Arithmetic Backbone of Aperiodic Tilings: Eigenvalues, Modular Group, and Cyclotomic Fields — E8 Intelligence Research

FINDING: Penrose tilings and related aperiodic substitution tilings are governed by substitution matrices whose eigenvalues and algebraic structure connect directly to the modular group SL(2,Z) and cyclotomic fields, revealing a deep arithmetic backbone beneath their geometric non-periodicity. | MATH: The substitution matrix for Penrose tilings (e.g., the 2×2 matrix with entries from the golden ratio φ = (1+√5)/2) has eigenvalues λ₁ = φ² = φ+1 ≈ 2.618 and λ₂ = 1/φ² ≈ 0.382. These eigenvalues satisfy the characteristic equation λ² − 3λ + 1 = 0, whose discriminant is 5, linking to the quadratic field ℚ(√5). The modular group SL(2,Z) acts on the upper half-plane via Möbius transformations z ↦ (az+b)/(cz+d); the Penrose substitution matrix (e.g., [[2,1],[1,1]]) is an element of SL(2,Z) with trace 3, and its eigenvalues are exactly φ² and φ⁻². The cyclotomic aperiodic substitution tilings (CAST) from the arXiv paper (1606.06858) have vertices in the 2n-th cyclotomic field ℚ(ζ₂ₙ), where ζ₂ₙ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com