An Icosahedral Quasicrystal and E8 derived quasicrystals
F. Fang∗ and K. Irwin
Quantum Gravity Research, Los Angeles, CA, U.S.A.
W
e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing
the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal
can also be thought 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 Fibonacci icosagrid, and they can be enriched to form the Fibonacci
icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice.
It is known that the combined structure and dynamics of all gravitational and Standard
Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this,
the Fibonacci icosagrid is a good candidates, for representing states and interactions
between particles and fields in quantum mechanics. We coin the name Quasicrystalline
Spin-Network (QSN) for this quasicrystalline structure.
I. Introduction
Until Shechtman et al.1 discovered them in nature,
quasicrystals were a pure mathematical curiosity, for-
bidden to exist physically by the established rules of
crystallography. This discovery intrigued scientists from
verious disciplines2–5. First there was a surge in the
interest in studying the mathematical aspects of qua-
sicrystals6–11. 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 properties12,13 and quasicrystal growth14.
As a field, quasicrystallography is still very new. The
interesting mathematics that accompanies the field is
relatively unexplored and provides opportunity for dis-
coveries that could have far reaching consequences in
physics and other disciplines.
In this paper, we introduce an icosahedral quasicrystal,
the Fibonacci icosagrid, that has an inclusion mapping
to a golden ratio based composition of three-dimensional
slices of the four-dimensional Elser-Sloane quasicrystal
projected from E815,16. This paper focuses on the geo-
metric connections between the Fibonacci icosagrid and
the E8 lattice while the next two papers will focus on
looking at these connections in an algebraic way or using
the Dirichlet integers (cite Ray and Amrik paper).
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-project17 and
dual-grid methods7. Section III applies the Fibonacci
multigrid method in three-dimension and obtains an
icosahedral quasicrystal, the Fibonacci icosagrid. After
obtaining this quasicrystal, we discovered an alternative
way of generating it using five sets of tetragrids. This
new way of constructing this quasicrystal reviewed a
direct connection between this quasicrystal and a com-
pound quasicrystal that is introduced in Section IV.
Important properties of the Fibonacci icosagrid are dis-
cussed too. For example, its vertex configurations, edge-
crossing types and space-filling analog. Section IV talks
about the compound quasicrystal that 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 quasicrystal3,6,18, it is generally believed that for a
structure to be a quasicrystal, it shall 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
1
e1
e2
e4
e3
e5
Figure 1: An example of a pentagrid, with e1, e2,...,e5 being 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.
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
dimensional crystal3,7 and the dual grid method3,7. We
independently developed a new method, the Fibonacci
multigrid method, for generating a quasicrystal. This
method is a special case of the generalized dual grid
method discussed in Socolar, et al’s paper7, which fo-
cuses on the cell space instead of the grid space, on
which we are focused. We will 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. As you can see
from Fig. 1, a pentagrid is defined as
~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 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
Corresponding Author •fang@quantumgravityresearch.org
page 2 of 17
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 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
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
rhombs shown in Fig. 2B.
3. The last step is to place the rhombs 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.
Corresponding Author •fang@quantumgravityresearch.org
page 3 of 17
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 tiles7), 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
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
(therefore infinite number of unit cells) between the
vertices(Fig. 3A). We have found a modification that
we can apply to the periodic grid to make the grid
itself quasiperiodic. After we discovered this method
independently, we realized that Socolar et. al. have
already published this method in 1986, although they
mainly focued on the quasicrystal in the cell space. The
modification is to change the Eq. 3 from periodic to
quasiperiodic7):
xN = T (N + α+
1
ρ
⌊
N
µ
+ β
⌋
),
(4)
where α, β, and ρ ∈ R, µ are irrational, and ”bc” is the
floor function and denotes the greatest integer. 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
6.
Therefore we call the modified pentagrid a Fibonacci
Pentagrid (Socolar et. al. used the same term). 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. Us-
ing this method, we generated an important three-
dimensional icosahedral quasicrystal that will be dis-
cussed 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, 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 discovered two
interesting properties of the icosagrid when only the
regular tetrahedral cells (Fig. 5) are ”turned on” (recog-
nized or shown). 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 will be 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
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, put together with a golden compo-
sition procedure achieved in the following manner(Fig.
8):
1. 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. Pick the four tetrahedral cells of the same orienta-
tion (Fig. 8B) and make four copies.
3. 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 ro-
tation angle of Cos−1( 3τ−1
4
), the golden rotation19
(Fig. 8C).
4. 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. Now expand the tetragrid associated with each of
the 4-tetrahedra sets by turning on the tetrahedra
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).
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. You can
see that they 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 these tetrahedral cells
separate 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
Corresponding Author •fang@quantumgravityresearch.org
page 4 of 17
Figure 4: The norm vectors of the icosagrid: e1, e2,...,e10.
Figure 5: Icosagrid with regular tetrahedral cells shown.
minimum of five crystal groups (the maximum number
of vertex sharing tetrahedral clusters in a crystalline
arrangement that is divisible by 20 is 4). Therefore the
minimum number of 10 plane classes, compared with the
70 plane classes in the evenly distributed vertex-sharing
20-tetrahedra cluster as shown in Fig. 10. You can
also see that from the evenly distributed vertex-sharing-
20-tetrahedra cluster to the 20G, the golden twisting
”expels” 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 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. Also it does not satisfy the second
property of a quasicrystal mentioned in section II: for any
finite patch in the quasicrystal, you can find an infinite
number of identical patches at other locations, with
translational and/or rotational transformations. For
Corresponding Author •fang@quantumgravityresearch.org
page 5 of 17
A
B
C
Figure 6: Icosagrid (B) separated into two opposite chiralities: left A) and right C).
Figure 7: Tetragrid with tetrahedral cells of two different orientation (Yellow and Cyan) and Octahedral gaps.
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)20 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 ra-
tio. 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 qua-
sicrystal with icosahedral symmetry, considering only
Corresponding Author •fang@quantumgravityresearch.org
page 6 of 17
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).
A
B
C
Figure 9: A) The right twisted 20G, B) The superposition of the left-twisted and right-twisted 20G.
the point set and/or the connections of both chiralities.
The symmetry reduces to a chiral icosahedral symmetry
if considering 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 nature.
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 E815,21. The mapping
matrix of the cut-and-project method is given below15
Π = − 1√
5
[
τI
H
H
σI
]
, where I = I4 = diag {1, 1, 1, 1}, σ =
√
5−1
2
and
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
group22. 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-
nions23 and it is inherently both four-dimensional and
eight-dimensional.
Corresponding Author •fang@quantumgravityresearch.org
page 7 of 17
A
B
Figure 10: A) An evenly distributed vertex-sharing-20-
tetrahedra cluster and B) a twisted 20G with
maximum plane class reduction.
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 deeply 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
Fibonacci 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 will be 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 , 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
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 power: L→ LS and S → L. In other
words, both segments are inflated by the golden ratio
τ . Therefore the Fibonacci chain is closed by τ 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 (more arguments
needed here - see my email).
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.
V. Connections between the Fibonacci
icosagrid and E8
It is obvious from the above discussion that both
compound quasicrystals are subsets of the Fibonacci
icosagrid. We also found that the type II compound
quasicrystal is a subset of the type I compound qua-
sicrystal 18. Indeed, the Fibonacci icosagrid can be
obtained from the cross-sections of the Elser-Sloane qua-
sicrystal with a process called ”enrichment” - defined
as a procedure to add necessary planar grids to make
the cross-sections a complete Fibonacci tetragrid. After
this ”enrichment” process, the compound quasicrystal
Corresponding Author •fang@quantumgravityresearch.org
page 8 of 17
e1
e2
B
A
C
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.
Table 1: Correspondence between the icosians and the E8 root vectors
Icosians
E8 root 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)
will become the Fibonacci icosagrid. This relationship
connects the Fibonacci icosagrid to the Elser-Sloane
quasicrystal and thus to E8. More detailed algebraic
interpretation of these connections will be talked about
in our coming papers. In this paper we focus more on
the geometric relationships (shown in Fig. 19 ) between
the Fibonacci icosagrid, E8 and its quasicrystals. It is
known that E8 (or two sets of E8) encodes all gauge
symmetry transformations between particles and forces
of the standard model of particle physics and gravity.
We intend to model the fundamental particles and forces
with the Fibonacci icosagrid, which is a network of Fi-
bonacci chains. From now on, we name the Fibonacci
icosagrid the qualsicrystaline spin-network (do I need
to say more here? I could but I don’t want to bring up
too many arguable stuff.).
Finally we want to point out the importance of the
golden composition procedure for composing the com-
pound quasicrystal. It not only significantly reduces
the plane classes, compared with the evenly distributed
20-tetrahedra cluster, but also makes the composition
a quasicrystal. In other words, the reason we use the
Corresponding Author •fang@quantumgravityresearch.org
page 9 of 17
Figure 12: A sample list of vertex configuration in the Fibonacci icosagrid.
Corresponding Author •fang@quantumgravityresearch.org
page 10 of 17
Figure 13: Diffraction pattern down the five-fold axis of the point space obtained by taking the vertices of the tetrahedra.
A
B
Figure 14: A) Space-filling analog of the Fibonacci icosagrid, B) An icosahedral cross-section of the Elser-Sloane qua-
sicrystal.
golden composition process to compose the cross sections
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 11 of 17
Figure 15: Two-dimensional projection of the 8 types of golden-ratio-length 600-cell intersections.
A
B
Figure 16: A) Type I tetrahedral cross-section of the Elser-Sloane quasicrystal, B) The compound quasicrystal
Corresponding Author •fang@quantumgravityresearch.org
page 12 of 17
A
B
Figure 17: A) Type II tetrahedral cross-section with τ scaled tetrahedra, B) The more sparse CQC
A
B
C
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
Corresponding Author •fang@quantumgravityresearch.org
page 13 of 17
Fibonacci
icosagrid
Compound
quasicrystal
Golden
Composi7on
Tetragrid
IcosaGrid
Fibonacc
i Spacing

Subset of
Fibonacci
tetragrid
Elser-Sloane
QC
Golden
Composi7on
Fibonacci
tetragrid
Fibonacci
Spacing
3D Tetra-
Slicing
Enrichment Subset of Cut-an
d-proje
ct
or Hop
f Mapp
ing
E8
Golden Composi7on Figure 19: The relationships between FIG and CQC and how they are generated.
Corresponding Author •fang@quantumgravityresearch.org
page 14 of 17
B
C
D
E
A
Figure 20: Convergence of the outer 20Gs, from A to E as the angle of rotation in the golden composition slowly increased
from 0 to the golden angle.
Corresponding Author •fang@quantumgravityresearch.org
page 15 of 17
References
1 M. Baake and U. Grimm. Aperiodic Order. Cambrage
University Press, 2013.
2 H. F. Blichfeldt. Finite Collineation Groups. Univer-
sity of Chicago Press, 1917.
3 N. D. Bruijn. Algebraic theory of penrose’s non-
periodic tilings of the plane. Indagationes Mathemat-
icae (Proceedings), 84(1):39–66, 1981.
4 A. Connes. Non-Commutative Geometry. Springer,
1988.
5 J. Conway and N. Sloane. Sohere Packing, Lattices
and groups. Springer, 1998.
6 V. Elser and N. J. A. Sloane. A highly symmetric four-
dimensional quasicrystal. J. Phys. A, 20:6161–6168,
1987.
7 B. et al. Photonic band gap phenomenon and opti-
cal properties of artificial opals. Physical Review E,
55(6):7619, 1997.
8 F. Fang, K. Irwin, J. Kovacs, and G. Sadler. Cab-
inet of curiosities: the interesting geometry of the
angle = arccos((3 - 1)/4) cabinet of curiosities: the
interesting geometry of the angle = arccos((3 - 1)/4).
ArXiv:1304.1771, 2013.
9 I. Fisher, Z. Islam, A. Panchula, K. Cheon, M. Kramer,
P. Canfield, and A. Goldman. Growth of large-grain
r-mg-zn quasicrystals from the ternary melt (r= y, er,
ho, dy and tb). Philosophical Magazine B, 77(6):1601–
1615, 1998.
10 C. L. Henley. Cell geometry for cluster-based qua-
sicrystal models. Physical Review B, 43(1):993, 1991.
11 D. Levine and P. J. Steinhardt. Quasicrystals. i. defi-
nition and structure. Physical Review B, 34(2):596,
1986.
12 R. Lifshitz.
The definition of quasicrystals.
arXiv:cond-mat/0008152 [cond-mat.mtrl-sci], 2000.
13 J. Miekisz and C. Radin. The unstable chemical
structure of quasicrystalline alloys. Physics Letters A,
119(3):133–134, 1986.
14 R. V. Moody and J. Patera. Quasicrystals and icosians.
Journal of Physics A: Mathematical and General,
26(12):2829, 1993.
15 S. Poon. Electronic properties of quasicrystals an
experimental review. Advances in Physics, 41(4):303–
363, 1992.
16 N. Rivier. A botanical quasicrystal. Le Journal de
Physique Colloques, 47(C3):C3–299, 1986.
17 J. Sadoc and R. Mosseri. The e8 lattice and quasicrys-
tals. Journal of non-crystalline solids, 153:247–252,
1993.
18 M. L. Senechal. Quasicrystals and Geometry. Cam-
brige University Press, 1995.
19 D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn.
Metallic phase with long-range orientational order and
no translational symmetry. Physical Review Letters,
53(20):1951, 1984.
20 J. E. Socolar, T. Lubensky, and P. J. Steinhardt.
Phonons, phasons, and dislocations in quasicrystals.
Physical Review B, 34(5):3345, 1986.
21 J. E. Socolar and P. J. Steinhardt. Quasicrystals. ii.
unit-cell configurations. Physical Review B, 34(2):617,
1986.
Corresponding Author •fang@quantumgravityresearch.org
page 16 of 17
22 P. J. Steinhardt and S.Östlund. Physics of quasicrys-
tals. World Scientific, 1987.
23 W. Steurer and S. Deloudi. Crystallography of
Quasicrystals: Concepts, Methods and Structures.
Springer Science and Business Media, 2009.
Corresponding Author •fang@quantumgravityresearch.org
page 17 of 17