Platonic Solids: Euler's Formula, Golden Ratio, and Crystallographic Symmetry — E8 Intelligence Research

FINDING: The five Platonic solids are the only regular convex polyhedra, and their dualities and metric ratios encode the golden ratio and crystallographic symmetry groups. | MATH: Euler's formula V − E + F = 2; for Platonic solids: p, q with p·V = 2E, q·F = 2E; solutions: 3, 3, 4, 3, 3, 4, 5, 3, 3, 5. Key ratios: icosahedron/dodecahedron circumradius-to-edge = φ· (√3) /2 ≈ 1. 401; inradius-to-edge for dodecahedron = φ²/ (2√ (3−φ) ) ≈ 1. 1135; dihedral angles: tetrahedron arccos (1/3) ≈ 70. 53°, octahedron arccos (−1/3) ≈ 109. 47°, icosahedron arccos (−√5/3) ≈ 138. 19°, dodecahedron arccos (−1/√5) ≈ 116. 57°. | CONNECTION: The golden ratio φ = (1+√5) /2 ≈ 1. 618 appears explicitly in icosahedron and dodecahedron coordinates (e. g. , vertices at (0, ±1, ±φ) permutations). The icosahedral symmetry group Iₕ is isomorphic to A₅ × C₂, a crystallographic point group in 3D (order 120). The tetrahedron's symmetry is Td (order 24), cube/octahedron Oₕ (order 48) — all are finite subgroups of O (3) and re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com