---
tag: MCDZP31ZF3
kind: concept
name: Finite geometry
msc: ["51E99", "05B25"]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_geometry
same_as:
  wikidata: Q1077896
  wikipedia: Finite geometry
---

## Summary

A finite geometry is a geometric system, such as a projective or affine plane or space, with only finitely many points.
The main examples are the projective and affine spaces over [[finite fields|0ZXZ82FTX2]], but there are also many finite planes that do not come from fields.
Finite geometry links geometry with combinatorics, algebra and coding theory, through objects such as [[block designs|4JCSS2Z1BA]], ovals and spreads.

## Explanations

### L1 human

Ordinary geometry has infinitely many points, but the rules of incidence also make sense for finite sets.
The smallest example is the [[Fano plane|QQ8GW1ASZM]]: seven points and seven lines, each line with three points, and any two points on exactly one line.
Counting arguments then replace measurements of length and angle.
Typical questions are how many points such a geometry can have and which sets of points avoid having three on a line.