The 24-Cell's Rotational Symmetry as the Binary Tetrahedral Group in Spin(3) — E8 Intelligence Research

FINDING: The 24-cell is the unique self-dual regular 4-polytope; its rotational symmetry group is the binary tetrahedral group 2T ≅ SL(2,3), which sits inside Spin(3) ≅ SU(2) ≅ Sp(1), the double cover of SO(3). MATH: - 24-cell: Schläfli {3,4,3}, 24 vertices, 96 edges, 96 triangular faces, 24 octahedral cells. Self-dual (dual = itself). - Vertices = root system D₄ ∪ {±e₁,±e₂,±e₃,±e₄} = 24 roots of D₄ (the 24-cell's vertices are exactly the D₄ root system, scaled). Coxeter group W(D₄) has order 192; rotational subgroup order 96. - Rotational symmetry group: order 576 = 24 × 24 (self-duality manifests as |Aut| = |V|²). Full symmetry group order 1152 = 2 × 576. - Spin(3) ≅ SU(2) ≅ Sp(1) ≅ unit quaternions S³. Double cover: SU(2) → SO(3), kernel {±I}, so π₁(SO(3)) = ℤ₂. - Binary tetrahedral group 2T = {±1, ±i, ±j, ±k, (±1±i±j±k)/2} — order 24, the 24 unit Hurwitz quaternions. This is the preimage of A₄ (order 12) under SU(2)→SO(3). - 2T ≅ SL(2,3); A₄ ≅ rotational symmetry of tetrahedron; A Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com