Quaternion E8 Lattice Unifies Quasicrystal Order via Golden-Ratio Projections — E8 Intelligence Research

Abstract

FINDING: The E8 lattice, generated via quaternion-based orientational order parameters, provides a unifying 8D root system whose 3D icosahedral projections encode quasicrystalline solidification — directly linking quaternion multiplication to golden-ratio crystallography. | MATH: E8 root system: 240 roots, norm² = 2; quaternion order parameter q ∈ H (|q|=1) with multiplication rule q₁q₂ = (r₁r₂ − v₁·v₂, r₁v₂ + r₂v₁ + v₁×v₂); icosahedral group H₃ ⊂ SO(3) realized via quaternion units {±1, ±i, ±j, ±k, (±1±i±j±k)/2} — the 24-cell (D₄) — whose 3D projection yields icosahedral vertices with edge ratios 1:φ (φ = (1+√5)/2 ≈ 1.618); E8 decomposes as (A₄⊕A₄) ⊕ (A₂⊕A₂⊕A₂) under H₃ folding, with Coxeter-Dynkin diagram branching at E₈. | CONNECTION: Golden ratio appears explicitly: the 120 icosian quaternions (norm 1, integer coefficients in ℤ[φ]) form the binary icosahedral group 2I, whose 3D vertex coordinates are scaled by 1/√(2+φ) — yielding edge lengths in ratio 1:φ. The E8 projection onto 3D Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com