An Icosahedral Quasicrystal asa Golden Modification of theIcosagrid and its Connection tothe E8 LatticeDr. Fang Fang∗, Klee Irwin Quantum Gravity Research

∗fang@QuantumGravityResearch.org

We present an icosahedral quasicrystal as a modification of the icosagrid, a multi-grid with 10 plane sets that are arranged with icosahedral symmetry. We usethe Fibonacci chain to space the planes, thereby obtaining a quasicrystal with

icosahedral symmetry. It has a surprising correlation to the Elser-Sloane quasicrystal1,a 4D cut-and-project of the E8 lattice. We call this quasicrystal the Fibonacci modifiedicosagrid quasicrystal. We found that this structure totally embeds another quasicrystalthat is a compound of 20 3D slices of the Elser-Sloane quasicrystal. The slices, which con-tain only regular tetrahedra, are put together by a certain golden ratio based rotation2.Interesting 20-tetrahedron clusters arranged with the golden ratio based rotation appearrepetitively in the structure. They are arranged with icosahedral symmetry. It turns outthat this rotation is the dihedral angle of the 600-cell (the super-cell of the Elser-Sloanequasicrystal) and the angle between the tetrahedral facets in the E8 polytope known asthe Gosset polytope.

Introduction

Until Shechtman et al.3 discovered them in nature,quasicrystals (QC) had remained nothing more than amathematical curiosity, forbidden to exist physically bythe established rules of crystallography. This discoveryserved to stoke the fires of intellectual endeavour in theminds of scientists from many disciplines4–6. Enthusi-asm for the mathematical exploration of QCs largelydied down after an initial peak in interest. These initialefforts studied, among other things, the two methods ofgenerating QCs (cut and project method from a higherdimensional crystal or the multigrain method7,8), thevertex types9, the combinatorics(non-commutative ge-ometry)of QCs10 and the phason dynamics11. Since

then, and in recent years, the focus of the majorityof research in the field has shifted toward the physicalaspects of QC. Namely their electronic and optical prop-erties12,13 and their growth14. As a field, quasicrystal-lography is still very new. The interesting mathematicsthat accompanies the field is relatively unexplored andprovides opportunity for discoveries that could have farreaching consequences in physics and other disciplines.

In this paper, we introduce the Fibonacci Icosagrid(FIG) QC along with an unexpected mapping to agolden ratio based composition of 3D slices of the Elser-Sloanes 4D QC projected from E8. E8 encodes all gaugesymmetry transformations between particles and forcesof the standard model of particle physics and gravity.

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Therefore, this novel quasicrystal may be useful fora loop quantum gravity type approach to unificationphysics.

The Fibonacci Icosagrid

Icosagrid and FCC

The 3D icosagrid, a portion of which is shown in Figure1A, is made of 10 sets of equidistant planes that areparallel to the facets of an icosahedron. An alternativeway of looking at the icosagrid is as five FCC latticesrotated in relation to one another with a golden ratiobased angle, Cos−1(3φ−14 )2, henceforth referred to asthe Golden Angle.

The FCC lattice can be thought of as a space-fillingcombination of regular tetrahedral and octahedral cells.In Figure 1A just the tetrahedral cells have been coloredin. The above mentioned rotation is achieved in thefollowing manner:

1. Take one tetrahedron from an FCC lattice andconnect to it another so that one of their faces iscompletely shared.

2. Pick one of the vertices of this shared face and fixit into place.

3. Now, keeping the vertex fixed and the faces touch-ing, rotate the two touching faces from one anotherto create the Golden Angle between them.

4. Repeat the process three more times adding tetra-hedra such that the same vertex remains fixed andthe rotations are always in the same direction.

By filling the space with the FCC lattices associatedwith each of the five tetrahedra, the object shown inFigure 1A is obtained, the Icosagrid. The core of theicosagrid is a chiral 20-tetrahedron cluster (20G) shownin Figure 1B.

The 20G is an interesting object. It can be thoughtof as five 4-groups (Figure 2) of tetrahedra rotated fromone another in the above fashion. Perhaps the mostfascinating feature of the 20G is its inherent chirality.When we color in the tetrahedral cells of the FCClattice we make a choice between two orientations oftetrahedra. This choice translates into right or lefthanded 20Gs. Figure 9 A and C show the right and lefttwisted 20Gs respectively. Both share the same pointset but have different connections ”turned on”.

Figure 1: A) The 10 equidistant plane sets or alternativelythe five FCC lattices rotated from one another,B) The center of the cluster in figure 1 - A chiral20 tetrahedron cluster with relative face rotationalangle of Cos−1( 3φ−1

4 )

Figure 2: One of the five 4-groups of tetrahedra that makeup the 20G

FIG QC derived from the icosagrid

The icosagrid (Figure 1A) is not a quasicrystal due tothe arbitrary closeness of its points. In order to createa quasicrystal from the structure a modified spacing isintroduced. The quasicrystal in Figure 3B is obtainedby modifying the spacing between the parallel planesin each FCC set of the Icosagrid to be the Fibonaccichain,

SLLSLSLLS...

where S = 1 and L = φ (see figure 3A). We call thisobject the Fibonacci Icosagrid, FIG. The diffractionpattern of the five-fold axis is shown in figure 4. Clearly,the FIG is a 3D quasicrystal with icosahedral symmetry.One of its important features, the central 20G, will bediscussed in detail in a later section.

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Figure 3: A) A slice of the Fibonacci modified FCC viewedfrom a vector perpendicular to the faces of thetetrahedra, B) The five sets of FCC rotated fromone another and to create the compounded qua-sicrystal.

Figure 4: Diffraction pattern down the five-fold axis of thepoint space obtained by taking the vertices of thethe tetrahedra.

The E8 Quasicrystals

Elser-Sloane quasicrystal

The Elser-Sloane quasicrystal (ESQC) is a 4D quasicrys-tal obtained as a projection of the eight-dimensional lat-tice E8. It is a highly symmetric ([3, 3, 5]) quasicrystalmade of intersecting 600-cells. The projection mappingmatrix is given below

Π = − 1√5

[φ−1I HH φI

]Where I = I4 = diag {1, 1, 1, 1}, φ =

√5−12 and

H =1

2

−1 −1 −1 −1

1 −1 −1 11 1 −1 −11 −1 1 −1

.Compound quasicrystal

The ESQC has two cross-sections that are 3D qua-sicrystals with tetrahedral symmetry. The first, Type

I, (shown in Figure 5A) has larger tetrahedra and fourtetrahedra at its center, while the second, Type II,(shown in Figure 6A) has smaller tetrahedra that are1/φ times the length of those in the Type I and hasonly one at its center.

Five copies of Type I, rotated from one another bythe Golden Angle generate an icosahedrally symmetriccompound quasicrystal (CQC) (figure 5B). And 20copies of Type II rotated by the same angle make asimilar, more sparse, CQC (Figure 6B). They sharethe same core structure as the FIG, the 20G shown inFigure 1B.

Figure 5: A) Cross-section of the Elser-Sloane quasicrystal,B) The compound quasicrystal

Figure 6: A) The second type of cross-section with φ scaledtetrahedra, B) The more sparse CQC

Convergence of the 20Gs with the goldenrotation

Why use the Golden Angle for composing the CQC?Besides creating a deep connection between the FIGand the ESQC, it converges the 5 or 20 slices of theESQC into a perfect non-crashing quasicrystal. Figure7 shows the steps of this convergence.

Mapping between the FIG and theCQCs

The FIG quasicrystal and CQC are built in completelydifferent ways. Yet, to our surprise, the FIG quasicrys-

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Figure 7: Convergence of the outer 20Gs

tal completely embeds the Type I and Type II CQCs.And the Type I CQC also completely embeds in TypeII. All tetrahedra shown in Figures 8 A-C are membersof the FIG quasicrystal. The cyan and red tetrahedrabelong to the CQCs and are subsets of the FIG. Thekey to this perfect mapping is the Fibonacci chain mod-ification of the FCC lattices (Figures 3A, 5A and 6A)and the golden ratio rotation.

Figure 8: A) Tetrahedra of the FIG, B) Cyan highlightedtetrahedra belong to the Type II compound, C)Red highlighted tetrahedra belong to the Type Icompound

The 20G, 600-cell compound and theirGolden Angle

Both the centers of the FIG and the CQC are 20Gs(Figure 1B). The right and left chiralities share thesame point set, as shown in Figure 9 A-C. The core ofthe ESQC is a 600-cell.

A 120 cell is a compound of five 600-cells rotatedfrom one another by the Golden Angle. Figures 10 Athrough C show a 3-dimensional projection of part ofthe compound, revealing the same chiral properties asfound in the 20G. We can say that a deep relationshipexists between these three structures: (1) FIG, (2) CQC,and (3), the 600-cell and its five compound.

Mapping between the ESQC, Tsai-Type QCand FIG

Projecting the center (600-cell) of the ESQC to 3D(Figure 11A) generates the point set of a Tsai-type qua-sicrystal (Figure 11B)15, except for the difference in thefirst layer. However, the full permutation cycle of thecenter tetrahedron in the Tsai-type quasicrystal forms

Figure 9: A) The right twisted 20G, B) The superpositionof both the left and right twisted 20 groups, C)the left twisted 20 group

Figure 10: Partial 3D projection of the five 600 cells whichcompound to make the 120 cell.

an icosahedrally symmetric pattern. This permutationset of the Tsai-type quasicrystal relates each tetrahedralposition to the others by the Golden Ratio Rotation inthe FIG. Accordingly, the Tsai-type quasicrystal maybe a subspace of the FIG quasicrystal.

Figure 11: A) Projection of the ESQC to 3D, B) The shellsof the Tsai type quasicrystal

Summary

This paper has introduced the surprisingly perfect map-ping between two 3-dimensional icosahedral quasicrys-tals, with one being built by modifying the icosagridand the other a composite of slices of the Elser-Sloanequasicrystal. The obvious link between these two qua-sicrystals is that they both can be thought of as com-posites of FCC lattices — with their 1D periodic chainsmodified to the Fibonacci chain. The fundamentalfeature that these two quasicrystals share, the 20G,resembles the 600-cell 5-compound in 4D. We conjec-ture that the FIG may have applications in describing

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the underlying mathematics of physical quasicrystalssuch as the Tsai-Type. Furthermore, we suggest thatexploring such aperiodic point spaces based on higherdimensional crystals may have applications for quantumgravity theory.

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