Aperiodic Tiles from Quadratic Irrational Sturmian Words and the Golden Ratio Hat Tile — E8 Intelligence Research

Abstract

FINDING: Aperiodic tile sets constructed from Sturmian words with quadratic irrational slopes, with scaling constants as units of real quadratic fields; exponent of repetition invariant under suffix for Sturmian words; aperiodicity of the hat tile proven via golden ratio. MATH: - Sturmian word slope \( \theta \) quadratic irrational → characteristic word \( c \) has exponent of repetition \( \mathrm{rep}(c) = \mathrm{rep}(y) \) for any suffix \( y \). - Scaling constant \( \lambda \) is a unit of a real quadratic field \( \mathbb{Q}(\sqrt{D}) \), i.e., \( \lambda = a + b\sqrt{D} \) with norm \( N(\lambda) = \pm 1 \). - Hat tile aperiodicity uses golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \), with associated ratios \( 1/\phi \approx 0.618 \), \( \phi^2 \approx 2.618 \). - Hexagonal aperiodic tilings with single edge-length, decorated periodic lattice. CONNECTION: - Golden ratio \( \phi \) and its reciprocal \( 1/\phi \) (0.618) appear directly in hat tile proof. - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com