FINDING: The E₈ lattice's 240 minimal vectors (root system) map directly to a 4D quasicrystal tiling via the H₄ Coxeter group, encoding golden ratio ratios in 8D → 4D projection. MATH: - E₈ root system: 240 vectors, each of squared length 2, forming the E₈ lattice. - H₄ Coxeter group: finite group of order 14400, generated by reflections in 4D, with Coxeter diagram ⬤—⬤—⬤—⬤ (edge labels 5). - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears as the ratio of lengths in H₄ root vectors (e.g., 1, φ, φ²). - Projection: E₈ → H₄ via folding of Coxeter-Dynkin diagrams (E₈'s diagram contains H₄ as a subdiagram when nodes are identified). - Quasicrystal tiling: The 240 E₈ vectors project to 120 vertices of a 4D 600-cell (H₄ polytope), whose 3D shadow yields icosahedral quasicrystal patterns with 5-fold symmetry. CONNECTION: - Golden ratio φ = 1.618 (and its reciprocal 0.618) governs the H₄ root system lengths and the 600-cell's edge ratios. - 5-fold symmetry (forbidden in periodic crys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Andrew Stewart Caldin (Fri,) studied this question.