E₈ Roots Projected via H₃ Symmetry Yield Icosahedral Quasicrystals — E8 Intelligence Research

Abstract

FINDING: E₈ lattice roots projected to 3D generate icosahedral quasicrystalline order via H₃ Coxeter group symmetry, with quaternion order parameters governing solidification. MATH: - E₈ root system: 240 vectors in ℝ⁸, norm² = 2, forming the unique even unimodular lattice (D₄⊕D₄ or E₈). - Projection: Choose 3D subspace spanned by eigenvectors of a Coxeter element of H₃ (order 5 rotation in 3D). The 240 roots project to vertices of a 3D quasicrystal with icosahedral point group (H₃, order 120). - Quaternion orientational order parameter: q ∈ ℍ, |q|=1, with icosahedral symmetry encoded in the binary icosahedral group (2I, order 120) — a finite subgroup of SU(2). - Solidification: Landau-type free energy F[Q] where Q is a rank-3 tensor order parameter; icosahedral phase emerges from isotropic liquid via E₈ lattice projection (arXiv:1905.12165v2). - Key constants: Golden ratio φ = (1+√5)/2 ≈ 1.618; its inverse φ⁻¹ ≈ 0.618; φ² ≈ 2.618; φ⁻² ≈ 0.382. These appear in H₃ Coxeter pla Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com