E8 Lattice and Binary Icosahedral Group Link via Quaternion Order in Quasicrystal Solidification — E8 Intelligence Research

Abstract

FINDING: The E8 lattice and binary icosahedral group 2I are linked via quaternion order parameters, providing a group-theoretic mechanism for icosahedral quasicrystal solidification from isotropic liquids. | MATH: E8 root system: 240 vertices, 6720 edges; Coxeter number h=30; theta series: \(\Theta_{E_8}(q) = 1 + 240 \sum_{n=1}^{\infty} \sigma_3(n) q^{2n}\); quaternion orientational order parameter Q ∈ ℍ; 2I ⊂ SU(2) has order 120; E8 lattice can be constructed from 2I via the McKay correspondence (dimensions of irreps: 1,2,3,4,5,6,4,3,2,1 sum to 120). | CONNECTION: Icosahedral symmetry (H3 Coxeter group) is a subgroup of E8; golden ratio φ = (1+√5)/2 ≈ 1.618 appears in 2I character values and E8 coordinates; quasicrystal diffraction patterns exhibit 5-fold symmetry forbidden in periodic crystals; base-60 not directly present but 120 (2×60) appears as order of 2I. | DEPTH: 9 — Directly links exceptional Lie algebra E8 (8D) to 3D icosahedral quasicrystals via quaternion algebra, unifying Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com