crystals
Article
The Curled Up Dimension in Quasicrystals
Fang Fang 1,*
, Richard Clawson 1,2
and Klee Irwin 1






Citation: Fang, F.; Clawson, R.;
Irwin, K. The Curled Up Dimension
in Quasicrystals. Crystals 2021, 11,
1238. https://doi.org/10.3390/cryst
11101238
Academic Editor: Enrique Maciá
Barber
Received: 11 August 2021
Accepted: 11 October 2021
Published: 14 October 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1 Quantum Gravity Research, Los Angeles, CA 90290, USA; Richard@QuantumgravityResearch.org (R.C.);
Klee@QuantumGravityResearch.org (K.I.)
2
Faculty of Health, Engineering and Sciences, University of Southern Queensland,
Toowoomba, QLD 4350, Australia
* Correspondence: Fang@QuantumGravityResearch.org; Tel.: +1-310-574-6934
Abstract: Most quasicrystals can be generated by the cut-and-project method from higher dimen-
sional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced
with aperiodic order, due to the irrationality of the projection. However, perfect periodic order is
discovered in the perpendicular space when gluing the cut window boundaries together to form
a curved loop. In the case of a 1D quasicrystal projected from a 2D lattice, the irrationally sloped
cut region is bounded by two parallel lines. When it is extrinsically curved into a cylinder, a line
defect is found on the cylinder. Resolving this geometrical frustration removes the line defect to
preserve helical paths on the cylinder. The degree of frustration is determined by the thickness of the
cut window or the selected pitch of the helical paths. The frustration can be resolved by applying a
shear strain to the cut-region before curving into a cylinder. This demonstrates that resolving the
geometrical frustration of a topological change to a cut window can lead to preserved periodic order.
Keywords: quasicrystals; aperiodic order; periodic order; perpendicular space; geometric frustration;
curled-up dimensions
1. Introduction
With the announcement by Shechtman et al. [1] in 1984 of a physical material exhibit-
ing crystallographically forbidden symmetries in its diffraction patterns, and the following
theoretical description by Levine et al. [2], the study of aperiodically ordered structures
(dubbed quasicrystals) moved from what was something of a niche specialty to a bur-
geoning field in both mathematics and material science. Like crystals, quasicrystals have
long range order, usually the symmetry of some regular polyhedron, and they have sharp
diffraction patterns, generally because they are built up from a finite number of primitive
structural units. But unlike crystals, these structural units are not periodically repeated,
so quasicrystals lack symmetry under any translation group. This allows for rotational
symmetries incompatible with periodicity, e.g. icosahedral, while undermining the main
theoretical tools (such as the Bloch theorem and standard Brillouin zones) that are used to
understand crystalline materials [3–5].
In time, theory was adapted to the new structures and great progress was made in
understanding their properties, in particular, with hydrodynamic [6] and effective field
theory [7] models. The foundation of this, of course, is having tractable models of their
basic geometry, and a key to this work was the cut-and-project description (also known
variously as the superspace, or model set, description) where points of a quasicrystal are
identified with projections of points belonging to a periodic lattice in a higher dimensional
space [3,8].
In other work, the rotational symmetry’s incompatibility with periodicity is identified
as a type of geometric frustration. This may be relieved locally by curving into a higher
dimension [9,10], or globally with the idea of distortion (nonmetricity) and disclination [11]
when focusing only on the quasicrystal space. In all these cases, the quasicrystal, which
Crystals 2021, 11, 1238. https://doi.org/10.3390/cryst11101238
https://www.mdpi.com/journal/crystals
Crystals 2021, 11, 1238
2 of 9
lacks translational periodicity itself, is seen as a distortion of some periodic structure, and
this connection provides a useful tool for modeling and analysis. This paper introduces
a method to see directly the periodicity in the quasicrystal by looking at its dual space,
the perpendicular space of the cut-and-project description. The periodicity in the per-
pendicular space not only gives a straightforward solution for the geometric frustration
in the quasicrystal space, but also defines a new kind of “closeness” for the point set in
the quasicrystal.
Section 2 reviews the cut-and-project method of generating a quasicrystal and the
duality between the quasicrystal and its perpendicular dual space. They both appear aperi-
odic if looked at individually. Section 3 points out that when looking at the perpendicular
space based on the sequential order of the quasicrystal points, if one glues the opposite
boundaries of the perpendicular space together, the order appears to be perfectly periodic.
The reverse can also be said, that the periodic order can be revealed in the quasicrystal if
looking at it according to the order of its perpendicular space. Section 4 suggests a solution
to the geometric frustration in the the P⊥ + P‖ space, by curving the perpendicular space
into a loop, and adding shear to align the frustrated boundary. Section 5 briefly summarizes
and suggests continued research.
2. Quasicrystals and the Perpendicular Space in the Cut-and-Project Approach
The method of cut-and-project from a higher-dimensional parent lattice is one of
the main tools for generating and studying quasicrystals [3,12,13]. Indeed, while other
methods such as inflation rules, matching rules, or generalized dual multigrids are often
used, in nearly all cases the quasicrystal also admits of a cut-and-project construction
(see for example [3–5]). One might even go so far as to say “all” rather than “nearly
all” quasicrystals can be so constructed, but the term “quasicrystal” is used in a variety
of contexts and there does not appear to be consensus on a universal formal definition
whereby one could make such a claim rigorous. (Exceptions to the claim are suggested by
Baake and Grimm’s discussion of aperiodic order beyond cut-and-project sets (or model
sets), such as the Thue–Morse and Rudin–Shapiro chains Chapter 10 of the [14], and by
Burdik et al’s study of quasicrystals based on so-called β-integers [15].) Nevertheless, the
cut-and-project method applies to at least a very general class of quasicrystals, which are
the subject of this note.
To illustrate the cut-and-project, we take a one dimensional quasicrystal, the Fibonacci
chain, as an example. Start with the Z2 lattice (Figure 1a). The 1D quasicrystal space, P‖, is
at an angle θ = arctan(Φ), where Φ is the golden ratio, to the root vectors of the Z2 lattice.
Its perpendicular space P⊥, by definition perpendicular to P‖, contains a finite segment
called the cut window. The cut window is extruded along the direction of P‖ to form the
cut region, an infinite region of fixed width which captures all the points in the Z2 lattice
that project to P‖ as the point set of the Fibonacci chain (points on solid line in Figure 2a).
Note, incidentally, that one obtains the same Fibonacci chain if one sets the positive P‖ at
the angle θ with the −l1 lattice vector, so that it slopes up to the right in Figure 1a instead
of down to the right.
Although the points in the cut region extend without limit in the infinite P‖, they are
bounded in P⊥ within the finite span of the cut window, as mentioned above. Therefore, the
point set in the cut window is dense. There is a dual nature between the points’ projections
in these two spaces. Points that are closer in the quasicrystal space tend to space away
from each other in the perpendicular space, and vice-versa. More importantly, there is a
pattern to the spacing.
Crystals 2021, 11, 1238
3 of 9
P⊥
P‖
θ
l1
l2
cut window
XXy
(a)
P⊥
P‖
θ
sin θ
cos θ
l1
l2
(b)
Figure 1. Cut-and-project scheme for Fibonacci chain. (a) Z2 parent lattice with lattice vectors l1 and l2,
showing P⊥, P‖, and cut region bounded by dashed lines. tan θ = Φ, the golden ratio. (b) Detail of
parent lattice (rotated so P‖ is horizontal) showing distances between points as projected in P⊥.
3 Periodic order in the curled up perpendicular space
The P⊥ distance between points is shown in Figure 1b, with cos θ and sin θ as the short and long
lengths, respectively. Following a sequence of points from left to right in the quasicrystal space as
shown in Fig. 2a, and comparing with Figure 1b, we see that when a position increment in P⊥ is
positive, the displacement is cos θ, while if the increment is negative it is − sin θ.
Now if we consider the cut window as a loop (gluing the bottom and top boundary together), we
can take all displacements as positive, as shown by the colored vertical arrows in Figure 2b. Formally,
with y as the P⊥ coordinate and {yt, yb} as the top and bottom cut window boundaries, we count the
perpendicular space interval between successive points as
∆n =
{
yn+1 − yn
yn+1 > yn
∆nt + ∆(n+1)b
yn+1 < yn
(1)
∆nt = yt − yn, ∆(n+1)b = y(n+1) − yb.
(2)
This amounts to using a sort of periodic coordinate for P⊥. When the displacement is undivided,
it is a short segment of length cos θ, as seen in Figure 1b. When the displacement is divided by
crossing the cut window boundary, a sum of two terms occurs. This sum is the complement, within
the cut window, of the long segment sin θ. The total length of the cut window is the P⊥ projection
of a unit cell, cos θ + sin θ, so this complement has length (sin θ + cos θ)− sin θ = cos θ. Thus, the
positive displacement in periodic coordinates is always cos θ for both divided and undivided segments,
∆1 = ∆2t + ∆2b = ∆3 = ∆4 = ∆5t + ∆5b = . . . . As a fraction of the cut window, this increment is
∆n
|cut window| =
cos θ
cos θ + sin θ
=
1
1 + tan θ
=
1
1 +Φ
=
1
Φ2
.
(3)
When the cut window is curved into a loop, the angular distance between these successive points is
therefore always the same, 2/Φ2, a periodicity incommensurate with the full circle. This is shown
(a)
P⊥
P‖
θ
l1
l2
cut window
XXy
(a)
P⊥
P‖
θ
sin θ
cos θ
l1
l2
(b)
gure 1. Cut-and-project scheme for Fibonacci chain. (a) Z2 parent lattice with lattice vectors l1 and l2,
owing P⊥, P‖, and cut region bounded by dashed lines. tan θ = Φ, the golden ratio. (b) Detail of
rent lattice (rotated so P‖ is horizontal) showing distances between points as projected in P⊥.
eriodic order in the curled up perpendicular space
he P⊥ distance between points is shown in Figure 1b, with cos θ and sin θ as the short and long
s, respectively. Following a sequence of points from left to right in the quasicrystal space as
in Fig. 2a, and comparing with Figure 1b, we see that when a position increment in P⊥ is
e, the displacement is cos θ, while if the increment is negative it is − sin θ.
ow if we consider the cut window as a loop (gluing the bottom and top boundary together), we
e all displacements as positive, as shown by the colored vertical arrows in Figure 2b. Formally,
as the P⊥ coordinate and {yt, yb} as the top and bottom cut window boundaries, we count the
dicular space interval between successive points as
∆n =
{
yn+1 − yn
yn+1 > yn
∆nt + ∆(n+1)b
yn+1 < yn
(1)
∆nt = yt − yn, ∆(n+1)b = y(n+1) − yb.
(2)
his amounts to using a sort of periodic coordinate for P⊥. When the displacement is undivided,
short segment of length cos θ, as seen in Figure 1b. When the displacement is divided by
g the cut window boundary, a sum of two terms occurs. This sum is the complement, within
window, of the long segment sin θ. The total length of the cut window is the P⊥ projection
it cell, cos θ + sin θ, so this complement has length (sin θ + cos θ)− sin θ = cos θ. Thus, the
e displacement in periodic coordinates is always cos θ for both divided and undivided segments,
2t + ∆2b = ∆3 = ∆4 = ∆5t + ∆5b = . . . . As a fraction of the cut window, this increment is
∆n
|cut window| =
cos θ
cos θ + sin θ
=
1
1 + tan θ
=
1
1 +Φ
=
1
Φ2
.
(3)
he cut window is curved into a loop, the angular distance between these successive points is
(b)
Figure 1. Cut-and-project scheme for Fibonacci chain. (a) Z2 parent lattice with lattice vectors l1 and l2, showing P⊥, P‖,
and cut region bounded by dashed lines. tan θ = Φ, golden ratio. (b) Detail of parent lattice (rotated so P‖ is horizontal)
showing distances between points as projected in P⊥.
3. Periodic Order in the Curled Up Perpendicular Space
The P⊥ distance between points is shown in Figure 1b, with cos θ and sin θ as the short
and long lengths, respectively. Following a sequence of points from left to right in the
quasicrystal space as shown in Figure 2a, and comparing with Figure 1b, we see that when
a position increment in P⊥ is positive, the displacement is cos θ, while if the increment is
negative, it is − sin θ.
Figure 2. a. Cut region in Z2 parent lattice (the region bounded by the top and bottom dashed lines)
and projection in P‖ (on solid line). b. The periodicity after identifying top and bottom boundaries of
P⊥, showing qual increments in P⊥ (xamples ∆1, ∆2b, and ∆2t labeled). c. Same priodicity illustrated
aftwrapping P⊥ into a loop. Dashed circle is P⊥; colors match the points and increments in a and b
with the points and arcs in c.
Figure 1b becomes inverted, with sin θ above and cos θ below. In that case, the positive increment ∆n
100
would be the larger sin θ instead of the smaller cos θ, and Eq. (3) would yield 1/Φ instead of 1/Φ2.
101
The points on the circle would still make the same pttern, but in the reverse direction, becaue an
102
angle on the circle of +2π/Φ is equivalent to an angle of −2π/Φ2, since Φ−1 +Φ−2 = 1.
Figure 2. (a) Cut region in Z2 paret lattice (region bounded by top and bottom dashed lines) and projection in P‖ (on solid
line). (b) Periodicity after identifying top and ottom bounaries of P⊥, showing equal increments in P⊥ (examples ∆1, ∆2b,
and ∆2t labeled). (c) Same periodicity illustrated after wrapping P⊥ into a oop. Dashed circle is P⊥; colors match points and
incremets in (a,b) with pints and arcs in (c). (d) Displaying the successive arc sgments and arrow tips with increasing
radius spreads t
circle into an anulus, making the self-similarity of the structure more visually apparent.
Crystals 2021, 11, 1238
4 of 9
Now, if we consider the cut window as a loop (gluing the bottom and top boundary
together), we can take all displacements as positive, as shown by the colored vertical arrows
in Figure 2b. Formally, with y as the P⊥ coordinate and {yt, yb} as the top and bottom
cut window boundaries, we count the perpendicular space interval between successive
points as
∆n =
{
yn+1 − yn
yn+1 > yn
∆nt + ∆(n+1)b
yn+1 < yn
(1)
∆nt = yt − yn, ∆(n+1)b = y(n+1) − yb.
(2)
This amounts to using a sort of periodic coordinate for P⊥. When the displacement is
undivided, it is a short segment of length cos θ, as seen in Figure 1b. When the displacement
is divided by crossing the cut window boundary, a sum of two terms occurs. This sum is
the complement, within the cut window, of the long segment sin θ. The total length of the
cut window is the P⊥ projection of a unit cell, cos θ + sin θ, so this complement has length
(sin θ + cos θ)− sin θ = cos θ. Thus, the positive displacement in periodic coordinates is
always cos θ for both divided and undivided segments, ∆1 = ∆2t + ∆2b = ∆3 = ∆4 =
∆5t + ∆5b = . . . . As a fraction of the cut window, this increment is
∆n
|cut window| =
cos θ
cos θ + sin θ
=
1
1 + tan θ
=
1
1 +Φ
=
1
Φ2
.
(3)
When the cut window is curved into a loop, the angular distance between these
successive points is therefore always the same, 2π/Φ2, a periodicity incommensurate with
the full circle. This is shown in Figure 2c for the first few points; repeated for many points,
it becomes like a phyllotaxis pattern as in Figure 2d. (One should remember, however,
that the disk-like nature of this representation is just for illustrative purposes, to exhibit
the structure of the pattern; the actual perpendicular space, when curved into a loop, is
just a circle, not a disk or annulus.) Incidentally, if one uses the alternate construction
mentioned for the Fibonacci chain, making P‖ slope up to the right in Figure 1a, then
Figure 1b becomes inverted, with sin θ above and cos θ below. In that case, the positive
increment ∆n would be the larger sin θ instead of the smaller cos θ, and Equation (3) would
yield 1/Φ instead of 1/Φ2. The points on the circle would still make the same pattern, but
in the reverse direction, because an angle on the circle of +2π/Φ is equivalent to an angle
of −2π/Φ2, since Φ−1 +Φ−2 = 1.
Another interesting pattern, in addition to this periodic angular advancement, is
that every new added point lands in the longest empty section of the circle to maximize
the uniformity of distribution of the points on the circle. In a way, the adjacency in the
quasicrystal space results in a periodic repulsion in P⊥, or, alternatively, a new “closeness”
can be defined in the looped P⊥ in terms of the periodicity. Specifically, we may define a
“periodic distance” in P⊥ as the number of integral 2πΦ2 periods separating points. Then,
points that are periodically close in P⊥ are physically close in P‖, and vice versa. This
contrasts with the ordinary Euclidean closeness in P⊥ which induces a sort of “repulsion”
in P‖ (resembling the same-sign charge distribution), and vice versa.
This periodicity in P⊥ can be extended to many higher dimensional quasicrystals,
where P⊥ is also higher dimensional. In this case, the cut window is often a polytope and
the cut region is the prism formed by extruding that polytope in the directions of P‖. The
cut window usually has matching opposite faces (i.e., congruent and parallel). (Exceptional
cases include cut windows with fractal boundaries [14,16], to which this analysis would
not be apply.) By focusing on the one-dimensional subspace normal (within P⊥) to a pair of
opposite faces, we can see the same periodicity as described above. This corresponds to the
quasiperiodic structure along a one-dimensional line in the quasicrystal space. Perhaps it is
even possible to wrap the entire cut window polytope into a closed manifold by identifying
each face with its opposite, just as we wrapped the 1D cut window of the Fibonacci chain
into a circle.
Crystals 2021, 11, 1238
5 of 9
Returning to our 1D case, the distribution of points in P⊥ also has a self-similar nature,
which resembles the self-similar nature of the P-adic numbers [17]. This is in fact the
self-similarity of the inflation/deflation structure of Fibonacci chains under substitution
rules. It can be seen on the circle of P⊥ as illustrated in the sequence of Figure 3. With just
the first two points, the circle is divided into two unequal arcs L and S whose length ratio
is golden. As mentioned above, the next point lands on the largest gap, the single L arc.
It divides that into two arcs related by the golden ratio, so that the longer of them is the
same arc length as the S arc that was not divided. This yields a circle with three arcs of only
two distinct lengths, again in the golden ratio, where the previous S is the new L. Since
there are now two L arcs, adding the next two points will divide each of the L’s, and again
we find the entire circle divided into arcs of only two distinct lengths whose ratio is golden,
the previous S being the new L. With each successive step, where the number of points
added in a step is the number of L arcs on the circle, we get an inflation with S→ L and
L → LS. This proceeds without limit, yielding an endless (approximate) self-similarity
of the Fibonacci structure of the points as projected onto the cut window, which is closed
upon itself into a circle. By displaying successive points on circles of increasing radii
(Figure 2d) one spreads the circle into a thick annulus, and the self-similar structure is
made more visually striking.
Figure 3. Self similarity of point distribution on the P⊥ loop. In a given step, there are n long arcs (L) of
angle 2πΦn , the short arcs (S) having angle
2π
Φn+1 . In the next step, n new points are added by incrementing
the angle in steps of 2πΦ2 . These all land in L arcs, dividing them into LS arcs at the new scale, with the
new L having angle 2π
Φn+1 . The number of L arcs in successive steps are the Fibonnaci numbers 1, 2, 3,
5. . . .
Returning to our 1D case, the distribution of points in P⊥ also has a self-similar nature, which
122
resembles the self-similar nature of the P-adic numbers [17]. This is in fact the self-similarity of the
123
inflation/deflation structure of Fibonacci chains under substitution rules. It can be seen on the circle of
124
P⊥ as illustrated in the sequence of Figure 3. With just the first two points, the circle is divided into
125
two unequal arcs L and S whose length ratio is golden. As mentioned above, the next point will land
126
on the largest gap, the single L arc. It divides that into two arcs related by the golden ratio, so that the
127
longer of them is the same arc length as the S arc that was not divided. This yields a circle with three
128
arcs of only two distinct lengths, again in the golden ratio, where the previous S is the new L. Since
129
there are now two L arcs, adding the next two points will divide each of the L’s, and again we find the
130
entire circle divided into arcs of only two distinct lengths whose ratio is golden, the previous S being
131
the new L. With each successive step, where the number of points added in a step is the number of L
132
arcs on the circle, we get an inflation with S→ L and L→ LS. This proceeds without limit, yielding
133
an endless (approximate) self-similarity of the Fibonacci structure of the points as projected onto the
134
cut window, which is closed upon itself into a circle. By displaying successive points on circles of
135
increasing radii (Fig. 2d) one spreads the circle into a thick annulus, and the self-similar structure is
136
made more visually striking.
137
The self-similarity is not exact, because the initial point set cannot be partitioned into subsets that
138
all reproduce the original at a fixed scale, but rather into two groups of subsets that reproduce it at two
139
Figure 3. Self similarity of point distribution on the P⊥ loop. In a given step, there are n long arcs (L) of angle 2πΦn , short arcs
(S) having angle 2π
Φn+1 . In next step, n new points are added by incrementing the angle in steps of
2π
Φ2 . These all land in L
arcs, dividing them into LS arcs at new scale, with new L having angle 2π
Φn+1 . The number of L arcs in successive steps are
Fibonnaci numbers 1, 2, 3, 5. . . .
The self-similarity is not exact because the initial point set cannot be partitioned into
subsets that all reproduce the original at a fixed scale, but rather into two groups of subsets
that reproduce it at two different scales. Specifically, at any level of inflation, the Fibonnaci
word can be partitioned into a set of L’s and S’s. Every L, when filled in by inflation,
reproduces the pattern of the original circle at a scale reduced by some Φn, while each S
when filled in r produces the original pattern at a scale reduced by Φn+1. The concept
of multifractal analysis suggests that we calculate the similarity dimension of each of
these subgroups separately. (We refer to Strogatz Chapter 11 of the [18] for an elementary
introduction to the analysis of fractal dimension).
For a number of partitions N(r) at scale factor r, the similarity dimension is
D =
ln N(r)
ln r
.
(4)
In the Fibonacci chain at the nth level, the number of L’s (reduction factor Φn) is the
Fibonacci number Fn, while the number of S’s (reduction factor Φn+1) is Fn−1. The, using
Binet’s frmula, Fn = (Φn −Φ−n)/
√
5, te r spective similarity dimensions for these two
groups are
Crystals 2021, 11, 1238
6 of 9
DL =
ln Fn
lnΦn
=
ln(Φn −Φ−n)− ln
√
5
lnΦn
→ 1
for large n
DS =
ln Fn−1
lnΦn+1
=
ln(Φn−1 −Φ−(n−1))− ln
√
5
lnΦn+1
→ 1
for large n.
(5)
This seems a fairly trivial result for the fractal dimension. It is a reflection of the fact
that unlike, e.g., the Cantor set, for any level of inflation we find that each interval filled by
a copy of the set is in turn completely filled by the reduced copies at their respective scales.
Among other things this means that the full point set is dense on the circle.
A calculation of the Minkowski dimension also yields unity, if the boxes are taken to
cover either the long or short arcs. For box size e = Φ−n the number of boxes is N(e) = Fn,
and the Minkowski dimension ln N(e)/ ln(e−1) gives the same expressions as DL and DS
above. This reflects the fact that unlike e.g. the Koch curve, the points of the Fibonacci
inflations all lie on the circle or, effectively, on a finite interval in R (where, as mentioned
before, they are dense). The correlation dimension [19] and a numerical calculation of
the pointwise dimension also give the same results. We thus have a set with an inexact,
nontrivial, rigorously structured type of self-similarity, which neither leaves finite gaps in
the topological space it lives in, nor “spreads” itself in the usual fractal sense into a larger
topological space. Consequently, its fractal dimension is trivial, in the sense that it matches
the topological dimension.
4. “Frustration” in the P⊥+ P‖ Space and Shear as a Solution
Geometric frustration refers in general to the conflict between some nontrivial local
order and the global environmental order, or as described by Sadoc et al. [9], the condition
that local order cannot propagate “freely” throughout the space. An example is the local
maximum sphere packing arrangement in a global maximum arrangement, or the breaking
of a periodic pattern due to some incompatible local order. Geometric frustration is a
common phenomenon in quasicrystals due to their aperiodic nature [10]. For example,
the icosahedral symmetry of many 3D quasicrystals is not compatible with a crystalline
propagation to fill flat space, but the frustration can be resolved by curving the space into
the fourth dimension and propagating the order on a 3-sphere in 4D.
In curling up a dimension in P⊥, we find a new kind of geometric frustration, as well
as a way to resolve it. To see this, we start by looking at the “parent” space, the projection
space that is the sum of P‖ and P⊥. The 1D Fibonacci chain is the projected image in P‖ of
points in the cut region of the 2D parent space, as shown in Figure 4. Firstly, we identify a
linear trajectory in the cut region connecting points that are adjacent in P‖ (gray segments
shown in Figure 4(a1)). As discussed in the previous section, the step advancement in
P⊥ is periodic. In P‖, however, there is a “frustration” for every cycle in P⊥, as shown
Figure 4(a1) by the intervals labeled d. This frustration breaks the periodicity in P‖.
As we curve P⊥ into a circle leaving P‖ invariant, the cut window trajectory becomes
helical, with the linear axial component as the propagation in P‖ and the angular (phase,
clock) component as the propagation in P⊥. The frustration manifests as a line defect
shown in Figure 4(b1), where the helical path breaks at the line where the top and bottom
boundaries of P⊥ meet. The breaks can be eliminated and the trajectory segments aligned
into a continuous helix by shearing the cut region along the axial direction so that a
rectangle is deformed into a parallelogram (Figure 4(a2)), thereby vertically aligning the
endpoints of the trajectory segments before curving the P⊥ into a cylinder (Figure 4(b2)).
Different choices of discrete shear strain are possible to join one section with the others.
A shear that connects a section with its first, second, third etc., neighbors on either side,
respectively, creates a single, double, triple helix, etc. (Figure 5). The shear may even create
isolated loops, if it connects each section back to itself. For cut windows of different sizes,
the minimum shear needed to produce a single helix increases as the cut window gets
thicker (Figure 6).
Crystals 2021, 11, 1238
7 of 9
topological dimension.
163
4 “Frustration” in the P⊥ + P‖ space and shear as a solution
164
Figure 4. The cut region (a1) wrapped into a cylinder (b1), with an exaggerated view to better see
the mismatch (c1). When introducing shear, the cut region (a2) becomes a parallelogram and the
mismatches in the cylinder close to form a true helix (b2), (c2).
Geometric frustration refers in general to the conflict between some nontrivial local order and
165
the global environmental order, or as described by Sadoc et al [9], the condition that local order
166
cannot propagate “freely” throughout the space. An example is the local maximum sphere packing
167
arrangement in a global maximum arrangement, or the breaking of a periodic pattern due to some
168
incompatible local order. Geometric frustration is a common phenomenon in quasicrystals due to
169
their aperiodic nature [10]. For example, the icosahedral symmetry of many 3D quasicrystals is not
170
compatible with a crystalline propagation to fill flat space, but the frustration can be resolved by
171
curving the space into the fourth dimension and propagating the order on a 3-sphere in 4D.
172
In curling up a dimension in P⊥, we find a new kind of geometric frustration, as well as a way to
173
resolve it. To see this, we start by looking at the “parent” space, the projection space that is the sum of
174
P‖ and P⊥. The 1D Fibonacci chain is the projected image in P‖ of points in the cut region of the 2D
175
parent space, as shown in Fig. 4. First we identify a linear trajectory in the cut region connecting points
176
that are adjacent in P‖ (gray segments shown in Fig. 4a1). As discussed in the previous section, the
177
•fang@quantumgravityresearch.org
page 6 of 9
Figure 4. Cut region (a1) wrapped into a cylinder (b1), with an exaggerated view to better see mismatch (c1).
When introducing shear, cut region (a2) becomes a parallelogram and mismatches in cylinder close to form a true helix
(b2, c2).
step advancement in P⊥ is periodic. In P‖, however, there is a “frustration” for every cycle in P⊥, as
178
shown Fig. 4a1 by the intervals labeled d. This frustration breaks the periodicity in P‖.
179
As we curve P⊥ into a circle leaving P‖ invariant, the cut window trajectory becomes helical, with
180
the linear axial component as the propagation in P‖ and the angular (phase, clock) component as the
181
propagation in P⊥. The frustration manifests as a line defect shown in Fig. 4b1, where the helical path
182
breaks at the line where the top and bottom boundaries of P⊥ meet. The breaks can be eliminated
183
and the trajectory segments aligned into a continuous helix by shearing the cut region along the axial
184
direction so that a rectangle is deformed into a parallelogram (Fig. 4a2), thereby vertically aligning the
185
endpoints of the trajectory segments before curving the P⊥ into a cylinder (Fig. 4b2).
186
Different choices of discrete shear strain are possible to join one section with the others. A shear
187
that connects a section with its first, second, third etc., neighbors on either side, respectively, creates a
188
single, double, triple helix, etc. (Fig. 5). The shear may even create isolated loops, if it connects each
189
section back to itself. For cut windows of different sizes, the minimum shear needed to produce a
190
single helix increases as the cut window ges thicker (Fig. 6).
191
Figure 5. When the cut reion is curvd into a cylider, different choices of shear c
ect differet
sections to make different helices.
helices are created when the hear conects each sections with its
nth neighbors, rather than just its immediate neighbors.
Figure 6. Comparing cut windows of different thickness: a. thinner window requires a smaller shear
correction d to connect a section with its first neighbors and make a single helix; b. thicker window
requires a larger shear correction d′ for a single helix.
Figure 5. When the cut region is curved into a cylinder, different choices of shear connect different sections to make
different helices. n helices are created when the shear connects each sections with its nth neighbors, rather than just its
immediate neighbors.
step advancement in P⊥ is periodic. In P‖, however, there is a “frustration” for every cycle in P⊥, as
178
shown Fig. 4a1 by the intervals labeled d. This frustration breaks the periodicity in P‖.
179
As we curve P⊥ into a circle leaving P‖ invariant, the cut window trajectory becomes helical, with
180
the linear axial component as the propagation in P‖ and the angular (phase, clock) component as the
181
propagation in P⊥. The frustration manifests as a line defect shown in Fig. 4b1, where the helical path
182
breaks at the line where the top and bottom boundaries of P⊥ meet. The breaks can be eliminated
183
and the trajectory segments aligned into a continuous helix by shearing the cut region along the axial
184
direction so that a rectangle is deformed into a parallelogram (Fig. 4a2), thereby vertically aligning the
185
endpoints of the trajectory segments before curving the P⊥ into a cylinder (Fig. 4b2).
186
Different choices of discrete shear strain are possible to join one section with the others. A shear
187
that connects a section with its first, second, third etc., neighbors on either side, respectively, creates a
188
single, double, triple helix, etc. (Fig. 5). The shear may even create isolated loops, if it connects each
189
section back to itself. For cut windows of different sizes, the minimum shear needed to produce a
190
single helix increases as the cut window gets thicker (Fig. 6).
191
Figure 5. When the cut region is curved into a cylinder, different choices of shear connect different
sections to make different helices. n helices are created when the shear connects each sections with its
nth nighbors, rather than jut its immediate eighbors.
Figure 6. Comparing cut windows of different thickness: a. thinner window requires a smaller shear
correction d to connect a section with its first neighbors and make a single helix; b. thicker window
requires a larger shear correction d′ for a single helix.
5 Summary and discussion
192
Figure 6. Comparing cut windows of different thickness: (a) thinner window requires a smaller shear correction d to connect
a section with its first neighbors and make a single helix; (b) thicker window requires a larger shear correction d′ for a
single helix.
Crystals 2021, 11, 1238
8 of 9
5. Summary and Discussion
In an attempt to resolve the geometric frustration in quasicrystals, which is a natural
result of their aperiodic nature, we discovered the inherent periodicity associated to a
quasicrystal when considering its dual space, the perpendicular space (P⊥) of the cut-and-
project method. The periodicity in P⊥ can be better seen when curving P⊥ and gluing its
upper and lower boundaries together. Using the Z2 lattice projecting to a 1D Fibonacci
chain (in P‖) as an example, P⊥ is a segment, which after being curved becomes a loop.
The periodicity of the points, incommensurate with the period of the P⊥ loop, fills the loop
with a dense point set that exhibits a 2-scale self similarity, but the fractal dimension of this
set is just unity, matching the topological dimension of the loop itself.
After putting P‖ and the curved P⊥ together, the geometric frustration, or the disjoints
in the Z2 to Fibonacci chain case, are clearly shown in this cylindrical space. A shear needs
to be introduced to connect the disjoints, or resolve the geometric frustration. Different
degrees of shear can result in different numbers of helices in the cylindrical space. The
concept generalizes naturally to higher-dimensions when the cut window is a polytope
with parallel opposing facets.
This paper also discussed a new kind of “closeness” based on the periodic nature of P⊥.
Points that are ordinarily close in P⊥ are far apart in P‖, but points that are “periodically”
close in P⊥ (separated by few integral periods) are also close in P‖.
The author will expand the research in higher dimensional quasicrystals and also
study the curved P⊥ as the phase space (in complex plane) of the quasicrystal to under-
stand deeper the interplay between P⊥ and P‖ and the connection between periodicity
and aperiodicity.
Author Contributions: F.F. discovered the periodicity in the perpendicular space of quasicrystals; F.F.
and R.C. identified its connection to the frustration in the curled-up cut region; K.I. led the overall
project in research on geometric frustration in quasicrystals. F.F. and R.C. wrote the paper. All authors
have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments: The authors thank Dugan Hammock, David Chester, and Raymond Aschheim
for their help in optimizing the code and fruitful discussions. We also thank the anonymous reviewers
for many helpful comments.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
MDPI Multidisciplinary Digital Publishing Institute
DOAJ Directory of open access journals
References
1.
Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic Phase with Long-Range Orientational Order and No Translational
Symmetry. Phys. Rev. Lett. 1984, 53, 1951–1953. [CrossRef]
2.
Levine, D.; Steinhardt, P.J. Quasicrystals: A New Class of Ordered Structures. Phys. Rev. Lett. 1984, 53, 2477–2480. [CrossRef]
3.
Janot, C. Quasicrystals: A Primer; Oxford University Press: Oxford, UK, 2012.
4.
Levine, D.; Steinhardt, P.J.; Quasicrystals. I. Definition and structure. Phys. Rev. B 1986, 34, 596. [CrossRef] [PubMed]
5.
Baake, M. A guide to mathematical quasicrystals. In Quasicrystals; Suck, J.B., Schreiber, M., Häussler, P., Eds.; Springer Series in
Materials Science; Springer: Berlin/Heidelberg, Germany, 2002; Volume 55._2. [CrossRef]
6.
de Boissieu, M. Phonons, phasons and atomic dynamics in quasicrystals. Chem. Soc. Rev. 2012, 41, 6778–6786. [CrossRef]
7.
Baggioli, M.; Landry, M. Effective Field Theory for Quasicrystals and Phasons Dynamics. SciPost Phys. 2020, 9, 62. [CrossRef]
8.
Janssen, T.; Janner, A. Aperiodic crystals and superspace concepts. Acta Crystallogr. B 2014, 70, 617–651. [CrossRef] [PubMed]
9.
Sadoc, J.F.; Mosseri, R. Geometric Frustration; Cambridge University Press: Cambridge, MA, USA, 1999.
10.
Fang, F.; Clawson, R.; Irwin, K. Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between
Discrete Curvature and Twist Transformations. Mathematics 2018, 6, 89. [CrossRef]
11. Kleinert, H. Gauge Fields in Condensed Matter; World Scientific: Singapore, 1989.
Crystals 2021, 11, 1238
9 of 9
12.
Senechal, M.J. Quasicrystals and Geometry; Cambridge University Press: Cambridge, MA, USA, 1995.
13.
Socolar, J.; Steinhardt, P.J. Quasicrystals. II. Definition and structure. Phys. Rev. B 1986, 34, 617. [CrossRef] [PubMed]
14.
Baake, M.; Grimm, U. Aperiodic Order Volume I; Cambridge University Press: Cambridge, MA, USA, 2013.
15.
Burdik,Č.; Frougny, C.; Gazeau, J.P.; Krejcar, R. Beta-integers as natural counting systems for quasicrystals. J. Phys. A Math. Gen.
1998, 31, 6449–6472. [CrossRef]
16.
Baake, M.; Klitzing, R.; Schlottmann, M. Fractally shaped acceptance domains of quasiperiodic square-triangle tilings with
dedecagonal symmetry. Phys. A Stat. Mech. Its Appl. 1992, 191, 554–558. [CrossRef]
17.
Lapidus, M.; Hung, L. Self-similar P-adic fractal strings and their complex dimensions. P-Adic Num. Ultrametr. Anal. Appl. 2009,
1, 167–180. [CrossRef]
18.
Strogatz, S.H. Nonlinear Dynamics and Chaos; Perseus Books, Reading; Cambridge, MA, USA, 1994.
19. Grassberger, P.; Procaccia, I. Measuring the strangeness of strange attractors. Phys. D Nonlinear Phenom. 1983, 9, 189–208.
[CrossRef]