Truncated Octahedron as BCC Voronoi Cell: A3 Root and Weight Geometry — E8 Intelligence Research

FINDING: The truncated octahedron is confirmed as the Voronoi cell of the body-centered cubic (BCC) lattice, with 8 hexagonal faces (perpendicular to A3 root directions) and 6 square faces (perpendicular to A3 weight directions), forming a 14-faced Archimedean solid with uniform edge length. | MATH: Faces = 8 hexagons + 6 squares = 14; Edges = 36 (all equal length); Vertices = 24 (each vertex: 1 square + 2 hexagons); Euler characteristic: V − E + F = 24 − 36 + 14 = 2. The A3 root system has 12 roots (±e_i ± e_j, i≠j) — these are the normals to the 6 square faces (paired as ±). The 8 hexagonal face normals are the A3 weight vectors (1/2(±e_1 ± e_2 ± e_3 ± e_4) with even sign product) — 8 directions, matching the 8 hexagons. Edge length ratio: if octahedron edge = a, truncated edge = a/√2. The dual is the tetrakis hexahedron (Catalan solid). | CONNECTION: The A3 root system is the symmetry of the tetrahedron (order 24), and the truncated octahedron's hexagonal faces are perpendicular to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com