Distribution of cell volumes in a Voronoi partition

Abstract

The distribution of cell volumes in a Voronoi partition of three-dimensional Euclidean space was obtained by a computer simulation method in which the volume of a cell is derived from the number of lattice points of a reference cubic lattice which are closer to the centre of that cell than to the centre of any other cell. The cell centres have a Poisson distribution. It has been found that the distribution of cell volumes is fairly well fitted by a gamma distribution function with parameter α ≃ 5·56, which is the value of α that gives the exact second moment of the volume distribution, as calculated by Gilbert in 1962. This is an extension to three-dimensions of the applicability of the gamma function to describe the area distribution in a two-dimensional partition (with α ≃ 3·6) and the length distribution in a one-dimensional partition (with the exact value α = 2).