# Generalization of a projective plane?

In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties:

1. Every two points are incident with a unique line
2. Every two lines are incident with a unique point
3. There are four points, no three colinear

I am trying to figure out a generalization of this concept. Here one finds the following:

Projective planes may be thought of as projective geometries of "geometric" dimension two. Higher dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane

However I cannot find a similar definition. What would be a natural generalization of the above three points to the general dimension $d\geq2$?

• I can't create a tag, but I was wondering if it would make sense to have a "finite-geometry" tag. – geo909 Apr 8 '14 at 20:47
• en.wikipedia.org/wiki/… – Qiaochu Yuan Apr 9 '14 at 5:13
• Thanks, Quiaochu! I'll try to understand now how this generalizes the definition of a projective plane. – geo909 Apr 9 '14 at 21:53