Q₈: Non-Abelian Group Linking Binary Icosahedral Group and E₈ Root System — E8 Intelligence Research

FINDING: Quaternion group Q₈ is a non-abelian group of order 8, foundational to binary polyhedral groups and linked to the E₈ root system via the binary icosahedral group. MATH: Q₈ = {±1, ±i, ±j, ±k} with i² = j² = k² = ijk = −1. Order 8, non-abelian. The binary icosahedral group (order 120) is a double cover of the icosahedral group and embeds in SU(2). The E₈ root system has 240 roots, and its Weyl group order is 696,729,600. The binary icosahedral group is a subgroup of the unit quaternions, and its McKay correspondence yields the E₈ extended Dynkin diagram. CONNECTION: The binary icosahedral group's quaternion representation directly generates the E₈ root system via the McKay correspondence. The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the icosahedron's geometry and in the coordinates of E₈ roots. The ratio 0.618 (1/φ) and 2.618 (φ²) are implicit in the icosahedral symmetry. The 120 elements of the binary icosahedral group correspond to the 120 vertices of the 600-cell (4D re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com