---
tag: 9QW65PGEPW
kind: concept
name: Erlangen program
msc: [51N25, 51-03, 20H15]
sources:
  - url: https://en.wikipedia.org/wiki/Erlangen_program
same_as:
  wikidata: Q315296
  wikipedia: Erlangen program
---

## Summary

The Erlangen program is a view of geometry that Felix Klein stated in 1872.
In it, a geometry is the study of the properties of a space that are invariant under a given [[group|G5S1M9]] of transformations.
Euclidean, affine and projective geometry then differ only in their groups: the [[Euclidean group|0H3YY8CPKH]], the [[affine group|G6GMSEZDB1]] and the projective group.
A larger group has fewer invariants, so projective geometry has fewer theorems than affine geometry, which has fewer than [[Euclidean geometry|KRZD96GSVC]].

## Explanations

### L1 human

Klein asked what makes a fact "geometric".
His answer: a fact is geometric if it survives the moves that the geometry allows.
In Euclidean geometry the moves are rigid motions, so lengths and angles count.
In affine geometry the moves may also stretch and shear, so lengths are lost, but parallel lines stay parallel and midpoints stay midpoints.