Binary Icosahedral Group: Quaternion Double Cover with Golden Ratio in SU(2) — E8 Intelligence Research

FINDING: Binary icosahedral group is a double cover of the icosahedral rotation group, and its quaternion representation yields a 120-element subgroup of unit quaternions. This group is intimately linked to the golden ratio φ, as the vertices of the regular icosahedron (and its dual dodecahedron) are coordinates in the golden ratio field ℚ(√5). The quaternion norm condition for these elements is |q| = 1, and the group is a finite subgroup of SU(2). This structure appears in topological quantum state spaces as the symmetry group of certain non-Abelian anyons and topological phases (e.g., Fibonacci anyons). | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; its reciprocal φ⁻¹ = (√5-1)/2 ≈ 0.618. Icosahedral vertices in quaternion form: (0, ±1, ±φ, ±φ⁻¹) up to permutations and sign choices. Binary icosahedral group order = 120. Quaternion norm: |q| = √(a²+b²+c²+d²) = 1. | CONNECTION: The golden ratio φ and its inverse φ⁻¹ appear as coordinates of the icosahedron/dodecahedron vertices, linking to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com