Eigenvalues of Penrose Substitution Matrices Govern Inflation and Spectral Properties — E8 Intelligence Research

Abstract

FINDING: The core mathematical engine of Penrose tilings is the substitution matrix — its dominant eigenvalue (the square of the golden ratio, τ² = 2.618…) governs inflation/deflation, while its other eigenvalues encode spectral properties (pure point vs. singular continuous) of the tiling's diffraction. | MATH: For the kite-and-dart (or rhombus) Penrose substitution, the substitution matrix M has eigenvalues: λ₁ = τ² = (3+√5)/2 ≈ 2.618, λ₂ = τ⁻² = (3−√5)/2 ≈ 0.382, and λ₃ = −1/τ ≈ −0.618. The Perron–Frobenius eigenvalue λ₁ = τ² gives the inflation factor (each tile inflates to τ² copies). The trace of M = 2, and det(M) = −1 (unimodular, preserving area up to scale). The cyclotomic CAST framework (arXiv:1606.06858) generalizes this: substitution matrices act on the 2n-th cyclotomic field ℚ(ζ₂ₙ), with minimal inflation multipliers being algebraic integers — for Penrose, n=5, ζ₁₀, and the multiplier is τ². | CONNECTION: Direct golden-ratio harmonics: λ₁ = 2.618 = τ², λ₂ = 0.382 = τ⁻², λ₃ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com