MODELS OF NON-EUCLIDEAN GEOMETRY

Section 1.2 Incidence Geometries

An incidence geometry satisfies the following axioms:

1. Two points determine a unique line.

2. Each line contains at least two points. ¹

3. There exist at least three non-collinear points.

A perhaps unexpected model for incidence geometry consists of four “points”, called \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) \(D\text{,}\) and six “lines”, called \(AB\text{,}\) \(AC\text{,}\) \(AD\text{,}\) \(BC\text{,}\) \(BD\text{,}\) \(CD\text{,}\) with the obvious conventions as to which lines contain which points.

This symbolic example shows that geometric models do not in fact need to seem very geometric!