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 \ ( (2 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 + 3 2. 732 \) and its relation to φ through \ ( (30°) = 3/2 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
Andrew Stewart Caldin (Wed,) studied this question.