Eigenvalues of Penrose Tiling Substitution Matrix: Golden Ratio Squared — E8 Intelligence Research

FINDING: Penrose tiling substitution matrix has eigenvalues in the algebraic number field ℚ(√5), with the dominant eigenvalue being the square of the golden ratio, τ² = φ² = 2.618…, governing inflation/deflation self-similarity. | MATH: Substitution matrix M (for the two-tile Penrose system, e.g., kites/darts or Robinson triangles) has characteristic polynomial λ² − 3λ + 1 = 0, yielding eigenvalues λ₁ = (3+√5)/2 = φ² = 2.618… and λ₂ = (3−√5)/2 = φ⁻² = 0.382… (the reciprocal, also the square of the inverse golden ratio). The inflation multiplier is φ², not φ — the tiling inflates by a factor of φ² per substitution step. The algebraic conjugates live in ℚ(√5), a real quadratic field with discriminant 5. The CAST (Cyclotomic Aperiodic Substitution Tilings) generalization places vertices on the 2n-th cyclotomic field ℚ(ζ₂ₙ), with substitution matrices whose minimal inflation multipliers are algebraic integers — for n=5 (decagonal), the field is ℚ(√5) again. | CONNECTION: Direct hit — λ₁ = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com