Aperiodic Monotile Family with Hexagonal Symmetry and Sturmian Construction — E8 Intelligence Research

Abstract

FINDING: Aperiodic monotile "hat" discovered with vertex angles 60°, 120°, 180° forming hexagonal symmetry in aperiodic tiling; continuum of such tiles parameterized by (a,b) with Sturmian word construction for quadratic irrational slopes. MATH: Vertex angles: 60°, 120°, 180° (sum 360°). Tile shape derived from union of 8 kites (or 5 hexagons). Continuum parameter (a,b) yields infinite family of aperiodic monotiles. Sturmian words with slope = quadratic irrational (e.g., golden ratio φ = (1+√5)/2 ≈ 1.618) generate aperiodic tile sets. Scaling constant = unit in real quadratic field ℚ(√D). CONNECTION: Hexagonal symmetry (crystallographic point group 6mm) but aperiodic — breaks translational symmetry. Vertex angles are multiples of 60°, linking to base-60 geometry. Quadratic irrational slopes (e.g., φ) connect to golden ratio 1.618 and its reciprocal 0.618. The Sturmian construction uses continued fractions of quadratic irrationals, relating to 0.382 = 1 - 1/φ, 0.786 = √φ - 1, 2.618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com