Chapter
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
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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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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.
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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.
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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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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.
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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.
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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.
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Electron Crystallography
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