Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research

FINDING: Platonic solids are governed by the finite Coxeter groups A₃, B₃, H₃ (crystallographic and non-crystallographic), whose root systems and Weyl groups encode all five regular polyhedra; recent work extends this to molecular realization (e.g., Platonic hydrocarbons) and even to convergence in AI latent spaces. | MATH: The five Platonic solids correspond to rank-3 Coxeter groups: tetrahedron = A₃ (order 24), cube/octahedron = B₃ (order 48), icosahedron/dodecahedron = H₃ (order 120, non-crystallographic). Key invariants: Schläfli symbols {3,3}, {4,3}, {3,4}, {5,3}, {3,5}. Dual pairs: (cube, octahedron), (icosahedron, dodecahedron), tetrahedron self-dual. Coxeter numbers: h(A₃)=4, h(B₃)=6, h(H₃)=10. Golden ratio φ = (1+√5)/2 = 1.618… appears explicitly in H₃: coordinates of icosahedron vertices involve φ; dodecahedron edge/radius ratios involve φ. The non-crystallographic nature of H₃ means no full lattice in 3D has 5-fold symmetry, but molecular realizations (e.g., dodecahedrane C₂

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com