Quaternionic E₈ Icosians Construct the 600-Cell and Snub 24-Cell — E8 Intelligence Research

FINDING: The 600-cell (hexacosichoron) and its snub 24-cell derivative are directly constructible from quaternionic representations of the E₈ root system, with vertices normalized to unit quaternions (icosians), linking 4D polytopal symmetry to the golden ratio. MATH: - 600-cell has 120 vertices, 720 edges, 1200 faces, 600 cells; its symmetry group is the binary icosahedral group \(2I\) of order 120. - Quaternionic vertices of the 600-cell are the 120 icosians: \(1/2(±1 ± i ± j ± k)\) (16 units) + 8 permutations of \((±1,0,0,0)\) + 96 even permutations of \(1/2(±φ, ±1, ±1/φ, 0)\) where \(φ = {1+√5}{2} ≈ 1.618\). - The snub 24-cell (from arXiv:0906.2109) has 96 vertices, symmetry group \(W(D_4):C_3\) of order 576, derived from the \(E_8\) root system via quaternionic projection. Its vertices are unit quaternions satisfying specific norm-1 constraints from the \(E_8\) lattice. - Key c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com