Projecting 6D Lattices to 3D: Unifying Quasicrystals via Coxeter Pairs — E8 Intelligence Research

FINDING: 3D Penrose tilings and icosahedral quasicrystals are generated by projecting a 6D hypercubic lattice onto 3D, with the Ammann lines/planes arising from Coxeter pairs (H₃-related) — unifying aperiodic order with root system geometry. | MATH: 6D hypercubic lattice ℤ⁶ projected onto the 3D irrep of the icosahedral group H₃ (order 120). The cut-and-project method uses a 3D window (acceptance region) in the perpendicular 3D space. Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618; its inverse φ⁻¹ = φ−1 ≈ 0.618; φ² = φ+1 ≈ 2.618. Ammann tiles in 3D have volumes in ratios involving φ (e.g., prolate/oblate rhombohedra with volumes ∝ φ and 1). Coxeter pairs (e.g., (A₅, H₃) or (D₆, H₃)) give the projection matrices whose entries are φ and 1/φ. | CONNECTION: Direct link to H₃ root system — the 30 vertices of the icosidodecahedron (or 12+20 of icosahedron+dodecahedron) are the H₃ roots. The 6D lattice projection yields 3D Penrose tilings whose vertices lie on a set of 5-fold, 3-fold, and 2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com