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An Icosahedral Quasicrystal and E8derived quasicrystals
F. Fang∗and K. Irwin
Quantum Gravity Research, Los Angeles, CA, U.S.A.
We present the construction of an icosahedral quasicrystal, a quasicrystalline spin
network, obtained by spacing the parallel planes in an icosagrid with the Fibonacci
sequence. This quasicrystal can also be thought of as a golden composition of five
sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals
that are golden compositions of the three-dimensional tetrahedral cross-sections of the
Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice.
These compound quasicrystals are subsets of the quasicrystalline spin network, and
the former can be enriched to form the later. This creates a mapping between the
quasicrystalline spin network and the E8lattice.
I. Introduction
Until Shechtman et al.
22
discovered a quasicrystal
in nature, crystallographic rules prohibited their
existence. This discovery intrigued scientists from
various disciplines, such as math, physics, material
science, chemistry and biology
1, 14, 19, 21, 25
. In the early
years, there was a surge in the interest in studying the
mathematical aspects of quasicrystals
4, 10, 12, 15, 17, 23, 24
.
Then in recent years, the focus of the majority of
research in the field has shifted toward the physical
aspects of quasicrystals, i.e. their electronic and optical
properties
7, 18
and quasicrystal growth
9
. The interesting
mathematical properties of quasicrystals are relatively
unexplored and investigation in this field provides
opportunity for discoveries that could have far reaching
consequences in physics and other disciplines.
In this paper, we introduce an icosahedral quasicrys-
tal, the quasicrystalline spin network, that is a superset
of a golden ratio based composition of three-dimensional
slices of the four-dimensional Elser-Sloane quasicrystal
projected from E8
6, 16
. This paper focuses on the geomet-
ric connections between the quasicrystalline spinwork
and the E8 lattice.
This paper includes six sections. In section II, we
briefly introduce the definition of a quasicrystal and
the usual methods for generating quasicrystals math-
ematically. Then using Penrose tilling as an example,
we introduce a new method - the Fibonacci multigrid
method - in additional to the the cut-and-project
3
and
dual-grid methods
24
. Section III applies the Fibonacci
multigrid method in three-dimension and obtains an
icosahedral quasicrystal, the Fibonacci icosagrid which
later called quasicrystalline spin network. After obtain-
ing this quasicrystal, we discovered an alternative way
of generating it using five sets of tetragrids. This new
way of constructing this quasicrystal revealed a direct
connection between this quasicrystal and a compound
quasicrystal that is introduced in Section IV. Impor-
tant properties of the Fibonacci icosagrid are discussed.
For example, its vertex configurations, edge-crossing
types and space-filling analog. Section IV introduces
about the compound quasicrystal that is obtained from
a projection/slicing/composition of the E8 lattice. It
starts with a brief review of the Elser-Sloane quasicrystal
and then introduces two three-dimensional compound
quasicrystals that are composites of three-dimensional
tetrahedral slices of the Elser-Sloane quasicrystal. We
also introduce an icosahedral slice of the Elser-Sloane
quasicrystal that is itself a quasicrystal with the same
unit cells as the space-filling analog of the Fibonacci
icosagrid. Section V compares the Fibonacci icosagrid
to the compound quasicrystals obtained from the Elser-
Sloane quasicrystal and demonstrates the connection
between them, suggesting a possible mapping between
Fibonacci icosagrid and the E8 lattice. The last section,
Section VI summarizes the paper.
II. Fibonacci Multigrid Method
While there is still no commonly agreed upon definition
of a quasicrystal
12, 13, 21
, it is generally believed that
for a structure to be a quasicrystal, it should have the
following properties:
1. It is ordered but not periodic.
2. It has long-range quasiperiodic translational order
and long-range orientational order. In other words,
for any finite patch within the quasicrystal, you
can find an infinite number of identical patches at
other locations, with translational and rotational
transformation.
3. It has finite types of prototiles/unit cells.
4. It has a discrete diffraction pattern.
Mathematically, there are two common ways of gen-
erating a quasicrystal: a cut-and-project from a higher
1
arXiv:1511.07786v3 [math.MG] 24 Jun 2016
e1
e2
e4
e3
e5
Figure 1: An example of a pentagrid, with e1,e2,...,e5being the norm of the grids.
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
A B
Figure 2:
A) Identify the intersections in a sample patch in pentagrid, B) construct a dual quasicrystal cell, here the
prolate and oblate rhombs, at each intersect point and then place them edge-to-edge while maintaining their
topological connectedness.
dimensional crystal
21, 24
and the dual grid method
21, 24
.
We independently developed a new method, the Fi-
bonacci multigrid method, for generating a quasicrystal.
This method is a special case of the generalized dual
grid method discussed in Socolar, et al
24
, which focuses
on the cell space instead of the grid space, on which
we are focused. We introduce this method in the next
few paragraphs, using the the pentagrid and its dual
quasicrystal, Penrose tiling, as an example.
The commonly used dual grid for generating a Penrose
tiling is called the pentagrid and it is a periodic grid.
Fig. 1 gives an example of a pentagrid. A pentagrid is
defined as (Fig. 1)
~x ·~e =xN, N ∈Z, , (1)
with
en=cos 2πn
5,sin 2πn
5, n = 0,1, ..., 4 (2)
xN=T(N+γ),(3)
where
en, n
= 0
,
1
, ...
4
,
are the norms of the parallel grid
lines,
T
is constant and it specifies the equal spacing
between the parallel grid lines. In other words, the
period of
xN
,
γ
is a real number and it corresponds to
the phase or offset of the grid with respect to the origin.
Fig. 2 shows the process of constructing a Penrose tiling
with this pentagrid:
1.
First we identify all the intersections in the penta-
grid (1
−
5 in Fig. 2A). There are only two types
of intersections in the grid (in Fig. 2A, 1 and 3 are
Corresponding Author •fang@quantumgravityresearch.org page 2 of 16
A
B
e1
e2
e4
e3
e5
e1
e2
e4
e3
e5
Figure 3: A) Pentagrid and B) Fibonacci pentagrid - quasicrystalized pentagrid using Fibonacci-sequence spacing.
the same and 2, 4 and 5 are the same), specified by
the angle of intersection.
2.
At each intersection point, a dual quasicrystal cell
can be constructed (Fig. 2B). The edges of the cell
are perpendicular to the grid lines that the edges
cross. Thus two types of intersections result in two
types of quasicrystal cells, the prolate and oblate
rhombi shown in Fig. 2B.
3.
The last step is to place the rhombi edge-to-edge
while maintaining their original topological connect-
edness. For example, as shown in Fig. 2B, although
the cells are translated to be placed edge-to-edge,
cell 1 is always connected to cell 2 through the
vertical edge, cell 2 is connected to cell 3 through
the other non-vertical edge, and so on.
If the values of the offset
γ
are properly chosen so that
there are no more than two lines intersecting at one point
(to avoid glue tiles
24
), the resulting quasicrystal will be
a Penrose tiling. As we can see from this procedure,
eventually, each vertex (intersection) in the pentagrid
will correspond to a cell in the Penrose tiling, and each
Corresponding Author •fang@quantumgravityresearch.org page 3 of 16
vertex in the Penrose tiling will correspond to a cell in
the pentagrid. Therefore the grid space and the cell
space are dual to each other.
Although its dual is a quasicrystal, the pentagrid
itself is not a quasicrystal due to the arbitrary closeness
(infinite number of unit cells) between the vertices(Fig.
3A). We independently discovered a modification that
can be applied to the periodic grid to make the grid itself
quasiperiodic. This method has already introduced by
Socolar et. al. in 1986. The modification is to change
the Eq. 3 from periodic to quasiperiodic24):
xN=T(N+α+1
ρN
µ+β),(4)
where
α
,
β
, and
ρ∈R
,
µ
are irrational, and
bc
is
the floor function and denotes the greatest integer less
than or equal to the argument. Since
N
can be all
integers, including negative ones, we can set all the other
variables in this expression to be positive without losing
its generality. This expression defines a quasiperiodic
sequence of two different spacings,
L
and
S
, with a
ratio of 1 + 1
/µ
. Changing
µ
,
α
and
β
can change the
relative frequency of the two spacings, the offsets of the
grid and the order of the sequence of the two spacings,
respectively.
This paper focuses on a special case of the quasiperi-
odic grid, where
ρ
=
µ
=
τ
and
τ
=
1+√5
2
is the golden
ratio. In this case,
xN
defines a Fibonacci sequence
12
.
The modified pentagrid is called Fibonacci Pentagrid.
Not only is its dual a quasicrystal(a Penrose tiling), but
it can be shown that the Fibonacci pentagrid itself is
also a quasicrystal. You can clearly see in Fig. 3B
that there are finite types of unite cells in the Fibonacci
pentagrid and the arbitrary closeness disappeared.
We call this method the Fibonacci multigrid. Using
this method, we generate a three-dimensional icosahe-
dral quasicrystal discussed in the following sections.
III. Fibonacci IcosaGrid
3.1 Icosagrid and tetragrid
An icosagrid is a three-dimensional planar periodic 10-
grid where the norm vectors of the ten planar grid,
en, n
= 1
,
2
, ...
10, coincide with the ten threefold sym-
metry axis of an icosahedron as shown in Fig. 4. With
this specification of these norm vectors, an icosagrid can
be generated using Eq. 1-3 with
γ
= 0. This icosagrid
dissects the three-dimensional space into infinite types of
three-dimensional cells. However, we present properties
of the icosagrid when only the regular tetrahedral cells
(Fig. 5) are ”turned on”. The first of these properties
is that this structure can be separated into two chiral
structures with opposite handedness. Fig. 6 shows how
these two structures with opposite chiralities can be
separated from the icosagrid. The second property is
that the icosagrid can be built in an alternative way
using five sets of tetragrids. The details are discussed
in the following paragraphs.
Similar to an icosagrid, a tetragrid is defined as a
three-dimensional planar periodic 4-grid where the norm
vectors of the four planar grid,
en, n
= 1
,
2
, ...
4, coincide
Figure 7:
Tetragrid with tetrahedral cells of two different
orientation (Yellow and Cyan) and Octahedral
gaps.
with the four threefold symmetry axis of a tetrahedron.
When
γ
= 0, the tetragrid becomes a periodic FCC
lattice which can be thought of as a space-filling com-
bination of regular tetrahedral cells of two orientations
and octahedral cells (Fig. 7). Tetrahedra of the two ori-
entations share the same point set (tetrahedral vertex).
The icosagrid can be thought of as a combination of five
sets of tetragrids, composed together with a golden com-
position procedure achieved in the following manner(Fig.
8):
1.
We start from the origin in the tetragrid and identify
the eight tetrahedral cells sharing this point with
four being in one orientation and the other four in
the dual orientation (yellow tetrahedra in Fig. 8A).
2.
We pick the four tetrahedral cells of the same ori-
entation (Fig. 8B) and make four copies.
3.
We place two copies together so that they share
their center point and the adjacent tetrahedral faces
are parallel, touching each other and with a relative
rotation angle of
Cos−1
(
3τ−1
4
), the golden rotation
8
(Fig. 8C).
4.
We repeat the process three more times to add the
other three copies to this structure (Fig. 8C, D,
E). A twisted 20-tetrahedra cluster, the 20-group
(20G)(Fig. 8F), is formed.
5.
We now expand the tetragrid associated with each
of the 4-tetrahedra sets by turning on the tetrahe-
dra of the same orientation as the existing four. An
icosagrid of one chirality is achieved(Fig. 6A). Sim-
ilarly, if the tetrahedral cell of the other orientation
are turned on, an icosagrid of the opposite chirality
will be achieved (Fig. 6C).
Corresponding Author •fang@quantumgravityresearch.org page 4 of 16
Figure 4: The norm vectors of the icosagrid: e1,e2,...,e10.
Figure 5: Icosagrid with regular tetrahedral cells shown.
In either case of the handedness, there is a 20G at
the center of the structure. Fig. 9A and C show the
two chiralities (or handedness). We call one left-twisted
and other other right-twisted of the 20G respectively
and Fig. 9 A shows the superposition of both chiralities.
The two chiralities share the same point set (the 61
tetrahedral vertices in the 20G, the pink points shown
in Fig. 9) but have different connections turned on
(blue or red as shown in Fig. 9). The fact that tun-
ing on these tetrahedral cells split the icosagrid into
two chiralities is interesting and provoked the further
investigation into this structure. From this point on, we
refer to the icosagrid as this set of tetrahedral cells (or
tetrahedra). In terms of tetrahedral packing, the center
20G is a dense packing of 20 tetrahedra with maximally
reduced plane class (parallel plane set). It groups the 20
tetrahedra into a minimum of five crystal groups (the
maximum number of vertex sharing tetrahedral clusters
Corresponding Author •fang@quantumgravityresearch.org page 5 of 16
Figure 6: Icosagrid (B) separated into two opposite chiralities: left A) and right C).
B C
D E F
A
Figure 8:
(A) A small tetragrid local cluster with eight tetrahedral cells, four ”up” and four ”down”. B-F) The golden
composition process).
Figure 9: A) The right twisted 20G, B) The superposition of the left-twisted and right-twisted 20G.
in a crystalline arrangement that is divisible by 20 is 4).
Therefore the plane classes are reduced to a minimum
number of 10, compared with the 70 plane classes in the
evenly distributed vertex-sharing 20-tetrahedra cluster
as shown in Fig. 10. One can also see that from the
evenly distributed vertex-sharing-20-tetrahedra cluster
to the 20G, the golden twisting ”shifts” the gaps between
the tetrahedral faces to the canonical pentagonal cones,
one of which is marked with a white circled arrow in Fig.
10B. In other words, these canonical pentagonal cones
Corresponding Author •fang@quantumgravityresearch.org page 6 of 16
Figure 10:
A) An evenly distributed vertex-sharing-20-
tetrahedra cluster and B) a twisted 20G with
maximum plane class reduction.
are a signature of the maximum plane class reduction
in the vertex-sharing 20-tetrahedra packing.
The Fibonacci icosagrid
The icosagrid, like the pentagrid, is not a quasicrystal
due to the arbitrary closeness and therefore the infinite
number of cell shapes it posesses. Also it does not
satisfy the second property of a quasicrystal mentioned
in section II: for any finite patch in the quasicrystal,
one can find an infinite number of identical patches at
other locations, under translation and/or rotation. For
example, there is no other 20G possible in an icosagrid
of infinite size. In order to convert the icosagrid to a
quasicrystal, just as how we converted the pentagrid
to the Fibonacci pentagrid, we use Eq. 4 instead of
Eq. 3 for
xN
. As a result, the spacings between the
parallel planes in the icosagrid becomes the Fibonacci
sequence. A 2D projection of one of the tetragrids
before and after this modification is shown in Fig. 11A
and Fig. 11B respectively. Each tetragrid becomes
a Fibonacci tetragrid and the icosagrid becomes the
Fibonacci icosagrid (Fig. 11C).
In the Fibonacci icosagrid, 20Gs appear at the vari-
ous locations beside the center of the structure (marked
with white dotted circles in Fig. 11C). The arbitrary
closeness is removed and there are finite types of lo-
cal clusters. We investigated the local clusters with
the nearest neighbor configuration around a vertex (the
vertex configuration)
26
and the detailed results will be
published soon. We will briefly discuss about the results
in this paper. We introduce a term, degree of connection,
to the vertex configurations. It is defined as the number
of unit length connections a vertex has. The minimum
degree of connection for the vertices in the Fibonacci
icosagrid is three and the maximum degree of connection
is 60. Also since the Fibonacci icosagrid is a collection
of five-tetrahedra sets (the tetrahedra are of the same
orientation in each set), there are only 30 unit-length
edge classes (tetrahedra of each orientation have six edge
classes). From the above facts, it is not hard to deduce
that there is a finite amount of vertex configurations in
the Fibonacci icosagrid. A sample of the vertex configu-
rations is shown in Fig. 12. Our following paper will also
discuss how all the vertices of the Fibonacci icosagrid
live in the Dirichlet integer space–integers of the form
a
+
bτ
where
a
and
b
are integers. We have noticed that
the edge-crossing points (edge intersection points) in
the Fibonacci icosagrid also live in the Direchlet inte-
ger space. We define the edge-crossing configurations
as with
p/q
where
p
is the ratio of the segments the
edge-crossing point divide the first edge into and
q
is the
ratio for the second edge. There is a finite amount of
types of the edge-crossing configurations and the value
for
p
and
q
are simple expressions with the golden ratio.
The diffraction pattern of the 5-fold axis is shown in Fig.
13. The 2-fold and 3-fold diffraction patterns are also
seen from the Fibonacci icosagrid. It indicates that the
Fibonacci icosagrid is a three-dimensional quasicrystal
with icosahedral symmetry, considering only the point
set and/or the connections of both chiralities. The sym-
metry reduces to a chiral icosahedral symmetry if were
considered the connections of only one handedness.
The quasicrystal in the cell space of the Fibonacci
icosagrid is similar to the Ammann tiling which will
not be discussed in this paper. We have investigated
another way to generate a space-filling analog of the
Fibonacci icosagrid that is isomorphic to a subspace of
the quasicrystal in the cell space of the Fibonacci grid.
Most of the vertices of this quasicrystal are the centers
of the regular tetrahedral cells in the Fibonacci icosagrid.
There are three types of intersecting polyhedral cells: the
icosahedron, dodecahedron and the icosidodecahedron.
Fig. 14A shows the point set of this quasicrystal and
some of the polyhedral cells. This kind of quasicrystal
is very common in nature11,27 .
IV. The Quasicrystals derived from E8
Elser-Sloane quasicrystal
The Elser-Sloane quasicrystal is a four-dimensional qua-
sicrystal obtained via cut-and-project or Hopf mapping
from the eight-dimensional lattice E8
6, 20
. The mapping
matrix of the cut-and-project method is given below6
Π = −1
√5τI H
H σI,
where I=I4=diag {1,1,1,1},σ=√5−1
2and
H=1
2
−1−1−1−1
1−1−1 1
1 1 −1−1
1−1 1 −1
.
The point group of the resulting quasicrystal is
H4
= [3
,
3
,
5], the largest finite real four-dimensional
Corresponding Author •fang@quantumgravityresearch.org page 7 of 16
Figure 11:
A) A 2D projection of a tetragrid, B) a 2D projection of a Fibonacci tetragrid, C) a sample Fibonacci IcosaGrid.
Notice that 20Gs formed up at the locations marked with while dotted circles.
group
2
. It is the symmetry group of the regular four-
dimensional polytope, the 600-cell.
H4
can be shown
to be isomorphic to the point group of
E8
using quater-
nions
5
and it is inherently both four-dimensional and
eight-dimensional.
The unit icosians, a specific set of Hamiltonian quater-
nions with the same symmetry as the 600-cell, form the
120 vertices of the 600-cell with unit edge length. They
can be expressed in the following form:
(±1,0,0,0) ,1
2(±1,±1,±1,±1) ,1
2(0,±1,±1,±σ, ±τ)
with all even permutations of the coordinates. The
quarternionic norm of an icosian
q
=
(a, b, c, d)
is
a2
+
b2
+
c2
+
d2
, which is a real number of the form
A
+
B√5
, where
A, B ∈Q
. The Euclidean norm of
q
is
A
+
B
and it is greater than zero. With respect
to the quarternionic norm, the icosians live in a four-
dimensional space. It can be shown that the Elser-Sloan
quasicrystal is in the icosian ring, finite sums of the 120
unit icosians. Under the Euclidean norm, the icosian
ring is isomorphic to an
E8
lattice in eight dimension.
There are 240 icosians of Euclidean norm 1 with 120
being the unit icosians and the other 120 being
σ
times
the unit icosians. How these icosians corresponds to the
240 minimal vectors,
en, n
= 1
,
2
, ...,
8 of the
E8
lattice
is shown in Table 1.
One type of three-dimensional cross-sections of the
Elser-Sloane quasicrystal forms quasicrystals with icosa-
hedral symmetry and the other type forms quasicrystals
with tetrahedral symmetry. The Fibonacci icosagrid is
related to both types of quasicrystals. The icosahedral
cross-section of the Elser-Slaone quasicrystal has the
same types of unit cells (Fig. 14B) and is of the same
symmetry group as the space-filling analog of the Fi-
bonacci icosagrid (Fig. 14A) introduced in the earlier
section. The five-compound of the tetrahedral quasicrys-
tals turned out to be a subset of the Fibonacci icosagrid.
Details are discussed in the following sections.
Compound quasicrystals
The center of the Esler-Slaone quasicrystal is a unit-
length 600-cell. Since it is closed under
τ
scaling, there
is an infinite number of concentric 600-cells that are sizes
of powers of the golden ratio. There is also an infinite
number of unit-length 600-cells at different locations
in the quasicrystal but none of them intersect each
other. For the golden-ratio-length 600-cells, they do
touch or intersect with each other in 8 different ways
(Fig. 15). Each 600-cell has 600 regular tetrahedral
facets. The Elser-Sloane quasicrystal has two kinds of
three-dimensional tetrahedral cross-sections in relation
Corresponding Author •fang@quantumgravityresearch.org page 8 of 16
Figure 13:
Diffraction pattern down the five-fold axis of the point space obtained by taking the vertices of the tetrahedra.
Table 1: Correspondence between the icosians and the E8root vectors
Icosians E8root vectors
(1,0,0,0) e1+e5
(0,1,0,0) e1+e5
(0,0,1,0) e1+e5
(0,0,0,1) e1+e5
(0, σ, 0,0,)1
2(−e1−e2+e3+e4+e5+e6−e7−e8)
(0,0, σ, 0) 1
2(−e1−e2−e3+e4+e5+e6+e7−e8)
(0,0,0, σ)1
2(−e1−e2−e3−e4+e5+e6+e7+e8)
to the center 600-cell. The first, type I, (shown in Fig.
16A) is a cross-section through the equator of the center
600-cell. It has four vertex sharing tetrahedra at its
center with their edge length
τ
times the edge length of
the 600-cell. The second, type II, (shown in Fig. 17A)
is a cross-section through a facet of the 600-cell. As a
result, it has smaller unit-length tetrahedral cells, with
only one at the center of the cross-section. Both cross-
sections are quasicrystals. As we can see from Fig. 16A
and Fig. 17A, the type I cross-section is a much denser
packing of regular tetrahedra compared with type II.
They both appear as a subset of the Fibonacci tetragrid.
More rigorous proof will be included in future papers
but here we will present a brief proof in the following
paragraph.
The substitution rule for the Fibonacci sequence is
the golden ratio:
L→LS
and
S→L
. In other words,
both segments are inflated by the golden ratio
τ
scaling.
Therefore the Fibonacci chain is closed by
τ
scaling
and we can call it a
τ
chain. As we mentioned earlier,
the Elser-Sloane quasicrystal is also closed by
τ
. A
cross-section, as a subset of the quasicrystal will have
one-dimensional chains that are subsets of the
τ
chain.
Then it is not hard to prove that these tetrahedral
cross-sections are a subset of the Fibonacci tetragrid.
Using the same golden composition method as used
in the construction of the Fibonacci icosagrid with the
Fibonacci tetragrid, a compound quasicrystal of type
I (Fig. 16B) can be generated with five copies of type
I cross-sections. And similarly, a type II compound
quasicrystal (Fig. 16B) can be generated with 20 copies
of slices at type II tetrahedral cross-sections. Both
compound quasicrystals recovered their lost five-fold
symmetry from the tetrahedral cross-section and became
icosahedrally symmetric.
Corresponding Author •fang@quantumgravityresearch.org page 10 of 16
Figure 14:
A) Space-filling analog of the Fibonacci icosagrid, B) An icosahedral cross-section of the Elser-Sloane qua-
sicrystal.
Figure 15: Two-dimensional projection of the 8 types of golden-ratio-length 600-cell intersections.
V. Connections between the Fibonacci
icosagrid and E8
It is clear from the above discussion that both compound
quasicrystals are subsets of the Fibonacci icosagrid.
More over, the type II compound quasicrystal is a subset
of the type I compound quasicrystal 18. Indeed, the Fi-
bonacci icosagrid can be obtained from the cross-sections
of the Elser-Sloane quasicrystal with a process called
”enrichment” - defined as a procedure to add necessary
planar grids to make the cross-sections a complete Fi-
bonacci tetragrid. After this ”enrichment” process, the
compound quasicrystal becomes the Fibonacci icosagrid.
This establishes the connection between the Fibonacci
icosagrid, the Elser-Sloane quasicrystal and
E8
. In fact,
we intend to model the fundamental particles and forces
with the Fibonacci icosagrid, which is a network of
Fibonacci chains. Hitherto, we name the Fibonacci
icosagrid the qualsicrystalline spin-network.
Finally we want to point out the importance of the
Corresponding Author •fang@quantumgravityresearch.org page 11 of 16
A B
Figure 16: A) Type I tetrahedral cross-section of the Elser-Sloane quasicrystal, B) The compound quasicrystal
A B
Figure 17: A) Type II tetrahedral cross-section with τscaled tetrahedra, B) The more sparse CQC
Figure 18:
A) Tetrahedra of the FIG, B) Cyan highlighted tetrahedra belong to the Type II compound, C) Red highlighted
tetrahedra belong to the Type I compound
golden composition procedure for composing the com-
pound quasicrystal. It not only significantly reduces
the plane classes, compared to the evenly distributed
20-tetrahedra cluster, but also makes the composition
a quasicrystal. In other words, the reason we use the
golden composition process to compose the cross sections
Corresponding Author •fang@quantumgravityresearch.org page 12 of 16
together to form the compound quasicrystal is the same
for introducing the Fibonacci spacing in the Icosagrid,
to convert the structure into a perfect quasicrystal. Fig.
20 shows the steps of this convergence. An interesting
fact is that the dihedral angle of the 600-cell is the
golden angle plus 60 degrees. Notice that the process,
golden composition, which is manually introduced here
to make the compound quasicrystal a perfect quasicrys-
tal, comes up naturally in the Fibonacci icosagrid when
derived from the icosagrid. While the other process,
Fibonacci spacing, which is manually introduced to the
icosagrid to convert it to a quasicrystal, emerges nat-
urally in the Elser-Sloane quasicrystal, therefore the
three-dimensional cross-sections of the Esler-Sloane qua-
sicrystal.
VI. Conclusions
This paper has introduced a method, the Fibonacci
multigrid method, to convert the grid space into a qua-
sicrystal. Using this method, we generated an icosahe-
dral quasicrystal Fibonacci icosagrid. We have shown a
mapping between the Fibonacci icosagrid and the qua-
sicrystals derived from
E8
. The compound quasicrystals
derived from the Elser-Sloane quasicrystal thus from
E8
too, is a subset of the Fibonacci icosagrid and they
can be ”enriched” to become the Fibonacci icosagrid.
We conjecture that exploring such an aperiodic point
space based on higher dimensional crystals may have
applications for quantum gravity theory.
Corresponding Author •fang@quantumgravityresearch.org page 13 of 16
Fibonacci(
icosagrid(
Compound(
quasicrystal(
Golden(
Composi7on(
Tetragrid(
IcosaGrid(
Fibonacci(Spacing(
Subset(of(
Fibonacci(
tetragrid(
Elser-Sloane(
QC(
Golden(
Composi7on(
Fibonacci(
tetragrid(
Fibonacci
Spacing
3D(Tetra-
Slicing(
Enrichment(
Subset(of(
Cut-and-project((
or(Hopf(Mapping(
E8(
Golden(
Composi7on(
Figure 19: The relationships between FIG and CQC and how they are generated.
Corresponding Author •fang@quantumgravityresearch.org page 14 of 16
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