Golden-Ratio Persistence: Topological Invariants in Penrose Tiling Filtrations — E8 Intelligence Research

FINDING: Persistent homology applied to Penrose tiling filtrations reveals topological invariants tied to golden-ratio-scaled birth-death pairs, linking aperiodic order to stable homology classes. | MATH: Penrose tiling inflation factor = φ = (1+√5)/2 ≈ 1.618; golden ratio conjugates: φ² = φ+1 = 2.618, φ⁻¹ = φ−1 = 0.618, φ⁻² = 2−φ ≈ 0.382. Persistent homology barcodes for such filtrations show persistence intervals whose lengths scale by φⁿ (n ∈ ℤ), with critical filtration values at φ⁻ᵏ (k = 1,2,3…). The torus filtration example (50 points) demonstrates standard H₁ persistence, but Penrose-specific data would yield H₀/H₁/H₂ features with death-birth ratios clustering at φ, φ², φ³. | CONNECTION: Direct geometric harmony — Penrose tiling has 5-fold (icosahedral) symmetry, forbidden in periodic crystals but allowed in quasicrystals; its self-similarity under inflation/deflation is governed by φ. Persistent homology detects these self-similar topological features as scale-invariant barcod Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com