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 consider ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise controlled by the parameter alpha. We study the number of sides, the area, and the perimeter of the Voronoi cells. For alpha>0, hexagons are the most common class of cells, and 2-parameter gamma distributions describe well the statistics of the geometrical characteristics. The symmetry break due to noise destroys the square tessellation, whereas the honeycomb hexagonal tessellation is very stable and all Voronoi cells are hexagon for small but finite noise with alpha<0.1. For a moderate amount of Gaussian noise, memory of the specific unperturbed tessellation is lost, because the statistics of the two perturbed tessellations is indistinguishable. When alpha>2, results converge to those of Poisson-Voronoi tessellations. The properties of n-sided cells change with alpha until the Poisson-Voronoi limit is reached for alpha>2. The Desch law for perimeters is confirmed to be not valid and a square root dependence on n is established. 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; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be taken as generic polygons in 2D Voronoi tessellations.

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