---
tag: DANFNN8C4Y
kind: concept
name: Projective plane
msc: [51A05, "51E15"]
specializes: [FGKNAFNHA3]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_plane
same_as:
  wikidata: Q1475760
  wikipedia: Projective plane
---

## Summary

A projective plane is an [[incidence structure|FGKNAFNHA3]] in which two points lie on exactly one line and two lines meet in exactly one point.
It is the plane of perspective drawing, where parallel lines meet at a point at infinity.
The real projective plane and the planes $\mathbb{P}^2(k)$ over fields are examples, and there are also planes that come from no field.
Finite projective planes are central objects of combinatorics.

## Explanations

### L1 human

In a perspective drawing, railway tracks meet at a point on the horizon.
A projective plane makes this a rule: any two lines meet, and the horizon becomes a line like the others.
Through two points there is still exactly one line.
The smallest projective plane, the Fano plane, has $7$ points and $7$ lines, with $3$ points on each line and $3$ lines through each point.