---
tag: HPX85PA8NX
kind: concept
name: Desarguesian plane
msc: [51A30, 51A05]
specializes: [DANFNN8C4Y]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_plane
---

## Summary

A Desarguesian plane is a [[projective plane|DANFNN8C4Y]] in which [[Desargues's theorem|5Z19Y02MME]] holds.
These are exactly the planes $\mathbb{P}^2(D)$ built from a division ring $D$, so they are the planes that come from linear algebra.
Every finite Desarguesian plane is $\mathbb{P}^2(\F_q)$ for a finite field $\F_q$.
Planes in which the theorem fails are called [[non-Desarguesian planes|HYE5FT4M34]].

## Explanations

### L1 human

Some projective planes are built from numbers, by taking lines through the origin in a three-dimensional space.
Others are built by more exotic rules.
Desargues's theorem about two triangles in perspective is the test that tells them apart: it holds exactly in the planes built from numbers.
The real projective plane and the Fano plane with $7$ points are Desarguesian.