Golden Rhombus and Rhombic Dodecahedron: FCC Voronoi Cells and Icosahedral Symmetry Breaking — E8 Intelligence Research

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