E8 Lattice and Binary Icosahedral Group Linked via Quaternion Order Parameters — E8 Intelligence Research

Abstract

FINDING: The E8 root system (Gosset 4_21 polytope) and the binary icosahedral group 2I are linked via quaternion orientational order parameters, providing a framework for icosahedral quasicrystal solidification from the 8-dimensional E8 lattice. MATH: - E8 lattice: 240 vertices, 6720 edges; root system of exceptional Lie group E8. - Binary icosahedral group 2I: order 120, double cover of icosahedral group (order 60). - Quaternion order parameter: \( q \in \mathbb{H} \), with norm 1, used to classify orientational symmetry breaking. - Theta function for E8: \( \Theta_{E8}(\tau) = 1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^n \), where \( q = e^{2\pi i \tau} \), \( \sigma_3(n) \) sum of cubes of divisors. - Modular form of weight 4 for SL(2,Z). CONNECTION: - Icosahedral symmetry (order 60) is a finite subgroup of SO(3); its double cover 2I is a finite subgroup of SU(2). - The E8 lattice projects to 3D icosahedral quasicrystals via the cut-and-project method using the icosia Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com