Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey number was established as Rₜrans (3, 3) =5. Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits =lim┬ (N) ⁡〖Ng/Nᵣ 〗, where N is the total number of green and red seeds, Ng and Nᵣ, were found =0. 2720. 001 (Voronoi) and = 0. 470. 02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1. 6900. 001 and for the Poisson line tessellation as S =1. 2650. 015.