Golden Rhombus and Rhombic Dodecahedron: FCC Voronoi Cells and Icosahedral Symmetry Breaking — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The rhombic dodecahedron appears as the Voronoi cell of the face-centered cubic (FCC) lattice, and its golden-ratio variant arises from icosidodecahedral radial expansion, linking it to icosahedral symmetry breaking. | MATH: Rhombic dodecahedron = Voronoi cell of FCC lattice (coordination number 12, packing density π/(3√2) ≈ 0.7405). Its 12 rhombic faces have diagonals in ratio √2:1 (not golden). However, the *golden rhombus* (diagonals in ratio φ:1, φ = (1+√5)/2 ≈ 1.618) forms the rhombic triacontahedron, not the dodecahedron. The icosidodecahedron has 30 vertices, 20 triangles, 12 pentagons; its radial expansion to a rhombic dodecahedron of icosidodecahedra implies a lattice with icosahedral point group (I_h, order 120) — a *quasicrystalline* precursor. The golden ratio appears in the icosidodecahedron's edge-to-circumradius ratio: edge = 2R sin(π/5) = R·√(2−2/φ) ≈ 0.618R·√2? Actually: edge = 2R sin(36°) = R·√(2−φ) ≈ 0.618R? No — sin(36°) = √(10−2√5)/4 ≈ 0.5878, so edge ≈ 1.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Rhombus and Rhombic Dodecahedron: FCC Voronoi Cells and Icosahedral Symmetry Breaking — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409581},
url = {https://zenodo.org/records/23409581},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Rhombus and Rhombic Dodecahedron: FCC Voronoi Cells and Icosahedral Symmetry Breaking — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409581
UR - https://zenodo.org/records/23409581
AB - FINDING: The rhombic dodecahedron appears as the Voronoi cell of the face-centered cubic (FCC) lattice, and its golden-ratio variant arises from icosidodecahedral radial expansion, linking it to icosahedral symmetry breaking. | MATH: Rhombic dodecahedron = Voronoi cell of FCC lattice (coordination number 12, packing density π/(3√2) ≈ 0.7405). Its 12 rhombic faces have diagonals in ratio √2:1 (not golden). However, the *golden rhombus* (diagonals in ratio φ:1, φ = (1+√5)/2 ≈ 1.618) forms the rhombic triacontahedron, not the dodecahedron. The icosidodecahedron has 30 vertices, 20 triangles, 12 pentagons; its radial expansion to a rhombic dodecahedron of icosidodecahedra implies a lattice with icosahedral point group (I_h, order 120) — a *quasicrystalline* precursor. The golden ratio appears in the icosidodecahedron's edge-to-circumradius ratio: edge = 2R sin(π/5) = R·√(2−2/φ) ≈ 0.618R·√2? Actually: edge = 2R sin(36°) = R·√(2−φ) ≈ 0.618R? No — sin(36°) = √(10−2√5)/4 ≈ 0.5878, so edge ≈ 1. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -