Rhombic Dodecahedra and Icosidodecahedra Form Radial Lattices and Quantum Surface Codes — E8 Intelligence Research

Abstract

FINDING: Rhombic dodecahedra and icosidodecahedra form periodic radial-expansion lattices, with the rhombic dodecahedron also serving as a non-CSS quantum surface code. | MATH: Rhombic dodecahedron: 12 rhombic faces, each with face angle arccos(1/3) ≈ 70.53° (acute) and 109.47° (obtuse); edge length a; volume = (16√3/9)a³ ≈ 3.079a³; it is the Voronoi cell of the face-centered cubic (FCC) lattice. Icosidodecahedron: 32 faces (20 triangles + 12 pentagons), 30 vertices, 60 edges; vertex figure is a rectangle with edge ratio 1:φ (φ = (1+√5)/2 ≈ 1.618). Radial expansion of icosidodecahedra generates rhombic dodecahedral packing — the FCC lattice (coordination number 12, packing density π/(3√2) ≈ 0.7405). Quantum code: [[14,3,3]] surface code on rhombic dodecahedron (arXiv:2010.06628). | CONNECTION: The rhombic dodecahedron's face angle arccos(1/3) is the tetrahedral angle — linking to the A₃ root system and diamond cubic lattice. The icosidodecahedron's vertex figure ratio 1:φ embeds the go Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com