# Cyclotomic Aperiodic Substitution Tilings with Dense Tile Orientations

## TL;DR

This paper extends the class of Cyclotomic Aperiodic Substitution Tilings to include those with Dense Tile Orientations, analyzing their inflation multipliers, symmetries, and local complexity for various parameters.

## Contribution

It introduces and characterizes CASTs with DTO, exploring their inflation multipliers, symmetries, and minimal irrational inflation factors for different cases.

## Key findings

- CASTs with DTO have irrational inflation argument
- Examples with dihedral symmetry for n=2 to 7
- Some examples exhibit finite local complexity

## 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 $\frac{k\pi}{2n}\neq\arg\left(\eta\right)\notin\mathbb{\pi Q}$. The minimal inflation multiplier $\eta_{min.irr.}$ is discussed for $n\geqq2$. 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).