexity - in each case the irregularity comes from different sources, known or unknown, but can be studied in a very coordinated fashion. Fractal geometry covers all those phenomena, irrespective of the source, and irrespective of whether they're phenomena in our real, ordinary space or phenomena in some abstract space in which one represents the behaviour of a dynamical system. I may just add in passing that it was quite striking to observe that in the context of chaos theory, which is a theory of order in deterministic systems that look very disorderly, the fractals were accepted with very little, how should I say, surprise or opposition. In a certain sense one's speaking of shapes that were not familiar to us; there was no, how should I say, set tradition of thinking about them; they had not been classified into our world in any particular fashion. Whereas in the real space one was associating well known, familiar, very well-defined complicated phenomena with mathematical esoterica, which was saying that these phenomena are simpler than they seem to be, and more tenable. This is not a matter of anything intrinsic, just simply the sociology of science; that one finds it easier to introduce innovations when there's no existing competitive view of the world present.

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, 5 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008