---
tag: 7F4HSNJJH8
kind: concept
name: Moufang plane
msc: [51A35, 17D05]
specializes: [DANFNN8C4Y]
sources:
  - url: https://en.wikipedia.org/wiki/Moufang_plane
same_as:
  wikidata: Q1950188
  wikipedia: Moufang plane
---

## Summary

A Moufang plane is a [[projective plane|DANFNN8C4Y]] in which the little Desargues theorem holds.
This is the special case of [[Desargues's theorem|5Z19Y02MME]] where the centre of perspective lies on the axis.
Equivalently, every line is a translation line: the plane has as many elations with that axis as possible.
Moufang planes are exactly the planes coordinatized by alternative division rings.
Every [[Desarguesian plane|HPX85PA8NX]] is a Moufang plane, and the [[octonionic projective plane|S26GYRVDR5]] is a Moufang plane that is not Desarguesian.

## Explanations

### L1 human

Desargues's theorem says that two triangles in perspective from a point are also in perspective from a line.
A Moufang plane is only asked to satisfy this when the point lies on the line.
This weaker rule still gives the plane many symmetries.
In coordinates, it allows multiplication that is not associative, as long as it is alternative, as for the octonions.