Fano's Geometry

Fano's geometry is a finite geometry described by Fano (1892, p. 114, §7). This geometry comes with five axioms, namely:

1. There exists at least one line.

2. Every line has exactly three points on it.

3. Not all the points are on the same line.

4. For two distinct points, there exists exactly one line on both of them.

5. Each two lines have at least one point on both of them.

Fano's geometry is categorical. Even so, there are several different though equivalent visual representations of Fano's geometry. Perhaps the most common is the so-called Fano plane which shows, among other things, that lines in Fano's geometry need not be straight.

Like many finite geometries, the number of provable theorems in Fano's geometry is small. One can show that in Fano's geometry, each two lines have exactly one point in common and that the geometry itself consists of exactly seven points and seven lines.