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.
Given a discrete set of points X in an Euclidean N-dimensional space, we have that for almost any point a of such a space there is one specific point x∈X which is closest to a. Some point x may be equally distant from two or more points of X. If X contains only two points, x 1 and x 2 , then the set of all points with the same distance from x 1 and x 2 is a hyperplane, which has codimension 1. The hyperplane bisects perpendicularly the segment from x 1 and x 2 . In general, the set of all points closer to a point x i ∈ X than to any other point x j ≠ x i , x j ∈ X is the interior of a convex (N-1)-polytope usually called the Voronoi cell for x i . The set of the (N-1)-polytopes Π i , each corresponding to, and containing, one point x i ∈X, is the Voronoi tessellation corresponding to X, and provides a partitioning of the considered N-dimensional space (Voronoi 1907(Voronoi , 1908)). As well known, the Delaunay triangulation (Delaunay 1934) gives the dual graph of the Voronoi tessellation (Okabe et al. 2000). Voronoi cells can be defined also for non-Euclidean metric spaces, but in the general case the existence of a Voronoi tessellation is not guaranteed.
Since the Voronoi tessellation creates a one-to-one optimal -in the sense of minimum distance -correspondence between a point and a polytope, 2D and 3D Voronoi tessellations have been considered for a long time for applications in several research areas, such as telecommunications (Sortais et al., 2007), biology (Finney 1975), astronomy (Icke 1996), forestry (Barrett 1997) atomic physics (Goede et al. 1997), metallurgy (Weaire et al., 1986), polymer science (Dotera 1999), materials science (Bennett et al., 1986). In solid-state physics, the Voronoi cells of the single component of a crystal are known as Wigner-Seitz unit cells (Ashcroft and Mermin 1976). In a geophysical context, Voronoi tessellations have been widely used to analyze spatially distributed observational or model output data (Tsai et al. 2004); in particular, they are a formidable tool for performing arbitrary space integration of sparse data, without adopting the typical procedure of adding spurious information, as in the case of linear or splines interpolations, etc. (Lucarini et al. 2007). Actually, in this regard that Thiessen and Alter, with the purpose of computing river basin water balances from irregular and sparse rain observations, discovered independently for the 2D case the tessellation introduced by Voronoi just few years earlier (Thiessen and Alter 1911). Moreover, recently a connection has been established between the Rayleigh-Bènard convective cells and Voronoi cells, with the hot spots (strongest upward motion of hot fluid) of the former basically coinciding with the points generating the Voronoi cells, and the locations of downward motion of cooled fluid coinciding with the sides of the Voronoi cells (Rapaport 2006).
The quest for achieving low computational cost for actually evaluating the Voronoi tessellation of a given discrete set of points X is ongoing and involves an extensive research performed within various scientific communities (Bowyer 1981;Watson 1981;Tanemura et al. 1983;Barber et al. 1996;Han and Bray 2006). The theoretical investigation of the statistical properties of general N-dimensional Voronoi tessellations, which has a great importance in applications, has proved to be a rather hard task, so that direct numerical simulation is the most extensively adopted investigative approach. For a review of the theory and applications of Voronoi tessellations, see Aurenhammer (1991) and Okabe et al. (2000).
A great deal of theoretical and computational work has focused on a more specific and tractable problem, that of studying the statistical properties of the geometric characteristics of Poisson-Voronoi tessellations. These are Voronoi tessellations obtained for a random set of points X generated as output of a homogeneous Poisson point process. This problem has a great relevance at practical level because it corresponds, e.g., to studying crystal aggregates with random nucleation sites and uniform growth rates. Exact results concerning the mean statistical properties of the interface area, inner area, number of vertices, etc. of the Voronoi cells have been obtained for 2+ dimensional Euclidean spaces (Meijering 1953;Christ et al., 1982;Drouffe and Itzykson 1984;Finch 2003;Calka 2003). Recently, some important results have been obtained for the 2D case (Hilhorst 2006). Several computational studies, performed considering a quite wide range of number of 2D and 3D Voronoi cells, have found results that basically agree with the theoretical findings, and, moreover, have shown that both 2-parameter (Kumar et al. 1992) and 3-parameter (Hinde and Miles 1980) gamma distributions fit up to a high degree of accuracy the empirical pdfs of the number of vertices, of the perimeter and of the area of the cells. Very extensive and more recent calculations have basically confir
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