Golden Rhombic Dodecahedron: A φ-Linked Bridge to the Icosidodecahedron — E8 Intelligence Research

Abstract

FINDING: The rhombic dodecahedron (RD) is a zonohedron with 12 rhombic faces; its dual is the cuboctahedron, and it tiles space (a Voronoi cell of the FCC lattice). The "golden" variant arises when its rhombi are golden rhombi (diagonals in ratio φ), which connects it to the icosidodecahedron via radial expansion. | MATH: - RD face rhombus: diagonals \(d_1, d_2\) with \(d_1/d_2 = \sqrt{2}\) (standard, non-golden). - Golden rhombus: \(d_1/d_2 = \varphi = (1+\sqrt{5})/2 \approx 1.618\), or equivalently \(d_2/d_1 = 1/\varphi \approx 0.618\). - Face angle of golden rhombus: \(\arccos(1/\sqrt{5}) \approx 63.435^\circ\) (acute) and \(116.565^\circ\) (obtuse) — note \(116.565^\circ = 180^\circ - 63.435^\circ\). - Golden RD: 12 faces, 24 edges, 14 vertices (8 of degree 3, 6 of degree 4). - Icosidodecahedron: 32 faces (20 triangles + 12 pentagons), 60 edges, 30 vertices; its vertex figure is a golden rectangle. - Radial expansion: RD of icosidodecahedra implies a packing where each Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com