---
tag: EC7XT2ZM5S
kind: concept
name: Affine plane
msc: [51A15, "51E15"]
specializes: [FGKNAFNHA3]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)
same_as:
  wikidata: Q382491
  wikipedia: Affine plane (incidence geometry)
---

## Summary

An affine plane is an [[incidence structure|FGKNAFNHA3]] of points and lines in which two points lie on exactly one line and the parallel axiom holds.
It abstracts the incidence properties of the ordinary Euclidean plane.
Every plane $k^2$ over a field is one, but there are also affine planes that come from no field.
Adding a line at infinity turns an affine plane into a [[projective plane|DANFNN8C4Y]].

## Explanations

### L1 human

Forget distances and angles in the plane, and keep only points, lines, and which point is on which line.
What remains are two rules: through two points there is one line, and through a point off a line there is exactly one parallel.
Any system of points and lines that obeys these rules, and is not too small, is an affine plane.
The smallest one has four points and six lines: every pair of points is a line, and the six lines fall into three pairs of parallels.