Sturmian Words and Quadratic Units: Aperiodic Tilings from Pisot Scaling — E8 Intelligence Research

Abstract

FINDING: Aperiodic tile sets can be constructed from Sturmian words with quadratic irrational slopes, yielding scaling constants that are units of real quadratic fields — directly linking aperiodic order to Pisot numbers and the golden ratio. | MATH: Sturmian word slope α ∈ Q(√d), d>0 squarefree; scaling constant λ is a unit of the real quadratic field (e.g., λ = φ = (1+√5)/2 for slope φ², or λ = 2+√3 for slope √3). The construction yields infinitely many aperiodic tile sets for any quadratic irrational slope; the scaling constant satisfies λ² - Tr(λ)λ + N(λ) = 0, with |N(λ)| = 1 (unit condition). | CONNECTION: Directly hits 1.618 (φ) and 2.618 (φ²) as scaling constants; also 0.618 (φ⁻¹) appears as the inverse unit. The quadratic field structure (Q(√5), Q(√2), Q(√3)) is the algebraic backbone of crystallographic root systems (A4, B4, H4) and 5-fold/8-fold/12-fold quasicrystalline symmetries. Sturmian words encode the golden-mean rotation (α = 1/φ²) — the most irrational number — which Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com