Quaternion Order Parameter Links 3D Quasicrystal Solidification to 8D E₈ Lattice — E8 Intelligence Research

Abstract

FINDING: The solidification of icosahedral quasicrystals is governed by a quaternion orientational order parameter, directly linking the 3D quasicrystal to the 8D E₈ lattice via a projection that preserves the binary icosahedral group symmetry. MATH: - E₈ lattice: 240 minimal vectors (roots) in 8D, with norm squared = 2. - Binary icosahedral group: order 120, double cover of the icosahedral rotation group (order 60). - Quaternion order parameter: \( q \in \mathbb{H} \), unit quaternions form \( S^3 \), which double-covers \( SO(3) \). The binary icosahedral group is a finite subgroup of \( S^3 \). - Projection: 3D quasicrystal = cut-and-project of E₈ lattice onto a 3D subspace, with the window defined by the Voronoi cell of the E₈ root system. - Key ratio: The golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) appears in the coordinates of E₈ roots (e.g., permutations of \( (\pm1, \pm\phi, 0, 0, 0, 0, 0, 0) \) scaled by \( 1/\sqrt{2} \)). CONNECTION: - Golden ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com