Golden Ratio and Graph Regularity Classify the Five Platonic Solids — E8 Intelligence Research

FINDING: The Platonic solids are fully classified by graph-theoretic regularity (vertex degree + face size), and the dodecahedron/icosahedron pair are duals whose vertex coordinates are generated by the golden ratio φ. | MATH: - Euler: V − E + F = 2. - Regularity: each vertex degree k, each face m-gon → kF = 2E = mV. - Solving with Euler yields only five (k,m) pairs: (3,3) tetrahedron, (3,4) cube, (4,3) octahedron, (3,5) dodecahedron, (5,3) icosahedron. - Dodecahedron vertex coordinates (unit edge): (0, ±φ⁻¹, ±φ), (±φ, 0, ±φ⁻¹), (±φ⁻¹, ±φ, 0) — where φ = (1+√5)/2 = 1.618…, φ⁻¹ = 0.618… - Icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) — same φ, permuted. - Truncation: vertex-truncating a dodecahedron by its dual (icosahedron) yields icosidodecahedron (32 faces, 60 edges, 30 vertices) — a uniform Archimedean solid. | CONNECTION: - φ and φ⁻¹ appear explicitly in both coordinate sets — the golden ratio is the sole irrational constant needed. - φ⁻¹ = 0.618…, φ² Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com