Quadratic Irrationals Link Sturmian Words to Aperiodic Tiling — E8 Intelligence Research

Abstract

FINDING: Sturmian words with quadratic-irrational slopes generate aperiodic tile sets whose scaling constants are units of real quadratic fields; the golden ratio directly proves aperiodicity of the "Einstein" hat tile. MATH: - Sturmian word slope θ ∈ ℚ(√d) (quadratic irrational) → characteristic word c_θ; exponent of repetition rep(x) is invariant under suffixing: rep(x) = rep(y) for all suffixes y of x. - Aperiodic tile set scaling constant = unit ε of real quadratic field ℚ(√d) (e.g., ε = (1+√5)/2 = φ ≈ 1.618 for d=5; ε = 2+√3 ≈ 3.732 for d=3). - Hat tile aperiodicity proof uses golden ratio: φ = (1+√5)/2; its continued fraction [1;1,1,1,…] yields the Sturmian slope 1/φ² = (3−√5)/2 ≈ 0.382. - Hexagonal aperiodic tilings with single edge-length: decorations of periodic lattice — implies substitution matrix eigenvalues are Pisot units (e.g., φ² = 2.618, φ³ = 4.236). CONNECTION: - Golden ratio φ = 1.618, φ−1 = 0.618, φ−2 = 0.382 — all appear as Sturmian slopes or scaling Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com