# Combinatorial constructions of intrinsic geometries

## TL;DR

This paper introduces a combinatorial approach to constructing intrinsic geometries using inverse sequences of finite graphs, enabling the definition of metric, geodesic, and curvature in limit spaces.

## Contribution

It presents a novel method for building intrinsic geometries from finite graphs and proposes new foundational notions for geometry based on these constructions.

## Key findings

- Finite graph sequences can approximate intrinsic geometries.
- New notions of metric, geodesic, and curvature are proposed.
- Examples illustrate the potential for nonstandard geometric foundations.

## Abstract

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the construction is sufficiently regular and uniform, then the notions of metric, geodesic and curvature can be defined in the space as the limits of their finite versions in the graphs. This gives rise to consider the graphs with metrics as finite approximations of the geometry of the space. On the basis of simple and generic examples, several nonstandard and novel notions are proposed for the Foundations of Geometry. They may be considered as a subject of a critical discussion.