Platonic Solids: The Five Convex Regular Polyhedra from Euler's Formula — E8 Intelligence Research

Abstract

FINDING: Platonic solids are the only five convex regular polyhedra, arising from constraints on vertex figures and face angles, and they appear in molecular geometry (e.g., bubble clusters, crystallographic lattices). MATH: Euler's formula: V - E + F = 2. For regular polyhedra: (p-2)(q-2) < 4, where p = sides per face, q = faces per vertex. Solutions: (3,3) tetrahedron, (4,3) cube, (3,4) octahedron, (5,3) dodecahedron, (3,5) icosahedron. Dihedral angles: tetrahedron ~70.53°, cube 90°, octahedron ~109.47°, dodecahedron ~116.57°, icosahedron ~138.19°. CONNECTION: Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in dodecahedron (face diagonals ratio φ) and icosahedron (edge-to-circumradius ratio φ/√3). Icosahedron and dodecahedron are duals; their symmetry group (I_h) is order 120, linked to root system H_3. Crystallographic symmetry: only tetrahedron, cube, octahedron appear in periodic lattices (point groups 432, 23, m3m); dodecahedron and icosahedron are non-crystallographic (5-fold symm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com