Topological Invariants of Aperiodic Tilings via Inflation and Noncommutative Geometry — E8 Intelligence Research

FINDING: Topological invariants of aperiodic tilings are defined via inflation symmetry and C*-algebra K-theory, linking quasicrystal order to noncommutative geometry. | MATH: Penrose tiling inflation factor = τ² = φ² = (1+√5)²/4 ≈ 2.618; K₀(K₁) of the hull C*-algebra yields Z² (or Z⁴ for 3D icosahedral) invariants; cut-and-project method: tiling = π(Γ ∩ (E∥ × V)) with Γ a lattice in E∥ ⊕ E⊥, window W ⊂ E⊥; de Bruijn's pentagrid: 5 families of parallel lines, intersection pattern → Penrose tiling. | CONNECTION: Direct: inflation ratio 2.618 = φ² (golden ratio squared); self-similarity eigenvalues include φ and φ⁻¹; the hull's transversal is a Cantor set with fractal dimension related to φ; 5-fold symmetry (forbidden in periodic crystals) is crystallographically encoded via the root lattice A₄ and its projection; base-60 not present, but the golden ratio's continued fraction [1;1,1,...] mirrors the self-similar substitution rule. | DEPTH: 8 — This unifies topology (K-theory), algebra (C Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com