Golden-Ratio Radial Lattices from Rhombic Dodecahedra and Icosidodecahedra — E8 Intelligence Research

FINDING: Rhombic dodecahedra and icosidodecahedra form periodic radial-expansion lattices with golden-ratio scaling, linked to D3/A3 root systems and crystallographic packing. | MATH: Rhombic dodecahedron: 12 rhombic faces, 24 edges, 14 vertices (6 degree-4, 8 degree-3); face diagonals ratio √2:1. Icosidodecahedron: 30 vertices, 60 edges, 32 faces (20 triangles, 12 pentagons); vertex figure is a golden rectangle. Radial expansion factor between successive polyhedral shells: φ = (1+√5)/2 ≈ 1.618; reciprocal φ⁻¹ = 0.618; φ⁻² = 0.382. Edge length of rhombic dodecahedron in a φ-expansion: aₙ = a₀·φⁿ. | CONNECTION: The rhombic dodecahedron is the Voronoi cell of the face-centered cubic (FCC) lattice (D3 root system); its dual is the cuboctahedron (A3 root system). The icosidodecahedron is the intersection of the dodecahedron and icosahedron, both of which are φ-based. Radial expansion by φ preserves the D3/A3 symmetry (octahedral group O_h, order 48) and generates a self-similar packing — a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com