The 600-Cell: Unique 4D Polytope with Quaternionic F4 Symmetry — E8 Intelligence Research

Abstract

FINDING: The 600-cell (hexacosichoron) is the sole 4D regular polytope with 120 vertices, 720 edges, 600 tetrahedral cells, and exhibits quaternionic symmetry tied to the F4 root system and the binary icosahedral group. | MATH: The 600-cell has Schläfli symbol {3,3,5}; its vertex figure is an icosahedron; its symmetry group is the Coxeter group H₄ of order 14400 = 120²; its vertices can be realized as the 120 unit quaternions forming the binary icosahedral group 2I, which is a double cover of the icosahedral group I (order 60). The F4 root system (24-cell) is a sublattice; the 600-cell's vertices include the 24-cell's 24 vertices plus 96 others derived from golden-ratio coordinates: (0, ±1, ±φ, ±φ⁻¹) and even permutations, where φ = (1+√5)/2 ≈ 1.618. | CONNECTION: The golden ratio φ appears explicitly in the coordinates of the 600-cell. The ratio of the circumradius to edge length is φ² = 2.618. The dihedral angle is 164.4775°, and the ratio of the inradius to circumradius involves φ. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com