FINDING: E8 lattice's 240 minimal vectors project to 4D quasicrystal with icosahedral symmetry, linking 8D root system to 3D quasicrystal solidification via quaternion order parameter. MATH: - E8 root system: 240 vectors in 8D, each squared length = 2. - Projection to 4D yields vertices of a quasicrystal with icosahedral symmetry (H₃ Coxeter group). - Quaternion orientational order parameter Q = q₀ + q₁ i + q₂ j + q₃ k, with |Q| = 1, used to classify isotropic → icosahedral quasicrystal phase transition. - Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618, its inverse φ⁻¹ ≈ 0.618, and φ² ≈ 2.618 appear in icosahedral vertex coordinates. - E8 lattice's Coxeter number h = 30; dual Coxeter number = 30; rank = 8. CONNECTION: - Icosahedral symmetry (H₃) is crystallographically forbidden in 3D periodic lattices but emerges in quasicrystals. - E8's 240 vectors project to 3D icosahedral vertices scaled by φ, φ⁻¹, and 1. - The quaternion order parameter's 4D rotation group Spin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Andrew Stewart Caldin (Tue,) studied this question.