**3. Quasicrystal dynamics**

The inherent nonlocal properties of quasicrystals allow us to study different dynamical models of self-interaction and interactions between different vertex configurations in quasicrystals, using the empires. Several game-of-life [21] algorithms have been previously studied on Penrose tiling, but they have either considered a periodic grid [26] or they have considered only local rules [27, 28]. Recently, for the first time, a game-of-life scenario has been simulated using nonlocal rules on a two-dimensional qusicrystal, the Penrose tiling, in [17]. In this simulation, for the

K vertex type, the emperor and its local patch are treated as a quasiparticle, a glider. The empire acts as a field and the interaction between two quasiparticles is modeled

*quasicrystal are all rhombohedrons, and the vertex configurations are analogous to those of the Penrose tiling. The empires are shown in three orientations below the vertex configurations. Other two vertex configurations*

*. The tiles of this*

*Vertex configurations and their empires for the Ammann tiling as projected from* <sup>6</sup> *to* <sup>3</sup>

as the interaction between empires.

*and their empires have been shown in [18] Figure 9.*

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

**Figure 3.**

**31**

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

#### **Figure 3.**

Furthermore, the method can be used also for non-cubic lattices, but it requires some adjustment, as the proper selection of the tiles becomes more complex. It has been shown [18] that the cut-window must be sub-divided into regions, which act as acceptance domains for individual tiles and can be used to compute both the relative frequencies of the vertex's configuration and the empire of a given tile configuration. The authors have used the method to compute the frequencies and

*Empire calculation for the vertex types S5 (left) and S (right). The red tiles represent the vertex patch and the green tiles represent the empire. The enlarged vertex patch is shown in the right corner for each case. The pictures*

*Empire calculation of the vertex types D (left), K (middle) and S4 (right). The red tiles represent the vertex patch and the green tiles represent the empire. The enlarged vertex patch is shown in the right corner for each*

*case. The pictures for all vertex types have been published in [16], Figures 13 and 14.*

configurations [23] and have compared their findings with previous results [23–25].

The inherent nonlocal properties of quasicrystals allow us to study different dynamical models of self-interaction and interactions between different vertex configurations in quasicrystals, using the empires. Several game-of-life [21] algorithms have been previously studied on Penrose tiling, but they have either considered a periodic grid [26] or they have considered only local rules [27, 28]. Recently, for the first time, a game-of-life scenario has been simulated using nonlocal rules on a two-dimensional qusicrystal, the Penrose tiling, in [17]. In this simulation, for the

, which has 36 vertex

sectors for an icosahedral projection of the *D*<sup>6</sup> lattice to <sup>3</sup>

*for all vertex types have been published in [16], Figures 13 and 14.*

**3. Quasicrystal dynamics**

**Figure 1.**

*Electron Crystallography*

**Figure 2.**

**30**

*Vertex configurations and their empires for the Ammann tiling as projected from* <sup>6</sup> *to* <sup>3</sup>*. The tiles of this quasicrystal are all rhombohedrons, and the vertex configurations are analogous to those of the Penrose tiling. The empires are shown in three orientations below the vertex configurations. Other two vertex configurations and their empires have been shown in [18] Figure 9.*

K vertex type, the emperor and its local patch are treated as a quasiparticle, a glider. The empire acts as a field and the interaction between two quasiparticles is modeled as the interaction between empires.

Several rules have been employed to describe both the self-interaction and the two-particle interactions. Firstly, the neighbors where the vertex patch is allowed to move are constrained by the higher dimensional projection, being the closest neighbors of the same vertex type in the perpendicular space. This approach differs from previous studies that consider the nearest neighbors situated in the local 2D representation [27, 28]. In **Figure 4**, one can see the K vertex type and 2D representation of the nearest neighbors in the perpendicular space that we have considered [17]. The distribution of the neighbors is interesting, as it surrounds an S vertex patch (sun) on one side and an S5 vertex patch (star) on the other side. We will expand on this configuration in the next section.

Besides the initial conditions, meaning the vertex type configuration and the initial position of the particles in the two-particle interaction case, the movement of the particles is influenced solely by their empires and their possibility space. When the empire changes, the possibility space changes as well, constraining the next move due to the new spatial configuration. The empire and the possibility space create a feedback loop of influence, which for an infinite quasicrystal propagates

One of the most interesting findings is that in the case of the two-particle interactions, the particles get locked in an oscillation type movement when they are in proximity. If we consider an analogy between the "least change" principle—the number of tiles that change between two steps is minimum—and a minimum energy principle, the system tends to reach a minimum energy state in oscillation. This is similar with the time crystal scenarios [29–32] where a system disturbed by a periodic signal reaches a quasistable state in which it oscillates at a period different from the period of the external kick. In this case, a quasiparticle will draw stability from its empire interaction with other quasiparticles' empires—a nonlocal induced

Quasicrystals projected from a higher dimensional lattice, <sup>5</sup> for example, show several properties dictated from the representation in the high dimension, like the symmetry and the vertex and empire distribution. As discussed previously, for the K vertex type, the nearest neighbors considered in the game-of-life scenario are also dictated by the higher dimensional lattice from which the tiling is projected, being

In **Figure 4**, we have shown the K vertex type with its eight neighbors that surround a sun and a star configuration. **Figure 5** shows the sun vertex patch surrounded by the five orientations of the K vertex type. This is a more complex structure that has five-fold symmetry. When choosing only the sun configurations that are bordered by the K-type vertices, we observe that these configurations come from two different regions in the perpendicular space. In **Figure 6**, we show the 2D distribution of the K-type vertices, plotted in two different colors, corresponding to the two distinct regions in the perpendicular space from which the vertices are projected. The vertices form interesting patterns on the Penrose tiling. **Figure 7**

*The sun configuration surrounded by the five different orientations of the K-type vertex patch.*

**4. Empires and higher dimensional representations**

the closest neighbors in the perpendicular space [17].

instantly at infinite distances.

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

stability.

**Figure 5.**

**33**

Secondly, the vertex never stays in the same position for two consecutive frames, being thus forced to move to one of the allowed neighbors in its immediate vicinity. Depending on the intrinsic configuration of the vertex patch, some vertices allow more freedom of movement than others. For example, a vertex with a fivefold symmetry will tend to perform a "circular" motion around its axis of symmetry, a rotation, while a vertex without the five-fold symmetry, like the K vertex, will have the possibility to propagate forward, the translational movement being a sequence of rotations around different centers.

Moreover, the particle moves following the "least change" rule, which states that the particle should move to the position (or one of the positions) where the result of that movement implies that the number of tiles changed in the empire is minimum. In other words, the particle will follow the path that requires the least number of changes in the tiles in the empire, while not being allowed to stay in the same position for two consecutive frames. When there is more than one choice that obeys the aforementioned rules, a random-hinge variable is introduced such that the particle will chose one of the favored positions. Due to the syntactical freedom provided by this choice, the path of a particle, unless constrained otherwise, is impossible to predict with 100% accuracy.

For the case of two-particle interactions, one more arbitrary constraint is introduced, where the local patches of the two particles are not allowed to overlap. A detailed discussion of the algorithm and the simulation setup can be found in [17].<sup>2</sup>

#### **Figure 4.**

*The K vertex type surrounded by the eight possible neighbors. The orange dots represent the position of the K vertices that surround the star and sun vertex patches.*

<sup>2</sup> Movies from the simulations can be watched on https://www.youtube.com/playlist?list=PL-kqKe jCypNT990P0h2CFhrRCpaH9e858.

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

Several rules have been employed to describe both the self-interaction and the two-particle interactions. Firstly, the neighbors where the vertex patch is allowed to move are constrained by the higher dimensional projection, being the closest neighbors of the same vertex type in the perpendicular space. This approach differs from previous studies that consider the nearest neighbors situated in the local 2D representation [27, 28]. In **Figure 4**, one can see the K vertex type and 2D representation of the nearest neighbors in the perpendicular space that we have considered [17]. The distribution of the neighbors is interesting, as it surrounds an S vertex patch (sun) on one side and an S5 vertex patch (star) on the other side. We

Secondly, the vertex never stays in the same position for two consecutive frames, being thus forced to move to one of the allowed neighbors in its immediate vicinity. Depending on the intrinsic configuration of the vertex patch, some vertices allow more freedom of movement than others. For example, a vertex with a fivefold symmetry will tend to perform a "circular" motion around its axis of symmetry, a rotation, while a vertex without the five-fold symmetry, like the K vertex, will have the possibility to propagate forward, the translational movement being a

Moreover, the particle moves following the "least change" rule, which states that the particle should move to the position (or one of the positions) where the result of that movement implies that the number of tiles changed in the empire is minimum. In other words, the particle will follow the path that requires the least number of changes in the tiles in the empire, while not being allowed to stay in the same position for two consecutive frames. When there is more than one choice that obeys the aforementioned rules, a random-hinge variable is introduced such that the particle will chose one of the favored positions. Due to the syntactical freedom provided by this choice, the path of a particle, unless constrained otherwise, is

For the case of two-particle interactions, one more arbitrary constraint is introduced, where the local patches of the two particles are not allowed to overlap. A detailed discussion of the algorithm and the simulation setup can be found in [17].<sup>2</sup>

<sup>2</sup> Movies from the simulations can be watched on https://www.youtube.com/playlist?list=PL-kqKe

*The K vertex type surrounded by the eight possible neighbors. The orange dots represent the position of the K*

will expand on this configuration in the next section.

*Electron Crystallography*

sequence of rotations around different centers.

impossible to predict with 100% accuracy.

jCypNT990P0h2CFhrRCpaH9e858.

*vertices that surround the star and sun vertex patches.*

**Figure 4.**

**32**

Besides the initial conditions, meaning the vertex type configuration and the initial position of the particles in the two-particle interaction case, the movement of the particles is influenced solely by their empires and their possibility space. When the empire changes, the possibility space changes as well, constraining the next move due to the new spatial configuration. The empire and the possibility space create a feedback loop of influence, which for an infinite quasicrystal propagates instantly at infinite distances.

One of the most interesting findings is that in the case of the two-particle interactions, the particles get locked in an oscillation type movement when they are in proximity. If we consider an analogy between the "least change" principle—the number of tiles that change between two steps is minimum—and a minimum energy principle, the system tends to reach a minimum energy state in oscillation. This is similar with the time crystal scenarios [29–32] where a system disturbed by a periodic signal reaches a quasistable state in which it oscillates at a period different from the period of the external kick. In this case, a quasiparticle will draw stability from its empire interaction with other quasiparticles' empires—a nonlocal induced stability.

#### **4. Empires and higher dimensional representations**

Quasicrystals projected from a higher dimensional lattice, <sup>5</sup> for example, show several properties dictated from the representation in the high dimension, like the symmetry and the vertex and empire distribution. As discussed previously, for the K vertex type, the nearest neighbors considered in the game-of-life scenario are also dictated by the higher dimensional lattice from which the tiling is projected, being the closest neighbors in the perpendicular space [17].

In **Figure 4**, we have shown the K vertex type with its eight neighbors that surround a sun and a star configuration. **Figure 5** shows the sun vertex patch surrounded by the five orientations of the K vertex type. This is a more complex structure that has five-fold symmetry. When choosing only the sun configurations that are bordered by the K-type vertices, we observe that these configurations come from two different regions in the perpendicular space. In **Figure 6**, we show the 2D distribution of the K-type vertices, plotted in two different colors, corresponding to the two distinct regions in the perpendicular space from which the vertices are projected. The vertices form interesting patterns on the Penrose tiling. **Figure 7**

shows a zoom-in region where only the sun configurations surrounded by K-type

surround sun configurations. When analyzing just the K-sun configurations projected from the same region in the perpendicular space we are looking at the empire distributions, considering the empires from the K vertices. We consider one sun configuration to be completed, when all the K vertices surrounding it have their empires turned on. One of the interesting findings is that regardless of the number of completed suns—K vertex empires turned on—no other K vertex type surrounding a sun is covered by these empires. The K vertex patch tiles closest to the center—the sun—remain uncovered by the empires from the other suns. This is valid also for the case in which the sun configurations are projected from the other region in the perpendicular space. Provided these configurations are from the same region in the perpendicular space, there are some tiles in the K vertices that are not

covered by empires coming from different suns—a "selective" nonlocality constrained by the higher dimensional representation. In **Figure 8**, we show the K-sun configurations side-by-side from both distinct regions in the perpendicular

In this chapter we have reviewed several properties of quasicrystals, their nonlocal empires, and the methods used to generate the quasicrystal configurations and the empires of their vertices. We have studied quasicrystals projected from higher dimensions, <sup>5</sup> to 2D (the Penrose tiling), <sup>6</sup> to <sup>3</sup> and <sup>6</sup> to <sup>3</sup> for the 3D case. For several vertex configurations, we have analyzed their empires—the nonlocal distribution of their forced tiles—in relation to the higher dimension representation. These nonlocal properties allow us to study the quasicrystal dynamics in a novel way, a nonlocal game-of-life approach, in which the empires and the possibility space dictate the movement and trajectory of the chosen quasiparticle configurations. Case studies of two-particle interactions based on nonlocal rules, while not exhaustive, are showing important similarities with other experimental physics discoveries, like time crystals. The research into the inherent nonlocality of quasicrystals proves very rich in describing the various quasicrystal configurations and their correlation with high dimensional representations. These studies open up a new, but very promising avenue of research that can bridge together different fields, like physics in high dimensions, gauge and group theory,

We acknowledge the many discussions had with Richard Clawson about this

project and we thank him for his useful comments and suggestions.

We have performed several studies of the empires of the K-type vertices that

vertices are plotted also in different colors.

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

phason dynamics and advanced material science.

**Acknowledgements**

**35**

space.

**5. Conclusions**

#### **Figure 6.**

*The K-type vertex distribution plotted in two different colors, one for each of the distinct regions in the perpendicular space from where these configurations are projected.*

#### **Figure 7.**

*Sun configurations surrounded by the five orientations of the K-type vertex plotted in two different colors corresponding to the two distinct regions in the perpendicular space from where these configurations are projected.*

#### **Figure 8.**

*Sun configurations surrounded by K vertex types from different regions in the perpendicular space displayed side-by-side. The black tiles represent the vertices with their empires turned on. The empires are colored in green.*

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

shows a zoom-in region where only the sun configurations surrounded by K-type vertices are plotted also in different colors.

We have performed several studies of the empires of the K-type vertices that surround sun configurations. When analyzing just the K-sun configurations projected from the same region in the perpendicular space we are looking at the empire distributions, considering the empires from the K vertices. We consider one sun configuration to be completed, when all the K vertices surrounding it have their empires turned on. One of the interesting findings is that regardless of the number of completed suns—K vertex empires turned on—no other K vertex type surrounding a sun is covered by these empires. The K vertex patch tiles closest to the center—the sun—remain uncovered by the empires from the other suns. This is valid also for the case in which the sun configurations are projected from the other region in the perpendicular space. Provided these configurations are from the same region in the perpendicular space, there are some tiles in the K vertices that are not covered by empires coming from different suns—a "selective" nonlocality constrained by the higher dimensional representation. In **Figure 8**, we show the K-sun configurations side-by-side from both distinct regions in the perpendicular space.

### **5. Conclusions**

**Figure 6.**

*Electron Crystallography*

**Figure 7.**

*projected.*

**Figure 8.**

**34**

*The K-type vertex distribution plotted in two different colors, one for each of the distinct regions in the*

*Sun configurations surrounded by the five orientations of the K-type vertex plotted in two different colors corresponding to the two distinct regions in the perpendicular space from where these configurations are*

*Sun configurations surrounded by K vertex types from different regions in the perpendicular space displayed side-by-side. The black tiles represent the vertices with their empires turned on. The empires are colored in green.*

*perpendicular space from where these configurations are projected.*

In this chapter we have reviewed several properties of quasicrystals, their nonlocal empires, and the methods used to generate the quasicrystal configurations and the empires of their vertices. We have studied quasicrystals projected from higher dimensions, <sup>5</sup> to 2D (the Penrose tiling), <sup>6</sup> to <sup>3</sup> and <sup>6</sup> to <sup>3</sup> for the 3D case. For several vertex configurations, we have analyzed their empires—the nonlocal distribution of their forced tiles—in relation to the higher dimension representation. These nonlocal properties allow us to study the quasicrystal dynamics in a novel way, a nonlocal game-of-life approach, in which the empires and the possibility space dictate the movement and trajectory of the chosen quasiparticle configurations. Case studies of two-particle interactions based on nonlocal rules, while not exhaustive, are showing important similarities with other experimental physics discoveries, like time crystals. The research into the inherent nonlocality of quasicrystals proves very rich in describing the various quasicrystal configurations and their correlation with high dimensional representations. These studies open up a new, but very promising avenue of research that can bridge together different fields, like physics in high dimensions, gauge and group theory, phason dynamics and advanced material science.

## **Acknowledgements**

We acknowledge the many discussions had with Richard Clawson about this project and we thank him for his useful comments and suggestions.

*Electron Crystallography*
