Coxeter Numbers and Exponents Unify Dodecagonal Quasicrystals from Higher-Dimensional Lattices — E8 Intelligence Research

FINDING: Dodecagonal quasicrystals arise from projections of higher-dimensional lattices (affine F4, B6, E6) with Coxeter numbers and exponents governing symmetry. MATH: - Coxeter number \( h \) for F4: 12; for B6: 12; for E6: 12. - Coxeter exponents: F4: 1,5,7,11; B6: 1,5,7,9,11; E6: 1,4,5,7,8,11. - 12-fold rotational symmetry (point group D12) is forbidden in periodic 2D/3D crystals but allowed in quasicrystals via projection from 6D (or higher) lattices. - Key ratio: \( \cos(\pi/12) = \sqrt{6} + \sqrt{2} \approx 0.9659 \); \( \sin(\pi/12) = \sqrt{6} - \sqrt{2} \approx 0.2588 \). CONNECTION: - 12-fold symmetry relates to golden ratio via \( \phi = 1.618 \) in pentagonal quasicrystals, but here the key geometric constant is \( \sqrt{3} \approx 1.732 \) (hexagonal/dodecagonal tiling). - Base-60 connection: 12 × 5 = 60; 12-fold symmetry aligns with base-60's divisibility by 12. - Crystallographic point group D12 (order 24) is a subgroup of affine Coxeter groups F4, B6, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com