A combinatorial approach to discrete geometry

Abstract

We present a paralell approach to discrete geometry: the first one introduces Voronoi cell complexes from statistical tessellations in order to know the mean scalar curvature in term of the mean number of edges of a cell. The second one gives the restriction of a graph from a regular tessellation in order to calculate the curvature from pure combinatorial properties of the graph. Our proposal is based in some epistemological pressupositions: the macroscopic continuous geometry is only a fiction, very usefull for describing phenomena at certain sacales, but it is only an approximation to the true geometry. In the discrete geometry one starts from a set of elements and the relation among them without presuposing space and time as a background.