Golden-Ratio Geometry in Coxeter Group H3: Precursor to 3D Quasicrystals — E8 Intelligence Research

FINDING: Coxeter group H3 — the symmetry group of the icosahedron/dodecahedron — is the finite reflection group with 30 roots, whose structure encodes golden-ratio geometry and is a direct precursor to 3D quasi-crystalline order. MATH: - H3 is generated by three reflections with orders: \( m₁₂=5, m₂₃=3, m₁₃=2 \). - Coxeter matrix: \[ {pmatrix} 1 & 5 & 2 \\ 5 & 1 & 3 \\ 2 & 3 & 1 {pmatrix} \] - 30 roots: 15 positive pairs, each pair corresponding to a symmetry axis of the icosahedron. - Cartan matrix for H3 (using the standard bilinear form) has determinant 4, and the eigenvalues relate to \( τ = (1+√5)/2 ≈ 1.618 \). - The root coordinates can be expressed in terms of \( τ \) and \( √2 \), e.g., the 30 roots are permutations of \( (± 1, ± τ, 0) \) and \( (± τ, ± 1, ± 1) \) up to scaling. - The Weyl group order: \( |H_3| = 120 = 2 × 60 \). - The Coxeter number \( h = 10 \), and the dual Coxeter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com