---
tag: 9B143EK6A8
kind: definition
name: Projective plane
msc: [51A05, "51E15"]
defines: DANFNN8C4Y
uses: [FGKNAFNHA3, 0AFNNTRTJE]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_plane
---

## Definition

A *projective plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:

1. Any two distinct points lie on exactly one common line.
2. Any two distinct lines have exactly one common point.
3. There are four points, no three of which are [[collinear|0AFNNTRTJE]].

## Remarks

Example: for a field $k$, the projective plane $\mathbb{P}^2(k)$ has as points the one-dimensional subspaces of $k^3$ and as lines the two-dimensional subspaces, with incidence given by inclusion.
Check: two distinct one-dimensional subspaces span exactly one plane.
Two distinct planes in $k^3$ meet in a line through $0$, by the dimension formula $2 + 2 - 3 = 1$.
The points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, $[1:1:1]$ have no three collinear.
For $k = \F_2$ this is the Fano plane, with $7$ points and $7$ lines.

Non-example: an [[affine plane|EC7XT2ZM5S]] satisfies clauses 1 and 3 but not clause 2, since it has parallel lines.

Clause 3 excludes degenerate cases, such as a single line with all points on it.
The axioms are self-dual: exchanging points and lines turns each clause into a statement that also holds, so the dual of a projective plane is a projective plane.
In a [[finite projective plane|2HWB7P1DZN]] every line has $n + 1$ points for some $n \ge 2$, the order of the plane.