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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.
This axiom is often phrased so as to require three 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!

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