---
tag: SPCYMFN5V2
kind: concept
name: Non-Euclidean geometry
msc: [51M10, 51M05]
sources:
  - url: https://en.wikipedia.org/wiki/Non-Euclidean_geometry
same_as:
  wikidata: Q233858
  wikipedia: Non-Euclidean geometry
---

## Summary

A non-Euclidean geometry is a geometry that keeps the other basic assumptions of [[Euclidean geometry|KRZD96GSVC]] but drops the [[parallel postulate|2PC3XY0PYZ]].
The two classical ones are [[hyperbolic geometry|GN4N94GQ0Z]], where a line has many parallels through an outside point, and [[elliptic geometry|H5GB19W2H3]], where it has none.
Their discovery showed that the parallel postulate cannot be proved from the other axioms.
They also became models for curved space in differential geometry and physics.

## Explanations

### L1 human

In school geometry, through a point off a line there is exactly one line that never meets it.
For two thousand years people tried to prove this from simpler rules and failed.
Non-Euclidean geometries are consistent worlds where this rule is false.
On a sphere, where great circles play the role of lines, any two lines meet, and the angles of a triangle add up to more than $180^\circ$.