Axiomatization of geometry employing group actions

TL;DR

This paper proposes a novel axiomatization of planar geometry using group actions, emphasizing topological concepts like connectedness and separation, and integrating modern ideas such as boundary at infinity and the Erlangen Program.

Contribution

It introduces a new axiomatic framework for geometry based on group actions and topological notions, unifying lines and planes, and connecting to modern mathematical and physical theories.

Findings

Axiomatization based on connectedness and separation.

Introduction of boundary at infinity and betweenness.

Alignment with the Erlangen Program and relativity concepts.

Abstract

The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness without any need to dwell on the definition of topology. In our approach line segments must be connected. Lines and planes are unified via the concept of separation: lines are separated into two components by each point, planes contain lines that separate them into two components as well. We add a subgroup of bijections preserving line segments and establishing unique isomorphism of basic geometrical sets, and the axiomatic structure is complete. Of fundamental importance is the Fixed Point Theorem that allows for creation of the concepts of length and congruency of line segments. The resulting structure is much more in sync with modern science than other axiomatic approaches to planar geometry. For instance, it leads naturally to the Erlangen Program in geometry. Our Conditions of Homogeneity and Rigidity have two interpretations. In physics, they correspond to the basic tenet that independent observers should arrive at the same measurement and are related to boosts in special relativity. In geometry, they mean uniqueness of congruence for certain geometrical figures. Another thread of the paper is the introduction of boundary at infinity, an important concept of modern mathematics, and linking of Pasch Axiom to endowing boundaries at infinity with a natural relation of betweenness. That way spherical geometry can be viewed as geometry of boundaries at infinity.