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Fractality and the end of regularity


Fractals and the distribution of galaxies
Benoît Mandelbrot Mathematician
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I would not like to make a long list of other fields of physics where fractals are important because it is all too easy to make a list of so many topics, that and that and that, without any connection. Let me just mention one and then, how should I say, wrap it up on several levels. The one that I will mention is the distribution of galaxies. Of course the galaxies that are close to us are not very uniformly distributed, and a very strange historical event occurs here. For many centuries individuals were thinking about matter, stellar matter, being collected in clumps of various sizes well before there was any evidence of this being the case. The main galaxy I suppose starting from the Milky Way, is quite recent - the realisation that we live in a certain clump of matter in the universe is quite recent, but well before it came, some individuals were dreaming about a universe in which matter existed very irregularly. Then galaxies came; then the prevailing wisdom among cosmologists became that the large structure of the universe is uniform, except perhaps very close to us, to our galaxy, where it is clumpy as we see very, very obviously. My analysis of the distribution of galaxies -distributions were well known in literature - made me conclude something quite different, that galaxies had a fractal distribution up to a certain depth in the universe, and then after that I couldn't say anything, but there was a considerable depth in the universe up to which galaxies were not uniformly distributed, but fractally distributed. The cross-over from fractality to uniformity is something which my data could not tackle. A friend and disciple of mine, a very distinguished professor in Rome, has shown that this cross-over occurs much farther out. But the reason I mention this example here in the same breath as DLA and turbulence is the following. In all three cases there's a phenomenon that follows well-defined simple equations. They are called partial differential equations. The basic partial differential equations of physics were discovered around 1800 by people who translated Newton's Law in a more and more refined fashion. The Laplace equation, Poisson equation, Fourier equation, Navier-Stokes' equations - they were all discovered in the early 19th century. They are differential equations, which means that they refer to functions that have at least two derivatives, and the whole of research on them has been devoted to phenomena whose solutions are smooth, with perhaps a few singularities involved in it. Now, very early on my work in physics, on physical phenomena to be more specific, raised the following question: how come in one universe, in one world, there's one part that everybody has been studying so successfully for so long that is ruled by smooth phenomena, two derivatives, Laplace, Poisson, Navier-Stokes and other equations. Then a kind of introduction of fractals which came as a foreign object, like a contradiction to everything else that we were used to. This view of fractality as being an interloper, as being something like an uncalled-for, unwelcome guest, was very, very widespread and several people concluded that fractality was only an approximation; that all these things were smooth and regular but it was convenient to make them fractal, that fractality was not true, what was here was just an illusion or a convenience. Now, very early on I had a very different thought. The thought was that fractality comes from the same sources as the smooth behaviour declared by those equations. For example, if you take a large number of point masses in a box - in practice one must continue the box periodically, but a big box, and a large number of masses. You put them in uniformly early on, you put interaction, which attracts as inverse square of distance, and then you push the whole thing by adding some energy. This experiment was done around IBM in the '60s, redone again and again and now is an experiment that is very easy to make. What happens is that very rapidly this matter, which was originally quite uniform, becomes fractally distributed, clumps and superclumps and so on. We did not put the clumps in on purpose, and the gravitational field outside of the masses is perfectly smooth and regular - two derivatives, Laplace equation. However, each mass is what's called a singularity of a field, and those singularities are not fixed, they can move. That is, the attraction of the other galaxies made the singularities move. Therefore the lack of attention in the past given to singularities was a mistake. One must not only solve the equation by having fixed singularities or slowly moving singularities, but by looking at how singularities are going to be displaced and how they'll be distributed. The evidence, which is so far purely numerical, is that the galaxies become fractally distributed. That is a very important point.

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: 5 minutes, 53 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008