Golden-Ratio Triplets Unify Platonic Duals and Rhombic Lattice Geometry — E8 Intelligence Research

Abstract

FINDING: The icosahedron and dodecahedron are duals whose vertex coordinates are generated directly from golden-ratio (φ) triplets; the rhombic dodecahedron emerges as a radial expansion of the icosidodecahedron, linking to cubic and hexagonal lattice periodicity. | MATH: Icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) where φ = (1+√5)/2 ≈ 1.618. Dodecahedron vertices (dual): (±1, ±1, ±1) and (0, ±1/φ, ±φ) — equivalently (±1, ±1, ±1) and (±φ⁻¹, ±φ, 0) with φ⁻¹ = φ−1 ≈ 0.618. Edge length ratio: for unit circumradius, icosahedron edge = 2/φ; dodecahedron edge = 2/√3 · (φ−1). Rhombic dodecahedron: 12 rhombic faces with diagonals in ratio √2:1 (not φ), but its vertices are the union of cube and octahedron vertices — a Voronoi cell of the body-centered cubic (BCC) lattice. | CONNECTION: φ appears explicitly in both Platonic solids' coordinates. The icosidodecahedron (Archimedean) has 30 vertices = midpoints of icosahedron edges, and its radial expansion yields the rhombic dodeca Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com