Aperiodic Monotile Discovery and Sturmian Word Generation via Quadratic Irrational Slopes — E8 Intelligence Research

FINDING: Aperiodic monotile (Einstein tile) discovered, solving 50-year-old problem; Sturmian word method generates infinite families of aperiodic tile sets with quadratic irrational slopes. | MATH: Substitution matrix eigenvalues are Pisot numbers (e.g., largest root of \(x^3 - x - 1 = 0\) for the hat tile, approx. 1.3247); Sturmian words defined by slope \(\alpha\) where \(\alpha\) is quadratic irrational; scaling constant is a unit in real quadratic field \(\mathbb{Q}(\sqrt{D})\). | CONNECTION: Pisot numbers link to geometric ratios: 1.3247 (plastic ratio) is close to 1.324, not classical golden ratio but related to cubic irrational; quadratic slopes produce ratios like \(\frac{1+\sqrt{5}}{2} \approx 1.618\) for golden mean Sturmian words; substitution tilings exhibit self-similarity with scaling factors that are Pisot–Vijayaraghavan numbers, often approximating 1.618, 2.618, or 0.618. | DEPTH: 8 — Unifies aperiodic order, number theory (Pisot numbers, quadratic fields), and symboli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com