A combinatorial approach to discrete geometry

TL;DR

This paper introduces two novel combinatorial methods for analyzing discrete geometry, focusing on curvature calculations from statistical tessellations and graph properties, challenging traditional continuous models.

Contribution

It presents the first parallel approaches to discrete geometry, linking Voronoi complexes and graph restrictions to curvature without assuming a background space-time.

Findings

Curvature can be derived from mean edges of Voronoi cells.

Graph restrictions from tessellations enable curvature calculation via combinatorial properties.

Discrete geometry offers a fundamental perspective beyond continuous approximations.

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.