# Platonic Solids

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

## Summary
Exactly 5 regular convex 3D solids exist. Proven complete in Euclid's Elements Book XIII.

## The five solids
| Solid | Faces | Shape | Vertices | Edges |
|---|---|---|---|---|
| Tetrahedron | 4 | Triangle | 4 | 6 |
| Cube | 6 | Square | 8 | 12 |
| Octahedron | 8 | Triangle | 6 | 12 |
| Dodecahedron | 12 | Pentagon | 20 | 30 |
| Icosahedron | 20 | Triangle | 12 | 30 |

## Why exactly five (angle-sum argument)
At each vertex, face angles meeting must sum to <360°. Only triangles (3,4,5 meeting), squares (3 meeting), pentagons (3 meeting) satisfy this — hexagons and beyond fail immediately.

## Euler's polyhedron formula
V−E+F=2 for all convex polyhedra — an early topological invariant (Euler characteristic).

## Duality
Cube↔octahedron, dodecahedron↔icosahedron, tetrahedron self-dual.

## Plato's Timaeus (c.360 BCE)
Associated 4 solids with classical elements (fire/earth/air/water) + dodecahedron with cosmos — philosophical speculation, not mathematics.

## Kepler's cosmological model (1596)
Nested the 5 solids to explain planetary orbit spacing — later abandoned once Kepler's own more accurate elliptical-orbit laws emerged.

## Modern science connections
Viral capsids use icosahedral symmetry; buckminsterfullerene C₆₀ relates to truncated icosahedron structure.

## Sources
### Tier 1
- Heath, T.L. (1956). *Euclid: The Thirteen Books of the Elements*. Vol. 3.
- Coxeter, H.S.M. (1973). *Regular Polytopes*. 3rd ed.
### Tier 2
- Field, J.V. (1988). *Kepler's Geometrical Cosmology*.

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