Aperiodic Monotiles and Hexagonal Symmetry: A √3-Based Continuum — E8 Intelligence Research

FINDING: The aperiodic monotile ("hat" tile) and its hexagonal-lattice relatives form a continuum of shapes whose inflation factors and substitution rules are rooted in √3 and cyclotomic fields, bridging aperiodic order with hexagonal crystallographic symmetry. MATH: - **Inflation factor**: The hat tile's substitution uses an inflation multiplier of **√3** (scaling by √3 in the hexagonal lattice). This is the key ratio — not golden-ratio based, but a **Pisot number** (√3 ≈ 1.73205), which is the algebraic integer satisfying x² − 3 = 0. - **Lattice**: The tiles are edge-to-edge on the **hexagonal lattice** (root lattice A₂), with vertices in the **2n-th cyclotomic field** ℚ(ζ₁₂) for n=6 (since 2n=12). The cyclotomic field ℚ(ζ₁₂) has degree φ(12)=4, and its ring of integers contains the hexagonal lattice basis {1, ζ₆} where ζ₆ = eiπ/3 = ½ + i√3/2. - **Continuum**: The Tile(a,b) family interpolates between the hat (a=b=1) and other monotiles; all are constructed from **5 hexag Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com