**1. Introduction**

Quasicrystals are objects with aperiodic order and no translational symmetry. These "peculiar" objects, deemed in and out of existence by theoretical considerations, have been discovered in 1982 by Shechtman [1], in agreement with previous predictions [2, 3]. Shechtman's discovery was honored with the award of the Nobel Prize in Chemistry in 2011. Believed to be rare in nature initially, up to this date there have been found roughly one hundred quasicrystal phases that exhibit diffraction patterns showing quasiperiodic structures in metallic systems [4, 5]. Quasicrystals were first constructed as aperiodic tilings defined by an initial set of prototiles and their matching rules; constructing these tilings meant aggregating tiles onto an initial patch so as to fill space, ideally without gaps or defects. Likewise, quasicrystals in physical materials are formed by atoms accumulating to one another according to the geometry of their chemical bonding. Curiously, the localized growth patterns would give rise to structures which exhibit long-range and nonlocal order, and mathematical constructions were later discovered for creating geometrically perfect, infinite quasiperiodic tilings of space. In material science, new electron crystallography methods and techniques have been developed to study the structure and geometrical patterns of quasicrystal approximants [6–8], revealing unique atom configurations of complicated quasicrystal approximant structures [7, 8].

Complex and varied in their structure, quasicrystals translate their intrinsic nonlocal properties into nonlocal dynamic patterns [9]. The empire problem is an investigation into the nonlocal patterns that are imposed within a quasicrystal by a finite patch, where just a few tiles can have a global influence in the tiling so as to force an infinite arrangement of other tiles throughout the quasicrystal [10]. Initial research into calculating empires—a term originally coined by Conway [11] focused on the various manifestations of the Penrose tiling [12] such as the decorated kites-and-darts, where Ammann bars would indicate the forced tiles [10], and the multi-grid method, where algebraic constraints can be employed [13]. More recently, the cut-and-project technique [14, 15]—where the geometry of convex polytopes comes into play—has been implemented into a most efficient method of computing empires [16]. The cut-and-project method offers the most generality and has been used to calculate empires for the Penrose tiling and other quasicrystals that are projections of cubic lattices *<sup>n</sup>* [16, 17]. Quasicrystals that are projections of non-cubic lattices (e.g., the Elser-Sloane tiling as a projection of the *E*<sup>8</sup> lattice to 4) are more restrictive and some modifications to the cut-and-project method have been made in order to correctly compute the tilings and their empires [18]. The cutand-project formalism can also be altered to calculate the space of all tilings that are allowed by a given patch, wherein the set of forced tiles of the patch's empire is precisely the mutual intersection of all tilings which contain that patch [15].

constructed with the Amman bars is, in fact, a Fibonacci grid [20]. When Amman lines intersect in a certain configuration, a tile or a set of tiles are forced. The multigrid method [13] describes a tiling that can be constructed using a dual of a pentagrid, a superposition of five distinct families of hyperplanes in the case of Penrose tiling. It can be used for generating empires in cases where the dual grid for the quasicrystal has a simple representation, but it is not effective when the quasi-

The most efficient method of generating empires in quasicrystals is the cut-andproject method. While the cut-and-project method and the multigrid one are mathematically equivalent in their use of generating empires, the cut-and-project method provides us with the possibility of recovering the initial mother lattice, even for defected quasicrystals [16]. The method has also been applied to projections of

When we project from a lattice Λ⊂*<sup>N</sup> π*∥, *π*<sup>⊥</sup> onto orthogonal subspaces ∥, ⊥,

For cubic lattices (<sup>Λ</sup> <sup>¼</sup> *<sup>N</sup>*) the cut-window is sufficient for determining the tiles which fill the tiling space ⊥, but it needs to be sub-divided into regions acting as acceptance domains for individual tiles, when projecting non-cubic lattices. To review some of the results of applying the cut-and-project method to the

calculation of empires, we will give several examples in 2D and 3D cases.

The 2D Penrose tiling, when projected from the cubic lattice <sup>5</sup>

For the 2D case, we will consider the Penrose tiling, a non-periodic tiling, a quasicrystal configuration that can be generated using an aperiodic set of prototiles.

types: D, J, K, Q, S, S3, S4 and S5 [21, 22]. In **Figure 1**, we show the empires for three of the vertex types that do not present five-fold symmetry (D, K, S4) computed with the cut-and-project method along with their possibility space [16]. The S4 vertex has the densest empire, while the D vertex has no forced tiles. The density of the empire depends on the size of the empire window, individual for each vetex type. In **Figure 2**, we show the empires and the possibility space of the five-fold symmetry vertex types, S5 (the star) and S (the sun) [16]. The empires of all 8

The cut-and-project method described above can be used also for computing the empires of a given patch in 3D, e.g., an Amman tiling defined by a projection of <sup>6</sup> to <sup>3</sup> [18]. In **Figure 3**, we show three orientations of the empires of two of the vertex types for this projection and we can see they differ in structure, as well as in density. Two other vertex types together with their empires are shown in [18].

<sup>1</sup> For a detailed description of this method and a comparison with the other methods, please refer to [16].

, has eight vertex

the *cut-window*, which is a convex volume W in the perpendicular space ⊥, determines the points selected to have their projections included in the tiling of the subspace ∥, acting thus as an acceptance domain for the tiling. Once the tiling is generated, all the tiles in a given local patch can be traced back to the mother lattice, giving a restriction on the cut-window. The *possibility-space-window* represents the union of all cut-windows that satisfy the restriction—all the tiles in the possibilityspace-window *can* legally coexist with the chosen patch. The *empire-window* represents the intersection of all the possible cut-windows—all the tiles inside the empire-window *must* coexist with the initial patch. This in turn acts as the cut-

crystal has a defect, as its dual is no longer a perfect multigrid.

non-cubic lattices, making it of general use [18].

*Empires: The Nonlocal Properties of Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.90237*

window for the forced tiles, the patch's empire.<sup>1</sup>

**2.2 Empires in 2D**

**2.3 Empires in 3D**

**29**

vertex types are displayed in [16].

The interest in the nonlocal nature of empires has led to further explorations into how multiple empires can interact within a given quasicrystal, where separate patches can impose geometric restrictions on each other no matter how far apart they may be located within the tiling. These interactions can be used to define rules (similar to cellular automata) to see what dynamics emerge from a game-of-life style evolution of the quasicrystal and for the first time such a simulation with nonlocal rules has been performed [17].

The empires can be used to recover information from the higher dimensional lattice from which the quasicrystal was projected, filtering out any defects in the quasicrystal and therefore providing an error self-correction tool for quasicrystal growth [16]. In terms of quasicrystal dynamics, the empires provide us with the opportunity of developing algorithms to study the behavior and interactions of quasicrystalline patches based on nonlocal rules—a very rich area of exploration.

In this chapter we review the nonlocal properties of quasicrystals and the studies done to generate and analyze the empires and we discuss some of the findings and their possible implications.

### **2. Empires in quasicrystals**

Empires represent thus all the tiles forced into existence at all distances by a quasicrystal patch. When it comes to analyzing the forced tile distribution in a quasicrystal, we differentiate between the local and nonlocal configurations. The tiles surrounding the vertex, that is the tiles that share one vertex, form the vertex patch. The local empire is the union of forced tiles that are in the immediate vicinity of an emperor, be it a vertex or a patch, where there are no "free" tiles in between the emperor and the forced tiles. The forced tiles that are at a distance from the emperor form the nonlocal part of the empire.

#### **2.1 Methods for generating empires**

Several methods for generating the empires in quasicrystals have been discussed in [16]. The Fibonacci-Grid method employs the Penrose tiling decoration using Amman bars that form a grid of five sets of parallel lines [10, 19]. The grid

#### *Empires: The Nonlocal Properties of Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.90237*

Complex and varied in their structure, quasicrystals translate their intrinsic nonlocal properties into nonlocal dynamic patterns [9]. The empire problem is an investigation into the nonlocal patterns that are imposed within a quasicrystal by a finite patch, where just a few tiles can have a global influence in the tiling so as to force an infinite arrangement of other tiles throughout the quasicrystal [10]. Initial research into calculating empires—a term originally coined by Conway [11] focused on the various manifestations of the Penrose tiling [12] such as the decorated kites-and-darts, where Ammann bars would indicate the forced tiles [10], and the multi-grid method, where algebraic constraints can be employed [13]. More recently, the cut-and-project technique [14, 15]—where the geometry of convex polytopes comes into play—has been implemented into a most efficient method of computing empires [16]. The cut-and-project method offers the most generality and has been used to calculate empires for the Penrose tiling and other quasicrystals that are projections of cubic lattices *<sup>n</sup>* [16, 17]. Quasicrystals that are projections of non-cubic lattices (e.g., the Elser-Sloane tiling as a projection of the *E*<sup>8</sup> lattice to 4) are more restrictive and some modifications to the cut-and-project method have been made in order to correctly compute the tilings and their empires [18]. The cutand-project formalism can also be altered to calculate the space of all tilings that are allowed by a given patch, wherein the set of forced tiles of the patch's empire is precisely the mutual intersection of all tilings which contain that patch [15].

The interest in the nonlocal nature of empires has led to further explorations into

The empires can be used to recover information from the higher dimensional lattice from which the quasicrystal was projected, filtering out any defects in the quasicrystal and therefore providing an error self-correction tool for quasicrystal growth [16]. In terms of quasicrystal dynamics, the empires provide us with the opportunity of developing algorithms to study the behavior and interactions of quasicrystalline patches based on nonlocal rules—a very rich area of exploration. In this chapter we review the nonlocal properties of quasicrystals and the studies done to generate and analyze the empires and we discuss some of the findings and

Empires represent thus all the tiles forced into existence at all distances by a quasicrystal patch. When it comes to analyzing the forced tile distribution in a quasicrystal, we differentiate between the local and nonlocal configurations. The tiles surrounding the vertex, that is the tiles that share one vertex, form the vertex patch. The local empire is the union of forced tiles that are in the immediate vicinity of an emperor, be it a vertex or a patch, where there are no "free" tiles in between the emperor and the forced tiles. The forced tiles that are at a distance from the

Several methods for generating the empires in quasicrystals have been discussed in [16]. The Fibonacci-Grid method employs the Penrose tiling decoration using Amman bars that form a grid of five sets of parallel lines [10, 19]. The grid

how multiple empires can interact within a given quasicrystal, where separate patches can impose geometric restrictions on each other no matter how far apart they may be located within the tiling. These interactions can be used to define rules (similar to cellular automata) to see what dynamics emerge from a game-of-life style evolution of the quasicrystal and for the first time such a simulation with

nonlocal rules has been performed [17].

their possible implications.

*Electron Crystallography*

**2. Empires in quasicrystals**

emperor form the nonlocal part of the empire.

**2.1 Methods for generating empires**

**28**

constructed with the Amman bars is, in fact, a Fibonacci grid [20]. When Amman lines intersect in a certain configuration, a tile or a set of tiles are forced. The multigrid method [13] describes a tiling that can be constructed using a dual of a pentagrid, a superposition of five distinct families of hyperplanes in the case of Penrose tiling. It can be used for generating empires in cases where the dual grid for the quasicrystal has a simple representation, but it is not effective when the quasicrystal has a defect, as its dual is no longer a perfect multigrid.

The most efficient method of generating empires in quasicrystals is the cut-andproject method. While the cut-and-project method and the multigrid one are mathematically equivalent in their use of generating empires, the cut-and-project method provides us with the possibility of recovering the initial mother lattice, even for defected quasicrystals [16]. The method has also been applied to projections of non-cubic lattices, making it of general use [18].

When we project from a lattice Λ⊂*<sup>N</sup> π*∥, *π*<sup>⊥</sup> onto orthogonal subspaces ∥, ⊥, the *cut-window*, which is a convex volume W in the perpendicular space ⊥, determines the points selected to have their projections included in the tiling of the subspace ∥, acting thus as an acceptance domain for the tiling. Once the tiling is generated, all the tiles in a given local patch can be traced back to the mother lattice, giving a restriction on the cut-window. The *possibility-space-window* represents the union of all cut-windows that satisfy the restriction—all the tiles in the possibilityspace-window *can* legally coexist with the chosen patch. The *empire-window* represents the intersection of all the possible cut-windows—all the tiles inside the empire-window *must* coexist with the initial patch. This in turn acts as the cutwindow for the forced tiles, the patch's empire.<sup>1</sup>

For cubic lattices (<sup>Λ</sup> <sup>¼</sup> *<sup>N</sup>*) the cut-window is sufficient for determining the tiles which fill the tiling space ⊥, but it needs to be sub-divided into regions acting as acceptance domains for individual tiles, when projecting non-cubic lattices.

To review some of the results of applying the cut-and-project method to the calculation of empires, we will give several examples in 2D and 3D cases.
