E8 Lattice from Quaternionic Icosians: Golden Ratio Roots Unify Lie Algebra and Quasicrystals — E8 Intelligence Research

Abstract

FINDING: The E8 lattice is constructible from quaternionic icosians (the binary icosahedral group), and its 240 roots encode golden-ratio coordinates; this structure underlies both exceptional Lie algebra E8 and icosahedral quasicrystal solidification via a quaternion orientational order parameter. MATH: - E8 root system: 240 roots in 8D, Gosset 4_21 polytope, 6720 edges. - Quaternionic construction: The binary icosahedral group (2I, order 120) generates the icosian ring; E8 roots = { (a+bi+cj+dk) ∈ 2I × 2I } with norm 2, yielding 240 vectors. - Golden ratio appears explicitly: coordinates of E8 roots are permutations of (±1, ±φ, 0, 0, 0, 0, 0, 0) and (±1, ±1, ±1, ±1, 0, 0, 0, 0) with even sign changes, where φ = (1+√5)/2 ≈ 1.618. - Quaternion orientational order parameter (from arXiv:1905.12165): classifies isotropic liquid → icosahedral quasicrystal transition; the order parameter lives in S^3 ⊗ S^3, whose Hopf fibration projects to S^2, and the E8 lattice emerges as the 8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com