E8 Decomposes into Two H4 Copies via Icosian Embedding, Linking 8D Lattice to 3D Quasicrystals — E8 Intelligence Research

Abstract

FINDING: The E8 root system decomposes into two orthogonal H4 copies via quaternion icosian embedding, linking 8D lattice symmetry to 3D icosahedral quasicrystal order parameters. MATH: - E8 root system: 240 roots in 8D, with Coxeter number 30, Weyl group order 696,729,600. - Decomposition: E8 ≅ H4 ⊕ H4 (two copies of the 4D 120-cell root system, each with 120 roots). - Quaternion icosian embedding: H4 roots are icosians — quaternions of form (a + b√5)/2 with a,b in the 24-cell lattice (Hurwitz integers). - Icosian ring: { (a₀ + a₁i + a₂j + a₃k + b₀φ + b₁φi + b₂φj + b₃φk)/2 | aₖ,bₖ ∈ ℤ, φ = (1+√5)/2 } — norm 1 elements give 120 units. - Quaternion orientational order parameter: q = q₀ + q₁i + q₂j + q₃k, |q| = 1, used to classify icosahedral quasicrystal solidification via Landau-type free energy. CONNECTION: - Golden ratio φ = 1.618 appears explicitly in the icosian ring coefficients — the H4 root system is built on φ, not on rational integers. - The 120-cell (H4) has Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com