FCC Voronoi Geometry: Rhombic Dodecahedron, Tetrahedral Angles, and Dual Cuboctahedron — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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
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BibTeX
@misc{caldin2026voronoi,
title = {{FCC Voronoi Geometry: Rhombic Dodecahedron, Tetrahedral Angles, and Dual Cuboctahedron — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431086},
url = {https://zenodo.org/records/23431086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - FCC Voronoi Geometry: Rhombic Dodecahedron, Tetrahedral Angles, and Dual Cuboctahedron — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431086
UR - https://zenodo.org/records/23431086
AB - 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
ER -