3D Icosahedral Quasicrystals from 6D Hypercubic Lattice via Golden Ratio Projection — E8 Intelligence Research

Abstract

FINDING: Explicit construction of 3D icosahedral quasicrystals via cut-and-project from 6D hypercubic lattice using golden ratio basis, with polyhedral radial expansions generating rhombic triacontahedra and hexecontahedra. MATH: - Golden ratio φ = (1+√5)/2 ≈ 1.618, with algebraic conjugates τ = φ, σ = 1-φ = -1/φ ≈ -0.618. - 6D hypercubic lattice Z⁶ projected onto 3D physical space E∥ via basis vectors involving φ: e.g., (1, φ, 0, -1, φ, 0) etc. - Cut window = projection of 6D unit cube onto orthogonal 3D space E⊥, yielding a rhombic triacontahedron (30 rhombic faces) as acceptance domain. - Rhombic hexecontahedron (60 rhombic faces) emerges as star-shaped "landing port" for icosahedral symmetry. - Polyhedral radial expansion: dodecahedra → icosidodecahedra → rhombic triacontahedra, with edge ratios following φ. CONNECTION: - Icosahedral symmetry group H₃ (order 120) directly linked to φ via its Coxeter presentation: [3,5] with Coxeter number 10. - Rhombic triacontahedr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com