# Cyclotomic Aperiodic Substitution Tilings

## Abstract

**:**

## 1. Introduction

- The tiling shall be aperiodic and repetitive (locally indistinguishable) to have an interesting (psychedelic) appearance.
- The tiling shall have a small inflation multiplier for reasons of economy. Large inflation multipliers either require large areas to be covered or many tiles of a small size to be used.
- The tiling shall yield “individual dihedral symmetry” ${D}_{n}$ or ${D}_{2n}$ with $n\ge 4$. In other words, it shall contain an infinite number of patches of any size with dihedral symmetry only by iteration of substitution rules on a single tile.Similar to G. Maloney we demand symmetry of individual tilings and not only symmetry of tiling spaces [8].

- A “tile” in ${\mathbb{R}}^{d}$ is defined as a nonempty compact subset of ${\mathbb{R}}^{d}$ which is the closure of its interior.
- A “tiling” in ${\mathbb{R}}^{d}$ is a countable set of tiles, which is a covering as well as a packing of ${\mathbb{R}}^{d}$. The union of all tiles is ${\mathbb{R}}^{d}$. The intersection of the interior of two different tiles is empty.
- A “patch” is a finite subset of a tiling.
- A tiling is called “aperiodic” if no translation maps the tiling to itself.
- “Prototiles” serve as building blocks for a tiling.
- Within this article the term “substitution” means, that a tile is expanded with a linear map—the “inflation multiplier”—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. Within this article we use the term for one substitutions only.
- We use ${\zeta}_{n}^{k}$ 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 $\overline{{\zeta}_{n}^{k}}={e}^{-\frac{2\mathtt{i}k\pi}{n}}$.
- $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}$ denotes the n-th cyclotomic field. Please note that $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}$ for $odd\phantom{\rule{0.277778em}{0ex}}n$.
- The maximal real subfield of $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}$ is $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}+\overline{{\zeta}_{n}}$.
- $\mathbb{Z}\left(\right)open="["\; close="]">{\zeta}_{n}$ denotes the the ring of algebraic integers in $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}$.
- $\mathbb{Z}\left(\right)open="["\; close="]">{\zeta}_{n}+\overline{{\zeta}_{n}}$ denotes the the ring of algebraic integers (which are real numbers) in $\mathbb{Q}\left(\right)open="("\; close=")">{\zeta}_{n}+\overline{{\zeta}_{n}}$.
- We use ${\mu}_{n,k}$ to denote the k-th diagonal of a regular n-gon with side length ${\mu}_{n,1}={\mu}_{n,n-1}=1$.
- $\mathbb{Z}\left(\right)open="["\; close="]">{\mu}_{n}$ denotes the ring of the diagonals of a regular n-gon.

## 2. Properties of Cyclotomic Aperiodic Substitution Tilings

**Definition**

**1.**

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 3. CASTs with Minimal Inflation Multiplier

#### 3.1. The Odd n Case

**Conjecture**

**1.**

**Remark**

**8.**

#### 3.2. The Even n Case

**Remark**

**9.**

## 4. CASTs with Inflation Multiplier Equal to the Longest Diagonal of a Regular Odd n-Gon

**Conjecture**

**2.**

**Conjecture**

**3.**

**Remark**

**10.**

**Remark**

**11.**

## 5. Rhombic CASTs with Symmetric Edges and Substitution Rules

**Definition**

**2.**

- There are two ways to place rhombs on the edge of substitution rules. We recall that the inner angles of the rhombs are integer multiples of $\frac{\pi}{n}$. We can place all rhombs on the edge so that the inner angles either with even or odd multiples of $\frac{\pi}{n}$ are bisected by the boundary of the supertile. We will call these two cases “even” and “odd edge configuration”, for details see Figure 7. A “mixed” configuration is not allowed, because it would force the existence of rhombs with inner angle equal to $\left(\right)open="("\; close=")">k+\frac{1}{2}$.
- We can choose the symmetry of the substitution rules and their edges. Possible choices are dihedral symmetry ${D}_{1}$ and ${D}_{2}$. Edges with dihedral symmetry ${D}_{1}$ can have the boundary of the supertile or its perpendicular bisector as line of symmetry. The smallest nontrivial solution for the latter case is the generalized Goodman-Strauss tiling [39]. Since this example does not provide individual dihedral symmetry ${D}_{n}$ or ${D}_{2n}$ in general, we will focus on the other case.Substitution rules of rhombs which appear on the edge of a substitution rule are forced to have the appropriate dihedral symmetry ${D}_{1}$ as well. This is also true for substitution rules of prototiles which lie on the diagonal, i.e., a line of symmetry of a substitution rule. The orientations of the edges have to be considered as well. These three conditions may force the introduction of additional rhomb prototiles and substitution rules. Additionally, the existence of edges with orientations may require additional preconditions.To avoid this problem, a general dihedral symmetry ${D}_{2}$ can be chosen for the substitution rules and their edges.
- Parity of the chosen n may require different approaches in some cases, similar to the example of the generalized Lançon-Billard tiling in Section 3.2 and Figure 4.

- The rhombs on the edges must not overlap. For this reason, for the tip of the substitution rule, only three configurations are possible as shown in Figure 8. Obviously, a tip as shown in Figure 8c is compliant to the cases 2 and 4 with odd edge configuration and Figure 8b to cases 1 and 3 with even edge configuration. Figure 8a requires the even edge configuration as well. Since all edges are congruent, it must be the start and the end of the same edge, which meet on that vertex. Since start and end of the edge are different, it can not have dihedral symmetry ${D}_{2}$. For this reason, the tip in Figure 8a is not compliant to case 3.
- Any rhomb ${R}_{k}$ with $n-3\ge k\ge 3$ on one edge implies the existence of a rhomb ${R}_{k-2}$ on the corresponding edge. In turn, rhomb ${R}_{k}$ on one edge is implied by a rhomb ${R}_{k+2}$ on the corresponding edge or a rhomb ${R}_{k}$ on the opposite edge. An example is shown in Figure 9.
- Any rhomb ${R}_{2}$ on one edge implies the existence of a line segment ${R}_{0}$ on the corresponding edge. In turn, rhomb ${R}_{2}$ on one edge is implied by a rhomb ${R}_{4}$ on the corresponding edge or a rhomb ${R}_{2}$ on the opposite edge.
- Any rhomb ${R}_{1}$ on one edge implies the existence of a rhomb ${R}_{1}$ on the corresponding edge. In turn, rhomb ${R}_{1}$ on one edge is implied by a rhomb ${R}_{3}$ on the corresponding edge or a rhomb ${R}_{1}$ on the opposite edge.
- Any line segment ${R}_{0}$ on one edge is implied by a rhomb ${R}_{2}$ on the corresponding edge or a line segment ${R}_{0}$ on the opposite edge.
- Any rhomb ${R}_{n-2}$ on one edge implies the existence of a rhomb ${R}_{n-4}$ on the corresponding edge. In turn, rhomb ${R}_{n-2}$ on one edge is implied by a rhomb ${R}_{n-2}$ on the opposite edge. (Rhomb ${R}_{n}$ does not exist, because the inner angle would be zero.)
- Any rhomb ${R}_{n-1}$ on one edge implies the existence of a rhomb ${R}_{n-3}$ on the corresponding edge. In turn, rhomb ${R}_{n-1}$ on one edge is implied by a rhomb ${R}_{n-1}$ on the opposite edge. (Rhomb ${R}_{n+1}$ does not exist, it would have an inner angle greater than π or smaller the 0.)
- If for a rhomb ${R}_{k},\phantom{\rule{0.277778em}{0ex}}n>k>0$ on one edge two related elements (rhomb or line segment) ${R}_{|k-2|}$ and ${R}_{k+2}$ exist on the corresponding edge, ${R}_{|k-2|}$ is closer to the tip than ${R}_{k+2}$.

- Any line segment ${R}_{0}$ on the edge implies the existence of a rhomb ${R}_{n-1}$ on the correspondent edge or a line segment ${R}_{0}$ on the opposite edge. As shown in Figure 10a,b, the existence of rhomb ${R}_{n-1}$ on the edge is not required to meet the KSK criterion.
- Any rhomb ${R}_{2}$ on the edge implies the existence of a rhomb ${R}_{n-3}$ on the correspondent edge or a rhomb ${R}_{2}$ on the opposite edge. As shown in Figure 10c,d, the KSK criterion is only met if at least one ${R}_{n-3}$ exists on the edge.$${\alpha}_{n-3}>{\alpha}_{n-1}\ge 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(cases\phantom{\rule{0.277778em}{0ex}}1b\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}3b)$$

- The line segment ${R}_{0}$ on the edge implies the existence of a line segment ${R}_{0}$ on the opposite edge only (rhomb ${R}_{n}$ does not exist).
- Any rhomb ${R}_{2}$ on the edge implies the existence of a rhomb ${R}_{n-2}$ on the correspondent edge or a rhomb ${R}_{2}$ on the opposite edge. So the KSK criterion is only met if at least one ${R}_{n-2}$ exists on the edge.$${\alpha}_{n-2}\ge 1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(cases\phantom{\rule{0.277778em}{0ex}}1a\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}3a)$$

- Any rhomb ${R}_{1}$ on the edge implies the existence of a rhomb ${R}_{n-2}$ on the correspondent edge or a rhomb ${R}_{1}$ on the opposite edge. So the KSK criterion is only met if at least one ${R}_{n-2}$ exists on the edge.$${\alpha}_{n-2}\ge 1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(cases\phantom{\rule{0.277778em}{0ex}}2b\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}4b)$$

- Any rhomb ${R}_{1}$ on the edge implies the existence of a rhomb ${R}_{n-1}$ on the correspondent edge or a rhomb ${R}_{1}$ on the opposite edge. So the KSK criterion is only met if at least one ${R}_{n-1}$ exists on the edge.$${\alpha}_{n-1}\ge 1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(cases\phantom{\rule{0.277778em}{0ex}}2a\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}4a)$$

**Remark**

**12.**

**Conjecture**

**4.**

**Remark**

**14.**

**Remark**

**15.**

## 6. Gaps to Prototiles Algorithm

- All prototiles have inner angles equal $\frac{k\pi}{n}$.
- All edges of all substitution rules are congruent and have dihedral symmetry ${D}_{2}$.
- As discussed in Section 5 the tiles on the edge have to be placed, so that the inner angles either with even or odd multiples of $\frac{\pi}{n}$ are bisected by the boundary of the supertile.
- The tiles on the edge are bisected by one or two lines of symmetry of the edge. This implies dihedral symmetry ${D}_{1}$ or ${D}_{2}$ of the corresponding substitution rules.
- The inflation multiplier η must fulfill the conditions in Theorem 1.
- The inflation multiplier η is defined by the sequence of tiles which are part of the edge.

- We start with the prototiles which appear on the edge of the substitution rule.
- We start the construction of the substitution rules by placing the prototiles on the edge.
- If the edge prototiles overlap the algorithm has failed. In this case, we may adjust the sequence of rhombs or other equilateral polygons on the edge and start another attempt.
- We try to “fill up” the substitution rules with existing prototiles under consideration of the appropriate dihedral symmetry ${D}_{1}$ or ${D}_{2}$. If gaps remain, they are defined as new prototiles and we go back to step (2). Please note, if a gap lies on one or two lines of symmetry, the substitution rule of the new prototile must also have the appropriate dihedral symmetry ${D}_{1}$ or ${D}_{2}$.
- If no gaps remain the algorithm was successful.

## 7. Extended Girih CASTs

- Regular decagon with inner angles $\frac{4\pi}{5}$
- Regular pentagon with inner angles $\frac{3\pi}{5}$
- Rhomb with inner angles $\frac{2\pi}{5}$ and $\frac{3\pi}{5}$
- Convex hexagon with inner angles $\frac{2\pi}{5}$,$\frac{4\pi}{5}$,$\frac{4\pi}{5}$,$\frac{2\pi}{5}$,$\frac{4\pi}{5}$,$\frac{4\pi}{5}$
- Convex hexagon with inner angles $\frac{3\pi}{5}$,$\frac{3\pi}{5}$,$\frac{4\pi}{5}$,$\frac{3\pi}{5}$,$\frac{3\pi}{5}$,$\frac{4\pi}{5}$
- Nonconvex hexagon with inner angles $\frac{2\pi}{5}$, $\frac{2\pi}{5}$, $\frac{6\pi}{5}$, $\frac{2\pi}{5}$, $\frac{2\pi}{5}$, $\frac{6\pi}{5}$

- All prototiles of an extended Girih CAST are equilateral polygons with the same side length.
- The inner angles of all prototiles are $\frac{k\pi}{n}$, $k\in \left(\right)open="\{"\; close="\}">2,3\dots \left(\right)open="("\; close=")">n-1,\left(\right)open="("\; close=")">n+2$.
- One of the prototiles may be a regular n-gon with inner angles $\frac{\left(\right)open="("\; close=")">n-2}{\pi}$.
- One of the prototiles may be a regular $2n$-gon with inner angles $\frac{\left(\right)open="("\; close=")">n-1}{\pi}$.

- All edges of the substitution rules are congruent and have dihedral symmetry ${D}_{2}$.
- All substitution rules except those for regular n-gons with $odd\phantom{\rule{0.277778em}{0ex}}n$ have dihedral symmetry ${D}_{2}$.
- The substitution rule of the regular n-gon with $odd\phantom{\rule{0.277778em}{0ex}}n$ has dihedral symmetry ${D}_{1}$.
- The substitution rule of the regular $2n$-gon has dihedral symmetry ${D}_{2n}$ for $odd\phantom{\rule{0.277778em}{0ex}}n$ and ${D}_{n}$ for $even\phantom{\rule{0.277778em}{0ex}}n$.

- In every corner of every substitution rule a regular $2n$-gon is placed.
- Edge and inflation multiplier have been derived from a periodic pattern of regular $2n$-gons and their inter space counterparts.

**Remark**

**16.**

**Remark**

**17.**

## 8. Summary and Outlook

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The diagonals ${\mu}_{11,k}$ of a regular hendecagon with side length 1 can be written as a sum of 22-nd roots of unity as described in Equation (9).

**Figure 2.**Cyclotomic Aperiodic Substitution Tilings (CAST) for the case $n=7$ with minimal inflation multiplier. The black tips of the prototiles mark their respective chirality.

**Figure 3.**CAST for the case $n=7$ with minimal inflation multiplier as described in [33] (Figure 1 and Section 3, 2nd matrix).

**Figure 6.**CAST for the case $n=11$ with inflation multiplier ${\mu}_{11,5}$. One vertex star within prototile ${P}_{4}$ has been chosen to illustrate the individual dihedral symmetry ${D}_{11}$.

**Figure 7.**The ”Edges” of a substitution rules are defined as the boundaries of the supertile (dashed line) and the rhombs bisected by it along one of their diagonals. The figure illustrates how the rhombs can be placed accordingly for even edge configuration (

**a**); and odd edge configuration (

**b**). The inner angles of the rhombs are integer multiples of $\frac{\pi}{n}$ and are denoted by small numbers near the tips (Example $n=7$).

**Figure 8.**Substitution rule for rhomb ${R}_{1}$ or ${R}_{n-1}$: The three possible configurations of the tip are shown in (

**a**), (

**b**) and (

**c**) (Example $n=7$).

**Figure 9.**Substitution rule for rhomb ${R}_{1}$ or ${R}_{n-1}$: Rhomb ${R}_{k}$ on the edge and its relatives at the corresponding edge (black) and the opposite edge (blue) (Example $n=7$, $k=4$).

**Figure 10.**Substitution rule for rhomb ${R}_{(n\pm 1)/2}$: The Kannan-Soroker-Kenyon (KSK) criterion requires the existence of rhomb ${R}_{n-3}$ on the edge as shown in (

**c**) and (

**d**). However, this is not the case for rhomb ${R}_{n-1}$ as shown in (

**a**) and (

**b**) (Case 1b, example $n=7$).

**Figure 11.**Substitution rule for rhomb ${R}_{1}$ or ${R}_{n-1}$: The possible configurations of the edges orientations are shown in (

**a**), (

**b**), (

**c**) and (

**d**) (Example $n=7$).

**Figure 12.**Substitution rule for rhomb ${R}_{1}$ or ${R}_{n-1}$: KSK criterion for corresponding edges with different orientations with ${\alpha}_{0}={\alpha}_{2}$ are shown in (

**a**), (

**b**), (

**c**) and ${\alpha}_{2k}={\alpha}_{2k+2};\phantom{\rule{0.277778em}{0ex}}k>1$ are shown in (

**d**), (

**e**) and (

**f**).

**Figure 13.**Orientations of edges (and orientations of lines of symmetry) for rhomb prototiles in case 1b (Example $n=11$).

**Figure 15.**Rhombic CAST examples for case 2a ($n=6$) and case 2b ($n=7$). The shown example for case 2b was slightly modified to reduce the number of prototiles to $\left(\right)$ as in case 1b. In detail, the edges of the rhomb prototiles have orientation as shown in Figure 13.

Sum of Diagonals | Sum of Roots of Unity | Conditions for n |
---|---|---|

${\mu}_{n,5}$ | $1+{\zeta}_{2n}^{2}+\overline{{\zeta}_{2n}^{2}}+{\zeta}_{2n}^{4}+\overline{{\zeta}_{2n}^{4}}$ | $n=11$ |

${\mu}_{n,4}$ | ${\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}+{\zeta}_{2n}^{3}+\overline{{\zeta}_{2n}^{3}}$ | $n\ge 9$ |

${\mu}_{n,3}$ | $1+{\zeta}_{2n}^{2}+\overline{{\zeta}_{2n}^{2}}$ | $n\ge 7$ |

${\mu}_{n,2}$ | ${\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}$ | $n\ge 5$ |

${\mu}_{n,2}+1$ | $1+{\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}$ | - |

3 | 3 | $n\ge 7$ |

2 | 2 | - |

1 | 1 | - |

Sum of Diagonals | Sum of Roots of Unity | Conditions for n |
---|---|---|

${\mu}_{n,6}$ | ${\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}+{\zeta}_{2n}^{3}+\overline{{\zeta}_{2n}^{3}}+{\zeta}_{2n}^{5}+\overline{{\zeta}_{2n}^{5}}$ | $n=12$ |

${\mu}_{n,5}$ | $1+{\zeta}_{2n}^{2}+\overline{{\zeta}_{2n}^{2}}+{\zeta}_{2n}^{4}+\overline{{\zeta}_{2n}^{4}}$ | $n\in \{10,12\}$ |

${\mu}_{n,4}$ | ${\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}+{\zeta}_{2n}^{3}+\overline{{\zeta}_{2n}^{3}}$ | $n\ge 8$ |

${\mu}_{n,3}$ | $1+{\zeta}_{2n}^{2}+\overline{{\zeta}_{2n}^{2}}$ | $n\ge 6$ |

${\mu}_{n,2}$ | ${\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}$ | $n\ge 4$ |

${\mu}_{n,4}+1$ | $1+{\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}+{\zeta}_{2n}^{3}+\overline{{\zeta}_{2n}^{3}}$ | $n=8$ |

${\mu}_{n,3}+1$ | $2+{\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}$ | $n\ge 6$ |

${\mu}_{n,2}+1$ | $1+{\zeta}_{2n}^{1}+\overline{{\zeta}_{2n}^{1}}$ | - |

3 | 3 | - |

2 | 2 | - |

1 | 1 | - |

Substitution Rules and Their Edges Have at Least Dihedral Symmetry D_{1} | Substitution Rules and Their Edges Have Dihedral Symmetry D_{2} | |
---|---|---|

Even edge configuration | Case 1a $(even\phantom{\rule{0.277778em}{0ex}}n)$ Case 1b $(odd\phantom{\rule{0.277778em}{0ex}}n)$ | Case 3a $(even\phantom{\rule{0.277778em}{0ex}}n)$ Case 3b $(odd\phantom{\rule{0.277778em}{0ex}}n)$ |

Odd edge configuration | Case 2a $(even\phantom{\rule{0.277778em}{0ex}}n)$ Case 2b $(odd\phantom{\rule{0.277778em}{0ex}}n)$ | Case 4a $(even\phantom{\rule{0.277778em}{0ex}}n)$ Case 4b $(odd\phantom{\rule{0.277778em}{0ex}}n)$ |

n | Minimal Rhomb Edge Sequence | Minimal Inflation Multiplier η_{min} |
---|---|---|

$4,5$ | $\overline{0-2}$ | ${\mu}_{n,2}+1$ |

$6,7$ | $\overline{0-2}-4-\overline{0-2}$ | ${\mu}_{n,3}+2{\mu}_{n,2}+1$ |

$8,9$ | $\overline{0-2}-4-\overline{0-2}-6-4-\overline{0-2}$ | ${\mu}_{n,4}+2{\mu}_{n,3}+2{\mu}_{n,2}+1$ |

$10,11$ | $\overline{0-2}-4-6-8-\overline{0-2}-4-\overline{0-2}-6-4-\overline{0-2}$ | ${\mu}_{n,5}+2{\mu}_{n,4}+2{\mu}_{n,3}+2{\mu}_{n,2}+1$ |

… | … | … |

n | Minimal Rhomb Edge Sequence | Minimal Inflation Multiplier η_{min} |
---|---|---|

$4,5$ | $1-\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}3\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">{\mu}_{n,2}+2$ |

$6,7$ | $1-3-\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1-5\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">{\mu}_{n,3}+2{\mu}_{n,2}+2$ |

$8,9$ | $1-3-5-\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1-\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}3-7\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-5-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">{\mu}_{n,4}+2{\mu}_{n,3}+2{\mu}_{n,2}+2$ |

$10,11$ | $1-3-5-7-1-3-1-5-9-3-1-7-5-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">{\mu}_{n,5}+2{\mu}_{n,4}+2{\mu}_{n,3}+2{\mu}_{n,2}+2$ |

… | … | … |

n | Minimal Rhomb Edge Sequence | Minimal Inflation Multiplier η_{min} |
---|---|---|

$4,5$ | $0-2-0-2-0$ | $2{\mu}_{n,2}+3$ |

$6,7$ | $0-2-4-0-2-0-2-0-4-2-0$ | $2{\mu}_{n,3}+4{\mu}_{n,2}+3$ |

$8,9$ | $0-2-4-6-0-2-4-0-2-0-2-0-4-2-0-6-4-2-0$ | $2{\mu}_{n,4}+4{\mu}_{n,3}+4{\mu}_{n,2}+3$ |

… | … | … |

n | Minimal Rhomb Edge Sequence | Minimal Inflation Multiplier η_{min} |
---|---|---|

$4,5$ | $1-3-1-1-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">2{\mu}_{n,2}+4$ |

$6,7$ | $1-3-5-1-3-1-1-3-1-5-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">2{\mu}_{n,3}+4{\mu}_{n,2}+4$ |

$8,9$ | $1-3-5-7-1-3-5-1-3-1-1-3-1-5-3-1-7-5-3-1$ | $\sqrt{{\mu}_{n,2}+2}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">2{\mu}_{n,4}+4{\mu}_{n,3}+4{\mu}_{n,2}+4$ |

… | … | … |

n | Inflation Multiplier | Corresponding Figure |
---|---|---|

7 | $2{\mu}_{7,3}+2{\mu}_{7,2}+1$ | Figure 18 |

7 | ${\mu}_{7,2}+2$ | Figure 19 |

4 | $\sqrt{{\mu}_{4,2}+2}$ | Figure 20 |

n | Inflation Multiplier | Corresponding Figure |
---|---|---|

4 | ${\mu}_{4,2}+1$ | Figure 21 |

5 | $\sqrt{{\mu}_{5,2}+2}({\mu}_{5,2}+1)$ | Figure 22 |

5 | $2({\mu}_{5,2}+1)$ | Figure 23 |

7 | ${\mu}_{7,2}+2{\mu}_{7,1}+2$ | Figure 24 |

n | Name | Inflation Multiplier | Patches with Individual Symmetry | Reference |
---|---|---|---|---|

5 | Penrose | ${\mu}_{5,2}$ | ${D}_{5}$ | [27,28,29,32] [2] (Chapter 10.3) [3] (Chapter 6.2) |

7 | Danzer’s 7-fold variant | ${\mu}_{7,2}$ | ${D}_{7}$ | [33] (Figure 1 and Section 3, 2nd matrix) Herein Figure 3 |

7 | Danzer’s 7-fold variant (two variants) | ${\mu}_{7,3}$ | ${D}_{1}$ | [32] (credited to L. Danzer) |

${D}_{7}$ | [33] (Figure 11) | |||

7 | Math Pages 7-fold | ${\mu}_{7,3}$ | - | [41] |

9 | Math Pages 9-fold | ${\mu}_{9,4}$ | ${D}_{9}$ | [41] |

5 | Lançon-Billard / Binary | $\sqrt{{\mu}_{5,2}+2}$ | - | [32,37,38] [3] (Chapter 6.5.1) |

6 | Shield | $\sqrt{{\mu}_{6,2}+2}$ | - | [32,59,60] [3] (Chapter 6.3.2) |

4 | Ammann-Beenker | ${\mu}_{4,2}+1$ | ${D}_{8}$ | [10,32,48,49] [2] (Chapter 10.4) [3] (Chapter 6.1) |

5 | Tie and Navette / Bowtie-Hexagon 1 | ${\mu}_{5,2}+1$ | - | [57,58] [22] (Section 8.2, Figure 8.3) [32] [56] (Figure 25) |

5 | Bowtie-Hexagon-Decagon 1 | ${\mu}_{5,2}+1$ | ${C}_{5}$ | [32] (credited to L. Andritz) |

7 | Danzer’s 7-fold (two variants) | ${\mu}_{7,2}+1$ | ${D}_{1}$ | [33] (Figure 12) [32] [3] (Chapter 6.5.2) |

7 | Franco-Ferreira-da-Silva 7-fold | ${\mu}_{7,2}+1$ | ${D}_{7}$ | [61] |

7 | Maloney’s 7-fold | ${\mu}_{7,2}+1$ | ${D}_{7}$ | [32] [62] (Figure 9) |

7 | Cyclotomic Trapezoids | ${\mu}_{n,2}+1$ | ${D}_{1}$ | [32,63] |

9 | ||||

11 | ||||

4 | Watanabe-Ito-Soma 8-fold | ${\mu}_{4,2}+2$ | ${D}_{8}$ | [32,64,65,66] |

4 | Generalized Goodman-Strauss rhomb | ${\mu}_{n,2}+2$ | ${D}_{1}$ | [39] [32] (credited to C. Goodman-Strauss and E. O. Harris) |

5 | ${C}_{5}$,${D}_{1}$ | |||

$\ge 6$ | ${D}_{1}$ | |||

6 | Watanabe-Soma-Ito 12-fold (variants) | ${\mu}_{6,2}+2$ | ${D}_{12}$ | [32,66] |

6 | Socolar | ${\mu}_{6,2}+2$ | ${D}_{2}$ | [10,32,67] |

6 | Stampfli-Gähler / Ship | ${\mu}_{6,2}+2$ | ${D}_{12}$ | [59,68,69] |

6 | Square Triangle | ${\mu}_{6,2}+2$ | ${D}_{6}$ | [70,71,72,73] [32] (credited to M. Schlottmann) [3] (Chapter 6.3.1) |

5 | Cromwell | ${\mu}_{5,2}+3$ | ${D}_{10}$ | [56] (Figures 12 and 13) |

5 | Topkapi Scroll | $2{\mu}_{5,2}+2$ | ${D}_{10}$ | Herein Figure 23, derived from [56] (Figures 14 and 15) and patterns shown in the Topkapi Scroll, in detail [53] (Panels 28, 31, 32, 34) |

5 | Bowtie-Hexagon-Decagon 2 | $2{\mu}_{5,2}+3$ | ${C}_{5}$ | [32] (credited to L. Andritz) |

5 | Bowtie-Hexagon-Decagon 3 | $3{\mu}_{5,2}+2$ | ${C}_{5}$ | [32] (credited to L. Andritz) |

5 | Darb-i Imam Shrine | $4{\mu}_{5,2}+2$ | ${D}_{10}$ | [54,55] [56] (Figure 21) |

7 | Franco’s 7-fold | ${\mu}_{7,3}+{\mu}_{7,2}+1$ | ${D}_{7}$ | [74] |

7 | Gähler-Kwan-Maloney 7-fold | ${\mu}_{7,3}+{\mu}_{7,2}+1$ | ${D}_{7}$ | [62] (Figure 10) |

7 | Socolar’s 7-fold | ${\mu}_{7,3}+2{\mu}_{7,2}+1$ | ${D}_{7}$ | [32] (credited to J. Socolar) |

9 | Franco-da-Silva-Inácio 9-fold | ${\mu}_{9,4}+{\mu}_{9,3}+{\mu}_{9,2}+1$ | ${D}_{9}$ | [75] |

11 | Maloney’s 11-fold | $3{\mu}_{11,5}+2{\mu}_{11,4}+2{\mu}_{11,3}+2{\mu}_{11,2}+1$ | ${D}_{11}$ | [8] |

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Pautze, S.
Cyclotomic Aperiodic Substitution Tilings. *Symmetry* **2017**, *9*, 19.
https://doi.org/10.3390/sym9020019

**AMA Style**

Pautze S.
Cyclotomic Aperiodic Substitution Tilings. *Symmetry*. 2017; 9(2):19.
https://doi.org/10.3390/sym9020019

**Chicago/Turabian Style**

Pautze, Stefan.
2017. "Cyclotomic Aperiodic Substitution Tilings" *Symmetry* 9, no. 2: 19.
https://doi.org/10.3390/sym9020019