The Truncated Octahedron as the A3 Weyl Polytope and Its Space-Filling via Affine Weyl Group — E8 Intelligence Research

Abstract

FINDING: The truncated octahedron is the Weyl orbit polytope of the A3 root system, and its space-filling property (the only uniform Archimedean solid to tile 3D space) is a direct consequence of the affine Weyl group A3~ (Coxeter group [3,4,3]) acting on the root lattice. | MATH: A3 root system: 12 roots (±e_i ± e_j, i≠j). Weyl group order |W(A3)| = 24 (S4). Truncated octahedron vertices: 24 = |W(A3)| / 1 (regular orbit). Edge length ratio: original octahedron edge : truncated edge = 1 : (√2 − 1) ≈ 1 : 0.4142. Face angles: squares (90°), hexagons (120°). Dihedral angle: arccos(−1/3) ≈ 109.47° (tetrahedral angle). Volume: 8√2 a³ (a = edge). Space-filling: Voronoi cell of body-centered cubic (BCC) lattice — dual to the A3* lattice. Coxeter-Dynkin diagram: A3 = o—o—o (rank 3). Affine extension A3~ adds one node, giving the tiling group. | CONNECTION: The truncated octahedron's hexagonal faces are perpendicular to the 6 directions of the A3 roots (±e_i ± e_j). The ratio of inradius to cir Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com