# Euclidean Geometry

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

## Summary
The geometry of flat space, built from Euclid's 5 postulates. Believed unique for 2000+ years until 19th-century non-Euclidean discoveries.

## Key facts
Triangle angles sum to 180°; exactly one parallel through external point; Pythagorean theorem holds.

## The parallel postulate's special status
Seemed provable from the other 4 for centuries — all attempts failed since it's genuinely independent.

## The 19th-century revolution
Bolyai and Lobachevsky (~1830, independently): hyperbolic geometry (multiple parallels). Gauss privately reached similar conclusions but didn't publish. Riemann (1854): elliptic geometry (no parallels).

## Beltrami's proof (1868)
Constructed explicit Euclidean model of hyperbolic geometry — proved it exactly as consistent as Euclidean geometry.

## Physical status
General relativity (1915): physical spacetime is actually non-Euclidean (curved by mass/energy). Euclidean geometry remains an excellent local approximation at everyday scales.

## Sources
### Tier 1
- Hartshorne, R. (2000). *Geometry: Euclid and Beyond*. Springer.
- Greenberg, M.J. (2008). *Euclidean and Non-Euclidean Geometries*. 4th ed.
### Tier 2
- Bonola, R. (1955 reprint). *Non-Euclidean Geometry*. Dover.

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