# Methods for Calculating Empires in Quasicrystals

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## Abstract

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## 1. Introduction—What Is the Empire Problem?

## 2. The Fibonacci-Grid Method

## 3. The Multigrid Method

## 4. The Cut-and-Project Method

## 5. Comparison between the Multigrid Method and the Cut-and-Project Method

#### 5.1. The Parallel between the Multigrid Method and the Cut-and-Project Method

#### 5.2. Empires in a Defected Quasicrystal

## 6. Summary and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Nagao, K.; Inuzuka, T.; Nishimoto, K.; Edagawa, K. Experimental observation of quasicrystal growth. Phys. Rev. Lett.
**2015**, 115, 075501. [Google Scholar] [CrossRef] [PubMed] - Grunbaum, B.; Shephard, G.C. Tilings and Patterns; W.H. Freeman and Company: New York, NY, USA, 1987. [Google Scholar]
- Conway, J.H. Triangle tessellations of the plane. Am. Math. Mon.
**1965**, 72, 915. [Google Scholar] - Effinger-Dean, L. The Empire Problem in Penrose Tilings. Bachelor Thesis, Williams College, Williamstown, MA, USA, 2006. [Google Scholar]
- Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl.
**1974**, 10, 266–271. [Google Scholar] - Socolar, J.E.S.; Steinhardt, P.J. Quasicrystals. II. Unit-cell configurations. Phys. Rev. B
**1986**, 34, 617–647. [Google Scholar] [CrossRef] - Senechal, M.J. Quasicrystals and Geometry; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Minnick, L. Generalized Forcing in Aperiodic Tilings. Bachelor Thesis, Williams College, Williamstown, MA, USA, 1998. [Google Scholar]
- Fang, F.; Irwin, K. An icosahedral quasicrystal and E8 derived quasicrystals. arXiv, 2015; arXiv:1511.07786. [Google Scholar]
- Gahler, F.; Rhyner, J. Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings. J. Phys. A Math. Gen.
**1986**, 19, 267–277. [Google Scholar] [CrossRef] - De Bruijn, N.G. Algebraic theory of Penrose’s non-periodic tiling. Proc. Kon. Ned. Akad. Wetensch.
**1981**, 84, 39–66. [Google Scholar] [CrossRef] - Henley, C.L. Sphere packing and local environments in Penrose tilings. Phys. Rev. B
**1986**, 34, 797–816. [Google Scholar] [CrossRef] - Sen, A.; Aschheim, R.; Fang, F.; Irwin, K. Emergence of aperiodic order in a random tiling network using Monte Carlo simulations. Manuscript in preparation.

**Figure 3.**An example of the resulting empire of the star vertex configuration. The blue bars are the first set of the forced bars and the red ones are the second set. The center patch is the given vertex configuration and the tiles surrounding it are the forced tiles that make up the empire.

**Figure 4.**An example of a pentagrid where ${\u03f5}_{0}$, ${\u03f5}_{1}$,…, ${\u03f5}_{4}$ are the grid vectors.

**Figure 5.**The grid as a set of parallel planes ${G}^{k}$ indexed by $k\in \mathbb{Z}$. $\overrightarrow{\u03f5}$ is the norm vector and $\gamma $ is the shift of the grid.

**Figure 6.**The dualizing procedure for generating a Penrose tiling from a pentagrid. To construct a Penrose tiling, first we identify the intersections in a sample patch in the pentagrid (

**a**) and then we construct a dual quasicrystal cell, here the prolate and oblate rhombuses, at each intersect point, placing them edge to edge while maintaining their topological connectedness (

**b**).

**Figure 7.**Calculating the coordinates of the vertices in the Penrose tiling using the indices of the corresponding mesh regions. (

**a**) open strips ${S}_{i}$ in grid ${G}_{i}$, where ${\u03f5}_{i}$ is the norm of the ${G}_{i}$ and ${\gamma}_{i}$ is the shift vector; (

**b**) a mesh region $M\left(x\right)$ containing a point x, which is given by the intersection of strips from all five grids; (

**c**) an intersection point in the grid and its corresponding tile in the tiling.

**Figure 8.**Results of the empire calculation for two different local vertex types in Penrose tiling, using the multigrid method. The green tiles are the forced tiles (empire) by the local vertex configuration (red), while the blue tiles are not forced.

**Figure 9.**Results of the empire calculation for the rest of vertex types in Penrose tiling, using the multigrid method. For each vertex type, an enlarged center vertex configuration is shown at the lower right corner of its empire.

**Figure 10.**A schematic diagram showing two ways of interpreting the cut-and-project method for generating a quasicrystal from a higher dimensional lattice. (

**a**) shows that the points are selected for projection as long as there is non-trivial intersection between their Voronoi cell and the quasicrystal space. The black points are the lattice points, and the hexagons are their Voronoi cells, E is the quasicrystal space, ${E}^{\perp}$ is the orthogonal space, and the solid blue and green segments are the projected tiles in the quasicrystal space; (

**b**) shows that the points are selected for projection as long as their projection on the orthogonal space falls inside of $\mathcal{W}$.

**Figure 12.**Illustration of the shifting of the cut-window in the direction of one of its norm vectors.

**Figure 13.**Results of the empire calculation for two different local vertex types in Penrose tiling, using the cut-and-project method; (

**a**) shows the empire of the vertex type S5 and (

**b**) shows the empire of the vertex type K. The green tiles are the forced tiles (empire) by the local vertex configuration (red), while the gray superimposed tiles are not forced. Notice that the gray tiles are projection of faces of full cubes from Z5 and they form networks of these projected cubic strings.

**Figure 14.**Results of the empire calculation of the rest of the vertex types in Penrose tiling, using the cut-and-project method. For each vertex type, an enlarged center vertex configuration is shown at the lower right corner of its empire.

**Figure 15.**(

**a**) Illustration of the cut-and-project method for obtaining a one dimensional quasicrystal, the Fibonacci word, from the ${Z}^{2}$ lattice; (

**b**) illustration of the cut-and-project method for obtaining the empires in the Fibonacci word; (

**c**) illustration of the reversing cut-and-project process, from a defected Fibonacci word to corresponding points in the two dimensional ${Z}^{2}$ lattice; (

**d**) illustration of the cut-and-project method for obtaining the empires in the defected Fibonacci word.

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**MDPI and ACS Style**

Fang, F.; Hammock, D.; Irwin, K.
Methods for Calculating Empires in Quasicrystals. *Crystals* **2017**, *7*, 304.
https://doi.org/10.3390/cryst7100304

**AMA Style**

Fang F, Hammock D, Irwin K.
Methods for Calculating Empires in Quasicrystals. *Crystals*. 2017; 7(10):304.
https://doi.org/10.3390/cryst7100304

**Chicago/Turabian Style**

Fang, Fang, Dugan Hammock, and Klee Irwin.
2017. "Methods for Calculating Empires in Quasicrystals" *Crystals* 7, no. 10: 304.
https://doi.org/10.3390/cryst7100304