Perron-Frobenius Eigenvectors Dictate Tile Frequencies in Penrose Tilings — E8 Intelligence Research

FINDING: Perron-Frobenius theory governs the dominant eigenvector of nonnegative matrices, which in Penrose tilings determines tile frequency ratios — linking spectral properties to algebraic number fields and quasi-crystalline order. | MATH: For a primitive nonnegative matrix \(A\), the Perron root \(λPF > 0\) is simple, with strictly positive eigenvector \(v\) satisfying \(A v = λPF v\). In Penrose tilings, the substitution matrix \(M\) (2×2 for the kite/dart or rhombus prototiles) has \(λPF = τ^2 = φ^2 = 2.618...\) (where \(φ = (1+√5)/2 = 1.618...\)). The eigenvector components give the area-weighted tile frequencies: for the rhombi, the ratio of thick to thin tiles is \(τ : 1\), i.e., \(1.618 : 1\). The algebraic conjugates of \(λPF\) (here \(-φ⁻² = -0.382...\)) appear in the cohomology/K-theory of the tiling hull. | CONNECTION: Direct geometric harmony: \(λPF = φ^2 = 2.618\) (your listed constant), and the Per Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com