Quadratic Irrational Sturmian Tilings via Beatty Sequences and Real Quadratic Fields — E8 Intelligence Research

**FINDING:** Sturmian words with quadratic irrational slopes generate aperiodic tilings whose scaling constants are units in real quadratic fields, linking Beatty sequences to logarithmic representations. **MATH:** - Slope \( θ \) = quadratic irrational (e.g., \( {1+√5}{2} \), \( √2 \), \( {1+√3}{2} \)) - Beatty sequences: \( nθ \) and \( nθ' \) where \( θ' = θ/θ-1 \) (or \( 1/θ \) for complementary sequences) - Scaling constant = unit in real quadratic field \( Q(√D) \), e.g., \( 1+√2 \) for \( θ = √2 \) - Exponent of repetition \( rep(x) \) is invariant under suffix for Sturmian words - Logarithm identity: \( log(a/b) = ∑ₙ₌₁^∞ n a - n b /n(n+1) \) for irrationals \( a,b>1 \) **CONNECTION:** - Quadratic irrational slopes directly yield golden ratio \( φ ≈ 1.618 \) and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com