Quadratic Field Units Generate Aperiodic Monotiles via Sturmian Sequences — E8 Intelligence Research

Abstract

FINDING: Real quadratic field units generate aperiodic monotiles via Sturmian sequences and continued fractions, linking tiling theory to algebraic number theory. MATH: - Quadratic irrational slope \( \alpha = \sqrt{D} \) or \( \frac{p+\sqrt{D}}{q} \) with discriminant \( D \) (non-square). - Fundamental unit \( \epsilon > 1 \) of real quadratic field \( \mathbb{Q}(\sqrt{D}) \) satisfies Pell's equation \( x^2 - D y^2 = \pm 1 \). - Sturmian word \( s(n) = \lfloor (n+1)\alpha + \beta \rfloor - \lfloor n\alpha + \beta \rfloor \) for irrational \( \alpha \). - Continued fraction expansion of \( \alpha \) is purely periodic for quadratic irrationals: \( \alpha = [a_0; \overline{a_1, \dots, a_k}] \). - Scaling constant for tile inflation is \( \epsilon \) (unit norm \( N(\epsilon) = \pm 1 \)). CONNECTION: - Golden ratio \( \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 \) is the fundamental unit of \( \mathbb{Q}(\sqrt{5}) \), generating the Fibonacci word (Sturmian) and Penrose t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com