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.

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).

Tables

Table 1. Table 1. Minimal inflation multipliers η m ​ i ​ n . i ​ r ​ r . subscript 𝜂 formulae-sequence 𝑚 𝑖 𝑛 𝑖 𝑟 𝑟 \eta_{min.irr.} of CASTs with DTO and vertices supported by the 2 ​ n 2 𝑛 2n -th cyclotomic field ℚ ​ ( ζ 2 ​ n ) ℚ subscript 𝜁 2 𝑛 \mathbb{Q}\left(\zeta_{2n}\right)

$n$		Minimal inflation multiplier ${\eta }_{min.irr.}$	Minimal Perron-Frobenius eigenvalue ${\lambda }_{1min.irr.}={\eta }_{min.irr.}\overline{{\eta }_{min.irr.}}$	Approach
$2$		$2+{\zeta }_4^1$	$5$	analytic
$3$		$2+{\zeta }_6^1$	$3$	analytic
$4$		$1+2{\zeta }_8^1+{\zeta }_8^3$	$6+{\zeta }_8^1+\overline{{\zeta }_8^1}=6+{\mu }_{4,2}$	numeric
$6$		$1+{\zeta }_{12}^1+{\zeta }_{12}^3$	$4+{\zeta }_{12}^1+\overline{{\zeta }_{12}^1}=4+{\mu }_{6,2}$	analytic
$8$		$1+{\zeta }_{16}^1+{\zeta }_{16}^4$	$3+{\zeta }_{16}^1+\overline{{\zeta }_{16}^1}+{\zeta }_{16}^3+\overline{{\zeta }_{16}^3}=3+{\mu }_{8,4}$	analytic
$12$		$1+{\zeta }_{24}^1+{\zeta }_{24}^4$	$4+{\zeta }_{24}^1+\overline{{\zeta }_{24}^1}+{\zeta }_{24}^3+\overline{{\zeta }_{24}^3}=4+{\mu }_{12,4}$	analytic
$oddn\geqq 5$		$1+{\zeta }_{2n}^1+{\zeta }_{2n}^{\left(n+1\right)/2}$	$3+{\zeta }_{2n}^1+\overline{{\zeta }_{2n}^1}=3+{\mu }_{n,2}$	analytic
	$5$	$1+{\zeta }_{10}^1+{\zeta }_{10}^3$	$3+{\zeta }_{10}^1+\overline{{\zeta }_{10}^1}=3+{\mu }_{5,2}$	analytic
	$7$	$1+{\zeta }_{14}^1+{\zeta }_{14}^4$	$3+{\zeta }_{14}^1+\overline{{\zeta }_{14}^1}=3+{\mu }_{7,2}$	analytic
	$9$	$1+{\zeta }_{18}^1+{\zeta }_{18}^5$	$3+{\zeta }_{18}^1+\overline{{\zeta }_{18}^1}=3+{\mu }_{9,2}$	analytic
	$\dots$
	$evenn\geqq 10$ $n\mathrm{mod}\mathrm{\hspace{0.33em}4}=2$	$1+{\zeta }_{2n}^1+{\zeta }_{2n}^{\left(n+2\right)/4}+{\zeta }_{2n}^{\left(3n+2\right)/4}$	$4+{\zeta }_{2n}^1+\overline{{\zeta }_{2n}^1}=4+{\mu }_{n,2}$	analytic
	$10$	$1+{\zeta }_{20}^1+{\zeta }_{20}^3+{\zeta }_{20}^8$	$4+{\zeta }_{20}^1+\overline{{\zeta }_{20}^1}=4+{\mu }_{10,2}$	analytic
	$14$	$1+{\zeta }_{28}^1+{\zeta }_{28}^4+{\zeta }_{28}^{11}$	$4+{\zeta }_{28}^1+\overline{{\zeta }_{28}^1}=4+{\mu }_{14,2}$	analytic
	$18$	$1+{\zeta }_{36}^1+{\zeta }_{36}^5+{\zeta }_{36}^{14}$	$4+{\zeta }_{36}^1+\overline{{\zeta }_{36}^1}=4+{\mu }_{18,2}$	analytic
	$\dots$
	$evenn\geqq 16$ $n\mathrm{mod}\mathrm{\hspace{0.33em}4}=0$	$1+{\zeta }_{2n}^1+{\zeta }_{2n}^{\left(n-2\right)/2}+{\zeta }_{2n}^{\left(n+4\right)/2}$	$4+{\zeta }_{2n}^1+\overline{{\zeta }_{2n}^1}+{\zeta }_{2n}^3+\overline{{\zeta }_{2n}^3}=4+{\mu }_{n,4}$	numeric for $16\leqq n\leqq 100$, conjecture for $n>100$
	$16$	$1+{\zeta }_{32}^1+{\zeta }_{32}^7+{\zeta }_{32}^{10}$	$4+{\zeta }_{32}^1+\overline{{\zeta }_{32}^1}+{\zeta }_{32}^3+\overline{{\zeta }_{32}^3}=4+{\mu }_{16,4}$	numeric
	$20$	$1+{\zeta }_{40}^1+{\zeta }_{40}^9+{\zeta }_{40}^{12}$	$4+{\zeta }_{40}^1+\overline{{\zeta }_{40}^1}+{\zeta }_{40}^3+\overline{{\zeta }_{40}^3}=4+{\mu }_{20,4}$	numeric
	$24$	$1+{\zeta }_{48}^1+{\zeta }_{48}^{11}+{\zeta }_{48}^{14}$	$4+{\zeta }_{48}^1+\overline{{\zeta }_{48}^1}+{\zeta }_{48}^3+\overline{{\zeta }_{48}^3}=4+{\mu }_{24,4}$	numeric
	$\dots$

Equations

$$\exists z\mid \frac{z}{\overline{z}}\eta \in \mathbb{Z}\left[{\zeta }_{2n}\right]\to {\lambda }_1=\eta \cdot \overline{\eta }\in \mathbb{Z}\left[{\zeta }_{2n}+\overline{{\zeta }_{2n}}\right]$$

$$\nexists z\mid \frac{z}{\overline{z}}\eta \in \mathbb{Z}\left[{\zeta }_{2n}\right]\to {\lambda }_1=\eta \cdot \overline{\eta }\notin \mathbb{Z}\left[{\zeta }_{2n}+\overline{{\zeta }_{2n}}\right]$$

$$\exists z\mid \frac{z}{\overline{z}}\eta \in \mathbb{Z}\left[{\zeta }_{2n}\right]\leftrightarrow {\lambda }_1=\eta \cdot \overline{\eta }\in \mathbb{Z}\left[{\zeta }_{2n}+\overline{{\zeta }_{2n}}\right]$$

$$\eta =\sum _{k=1}^l{\zeta }_{2n}^{{\alpha }_k}\in \mathbb{Z}\left[{\zeta }_{2n}\right]({\alpha }_k\in \mathbb{Z},{\alpha }_l-{\alpha }_1<2n,{\alpha }_k\le {\alpha }_{k+1})$$

$$\eta =\sum _{k=1-n}^n{a}_k{\zeta }_{2n}^k\in \mathbb{Z}\left[{\zeta }_{2n}\right](a_k\in {\mathbb{N}}_0,l=\sum _{k=1-n}^n{a}_k,\max (a_k)>0)$$

$${\lambda }_1=|{\eta }^2|=\eta \cdot \overline{\eta }=b_0+\sum _{k=1}^{\lfloor (n-1)/2\rfloor }{b}_k\left({\zeta }_{2n}^k+\overline{{\zeta }_{2n}^k}\right)(b_0,b_k\in {\mathbb{N}}_0)$$

$${\lambda }_1=\sum _{k=1}^{\lfloor n/2\rfloor }{c}_k{\mu }_{n,k}(c_k\in {\mathbb{N}}_0)$$

$$b_0=\sum _{k=1-n}^n{a}_k^2\ge \sum _{k=1-n}^n{a}_k=l$$

$$\max \left(c_k\right)\ge 1(k\ge 2;oddn\ge 5)$$

$$\min \left(\left(\max \left(c_k\right),oddk\right),\left(\max \left(c_k\right),evenk\right)\right)\ge 1(evenn\ge 4)$$

$$b_k\ge b_{k+2}(\lfloor (n-1)/2\rfloor \ge k+2>k\ge 0)$$

$$\max \left(b_1,b_2\right)\ge 1(oddn\ge 5)$$

$$\min \left(b_0,b_1\right)\ge 1(evenn\ge 4)$$

$${\lambda }_1=\eta \overline{\eta }=3+{\zeta }_{2n}^{\alpha {}_2-\alpha {}_1}+\overline{{\zeta }_{2n}^{\alpha {}_2-\alpha {}_1}}+{\zeta }_{2n}^{\alpha {}_3-\alpha {}_1}+\overline{{\zeta }_{2n}^{\alpha {}_3-\alpha {}_1}}+{\zeta }_{2n}^{\alpha {}_3-\alpha {}_2}+\overline{{\zeta }_{2n}^{\alpha {}_3-\alpha {}_2}}$$

Full text

1. Introduction

Aperiodic substitution tilings and their properties are investigated and discussed in view of their application in physics and chemistry. However, they are also of mathematical interest on their own. Without any doubt, they have great aesthetic qualities as well. See [8, 1, 2, 3] and references therein. An interesting variant are substitution tilings with dense tile orientations (DTO), i.e. the orientations of the tiles are dense on the circle. Well known examples are the Pinwheel tiling as described by Radin and Conway in [10, 11] and its relatives in [12, 5, 6]. Recently Frettlöh, Say-awen and de las Peñas described “Substitution Tilings with Dense Tile Orientations and n-fold Rotational Symmetry” [7] and continued the discussion in [13]. For this reason we extend the results in [9, Chapter 2] regarding Cyclotomic Aperiodic Substitution Tilings (CAST) to cases with DTO in Section 2. In Section 3 it is shown that every CAST with DTO has an inflation multiplier $\eta$ with irrational argument so that $\nicefrac{{k\pi}}{{2n}}\neq\arg\left(\eta\right)\notin\mathbb{\pi Q}$. Furthermore minimal inflation multipliers $\eta_{min.irr.}$ of CASTs with DTO are derived in Section 4. In Section 5 examples with minimal inflation multiplier and individual dihedral symmetry $D_{2n}$ are introduced for the cases $n\in\left\{2,3,4,5,6,7\right\}$ and compared to the results in [7] and [13].

For terms and definitions we stay close to [1], [3], [7], [13], [9] and references therein:

• •

A “tile” in $\mathbb{\mathbb{\mathbb{R}}}^{d}$ is defined as a nonempty compact subset of $\mathbb{\mathbb{\mathbb{R}}}^{d}$ which is the closure of its interior.

• •

A “tiling” $\mathcal{T}$ in $\mathbb{\mathbb{\mathbb{R}}}^{d}$ is a countable set of tiles, which is a covering as well as a packing of $\mathbb{\mathbb{\mathbb{R}}}^{d}$. The union of all tiles is $\mathbb{\mathbb{\mathbb{R}}}^{d}$. The intersection of the interior of two different tiles is empty.

• •

A “patch” $\mathcal{P\mathcal{\subset}T}$ is a finite subset of a tiling.

• •

A tiling $\mathcal{T}$ is called “aperiodic” if no translation maps the tiling to itself.

• •

“Prototiles” $P_{x}$ serve as building blocks for a tiling.

• •

Within this article the term “substitution” means, that a tile is expanded and rotated with a linear map - the complex “inflation multiplier” $\eta\subset\mathbb{C}$ - and dissected into copies of prototiles in original size - the “substitution rule”.

• •

A “supertile” is the result of one or more substitutions, applied to a single tile.

• •

The result of $k$ substitutions applied to a single tile is noted as “level-k supertile”.

• •

A substitution is primitive if there is a $k\in\mathbb{N}$ for every prototile $P_{x}$ such that the corresponding level-$k$ supertile contains equivalent copies of all prototiles.

• •

A substitution tiling $\mathcal{T}$ with $l\in\mathbb{\mathbb{N}}_{>0}$ prototiles and their substitution rules is the result of an infinite number of substitutions applied to a single prototile.

• •

A primitive substitution tiling $\mathcal{T}$ (or more accurately, the set of substitution rules of its prototiles) is partially characterized by its primitive substitution matrix $M\in\mathbb{\mathbb{N}}_{0}^{l\times l}$ with a Perron-Frobenius eigenvalue $\lambda_{1}=\eta\overline{\eta}$.

• •

A tiling has “dense tile orientations” (DTO) if the orientations of the tiles are dense on the circle.

• •

A tiling has DTO if some level-$k$ supertile contains two congruent tiles which are rotated against each other by an irrational angle $\varTheta\notin\mathbb{\pi Q}$.

• •

A tiling with “individual cyclical or dihedral symmetry” $C_{n}$ or $D_{n}$ contains an infinite number of patches of any size with cyclical or dihedral symmetry $C_{n}$ or $D_{n}$.

• •

A tiling $\mathcal{T}$ with DTO has “finite local complexity” (FLC) with respect to rigid motions if for any $r>0$ there are only finitely many pairwise non-congruent patches in $\mathcal{T}$ fitting into a ball of radius $r$.

• •

We use $\zeta_{n}^{k}\in\mathbb{C}$ to denote the $n$-th roots of unity so that $\zeta_{n}^{k}=e^{\frac{2\mathtt{i}k\pi}{n}}$ and its complex conjugate $\overline{\zeta_{n}^{k}}=e^{-\frac{2\mathtt{i}k\pi}{n}}$.

• •

$\mathbb{Q}\left(\zeta_{n}\right)$ denotes the $n$-th cyclotomic field. Since $\mathbb{Q}\left(\zeta_{n}\right)=\mathbb{Q}\left(\zeta_{2n}\right)$ for $odd\;n$ we discuss tilings in the $2n$-th cyclotomic fields $\mathbb{Q}\left(\zeta_{2n}\right)$ only.

• •

The maximal real subfield of $\mathbb{Q}\left(\zeta_{n}\right)$ is $\mathbb{Q}\left(\zeta_{n}+\overline{\zeta_{n}}\right)$.

• •

$\mathbb{Z}\left[\zeta_{n}\right]$ denotes the ring of algebraic integers in $\mathbb{Q}\left(\zeta_{n}\right)$.

• •

$\mathbb{Z}\left[\zeta_{n}+\overline{\zeta_{n}}\right]$ denotes the the ring of algebraic integers (which are real numbers) in $\mathbb{Q}\left(\zeta_{n}+\overline{\zeta_{n}}\right)$.

• •

We use $\mu_{n,k}$ to denote the $k$-th diagonal of a regular $n$-gon with side length $1$, in detail $\mu_{n,k}=\sum_{i=0}^{k-1}\zeta_{2n}^{2i-k+1}=\nicefrac{{sin(\nicefrac{{k\pi}}{{n}})}}{{sin(\nicefrac{{\pi}}{{n}})}}$ with $\left\lfloor n/2\right\rfloor\geq k\geq 1$ and $\mu_{n,n-k}=\mu_{n,k}$ with $n>k\geq 1$.

• •

$\mathbb{Z}\left[\mu_{n}\right]=\mathbb{Z}\left[\mu_{n,1},\mu_{n,2},\mu_{n,3}...\mu_{n,\left\lfloor n/2\right\rfloor}\right]$ denotes the ring of the diagonals of a regular $n$-gon.

• •

The minimal inflation multiplier $\eta_{min.irr.}$ is the inflation multiplier $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]\setminus\left\{0\right\}$ of a CAST with irrational argument $\arg\left(\eta\right)\notin\pi\mathbb{Q}$ and with minimal absolute value.

2. A

Revised Definition Of CASTs

We recall the definition of Cyclotomic Aperiodic Substitution Tilings as noted in [9]:

Definition 2.1**.**

A (substitution) tiling $\mathcal{T}$ in the complex plane is cyclotomic if the coordinates of all vertices are algebraic integers in $\mathbb{Z}\left[\zeta_{2n}\right]$, i.e. an integer sum of the $2n$-th roots of unity. As a result all vertices of all substituted prototiles and the inflation multiplier are algebraic integers as well.

Recently substitution tilings with dense tile orientations (DTO) and $n$-fold rotational symmetry have been described in [7] by Frettlöh, Say-awen and de las Peñas. The results in the paper imply that a substitution tiling $\mathcal{T}$ with inflation multiplier $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]\mid\left|\eta\right|>1$ and $\arg\left(\eta\right)\notin\mathbb{\pi Q}$ has vertices supported by the cyclotomic field $\mathbb{Q}\left(\zeta_{2n}\right)$ but not necessarily by the ring of algebraic integers $\mathbb{Z}\left[\zeta_{2n}\right]$. An example is shown in Figure 2.1. Obviously Definition 2.1 is too constrictive.

Theorem 2.1**.**

A substitution tiling $\mathcal{T}$ with vertices supported by the $2n$-th cyclotomic field $\mathbb{Q}\left(\zeta_{2n}\right)$, $l\in\mathbb{\mathbb{N}}_{>0}$ prototiles and a primitive positive substitution matrix $M\in\mathbb{\mathbb{N}}_{0}^{l\times l}$ with a Perron–Frobenius eigenvalue $\lambda_{1}=|\eta^{2}|=\eta\cdot\overline{\eta}$ has an inflation multiplier $\eta\mid\left|\eta\right|>1$ which can be written as $\frac{z}{\overline{z}}\eta\in\mathbb{Z}\left[\zeta_{2n}\right]$ for some $z\in\mathbb{Q}\left(\zeta_{2n}\right)\setminus\left\{0\right\}$ .

Proof.

It is known that the ring of integers $\mathcal{O}_{K}$ of an algebraic number field $K$ is the ring of all integral elements contained in $K$. Furthermore an integral element is a root of a monic polynomial with rational integer coefficients, such as the characteristic polynom of $M\in\mathbb{\mathbb{N}}_{0}^{l\times l}$. In other words, the Perron–Frobenius eigenvalue $\lambda_{1}$ of $M$ is an integral element of $K$ and so $\lambda_{1}\in\mathcal{O}_{K}\subset K$. Since the tiling $\mathcal{T}$ has vertices supported by the $2n$-th cyclotomic field $\mathbb{Q}\left(\zeta_{2n}\right)$ we can deduce that $\eta\in\mathbb{Q}\left(\zeta_{2n}\right)$ and $\lambda_{1}=\eta\cdot\overline{\eta}\in\mathbb{Q}\left(\zeta_{2n}+\overline{\zeta_{2n}}\right)$. We can note that $K\simeq\mathbb{Q}\left(\zeta_{2n}\right)$ and $\mathcal{O}_{K}\simeq\mathbb{Z}\left[\zeta_{2n}\right]$, hence $\lambda_{1}=\eta\cdot\overline{\eta}\in\mathbb{Z}\left[\zeta_{2n}+\overline{\zeta_{2n}}\right]$.

To complete the proof we have to investigate, for which inflation multipliers $\eta$ the Perron–Frobenius eigenvalue fulfills the conditions $\lambda_{1}=\eta\cdot\overline{\eta}\in\mathbb{Z}\left[\zeta_{2n}+\overline{\zeta_{2n}}\right]$:

[TABLE]

[TABLE]

As a result we can conclude:

[TABLE]

∎

The results in [9, Chapter 2] apply also for tilings in Theorem 2.1 and we can revise Definition 2.1 as follows:

Definition 2.2**.**

A substitution tiling $\mathcal{T}$ in the complex plane is cyclotomic if the coordinates of all vertices are supported by the $2n$-th cyclotomic field $\mathbb{\mathbb{Q}}\left(\zeta_{2n}\right)$.

For simplification we will limit the following discussion to the case that $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]$.

3. Irrational Angles

In the following chapter we well discuss under which conditions a CAST may yield DTO.

We recall, all vertices of $\mathcal{T}$ are supported by the $2n$-th cyclotomic field $\mathbb{\mathbb{Q}}\left(\zeta_{2n}\right)$. So each finite patch $\mathcal{P}$ which contains at least one copy of each level-$1$ supertile of each of the $l$ prototiles $P_{x}$ of $\mathcal{T}$ can be mapped to $\mathcal{P}^{\prime}$ with an integer multiplier so that the vertices of $\mathcal{P}^{\prime}$ are supported by $\mathbb{Z}\left[\zeta_{2n}\right]$. As a consequence each CAST can be described by a set of prototiles and substitution rules, or more accurately, level-$1$ supertiles with vertices supported by $\mathbb{Z}\left[\zeta_{2n}\right]$. A CAST with DTO contains congruent tiles and so also corresponding edges which are rotated against each other by an irrational angle $\varTheta\notin\mathbb{\pi Q}$. As consequence at least one prototile exists whose edges enclose irrational angles. This and the primitivity of the tiling imply that the inflation multiplier of a CAST with DTO must have an irrational argument so that $\arg\left(\eta\right)\notin\mathbb{\pi Q}$, but not necessarily vice versa. An example for the latter case is shown in Figure 3.1.

While in [7, Theorem 5] and [13, Theorem 5] a parallelogram criterion was used to identify irrational angles, we need to extend their results to a more general approach. In detail we need a criterion to decide which elements of $\mathbb{Z}\left[\zeta_{2n}\right]$ have rational or irrational arguments. The author believes (as the authors of the cited articles) that the following results must be known already, but he is also not aware of any reference.

Lemma 3.1**.**

Any $z\in\mathbb{Q}\left(\zeta_{2n}\right)\mid\left|z\right|=1$ on the unit circle which is not a root of unity in $\mathbb{Q}\left(\zeta_{2n}\right)$ has an irrational argument, i.e. $z\in\mathbb{Q}\left(\zeta_{2n}\right)\setminus\left\{\zeta_{2n}^{k}\right\}\leftrightarrow\arg\left(z\right)\notin\pi\mathbb{Q}$ for $\left|z\right|=1$.

Proof.

It is known that any $z\in\mathbb{C}\mid\left|z\right|=1,\;arg\left(z\right)\in\pi\mathbb{Q}$ is a root of unity of some $2n$-th cyclotomic field $\mathbb{Q}\left(\zeta_{2n}\right)$ so that $z=e^{\frac{2\mathtt{i}k\pi}{2n}}=\zeta_{2n}^{k}$ . The $2n$-th cyclotomic field $\mathbb{Q}\left(\zeta_{2n}\right)$ has by definition exactly $2n$ roots of unity $\zeta_{2n}^{k}\mid 0\leq k<2n$. As a consequence every other $z\in\mathbb{Q}\left(\zeta_{2n}\right)\setminus\left\{\zeta_{2n}^{k}\right\}\mid\left|z\right|=1$ on the unit circle must have an irrational argument.∎

Lemma 3.2**.**

For any $z\mathbb{\in Q}\left(\zeta_{2n}\right)$ with rational argument $\arg\left(z\right)\in\pi\mathbb{Q}$ the argument of $z$ is an integer multiple of $\frac{\pi}{2n}$, so that $\arg\left(z\right)=\frac{k\pi}{2n}$. In other words, the elements of $\mathbb{Q}\left(\zeta_{2n}\right)$ with rational argument can be found along $2n$ straight lines in the complex plane running though [math] and tilted by $\frac{k\pi}{2n}$ relative to the real axis.

Proof.

It is known that $\arg\left(z\right)=\frac{1}{2}\arg\left(\frac{z}{\bar{z}}\right)$. Because of Lemma 3.1 we know that if the argument of $z$ is rational or not, the same is true for $\frac{z}{\bar{z}}$ and vice versa, i.e. $\arg\left(\frac{z}{\bar{z}}\right)\notin\pi\mathbb{Q}\leftrightarrow\arg\left(z\right)\notin\pi\mathbb{Q}$ and $\arg\left(\frac{z}{\bar{z}}\right)\in\pi\mathbb{Q}\leftrightarrow\arg\left(z\right)\in\pi\mathbb{Q}$. The latter term is true if and only if $\frac{z}{\bar{z}}=\zeta_{2n}^{k}$. Since $\arg\left(\zeta_{2n}^{k}\right)=\frac{k\pi}{n}$ we can conclude that $\arg\left(z\right)\in\pi\mathbb{Q\leftrightarrow}\arg\left(z\right)=\frac{k\pi}{2n}$. ∎

Obviously $\eta$ can be written as sum of $l$ roots of unity in different ways:

[TABLE]

[TABLE]

Assumption 1**.**

There are multiple ways to describe $\eta$. We assume $\alpha_{k}$ and $a_{k}$ to be chosen so that the sum is irreducible, i.e. $l$ is minimal.

Proposition 3.1**.**

Every inflation multiplier with rational argument $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]\setminus\left\{0\right\}\mid\arg\left(\eta\right)\in\pi\mathbb{Q}$ can be noted as in Equation (3.1) with $\alpha_{k+1}-\alpha_{k}=\alpha_{l-k+1}-\alpha_{l-k}$ and vice versa.

Proof.

Lemma 3.2 implies that the absolute value of $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]\mid\arg\left(\eta\right)\in\pi\mathbb{Q}$ has the following properties: $\left|\eta\right|\mathbb{\in\mathbb{Z}}\left[\zeta_{2n}+\overline{\zeta_{2n}}\right]$ for $\arg\left(\eta\right)=\frac{k\pi}{2n}$ with $even\text{~}k\in\mathbb{Z}$ and $\left|\eta\right|\mathbb{\in\mathbb{Z}}\left[\zeta_{4n}+\overline{\zeta_{4n}}\right]\setminus\mathbb{\mathbb{Z}}\left[\zeta_{2n}+\overline{\zeta_{2n}}\right]$ for $\arg\left(\eta\right)=\frac{k\pi}{2n}$ with $odd~{}k\in\mathbb{Z}$. Multiplication of $\left|\eta\right|$ with $e^{\frac{ik\pi}{2n}}$ and rearrangement of the order of the summands will deliver an appropriate notation for every $k$. Vice versa for every $\eta$ in appropriate notation a $k\in\mathbb{Z}$ exists so that $\arg\left(\eta e^{-\frac{ik\pi}{2n}}\right)=0$, hence $\arg\left(\eta\right)=\nicefrac{{k\pi}}{{2n}}\in\pi\mathbb{Q}$. ∎

Proposition 3.1 and Assumption 1 can be combined to form a criterion to decide if $\eta$ has a rational or an irrational argument.

4. Minimal Inflation Multipliers

with Irrational Argument

In the following chapter we will discuss the minimal inflation multiplier with irrational argument so that $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]\mid\arg\left(\eta\right)\notin\mathbb{\pi Q}$.

We recall the results from [9, Chapter 2]: The Perron–Frobenius eigenvalue $\lambda_{1}\in\mathbb{N}_{0}\left[\mu_{n}\right]$ of the substitution matrix $M$ of $\mathcal{T}$ can be written as sum of pairs of complex conjugated roots of unity $\zeta_{2n}^{k}+\overline{\zeta_{2n}^{k}}$ or as sum of diagonals of a regular unit $n$-gon $\mu_{n,k}$ with additional conditions:

[TABLE]

[TABLE]

By expanding Equation (4.1) with (3.2) we can conclude:

[TABLE]

Aperiodicity of $\mathcal{T}$ and primitivity of its substitution matrix for $n\geq 4$ are ensured by the following conditions for the coefficients $c_{k}$:

[TABLE]

[TABLE]

The same properties can be ensured by the following conditions for the coefficients $b_{k}$:

[TABLE]

[TABLE]

[TABLE]

Lemma 4.1**.**

Since $\arg\left(\zeta_{2n}^{\alpha{}_{1}}+\zeta_{2n}^{\alpha{}_{2}}\right)=\frac{\arg\left(\zeta_{2n}^{\alpha{}_{1}}\right)+\arg\left(\zeta_{2n}^{\alpha{}_{2}}\right)}{2}\in\mathbb{\pi Q}$ with $\alpha{}_{1}\neq-\alpha{}_{2}$ the minimal inflation multiplier $\eta_{min.irr.}$ must be a sum of $l\geq 3$ roots of unity to ensure $\arg\left(\eta_{min.irr.}\right)\notin\mathbb{\pi Q}$.

For $odd\;n\geqq 5$ we use Equations (4.1), (4.3), (4.6), (4.7) and Lemma 4.1 to identify the minimal Perron-Frobenius eigenvalue $\lambda_{1\;min.irr.}=\eta_{min.irr.}\cdot\overline{\eta_{min.irr.}}$. We can conclude that $b_{0}\geqq 3$ and either $b_{1}\geqq 1$ or $b_{2}\geqq 1$. As a consequence the first candidates for $\lambda_{1\;min.irr.}$ to be checked are the terms $3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}$ and $3+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}$.

For $even\;n\geqq 4$ we use a similar approach using Equations (4.1), (4.3), (4.6), (4.8) and Lemma 4.1. Here we can conclude that $b_{0}\geqq 3$ and $b_{1}\geqq 1$, so the first candidate for $\lambda_{1\;min.irr.}$ to be checked is the term $3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}$.

If no appropriate $\eta$ exists for the initial candidates then in both cases Equation (4.6) provides a guideline to identify the next candidates by adding $\zeta_{2n}^{0}=1$ or an appropriate $\zeta_{2n}^{k}$+$\overline{\zeta_{2n}^{k}}$.

The initial candidates for $\lambda_{1}$ as well as all other candidates with $b_{0}=3$ can be checked with the following equation for $\lambda_{1}$. For this purpose we expand Equation (4.1) with (3.1) and $l=3$:

[TABLE]

The check itself can be completed by solving an equation system which may or may not have solutions for $\alpha_{1},\;\alpha_{2},\;\alpha_{3}$ which determine the inflation multiplier $\eta=3+\zeta_{2n}^{\alpha{}_{1}}+\zeta_{2n}^{\alpha{}_{2}}+\zeta_{2n}^{\alpha{}_{3}}$.

For $l\geqq 4$ we we will use a numerical approach. A computer algorithm is designed to decide if an $\eta=\sum_{k=1}^{l}\zeta_{2n}^{\alpha_{k}}$ exists for arbitrary $\lambda_{1}\in\mathbb{N}_{0}\left[\mu_{n}\right]$, $l\geqq 3$ and $n\geqq 2$ by checking all combinations of $\alpha_{1},\alpha_{2},\alpha_{3}...\alpha_{l}$ with some reasonable optimisations.

It is known that for $n\in\left\{2,3\right\}$ the sets of algebraic integers in the $4$th and $6$th cyclotomic field $\mathbb{Z}\left[\zeta_{4}\right]$ and $\mathbb{Z}\left[\zeta_{6}\right]$ form lattices - uniform, discrete and periodic point sets - in the complex plane $\mathbb{C}$. In detail $\mathbb{Z}\left[\zeta_{4}\right]$ is equivalent to the set of Euler integers and $\mathbb{Z}\left[\zeta_{6}\right]$ is equivalent to the Eisenstein integers. We investigate the lattice points close to [math] and find $\eta_{min.irr.}=\zeta_{2n}^{0}+\zeta_{2n}^{0}+\zeta_{2n}^{1}\mid n\in\left\{2,3\right\}$.

For $n=4$ we can show numerically that $\eta=\zeta_{8}^{0}+\zeta_{8}^{0}+\zeta_{8}^{1}$ and $\lambda_{1}=6+2\left(\zeta_{8}^{1}+\overline{\zeta_{8}^{1}}\right)$ are minima under preceding conditions for $l=3$. Furthermore we can show that for $l=4$ and $0\leqq\alpha_{1}<\alpha_{2}<\alpha_{3}<\alpha_{4}<2n$ no $\lambda_{1}$ exist which fulfills the preceding conditions, in detail $\arg\left(\zeta_{8}^{0}+\zeta_{8}^{1}+\zeta_{8}^{2}+\zeta_{8}^{3}\right)=\nicefrac{{3\pi}}{{8}}\in\pi\mathbb{Q}$ and $\left|\zeta_{8}^{0}+\zeta_{8}^{1}+\zeta_{8}^{3}+\zeta_{8}^{6}\right|^{2}=4-\left(\zeta_{8}^{1}+\overline{\zeta_{8}^{1}}\right)=4-\mu_{4,2}\notin\mathbb{N}\left(\mu_{4}\right)$. As a consequence at least two $\alpha{}_{k}$ must have identical values which is also true all $l\geqq 5$. As a consequence we note that $\lambda_{1}\geqq 6+\zeta_{8}^{1}+\overline{\zeta_{8}^{1}}$ for $l\geqq 4$. We check the lower boundary which is smaller than the minimum of $\lambda_{1}$ for $l=3$ and find $\eta_{min.irr.}=\zeta_{8}^{0}+\zeta_{8}^{1}+\zeta_{8}^{1}+\zeta_{8}^{3}$ with $\lambda_{1\;min.irr.}=6+\zeta_{8}^{1}+\overline{\zeta_{8}^{1}}$.

For $n=6$ we have $\zeta_{12}^{2}+\overline{\zeta_{12}^{2}}=1$ and so we can note $3+\zeta_{12}^{1}+\overline{\zeta_{12}^{1}}<3+\zeta_{12}^{1}+\overline{\zeta_{12}^{1}}+\zeta_{12}^{2}+\overline{\zeta_{12}^{2}}=4+\zeta_{12}^{1}+\overline{\zeta_{12}^{1}}<3+\zeta_{12}^{1}+\overline{\zeta_{12}^{1}}+2\left(\zeta_{12}^{2}+\overline{\zeta_{12}^{2}}\right)=5+\zeta_{12}^{1}+\overline{\zeta_{12}^{1}}$. For the first candidate we already know that no solution for $\eta$ exists. For the second and third candidate we find $\eta_{min.irr.}=\zeta_{12}^{0}+\zeta_{12}^{1}+\zeta_{12}^{3}$.

For $n=8$ we have $\zeta_{16}^{3}+\overline{\zeta_{16}^{3}}<1<\zeta_{16}^{2}+\overline{\zeta_{16}^{2}}$ and so we can note $3+\zeta_{16}^{1}+\overline{\zeta_{16}^{1}}<3+\zeta_{16}^{1}+\overline{\zeta_{16}^{1}}+\zeta_{16}^{3}+\overline{\zeta_{16}^{3}}<4+\zeta_{16}^{1}+\overline{\zeta_{16}^{1}}<3+\zeta_{16}^{1}+\overline{\zeta_{16}^{1}}+\zeta_{16}^{2}+\overline{\zeta_{16}^{2}}+\zeta_{16}^{3}+\overline{\zeta_{16}^{3}}$. For the first candidate we already know that no solution for $\eta$ exists. For the second candidate we find $\eta_{min.irr.}=\zeta_{16}^{0}+\zeta_{16}^{1}+\zeta_{16}^{4}$.

For $n=12$ we have $\zeta_{24}^{4}+\overline{\zeta_{24}^{4}}=1$ and so we can note $3+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}<4+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}<3+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}+\zeta_{24}^{3}+\overline{\zeta_{24}^{3}}<4+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}+\zeta_{24}^{3}+\overline{\zeta_{24}^{3}}=3+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}+\zeta_{24}^{3}+\overline{\zeta_{24}^{3}}+\zeta_{24}^{4}+\overline{\zeta_{24}^{4}}<5+\zeta_{24}^{1}+\overline{\zeta_{24}^{1}}$. For the first candidate we already know that no solution for $\eta$ exists. For the second candidate we can show that numerically. For the third term no solution exists. For the fourth and fifth candidates we find $\eta_{min.irr.}=\zeta_{24}^{0}+\zeta_{24}^{1}+\zeta_{24}^{4}$.

For $even\;n\geqq 10$ with $n\mathrm{\;mod\;}4=2$ we have $\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}>1$ and so we can note $3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<4+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}+\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}$. For the first candidate we already know that no solution for $\eta$ exists. For the second candidate we find $\eta_{min.irr.}=\zeta_{2n}^{0}+\zeta_{2n}^{1}+\zeta_{2n}^{(n+2)/4}+\zeta_{2n}^{(3n+2)/4}$.

For $even\;n\geqq 16$ with $n\mathrm{\;mod\;}4=0$ we have $\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}>1$ and so we can note $3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<4+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}+\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}<5+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<4+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}+\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}<6+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}+\zeta_{2n}^{3}+\overline{\zeta_{2n}^{3}}$. For the first and the third candidate no solution for $\eta$ exists. For the second and the forth candidate no solution for $\eta$ were found.

Conjecture 4.1**.**

For $even\;n\geqq 16$ with $n\;\mathrm{mod\;}4=0$* no $\eta\in\mathbb{Z}\left[\zeta_{2n}\right]$ exists so that $\lambda_{1}=\eta\overline{\eta}=4+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}$ or $\lambda_{1}=\eta\overline{\eta}=5+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}$.*

The status of Conjecture 4.1 is subject to further research. However, for $n\leqq 100$ it was confirmed numerically. For the fifth term we find $\eta_{min.irr.}=\zeta_{2n}^{0}+\zeta_{2n}^{1}+\zeta_{2n}^{(n+2)/4}+\zeta_{2n}^{(3n+2)/4}$.

For $n=5$ we have $1+\zeta_{10}^{2}+\overline{\zeta_{10}^{2}}=\zeta_{10}^{1}+\overline{\zeta_{10}^{1}}$ and so we can note $2+\zeta_{10}^{1}+\overline{\zeta_{10}^{1}}=3+\zeta_{10}^{2}+\overline{\zeta_{10}^{2}}<3+2\left(\zeta_{10}^{2}+\overline{\zeta_{10}^{2}}\right)<3+\zeta_{10}^{1}+\overline{\zeta_{10}^{1}}=4+\zeta_{10}^{2}+\overline{\zeta_{10}^{2}}<3+3\left(\zeta_{10}^{2}+\overline{\zeta_{10}^{2}}\right)$. For the first and second candidate term we find $\eta=\zeta_{10}^{0}+\zeta_{10}^{1}$ with rational argument: $\arg\left(\zeta_{10}^{0}+\zeta_{10}^{1}\right)=\nicefrac{{\pi}}{{1\text{0}}}$. For the third term no solution exists. For the fourth candidate we find $\eta_{min.irr.}=\zeta_{10}^{0}+\zeta_{10}^{1}+\zeta_{10}^{3}$.

For $odd\;n\geqq 7$ we have $\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}>1$ and so we can note $3+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}<3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<4+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}<3+2\left(\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}\right)$. For $odd\;n\geqq 9$ we can additionally note $3+\zeta_{2n}^{1}+\overline{\zeta_{2n}^{1}}<3+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}+\zeta_{2n}^{4}+\overline{\zeta_{2n}^{4}}$. As a consequence there are no smaller candidates than the initial candidates. For $3+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}$ we find $\alpha_{2}-\alpha_{1}=2$ and $\alpha_{3}-\alpha_{1}=\nicefrac{{n}}{{2}}+1\notin\mathbb{Z}$ for $odd\;n$, so it is not a valid solution. For $3+\zeta_{2n}^{2}+\overline{\zeta_{2n}^{2}}$ we find $\alpha_{2}-\alpha_{1}=1$ and $\alpha_{3}-\alpha_{1}=\nicefrac{{(n+1)}}{{2}}\in\mathbb{Z}$ for $odd\;n$.

As a consequence the solution $\eta_{min.irr.}=\zeta_{2n}^{0}+\zeta_{2n}^{1}+\zeta_{2n}^{\nicefrac{{(n+1)}}{{2}}}$ applies for $odd\;n\geqq 5$ in general.

A summary of the results for $\eta_{min.irr.}$ and $\lambda_{1}$ can be found in Table 1.

All results for $\eta_{min.irr.}$ meet the conditions in Proposition 3.1 and Assumption 1, so that $\arg\left(\eta_{min.irr.}\right)\notin\mathbb{\pi Q}$.

Remark 4.1*.*

Because of the symmetry properties of $\mathbb{Z}\left[\zeta_{2n}\right]$ in detail dihedral symmetry $D_{2n}$ and $\arg\left(\eta_{min.irr.}\right)\notin\pi\mathbb{Q}$ for this and the following cases in total always $4n$ solutions for $\eta_{min.irr.}$ exist. For simplification we just note one representative solution.

5. Examples of CASTs with DTO

and Minimal Inflation Multiplier

In this chapter we will introduce substitution rules of CASTs with DTO for $n\in\{2,3,4,5,6,7\}$ and minimal inflation multiplier as shown in Figures 5.1 - 5.6 and compare our results with those in [7] and [13].

Within this paper $n$ is used to describe the $2n$-th cyclotomic fields $\mathbb{Q}\left(\zeta_{2n}\right)$ which support the vertices of the tilings, while [7] and [13] use it to describe tilings invariant under $n$-fold rotation. To avoid confusion we use the notation $n^{*}$ for the latter purpose. With $\mathbb{Q}\left(\zeta_{n}\right)\mathbb{=Q}\left(\zeta_{2n}\right)$ for $odd\;n$ it is obvious that $n^{*}=n$ for $odd\;n^{*}$ and $n^{*}=2n$ for $even\;n^{*}$. This is especially important for comparing results, such as symmetry conditions and inflation multipliers:

The tilings in [7] and [13] yield individual cyclical symmetry $C_{{}^{n*}}$, while the tilings and substitution rules in this article yield individual dihedral symmetry $D_{2n}$.

The inflation multiplier of the tilings in [7] and [13] can be rewritten as $\eta=\zeta_{2n}^{0}+\zeta_{2n}^{0}+\zeta_{2n}^{1}$. By comparing the value with the results in Table 1 we note that $\left|\eta_{min.irr.}\right|<\left|\zeta_{2n}^{0}+\zeta_{2n}^{0}+\zeta_{2n}^{1}\right|$ for $n\geqq 4$ and $\left|\eta_{min.irr.}\right|=\left|\zeta_{2n}^{0}+\zeta_{2n}^{0}+\zeta_{2n}^{1}\right|$ for $n\in\left\{2,3\right\}$.

Furthermore the tilings in [7] and [13] were designed to yield finite local complexity (FLC). As noted in [4, Remark 2.6], [7] and [13, Lemma 7] a substitution tiling $\mathcal{T}$ with respect to a substitution with a finite prototile set consisting of polygons satisfying the condition that the tiles in the level-$k$ supertiles meet “full-edge to full-edge” for all $k\in\mathbb{\mathbb{N}}_{>0}$ for all prototiles $P_{x}$ implies that the tiling $\mathcal{T}$ has FLC.

The substitution rules of CASTs with $n\in\{2,3,4,5,6\}$ in Figures 5.1 - 5.5 satisfy this conditions perfectly. As the substitution rules in [7] and [13] they meet the “full-edge to full-edge” condition and all inflated edges of the same length are dissected in the same manner. In addition and contrast to [7] and [13] the dissections of the inflated edges herein are strictly mirror symmetric, so indication of orientations can be omitted in general.

As in [7] and [13] additional vertices (pseudo-vertices) are introduced to dissect all edges of length $2$ or $3$ into segments of length $1$. These edges can be found on the boundaries of the $P_{A}$-prototiles of the CASTs with $n\in\{2,3,4\}$ and the $P_{B}$-prototile of the CAST with $n=4$ as shown in Figures 5.1 - 5.3. Another pseudo-vertex has to be introduced to divide one of the long edges of the $P_{F}$-prototile of the CAST with $n=5$ into two segments of length $1$ and $\zeta_{10}^{1}+\overline{\zeta_{10}^{1}}$ as shown in Figure 5.4. Here an appropriate decoration is applied.

While in [7] and [13] the derivation of substitution rules starts with a regular $n^{*}$-gon prototile, we choose the rather irregular convex $P_{A}$-prototile for this purpose. Its boundary can be described as polygonal line of unit segments - analogous to roots of unity added up to the minimal inflation multiplier $\eta_{min.irr.}$ - and a straight “baseline” of length $\left|\eta_{min.irr.}\right|$ which connects start and end point.

The substitution rule for a $P_{A}$-prototile can easily be obtained by lining up $P_{A}$-tiles by their baselines - again analogous to roots of unity added up to the minimal inflation multiplier $\eta_{min.irr.}$

• and filling up the remaining gaps with rhombs, isosceles triangles or other appropriate polygons. The cases $n\in\{4,5\}$ require the introduction of asymmetric concave prototiles, in detail the $P_{B}$-prototile in the shape of a irregular concave hexagon in Figure 5.3 and the $P_{F}$-tile in the shape of a concave tetragon (a kite) in Figure 5.4. It may take two inflation steps applied to the newly introduced prototiles before a dissection into prototiles is possible and the substitution rule can be found. During the process additional gaps in new shapes, or more accurately, new prototiles may appear which also require additional substitution rules. As a consequence it may take several recursions and a process of trial and error until all prototiles and substitution rules have been found and the tiling yields the desired properties such as individual dihedral symmetry, primitivity, DTO and FLC.

On the strength of the author’s experience, the introduction of individual dihedral symmetry as well as FLC tend to increase the number of prototiles and substitution rules.

To ensure primitivity it is helpful to place pairs of $P_{A}$-tiles (with the baseline encased) into substitution rules of other prototiles. DTO can be ensured by placing two such pairs of $P_{A}$-tiles with different chiralities into one substitution rule.

It seems the limitation to $n\in\{2,3,4,5,6,7\}$ is a rather arbitrary choice. For this reason we note:

Conjecture 5.1**.**

CASTs with DTO, FLC, minimal inflation multiplier as shown in Table 1 and individual dihedral symmetry $D_{2n}$ exist for all $n\geqq 2$.

The status of Conjecture 5.1 is subject to further research. The author believes it is true, however a rigorous proof is not available yet.

6. Summary

The definition of Cyclotomic Aperiodic Substitution Tilings (CASTs) in [9, Chapter 2] was revised in a way that it covers CASTs with dense tile orientations (DTO) and so the results in [9, Chapter 2] apply. It was shown that every CAST with DTO has an inflation multiplier $\eta$ with irrational argument so that $\nicefrac{{k\pi}}{{2n}}\neq\arg\left(\eta\right)\notin\mathbb{\pi Q}$. We could derive the minimal inflation multiplier for such tilings for $odd\;n\geqq 3$ and $even\;n\geqq 2$ with $n\;\mathrm{mod\;}4=2$. For $even\;n\geqq 2$ with $n\mathrm{\;mod\;}4=0$ our results depend on the status of Conjecture 4.1. However, for $n\leqq 100$ it was confirmed numerically.

For $n\in\left\{2,3,4,5,6,7\right\}$ we found examples for CASTs with DTO, minimal inflation multiplier and individual dihedral symmetry $D_{2n}$. The examples for $n\in\left\{2,3,4,5,6\right\}$ also yield finite local complexity (FLC). It is very likely that CASTs with DTO, FLC, minimal inflation multiplier and individual dihedral symmetry $D_{2n}$ for exist for all $n\geqq 2$.

As a final remark the author would like to emphasise the great aesthetic qualities of CASTs with DTO - such as their ability to create a subtle impression of imperfection - which may motivate further research beyond their mathematical and geometrical properties.

Acknowledgment

The author would like to thank Michael Baake (Bielefeld University), Dirk Frettlöh (Bielefeld University), Uwe Grimm (The Open University, Milton Keynes), Reinhard Lück and Christian Georg Mayr (Technische Universität Dresden) for their continued support and encouragement.