# Non-Euclidean Geometry

**Type:** Concept  
**Domain:** Geometry  
**Codex URL:** /mathematics/geometry/non-euclidean-geometry/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Rejecting Euclid's parallel postulate yields consistent alternative geometries — later realized physically in general relativity.

## The three geometries
- Euclidean (flat): exactly 1 parallel; triangle angles = 180°
- Hyperbolic: infinitely many parallels; triangle angles < 180°
- Elliptic (spherical): no parallels; triangle angles > 180°

## Discovery (independently, c.1829-1832)
Lobachevsky (1829, published), Bolyai (1832, published), Gauss (earlier, unpublished — feared "outcry of the Boeotians").

## Beltrami's 1868 consistency proof
Constructed explicit Euclidean models of hyperbolic geometry (pseudosphere, disk/half-plane models) — proved hyperbolic geometry is consistent iff Euclidean geometry is, finally resolving the 2,000-year parallel-postulate question.

## Riemannian geometry (1854)
Riemann's habilitation lecture generalized to continuously varying curvature — later became the mathematical language of general relativity (Einstein, 1915).

## Cosmological flatness
Modern CMB measurements show observable universe geometry very close to flat — an empirical question enabled by, but distinct from, the mathematical discovery itself.

## Sources
### Tier 1
- Bonola, R. (1955). Non-Euclidean Geometry.
- Greenberg, M.J. (2007). Euclidean and Non-Euclidean Geometries. 4th ed.
### Tier 2
- Misner, Thorne and Wheeler (1973). Gravitation.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*