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 \( \theta \) = quadratic irrational (e.g., \( \frac{1+\sqrt{5}}{2} \), \( \sqrt{2} \), \( \frac{1+\sqrt{3}}{2} \)) - Beatty sequences: \( \lfloor n\theta \rfloor \) and \( \lfloor n\theta' \rfloor \) where \( \theta' = \frac{\theta}{\theta-1} \) (or \( 1/\theta \) for complementary sequences) - Scaling constant = unit in real quadratic field \( \mathbb{Q}(\sqrt{D}) \), e.g., \( 1+\sqrt{2} \) for \( \theta = \sqrt{2} \) - Exponent of repetition \( \mathrm{rep}(x) \) is invariant under suffix for Sturmian words - Logarithm identity: \( \log(a/b) = \sum_{n=1}^\infty \frac{ \lfloor n a \rfloor - \lfloor n b \rfloor }{n(n+1)} \) for irrationals \( a,b>1 \) **CONNECTION:** - Quadratic irrational slopes directly yield golden ratio \( \phi \approx 1.618 \) and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com