Quaternion Order Parameter Links E8 Lattice to Icosahedral Quasicrystal Solidification — E8 Intelligence Research

FINDING: Solidification of icosahedral quasicrystals modeled via quaternion orientational order parameter, linked to E8 lattice projection. MATH: Quaternion order parameter Q = q0 + q1 i + q2 j + q3 k, with |Q| = 1; E8 root lattice in 8D projects to 3D icosahedral tiling; phason window acceptance domain is a triacontahedron (golden rhombic polyhedron). Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618; its inverse φ⁻¹ ≈ 0.618; related ratios 0.382 (φ⁻²), 0.786 (√φ⁻¹). CONNECTION: Icosahedral symmetry (H3 Coxeter group) embedded in E8 (E8 contains H4 as subgroup); phason window geometry yields Ammann-Mackay tiling with edge lengths in ratio 1:φ; acceptance domain volume proportional to φ³. DEPTH: 8 — Directly links 8D lattice to 3D quasicrystal, unifying quaternion algebra, golden ratio, and crystallographic symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com