12-Fold Quasicrystals from Affine Coxeter Groups E6, F4, B6 via Lattice Projection — E8 Intelligence Research

FINDING: 12-fold quasicrystals from affine extensions of Coxeter groups E6, F4, B6 via higher-dimensional lattice projection. MATH: - Coxeter exponents: E6 (1,4,5,7,8,11), F4 (1,5,7,11), B6 (1,3,5,7,9,11). - Coxeter number h: E6=12, F4=12, B6=12. - 12-fold rotational symmetry in projection plane. - Projection from 6D (E6, B6) or 4D (F4) lattices into 2D/3D subspaces. - Key ratio: 12-fold symmetry implies angles of 30°, 60°, 90°, 120°, 150°; edge length ratios involve √3, 2, and golden ratio φ = (1+√5)/2 ≈ 1.618. - Affine extension: adds translation vectors generating aperiodic order. CONNECTION: - Coxeter number h=12 links directly to 12-fold rotational symmetry, a forbidden crystallographic symmetry in periodic crystals. - Golden ratio φ appears in the projection matrix entries for E6 and F4 quasicrystals (since E6 and F4 root systems contain φ-related coordinates). - Base-60 connection: 12-fold symmetry is a divisor of 60; 12 × 5 = 60, linking to Babylonian base-6 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com