FCC Voronoi Geometry: Rhombic Dodecahedron, Tetrahedral Angles, and Dual Cuboctahedron — E8 Intelligence Research

Abstract

FINDING: The rhombic dodecahedron is the Voronoi cell of the FCC lattice, whose geometry encodes tetrahedral coordination (arccos(1/√3)) and face-diagonal ratios tied to √2 and √3, with dual relationship to the cuboctahedron. | MATH: FCC Voronoi cell = rhombic dodecahedron (12 rhombic faces, 24 edges, 14 vertices). Face diagonal ratio: short/long = 1/√2 ≈ 0.7071. Tetrahedral angle between face normals: arccos(1/√3) ≈ 54.7356°. Dihedral angle of rhombic dodecahedron: arccos(−1/3) ≈ 109.471° (tetrahedral angle complement). Packing fraction FCC = π/(3√2) ≈ 0.7405. Reciprocal lattice of FCC = BCC (cube Voronoi). Rhombus acute angle = arccos(1/3) ≈ 70.528°, obtuse = 109.471°. | CONNECTION: The rhombic dodecahedron's face diagonal ratio 1/√2 = 0.7071 is NOT a golden ratio, but its dihedral angle 109.471° = 180° − 70.528°; note 70.528° = 2 × 35.264° (half of tetrahedral angle). The ratio √2 appears in FCC stacking (ABCABC), and √3 in face diagonal of cube. No direct φ (1.618) or 0.618/0.786/2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com