The Double Cover: Unifying Quantum Spin and Classical Rotation — E8 Intelligence Research

FINDING: SU(2) double-cover of SO(3) is the core mathematical bridge between quantum spin (half-angle rotations) and classical 3D rotations, with the Bloch sphere as its geometric realization. | MATH: SU(2) ≅ {unit quaternions} ≅ S³; SO(3) ≅ S³/{±1} (projective space); double cover map ρ: SU(2)→SO(3) with kernel {±I}; half-angle: rotation by θ in SO(3) ↔ rotation by θ/2 in SU(2); Bloch sphere: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, with θ∈[0,π], φ∈[0,2π). | CONNECTION: The Bloch sphere's θ/2 is the *same* half-angle structure as the golden-ratio-linked icosahedral rotations (π/5, 2π/5) — the icosahedral group A₅ ⊂ SO(3) lifts to the binary icosahedral group 2I ⊂ SU(2) of order 120, whose elements include rotations by π/5 (36°) and 2π/5 (72°), both multiples of the golden angle 2π/φ² ≈ 137.5° (or its complement 2π/φ ≈ 222.5°). The ratio 0.618 = 1/φ appears in the icosahedron's edge-to-circumradius ratio: a = (2/√(φ+2))·R ≈ 1.051R, but more directly, the dihedral angle of the icosahedro

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com