Fragmenting the universe. III: The construction and statistics of 3-D Voronoi tessellations

In this paper we present a study of the statistical properties of three-dimensional Voronoi tessellations. This unique tiling of space into polyhedral cells generated by a random distribution of seeds is a phenomological model for a cell-like or sponge-like distribution of galaxies. After an introduction with an historical background of the subject we will present some analytically known results on Voronoi tessellations. This is followed by a description of the algorithm to calculate three-dimensional Voronoi tessellations from a given distribution of nuclei in a box with periodic boundary conditions. By means of a Monte Carlo study we subsequently turn towards an extensive statistical analysis of several geometrical aspects of three dimensional Voronoi tessellations by means of a Monte Carlo study