# Quasicrystal Tilings in Three Dimensions and Their Empires

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Projection Method

#### 2.1. Overview of Projection Method

#### 2.2. Lattice Points and the Cut-Window

#### 2.3. Tiles and Regions of the Cut-Window

#### 2.4. Vertex Configurations and Sectors of $\mathcal{W}$

## 3. Frequencies of Vertex Configurations

## 4. Empires

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Step-by-step illustration of the cut-and-project process for the triangular lattice $\mathsf{\Lambda}={A}_{2}$ (

**a**) projecting onto a 1-dimensional tiling space ${\mathbb{E}}_{\parallel}$ (

**b**). The Voronoi cells for the lattice are all regular hexagons (

**c**) and it is only those vertices with Voronoi cells that intersect ${\mathbb{E}}_{\parallel}$ which are selected (

**d**). These vertices are then projected onto ${\mathbb{E}}_{\parallel}$ and define the vertices of the tiling of ${\mathbb{E}}_{\parallel}$ (

**e**). The lattice edges (dashed lines) connecting the selected lattice points are projected to ${\mathbb{E}}_{\parallel}$ to form the (1-dimensional) tiles which fill ${\mathbb{E}}_{\parallel}$ (

**f**).

**Figure 2.**A cut and projection ${A}_{2}\to {\mathbb{E}}^{1}$, where ${\mathbb{E}}_{\parallel}$ has a shallow enough slope that overlapping edges can arise when one is deciding which edges to select as tiles. (

**a**) The space ${\mathbb{E}}_{\parallel}$ intersects the Voronoi cells of lattices points ${\lambda}_{i}$ for $i=\{1,2,3,4,6\}$. It is not clear, however, which edges should be selected as tiles in ${\mathbb{E}}_{\parallel}$. For instance (

**b**) the edge from ${\lambda}_{1}^{\parallel}$ to ${\lambda}_{3}^{\parallel}$ clashes with the vertex ${\lambda}_{2}^{\parallel}$ and similarly (

**c**) the edge connecting ${\lambda}_{2}^{\parallel}$ to ${\lambda}_{4}^{\parallel}$ conflicts with the vertex ${\lambda}_{3}^{\parallel}$. The solution lies with the boundary facets of the Voronoi cells. Each lattice edge $\{{\lambda}_{a},{\lambda}_{b}\}$ is dual to the facet that forms the common boundary between $V({\lambda}_{a})$ and $V({\lambda}_{b})$. It is only when ${\mathbb{E}}_{\parallel}$ intersects with the boundary facet $V({\lambda}_{a})\cap V({\lambda}_{b})$ that the edge $\{{\lambda}_{a}^{\parallel},{\lambda}_{b}^{\parallel}\}$ is selected for the tiling. The boundary facets which intersect ${\mathbb{E}}_{\parallel}$ (

**d**) and the edges to which they are dual (

**e**) are the correct choice of edges which project to ${\mathbb{E}}_{\parallel}$ without overlapping (

**f**).

**Figure 3.**A tiling ${\mathcal{T}}^{({A}_{2})}$ defined as a projection ${A}_{2}\to {\mathbb{E}}^{1}$ of the triangular lattice to the line. The cut-window $\mathcal{W}={V}_{\perp}$ is an interval in ${\mathbb{E}}_{\perp}$ (thick red line) and dictates which lattice points $\lambda $ are selected for the tiling $\mathcal{T}$: if a lattice points falls within the cut-window ${\lambda}_{\perp}\in \mathcal{W}$ then it is included in the tiling $\lambda \in \mathcal{T}$.

**Figure 4.**(

**a**–

**c**) Three of the six tile types $\tilde{\mathbf{t}}$ (colored arrows) are shown for the ${\mathcal{T}}^{({A}_{2})}$ along with their corresponding regions $\mathfrak{r}=-{\tilde{\mathbf{t}}}_{\perp}^{\ast}\subset \mathcal{W}$ (thick red lines). Any lattice point $\lambda $ which falls into a region, ${\lambda}_{\perp}\in \mathfrak{r}$, will be adjacent to a tile $\mathbf{t}$ which is a translated copy $\mathbf{t}=(\tilde{\mathbf{t}}+\lambda )$ of the tile type $\tilde{\mathbf{t}}$ corresponding to $\mathfrak{r}$. (

**d**) The selected tiles are shown along with the whole cut-window.

**Figure 5.**Various tilings ${\mathcal{T}}^{({D}_{6})}$ generated as projections ${D}_{6}\to {\mathbb{E}}^{3}$ with randomly chosen shift vectors.

**Figure 6.**Tiles and corresponding cut-window regions for the tiling ${\mathcal{T}}^{({D}_{6})}$. Four tiles are shown (left) alongside their corresponding regions (right) in the cut-window. The tiles ${\mathcal{T}}^{({D}_{6})}$ are all tetrahedrons while their corresponding regions are pyramids with a rhombic base (

**a**–

**c**) or parallelepipeds (

**d**).

**Figure 7.**A vertex configuration $\tilde{\mathfrak{C}}$ (the combination of blue and green tile types) and its corresponding sector $\mathfrak{s}\subset {\mathbb{E}}_{\perp}$ (red) for the ${\mathcal{T}}^{({A}_{2})}$ tiling. Two tile types $\tilde{\mathbf{t}}$ (blue arrow) and ${\tilde{\mathbf{t}}}^{\prime}$ (green arrow) together make up the vertex configuration $\tilde{\mathfrak{C}}=\{\tilde{\mathbf{t}},{\tilde{\mathbf{t}}}^{\prime}\}$. Their corresponding regions in the cut-window, $\mathfrak{r}=-{\mathbf{t}}_{\perp}^{\ast}$ (thick blue) and ${\mathfrak{r}}^{\prime}=-{{\mathbf{t}}^{\prime}}_{\perp}^{\ast}$ (thick green) overlap in $\mathcal{W}$ and their intersection $\mathfrak{s}=\mathfrak{r}\cap {\mathfrak{r}}^{\prime}$ is the sector which corresponds to the configuration defined by the two tile types. A lattice point $\lambda $ which falls into this sector, ${\lambda}_{\perp}\in \mathfrak{s}$, will have a translated copy of $\tilde{\mathfrak{C}}$ as its vertex configuration, $\mathfrak{C}(\lambda )=\tilde{\mathfrak{C}}+\lambda $.

**Figure 8.**Sectors corresponding to twelve of the 36 vertex types of the tiling ${\mathcal{T}}^{({D}_{6})}$ (projection ${D}_{6}\to {\mathbb{E}}^{3}$) are shown. Each vertex configuration comes in 120 orientations (respecting the ${H}_{3}$ symmetry present in ${\mathbb{E}}_{\parallel}$) and so each configuration corresponds to 120 individual sectors in $\mathcal{W}$. Each group of 120 sectors, in turn, exhibits the same ${H}_{3}$ symmetry in the cut-window $\mathcal{W}\subset {\mathbb{E}}_{\perp}$.

**Figure 9.**Vertex configurations and their empires for the Ammann tiling (projection ${\mathbb{Z}}^{6}\to {\mathbb{E}}^{3}$). The tiles of this quasicrystal are all rhombohedrons, and the vertex configurations are analogous to those of the Penrose tiling. The empires (shown in three orientations to the right the vertex configurations) vary in both structure and density.

**Table 1.**The 36 vertex configurations $\mathfrak{C}$ for the tiling ${\mathcal{T}}^{({D}_{6})}$ are shown along with their corresponding cut-window sectors $\mathfrak{s}\left(\mathfrak{C}\right)$, their empires $\mathcal{E}\left(\mathfrak{C}\right)$, and their frequencies $\mathcal{F}\left(\mathfrak{C}\right)$.

$\mathfrak{C}$ | $\mathfrak{S}(\mathfrak{C})$ | $\mathcal{E}(\mathfrak{C})$ | $\mathcal{F}(\mathfrak{C})$ | $\mathfrak{C}$ | $\mathfrak{S}(\mathfrak{C})$ | $\mathcal{E}(\mathfrak{C})$ | $\mathcal{F}(\mathfrak{C})$ |
---|---|---|---|---|---|---|---|

$-38+17\sqrt{5}$ | $-360+161\sqrt{5}$ | ||||||

$843-377\sqrt{5}$ | $-521+233\sqrt{5}$ | ||||||

$233\sqrt{5}-521$ | $161-72\sqrt{5}$ | ||||||

$322-144\sqrt{5}$ | $233\sqrt{5}-521$ | ||||||

$161-72\sqrt{5}$ | $843-377\sqrt{5}$ | ||||||

$161-72\sqrt{5}$ | $47-21\sqrt{5}$ | ||||||

$305\sqrt{5}-682$ | $161-72\sqrt{5}$ | ||||||

$17\sqrt{5}-38$ | $123-55\sqrt{5}$ | ||||||

$47-21\sqrt{5}$ | $17\sqrt{5}-38$ | ||||||

$17\sqrt{5}-38$ | $17\sqrt{5}-38$ | ||||||

$9-4\sqrt{5}$ | $17\sqrt{5}-38$ | ||||||

$89\sqrt{5}-199$ | $322-144\sqrt{5}$ | ||||||

$34\sqrt{5}-76$ | $233\sqrt{5}-521$ | ||||||

$161-72\sqrt{5}$ | $17\sqrt{5}-38$ | ||||||

$161-72\sqrt{5}$ | $161-72\sqrt{5}$ | ||||||

$18-8\sqrt{5}$ | $-2+\sqrt{5}$ | ||||||

$-38+17\sqrt{5}$ | $9-4\sqrt{5}$ | ||||||

$-11+5\sqrt{5}$ | $9-4\sqrt{5}$ |

**Table 2.**Comparison of data computed for the ${\mathcal{T}}^{({D}_{6})}$ by authors and by Kramer et at. [18]. Here, $\tau =(1+\sqrt{5})/2$ is the Golden Ratio and the ambiguity between types 16.1 and 23.1 is notated.

$\mathfrak{C}$ | $\mathfrak{S}\left(\mathfrak{C}\right)$ | $\mathcal{F}\left(\mathfrak{C}\right)$ | $\mathcal{F}\left(\mathfrak{C}\right)$ (Kramer et al.) | Type (Kramer et al.) |
---|---|---|---|---|

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $24.1$ | ||

$-360+161\sqrt{5}$ | $\frac{1+{\tau}^{2}}{{\tau}^{13}}$ | $14.1$ | ||

$843-377\sqrt{5}$ | $\frac{2}{{\tau}^{14}}$ | $13.1$ | ||

$-521+233\sqrt{5}$ | $\frac{2}{{\tau}^{13}}$ | $17.1$ | ||

$-521+233\sqrt{5}$ | $\frac{2}{{\tau}^{13}}$ | $12.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $20.1$ | ||

$322-144\sqrt{5}$ | $\frac{2}{{\tau}^{12}}$ | $18.1$ | ||

$-521+233\sqrt{5}$ | $\frac{2}{{\tau}^{13}}$ | $21.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $16.1$ (or $23.1$) | ||

$843-377\sqrt{5}$ | $\frac{2}{{\tau}^{14}}$ | $19.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $15.1$ | ||

$47-21\sqrt{5}$ | $\frac{2}{{\tau}^{8}}$ | $11.4$ | ||

$-682+305\sqrt{5}$ | $\frac{1}{{\tau}^{15}}$ | $22.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{15}}$ | $23.1$ (or $16.1$) | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $11.1$ | ||

$123-55\sqrt{5}$ | $\frac{2}{{\tau}^{10}}$ | $9.1$ | ||

$47-21\sqrt{5}$ | $\frac{2}{{\tau}^{8}}$ | $9.3$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $2.2$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $10.4$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $1.2$ | ||

$9-4\sqrt{5}$ | $\frac{1}{{\tau}^{6}}$ | $11.3$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $2.1$ | ||

$-199+89\sqrt{5}$ | $\frac{2}{{\tau}^{11}}$ | $9.2$ | ||

$322-144\sqrt{5}$ | $\frac{2}{{\tau}^{12}}$ | $5.1$ | ||

$-76+34\sqrt{5}$ | $\frac{2}{{\tau}^{9}}$ | $10.1$ | ||

$-521+233\sqrt{5}$ | $\frac{2}{{\tau}^{13}}$ | $3.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $1.1$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $11.2$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $4.1$ | ||

$161-72\sqrt{5}$ | $\frac{1}{{\tau}^{12}}$ | $10.2$ | ||

$18-8\sqrt{5}$ | $\frac{2}{{\tau}^{6}}$ | $11.5$ | ||

$-2+\sqrt{5}$ | $\frac{1}{{\tau}^{3}}$ | $6.1$ | ||

$-38+17\sqrt{5}$ | $\frac{1}{{\tau}^{9}}$ | $10.3$ | ||

$9-4\sqrt{5}$ | $\frac{1}{{\tau}^{6}}$ | $7.1$ | ||

$11-5\sqrt{5}$ | $\frac{2}{{\tau}^{5}}$ | $7.2$ | ||

$9-4\sqrt{5}$ | $\frac{1}{{\tau}^{6}}$ | $8.1$ |

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**MDPI and ACS Style**

Hammock, D.; Fang, F.; Irwin, K. Quasicrystal Tilings in Three Dimensions and Their Empires. *Crystals* **2018**, *8*, 370.
https://doi.org/10.3390/cryst8100370

**AMA Style**

Hammock D, Fang F, Irwin K. Quasicrystal Tilings in Three Dimensions and Their Empires. *Crystals*. 2018; 8(10):370.
https://doi.org/10.3390/cryst8100370

**Chicago/Turabian Style**

Hammock, Dugan, Fang Fang, and Klee Irwin. 2018. "Quasicrystal Tilings in Three Dimensions and Their Empires" *Crystals* 8, no. 10: 370.
https://doi.org/10.3390/cryst8100370