The geometrical properties of irregular two-dimensional Voronoi tessellations

We describe a parameter used to quantify the regularity of two-dimensional Voronoi tessellations based upon assemblies of ‘hard-core’ discs. The value of this parameter may vary continuously from zero, for a completely random Poisson Voronoi tessellation, to one, for a fully ordered regular hexagonal honeycomb. For various values of this parameter, 105 Voronoi cells are simulated and the statistical distributions of the number of sides per cell, the cell vertex angles, the cell edge lengths, the cell perimeters and the cell areas are each derived. The mean perimeters, areas and numbers of sides in the neighbouring cells are also investigated for n-sided cells in these tessellations. We find that, for all except the cell vertex angle distributions, the data can be adequately described by fitting either to existing models or, in the cases of the mean perimeters and mean areas of n-sided cells, to models which we propose.