600-Cell Vertices as Binary Icosahedral Group Unit Quaternions — E8 Intelligence Research

FINDING: The 600-cell is the 4D regular polytope whose 120 vertices are exactly the unit quaternions of the binary icosahedral group 2I ≅ SL(2,5), the double cover of the icosahedral rotation group A5 ≅ I. MATH: - Binary icosahedral group: 2I = {±1, ±i, ±j, ±k, ½(±1±i±j±k), and the 96 even permutations of ½(±1±φ⁻¹±φ±0)} — order 120. - Isomorphism: 2I ≅ SL(2,5), the binary polyhedral group; its image in SO(3) is A5 (order 60). - 600-cell: 120 vertices, 720 edges, 1200 triangular faces, 600 tetrahedral cells. Vertex figure = icosahedron. Schläfli symbol {3,3,5}. - Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻¹ = φ−1 ≈ 0.6180339887. - Vertex coordinates (8 + 16 + 96 = 120): all permutations of (±1,0,0,0) [8]; ½(±1,±1,±1,±1) [16]; and even permutations of ½(±φ, ±1, ±φ⁻¹, 0) [96]. - Symmetry group: Coxeter group H4, order 14400 = 120². H4 is the largest finite reflection group in 4D. - Quaternion multiplication: unit quaternions q = w + xi + yj + zk with w²+x²+y²+z² = 1 form S³; the 120 ver Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com