From symmetry break to Poisson point process in 2D Voronoi tessellations: the generic nature of hexagons

We bridge the properties of the regular square and honeycomb Voronoi tessellations of the plane to those of the Poisson-Voronoi case, thus analyzing in a common framework symmetry-break processes and the approach to uniformly random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions is perturbed through a Gaussian noise whose adimensional strength is controlled by the parameter α. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For α>0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of statistical properties of the analyzed geometrical characteristics. The symmetry break induced by the introduction of noise destroys the square tessellation, which is structurally unstable, whereas the honeycomb hexagonal tessellation is very stable and all Voronoi cells are hexagon for small but finite noise with α<0.1. Several statistical signatures of the symmetry break are evidenced. Already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the two perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is confirmed to be not valid and a square root dependence on n, which allows an easy link to the Lewis law for areas, is established. Finally, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells coincides with the full ensemble mean; this might imply that the number of sides acts as a thermodynamic state variable fluctuating about n=6, and this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be taken as generic polygons in 2D Voronoi tessellations.