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

Abstract

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: \( m_{12}=5, m_{23}=3, m_{13}=2 \). - Coxeter matrix: \[ \begin{pmatrix} 1 & 5 & 2 \\ 5 & 1 & 3 \\ 2 & 3 & 1 \end{pmatrix} \] - 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 \( \tau = (1+\sqrt{5})/2 \approx 1.618 \). - The root coordinates can be expressed in terms of \( \tau \) and \( \sqrt{2} \), e.g., the 30 roots are permutations of \( (\pm 1, \pm \tau, 0) \) and \( (\pm \tau, \pm 1, \pm 1) \) up to scaling. - The Weyl group order: \( |H_3| = 120 = 2 \times 60 \). - 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