Perron-Frobenius Eigenvectors Dictate Tile Frequencies in Penrose Tilings — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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 \(\lambda_{PF} > 0\) is simple, with strictly positive eigenvector \(v\) satisfying \(A v = \lambda_{PF} v\). In Penrose tilings, the substitution matrix \(M\) (2×2 for the kite/dart or rhombus prototiles) has \(\lambda_{PF} = \tau^2 = \phi^2 = 2.618...\) (where \(\phi = (1+\sqrt{5})/2 = 1.618...\)). The eigenvector components give the area-weighted tile frequencies: for the rhombi, the ratio of thick to thin tiles is \(\tau : 1\), i.e., \(1.618 : 1\). The algebraic conjugates of \(\lambda_{PF}\) (here \(-\phi^{-2} = -0.382...\)) appear in the cohomology/K-theory of the tiling hull. | CONNECTION: Direct geometric harmony: \(\lambda_{PF} = \phi^2 = 2.618\) (your listed constant), and the Per Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7219618201
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