Golden-Ratio Frequencies in Penrose Tilings via Perron-Frobenius Eigenvectors — E8 Intelligence Research

FINDING: Penrose tilings and the Perron-Frobenius eigenvector govern the golden-ratio frequency of tile types in aperiodic substitution systems, linking 5-fold symmetry to non-periodic order. | MATH: Substitution matrix \( M \) (e.g., for Penrose kites/darts: \( M = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \)); Perron-Frobenius eigenvalue \( \lambda = \varphi^2 = 2.618\ldots \) (where \( \varphi = (1+\sqrt{5})/2 = 1.618\ldots \)); eigenvector \( \mathbf{v} = (\varphi, 1) \) giving tile frequency ratio \( \varphi : 1 \). Also \( \varphi^{-1} = 0.618\ldots \), \( \varphi^{-2} = 0.382\ldots \). | CONNECTION: Direct — the substitution matrix's dominant eigenvector yields the golden ratio as the exact frequency ratio of the two tile types. The eigenvalue \( \varphi^2 = 2.618 \) is the growth factor per substitution step. This is the same \( \varphi \) appearing in the Penrose tiling's 5-fold rotational symmetry (impossible in periodic crystals) and in the inflation/deflation self-simila Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com