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The four-thirds conjecture and proof that mathematics is still alive


Brownian motion and the four-thirds conjecture
Benoît Mandelbrot Mathematician
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I will perhaps continue, before I speak of the artistic aspects more specifically, on my contribution to mathematics outside of the solution to the old problem that was so old that nobody even cared about it, and the beginning of the revival of the study of iteration. What I said about difficult questions coming out immediately after the easy ones has its best example in the following construction. It is called random walk in the plane, or Brownian motion in the plane. Imagine the plane is covered with a lattice of lines, and you start at the origin and you go forward, back, left or right with equal probability. You start from a point, there's a very high probability that you return to where you started very quickly, but sometimes you take a very long time, and the cases where it takes you a very long time are representative of what happens in the continuous plane representation, in which random walk becomes Brownian motion. The point which sort of wanders around for a long time and eventually by definition, by requirement, returns to its origin, I call that the Brownian bridge, two dimensions. It covers a patch of a plane. I had this patch of a plane drawn by curiosity because Brownian motion is the best understood of the random fractals, but I always had a feeling that it was understood only intellectually and not visually and tactually. We drew this blob and my first impression was it looked like very very irregular islands. Now islands by that time were things I knew very well. That was the example that I used to make the ideas of fractal geometry understandable to the engineers and physicists I was dealing with. And my father had been such a map nut, my house was full with maps. So a very very irregular island, well, in my experience the very irregular islands have a dimension usually less than - maybe - four thirds. So my conjecture was that the dimension of the blob may be four thirds. Then I made calculations one can measure dimension. Dimension is the main measure of roughness for fractals, and we measured it by one fashion. We found four thirds with extraordinary precision, I think five decimals. I tried to prove it ...pardon, I forgot to say what the four thirds is. The four thirds is the boundary of it, which is all the points that are accessible outside. Within that island the Brownian motion goes back and forth and in a very irregular fashion, so there are many, many gaps in Brownian motion. But one looks at the whole cluster as a body and one looks at the boundary. The dimension is four thirds, visually and numerically. It became a conjecture in my book of 1982, The Fractal Geometry of Nature; there's a page on that and an invitation to my friends to prove it. Well, one friend told me, "I will give you the proof tomorrow." He didn't. Another one, "Next week." He didn't. Another, "Next month." He didn't. Another, "Next year." He did not. It has now been through a very large number of papers, a considerable literature, which incidentally solved many interesting questions and raised many interesting side issues that could be handled by mathematics, but the four thirds conjecture is still open, after sixteen years, which is close to infinity given the number of very brilliant people who have been trying. And again, what was my contribution here? Not to solving it, but to look at one new aspect of Brownian motion numerically, to let my eye wander, to look at variants of this cluster, modify it, change it, and see that this four thirds appearance brings together many considerations from physics, where four thirds is a magic number in many important phenomena, and altogether to establish a conjecture about one particular shape. Now that particular shape shows promise of becoming a way of exploring the plane, that is, there one has many aspects of a plane, topological, metric, etc., etc., etc. This maybe yet another aspect of the structure of the plane. But we don't know. It is totally open. It's a beginning, and to me what is so heart-warming is that my technique, which is entirely based always upon the eye and upon physical analogies, upon intuition, has been so successful, sometimes in closing an old-standing problem, and more often in opening entirely new ones which have become enormous explorations in which my contribution has been to show a few of the major facts one should be looking at.

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: 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: 6 minutes, 7 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008