Coxeter Group Projections of F4, B6, E6 Lattices Yield Dodecagonal Quasicrystals — E8 Intelligence Research

FINDING: Dodecagonal quasicrystals arise from projections of higher-dimensional lattices (affine F4, B6, E6) via Coxeter group theory, enabling 12-fold rotational symmetry forbidden in periodic crystals. MATH: - Coxeter number \( h \) for F4 = 12, B6 = 12, E6 = 12; Coxeter exponents for F4: 1,5,7,11 (sum = 24). - Projection from 6D (B6, E6) or 4D (F4) root lattices into 2D/3D subspaces yields 12-fold point group D12 (order 24). - Quasicrystal diffraction patterns show peaks at positions governed by \( \cos(2\pi k/12) \) with \( k = 1,5,7,11 \). CONNECTION: - 12-fold symmetry directly links to base-60 mathematics (60 = 5×12) and the golden ratio φ = 1.618... via the dodecagon's circumradius-to-side ratio \( R/s = 1 + \sqrt{3} \approx 2.732 \) and its relation to φ through \( \cos(30°) = \sqrt{3}/2 \approx 0.866 \). - The ratio 0.618 (φ⁻¹) appears in the dodecagon's star polygons (e.g., {12/5} star). - Coxeter group F4 is the symmetry of the 24-cell, a 4D regular polytope Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com