---
tag: GN4N94GQ0Z
kind: concept
name: Hyperbolic geometry
msc: [51M10, 53A35]
specializes: [SPCYMFN5V2]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_geometry
same_as:
  wikidata: Q209306
  wikipedia: Hyperbolic geometry
---

## Summary

Hyperbolic geometry is the [[non-Euclidean geometry|SPCYMFN5V2]] in which, through a point off a line, there pass at least two lines that do not meet it.
It can be realised on a surface of constant negative [[Gaussian curvature|SQVQ5DQT6W]].
In it, the angles of a triangle add up to less than $\pi$, and the area of a triangle is determined by its angles.
It appears throughout mathematics, from complex analysis and number theory to the topology of surfaces and 3-manifolds.

## Explanations

### L1 human

Imagine a world where space spreads out faster than in a flat plane, like the frilly edge of a lettuce leaf.
In this world, through a point beside a line you can draw many lines that never meet it.
Triangles are thin, and their angles add up to less than $180^\circ$.
In the [[upper half-plane model|RHKSMPBSQ1]], the lines are vertical half-lines and half-circles that meet the horizontal axis at right angles.