Cyclotomic Aperiodic Substitution Tilings with Dense Tile Orientations

Abstract

The class of Cyclotomic Aperiodic Substitution Tilings (CAST) whose vertices are supported by the $2n$-th cyclotomic field $\mathbb{Q}\left({\zeta }_{2n}\right)$ is extended to cases with Dense Tile Orientations (DTO). It is shown that every CAST with DTO has an inflation multiplier $\eta$ with irrational argument so that $\frac{k\pi }{2n}\ne \arg \left(\eta \right)\notin \pi \mathbb{Q}$. The minimal inflation multiplier ${\eta }_{min.irr.}$ is discussed for $n\geqq 2$. Examples of CASTs with DTO, minimal inflation multiplier ${\eta }_{min.irr.}$ and individual dihedral symmetry $D_{2n}$ are introduced for $n\in \left\{2,3,4,5,6,7\right\}$. The examples for $n\in \left\{2,3,4,5,6\right\}$ also yield finite local complexity (FLC).

18 pages, 9 figures

References in corpus (1)

• Cyclotomic Aperiodic Substitution Tilings