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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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