Platonic Solids: Five Convex Polyhedra Encoding Golden Ratio and Root Systems — E8 Intelligence Research

FINDING: Platonic solids are the only five convex regular polyhedra, and their dualities and metric ratios encode the golden ratio and crystallographic root systems; the brown dwarf survey is unrelated to sacred geometry. | MATH: Euler's formula V − E + F = 2 (tetrahedron: 4,6,4; cube: 8,12,6; octahedron: 6,12,8; dodecahedron: 20,30,12; icosahedron: 12,30,20). Dual pairs: cube↔octahedron, dodecahedron↔icosahedron, tetrahedron self-dual. Key constants: dodecahedron edge/diagonal ratio = φ = (1+√5)/2 ≈ 1.618; icosahedron circumradius/edge = √(10+2√5)/4 ≈ 0.951; dihedral angles: tetrahedron arccos(1/3) ≈ 70.53°, cube 90°, octahedron arccos(−1/3) ≈ 109.47°, dodecahedron arccos(−1/√5) ≈ 116.57°, icosahedron arccos(−√5/3) ≈ 138.19°. | CONNECTION: Dodecahedron and icosahedron contain φ explicitly (pentagonal faces, golden triangles). The tetrahedron, cube, and octahedron map to the A₃, B₃, and C₃ crystallographic root systems; the icosahedron/dodecahedron relate to the H₃ root system (non-cry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com