Platonic Solids

The five regular convex polyhedra - perfect symmetry in three dimensions

The Five Regular Polyhedra

Tetrahedron
Tetrahedron 4 faces Fire
Regular tetrahedron - simplest Platonic solid with triangular faces
Cube
Cube (Hexahedron) 6 faces Earth
Regular hexahedron - most familiar Platonic solid with square faces
Octahedron
Octahedron 8 faces Air
Regular octahedron - dual to the cube with triangular faces
Dodecahedron
Dodecahedron 12 faces Universe
Regular dodecahedron - complex Platonic solid with pentagonal faces
Icosahedron
Icosahedron 20 faces Water
Regular icosahedron - most complex Platonic solid with triangular faces

About the Platonic Solids

The Platonic solids are the only five regular convex polyhedra, representing perfect symmetry in three-dimensional space:

  • Philosophy - Classical elements of nature
  • Mathematics - Perfect symmetry and regularity
  • Crystallography - Natural crystal structures
  • Architecture - Sacred geometry
  • Art - Sculptural forms
  • Science - Molecular structures
Mathematical Properties
Euler's Formula
V - E + F = 2
(Vertices - Edges + Faces = 2)
Regularity
All faces congruent
All vertices identical
Duality
Tetrahedron ↔ Tetrahedron
Cube ↔ Octahedron
Dodecahedron ↔ Icosahedron
Golden Ratio
φ = (1 + √5)/2 ≈ 1.618
Key to dodecahedron & icosahedron
Ancient Wisdom: Plato associated each solid with a classical element: tetrahedron (fire), cube (earth), octahedron (air), icosahedron (water), and dodecahedron (universe).

Historical and Cultural Significance

Ancient Philosophy
  • Plato (428-348 BC): Associated with classical elements
  • Pythagoreans: Symbols of cosmic order
  • Timaeus: Building blocks of the universe
Mathematical Development
  • Euclid (ca. 300 BC): Proved there are exactly five
  • Theaetetus: First systematic study
  • Modern math: Group theory and symmetry
Natural Occurrence
  • Crystals: Pyrite (cube), fluorite (octahedron)
  • Viruses: Icosahedral capsids
  • Molecules: Methane (tetrahedron)
Modern Applications
  • Architecture: Sacred geometry, domes
  • Gaming: Polyhedral dice
  • Art: Sculptures and installations
Quick Reference
V - E + F = 2
Euler's Formula
5
Platonic Solids
φ
Golden Ratio
∞
Symmetry Groups
427 BC
Plato
Symmetry Groups
Td Tetrahedron: 24 symmetries
Oh Cube/Octahedron: 48 symmetries
Ih Dodecahedron/Icosahedron: 120 symmetries
Each solid has rotational and reflectional symmetries forming mathematical groups.
Dual Relationships
↔ Tetrahedron is self-dual
↔ Cube ↔ Octahedron
↔ Dodecahedron ↔ Icosahedron
In duality, vertices become faces and faces become vertices.
Classical Elements
🔥 Fire: Tetrahedron (sharp, hot)
🌍 Earth: Cube (stable, solid)
💨 Air: Octahedron (light, flowing)
💧 Water: Icosahedron (fluid)
🌌 Universe: Dodecahedron (cosmos)