Statistics of Three Dimensional Random Voronoi Tessellations

If S is a set of points in a Euclidean space R, and each point of the space is associated with the nearest point of S, then the space is divided into convex polyhedra, or cells. Such a partition is called a Voronoi tessellation, also known as a Dirichlet or Theissen tessellation. When S is generated randomly, the result is a random Voronoi tessellation. Such patterns turn up in the crystallization of metals [1,2], geography [3], pattern recognition [4], numerical interpolation [5], and many other subjects. A general scheme was derived in [6] for calculating statistics of random Voronoi tessellations for sets S generated by a Poisson point process of unit density. This scheme is applied in this paper to find statistics of random tessellations of three dimensional space and plane cross-sections of such tessellations. Meijering [1] derived the mean values of many quantities. These are given in Table 1. Gilbert [2] expressed the variance of the cell volume in terms of a double integral. This paper finds the variances and covariances for all these quantities in terms of integrals, which are evaluated numerically. Also found are the distributions of edge lengths of cells and cross-sections.