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: \(F_{n+2}=F_{n+1}+F_n\) → matrix \(M=\begin{pmatrix}1&1\\1&0\end{pmatrix}\), eigenvalues \(\lambda_\pm=\frac{1\pm\sqrt5}{2}\) = \(1.618...\) and \(-0.618...\) (i.e., \(\varphi\) and \(-\varphi^{-1}\)). - \(F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5}\) (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 \(\varphi^2=2.618...\) and \(\varphi^{-2}=0.382...\) — the inflation multiplier is \(\varphi^2\), 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