Address: Laboratoire de Physique
Théorique
Bâtiment 210
Université de ParisSud
91405 Orsay
France
Telephone: (0)1 6915 7787
Fax:
(0)1 6915 8287
Email:
Henk Hilhorst
Publications .pdf .ps (last update December 2014)
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References will be added./
/Last update October 29, 2010./
The three figures on the left represent what is called PoissonVoronoi tessellations. The red dots are the "centers", randomly and uniformly distributed
in the plane
with unit density [mathematicians say that they are a Poisson process
of intensity 1].
Figure 1 is just a statistically average tessellation. You see that the Voronoi cells come in many different shapes and sizes. In particular, they have a variable number of sides. The probability distribution of this number of sides  let's call it n  is a quantity of interest. 

Figure 1 

Figure 2 is a very exceptional tessellation in which there occurs a cell that is 24sided! Such events are extremely rare. They have been the subject of my research in the period 20052008. Let us call p_{n} the probability that a cell randomly picked from a tessellation have n=24 sides. For n=24 we have that p_{24} is only about 10^{18}. It is remarkable that most of the firstneighbor cells of the 24sided "central" cell are quite elongated. This feature will become more and more pronounced when we increase the number of sides n. 

Figure 2 

Figure 3 is even more exceptional than Figure 2. A cell occurs that is 48sided! The probability for a cell to have 48 sides is only about 10^{60}. But if it does occur, it will look like shown in the figure. 

Figure 3 