Golden Ratio Eigenvalues in Penrose Substitution Matrices — E8 Intelligence Research

FINDING: Penrose tiling substitution matrices encode inflation/deflation via eigenvalues tied to the golden ratio, with eigenvectors defining self-similar vertex lattices over cyclotomic fields. | MATH: The standard Penrose substitution matrix (for the two rhombi, thick and thin) has eigenvalues λ₁ = φ² = (3+√5)/2 ≈ 2.618 and λ₂ = 1/φ² = (3−√5)/2 ≈ 0.382. The inflation multiplier is φ (≈1.618), and the algebraic number field is ℚ(√5). For the CAST class (arXiv:1606.06858), substitution matrices act on ℤ[ζ₂ₙ] (2n-th cyclotomic field), with minimal inflation multipliers being algebraic integers — e.g., for n=5 (decagonal), the multiplier is φ; for n=7 (tetradecagonal), it involves 2cos(π/7) ≈ 1.8019. Eigenvectors of the substitution matrix give the relative frequencies of tile types: for Penrose, the Perron–Frobenius eigenvector is proportional to (φ, 1) for thick:thin rhombi, i.e., ratio φ:1. | CONNECTION: The eigenvalues 2.618 and 0.382 are exactly the golden ratio squared and its reci Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com