Eigenvalues of Penrose Tiles: From Fibonacci to Cyclotomic Fields — E8 Intelligence Research

FINDING: Fibonacci numbers computed via matrix exponentiation reveal the golden ratio as the dominant eigenvalue of the substitution matrix governing Penrose-tile inflation/deflation; cyclotomic aperiodic substitution tilings generalize this to higher-order algebraic fields. MATH: - Fibonacci recurrence: \(Fₙ₊₂=Fₙ₊₁+F_n\) → matrix \(M={pmatrix}1&1\\1&0{pmatrix}\), eigenvalues \(λ_±=1±√5/2\) = \(1.618...\) and \(-0.618...\) (i.e., \(φ\) and \(-φ⁻¹\)). - \(F_n = {φ^n - (-φ)⁻ⁿ}{√5}\) (Binet form). - Matrix exponentiation: \(M^n\) computed in \(O(log n)\) via fast doubling or eigen-decomposition. - Penrose tiling substitution matrix (e.g., for kites/darts or rhombi) has eigenvalues \(φ^2=2.618...\) and \(φ⁻²=0.382...\) — the inflation multiplier is \(φ^2\), and the Perron–Frobenius eigenvalue governs tile density ratios. - CAST (Cyclotomic Aperiodic Substitution Tilings): vertices Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com