Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings

Abstract

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

Biological hypercrystals

“…A Voronoi pattern results from the slicing of a plane with convex polygons so that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other one. Therefore, the partitioning of an infine plane into regions based on the distance to a specified discrete set of points (called seeds or nuclei) constructs the Voronoi tessellation [12,13]. The pattern stemming from the cells' membranes in the epithelial tissue shown in Figure 1A reasonably corresponds to the Voronoi tessellation arising from to the cells' nuclei distribution throughout the space.…”

Shannon Entropy of Ramsey Graphs with up to Six Vertices

“…It is noteworthy that the Voronoi tessellation actually was introduced first by Descartes [19]. Voronoi tessellations are usually quantified with the so-called Shannon entropy, which was discovered in 1948 by Claude Shannon, which is used as a measure of information, of uncertainty, and unlikelihood [22][23][24]. This measure is defined for any given probability distribution [22][23][24].…”

“…Voronoi tessellations are usually quantified with the so-called Shannon entropy, which was discovered in 1948 by Claude Shannon, which is used as a measure of information, of uncertainty, and unlikelihood [22][23][24]. This measure is defined for any given probability distribution [22][23][24]. The quantity introduced by Shannon has the same mathematical form as the entropy in statistical mechanics, thus, he labeled this measure, as allegedly suggested by von Neumann: "entropy".…”

“…Regrettably, numerous misinterpretations of the Shannon entropy, arising from the resemblance of the aforementioned formulae, appeared in the scientific literature [23]. Shannon Entropy, as applied to Voronoi tessellations, may be seen as a measure of "ordering" in a given tessellation [17][18][19][20][21][22]. Controversies of such an understanding of the Shannon entropy are discussed in ref.…”

“…Controversies of such an understanding of the Shannon entropy are discussed in ref. [22]. For any given set of points corresponding to the Voronoi tessellation diagram, the Shannon entropy (also known as the Shannon measure of information), denoted S, is defined by Equation ( 2):…”