---
tag: CSXT0BG2CM
kind: concept
name: Incidence geometry
msc: [51A05, "51E30"]
sources:
  - url: https://en.wikipedia.org/wiki/Incidence_geometry
same_as:
  wikidata: Q4582901
  wikipedia: Incidence geometry
---

## Summary

Incidence geometry is the part of geometry that studies points, lines and the relation "lies on", and nothing else.
Its objects are [[incidence structures|FGKNAFNHA3]] that satisfy axioms such as "two points lie on exactly one line".
[[Affine planes|EC7XT2ZM5S]], [[projective planes|DANFNN8C4Y]] and finite geometries are its main examples.
It asks which configurations the axioms force.
It also asks which algebra, such as a field or a division ring, can be rebuilt from the points and lines alone.

## Explanations

### L1 human

Ordinary geometry talks about lengths and angles.
Incidence geometry throws these away and keeps only the question of which points lie on which lines.
Surprisingly much survives: theorems such as those of Desargues and Pappus only speak about points being on lines.
It also allows small finite worlds, such as a geometry with $7$ points and $7$ lines, that behave like the plane in this respect.