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

FINDING: 12-fold quasicrystallography derived from affine Coxeter groups F4, B6, and E6 via higher-dimensional lattice projection into 2D/3D subspaces. | MATH: Coxeter number h(F4)=12, h(B6)=12, h(E6)=12; Coxeter exponents for F4: 1,5,7,11; for B6: 1,5,7,9,11,13; for E6: 1,4,5,7,8,11. Projection uses root system lattices (e.g., F4 lattice in 4D, B6 in 6D, E6 in 6D) onto a plane or space with 12-fold rotational symmetry (point group D12). Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618 appears in E6 root system scaling; ratios 0.382 (2-φ), 0.618 (φ-1), 1.618, 2.618 (φ²) emerge in projection scaling factors. | CONNECTION: 12-fold symmetry is forbidden in periodic 2D/3D crystals (crystallographic restriction theorem allows only 2,3,4,6-fold). D12 point group is non-crystallographic in 2D/3D but becomes crystallographic in 4D (F4) and 6D (B6, E6). The affine Coxeter groups generate quasicrystalline tilings with golden ratio proportions, linking directly to Penrose tilings (5-fold) and 12- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com