---
tag: K2YVMTGTBB
kind: concept
name: Order of a projective plane
msc: ["51E15", "05B25"]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_plane
---

## Summary

The order of a [[finite projective plane|2HWB7P1DZN]] is the number $n$ such that every line has $n + 1$ points.
It determines all the basic counts: each point is on $n + 1$ lines, and there are $n^2 + n + 1$ points and as many lines.
The plane $PG(2, q)$ over the field with $q$ elements has order $q$, and which other orders occur is a central open question of finite geometry.

## Explanations

### L1 human

In a finite projective plane all lines have the same size, so one number describes the size of the whole plane.
By convention it is one less than the number of points on a line.
The [[Fano plane|QQ8GW1ASZM]] has three points on each line, so its order is $2$.
The minus one makes the order of the plane over a field with $q$ elements equal to $q$.