Mathematical analysis reveals identical in-sphere to circumsphere radius ratios and explicit golden ratio constants in dodecahedra and icosahedra, highlighting deep geometric dualities.
FINDING: The five Platonic solids are the only convex regular polyhedra, and their dualities and in-sphere/circumsphere ratios encode golden-ratio constants. | MATH: Euler's formula V−E+F=2; tetrahedron {3,3}, cube {4,3}, octahedron {3,4}, dodecahedron {5,3}, icosahedron {3,5}; dual pairs (cube↔octahedron, dodecahedron↔icosahedron, tetrahedron self-dual); in-sphere/circumsphere radius ratios: tetrahedron 1/3, cube 1/√3, octahedron 1/√3, dodecahedron √(2/3)·(√5−1)/2 ≈ 0.7947, icosahedron √(3/5)·(√5−1)/2 ≈ 0.7947; dodecahedron face diagonal/edge = φ = (1+√5)/2 ≈ 1.618; icosahedron edge/circumradius = 4/√(10+2√5) ≈ 1.0515; dihedral angles: tetrahedron 70.53°, cube 90°, octahedron 109.47°, dodecahedron 116.57°, icosahedron 138.19°. | CONNECTION: Dodecahedron and icosahedron both contain φ explicitly — their in-sphere/circumsphere ratios are identical (≈0.7947) and involve (√5−1)/2 = 1/φ ≈ 0.618; the golden ratio appears in the dodecahedron's face diagonals and the icosahedron's vertex arra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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Andrew Stewart Caldin (2026) studied this question.