By Fang Fang, Sinziana Paduroiu, Dugan Hammock and Klee Irwin

In quasicrystals, any given local patch—called an emperor—forces at all distances the existence of accompanying tiles—called the empire—revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

### Tag Cloud

Empires: The Nonlocal Properties

of Quasicrystals

Fang Fang, Sinziana Paduroiu, Dugan Hammock

and Klee Irwin

Abstract

In quasicrystals, any given local patch—called an emperor—forces at all dis-

tances the existence of accompanying tiles—called the empire—revealing thus their

inherent nonlocality. In this chapter, we review and compare the methods currently

used for generating the empires, with a focus on the cut-and-project method, which

can be generalized to calculate empires for any quasicrystals that are projections of

cubic lattices. Projections of non-cubic lattices are more restrictive and some mod-

ifications to the cut-and-project method must be made in order to correctly com-

pute the tilings and their empires. Interactions between empires have been modeled

in a game-of-life approach governed by nonlocal rules and will be discussed in 2D

and 3D quasicrystals. These nonlocal properties and the consequent dynamical

evolution have many applications in quasicrystals research, and we will explore the

connections with current material science experimental research.

Keywords: quasicrystals, empire, nonlocality, Penrose tiling, cut-and-project

1. Introduction

Quasicrystals are objects with aperiodic order and no translational symmetry.

These “peculiar” objects, deemed in and out of existence by theoretical consider-

ations, 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 quasi-

periodic 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 compli-

cated quasicrystal approximant structures [7, 8].

1

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 deco-

rated 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 n [16, 17]. Quasicrystals that are projections of

non-cubic lattices (e.g., the Elser-Sloane tiling as a projection of the E8 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 cut-

and-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

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

2

Electron Crystallography

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-

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

The most efficient method of generating empires in quasicrystals is the cut-and-

project method. While the cut-and-project method and the multigrid one are math-

ematically 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⊂N π∥, π⊥ 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 possibility-

space-window can legally coexist with the chosen patch. The empire-window repre-

sents 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-

window for the forced tiles, the patch’s empire.1

For cubic lattices (Λ ¼ N) 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.

2.2 Empires in 2D

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.

The 2D Penrose tiling, when projected from the cubic lattice 5, has eight vertex

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) com-

puted 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

vertex types are displayed in [16].

2.3 Empires in 3D

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 6

to 3 [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].

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

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Empires: The Nonlocal Properties of Quasicrystals

DOI: http://dx.doi.org/10.5772/intechopen.90237

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

sectors for an icosahedral projection of the D6 lattice to 3, which has 36 vertex

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

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] algo-

rithms have been previously studied on Penrose tiling, but they have either consid-

ered 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.

Figure 1.

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.

Figure 2.

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

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

4

Electron Crystallography

The empire acts as a field and the interaction between two quasiparticles is modeled

as the interaction between empires.

Figure 3.

Vertex configurations and their empires for the Ammann tiling as projected from 6 to 3. 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.

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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 repre-

sentation of the nearest neighbors in the perpendicular space that we have consid-

ered [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.

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 five-

fold symmetry will tend to perform a “circular” motion around its axis of symme-

try, 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 intro-

duced, 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].2

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

jCypNT990P0h2CFhrRCpaH9e858.

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.

6

Electron Crystallography

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, 5 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

Figure 5.

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

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Empires: The Nonlocal Properties of Quasicrystals

DOI: http://dx.doi.org/10.5772/intechopen.90237

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.

8

Electron Crystallography

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 sur-

rounding 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

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, 5 to 2D (the Penrose tiling), 6 to 3 and 6 to 3 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 quasi-

particle configurations. Case studies of two-particle interactions based on nonlocal

rules, while not exhaustive, are showing important similarities with other experi-

mental 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.

9

Empires: The Nonlocal Properties of Quasicrystals

DOI: http://dx.doi.org/10.5772/intechopen.90237

Author details

Fang Fang*†, Sinziana Paduroiu†, Dugan Hammock and Klee Irwin

Quantum Gravity Research, Los Angeles, CA, USA

*Address all correspondence to: fang@quantumgravityresearch.org

†These authors contributed equally.

© 2019 TheAuthor(s). Licensee IntechOpen. This chapter is distributed under the terms

of theCreativeCommonsAttribution License (http://creativecommons.org/licenses/

by/3.0),which permits unrestricted use, distribution, and reproduction in anymedium,

provided the original work is properly cited.

10

Electron Crystallography

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