Platonic Solids, Duality, and Symmetry Groups in Polyhedral Geometry — E8 Intelligence Research

FINDING: Platonic solids are the only five convex regular polyhedra; their dual pairs (tetrahedron self-dual, cube/octahedron, dodecahedron/icosahedron) exhibit symmetry groups linked to root systems and crystallographic lattices. Truncation and rectification generate Archimedean solids and Catalan solids, preserving symmetry group structure. MATH: - Euler characteristic: V - E + F = 2 - Dual polyhedron: V ↔ F, E unchanged - Symmetry groups: Tetrahedral (A₄, order 12), Octahedral (S₄, order 24), Icosahedral (A₅, order 60) — all finite Coxeter groups - Truncation/rectification ratios: For cube → truncated cube, edge length ratio involves √2; for dodecahedron → truncated dodecahedron, golden ratio φ = (1+√5)/2 ≈ 1.618 appears in vertex coordinates - Key constants: φ (1.618), 1/φ (0.618), φ² (2.618), φ⁻² (0.382) — all present in dodecahedron/icosahedron geometry CONNECTION: - Dodecahedron/icosahedron dual pair directly embeds φ in face/vertex arrangements (12 pentagons, 20 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com