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Golden, Quasicrystalline, Chiral Packings of

Tetrahedra

Fang Fanga, Garrett Sadlera, Julio Kovacs, Klee Irwina, b

Quantum Gravity Research Group, Topanga, California 90290, USA

Since antiquity, the packing of convex shapes has been of great interest to many scientists

and mathematicians [1-7]. Recently, particular interest has been given to packings of

three-dimensional tetrahedra [8-20]. Dense packings of both crystalline [8, 10, 15, 17, 19]

and semi-quasicrystalline [14] have been reported. It is interesting that a semi-

quasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic

simulation approach [14]. However, this packing is not perfectly quasicrystalline and the

packing density, while dense, is not maximal. Here we suggest that a "golden rotation"

between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline

packing. Using this golden rotation, tetrahedra can be organized into "triangular",

"pentagonal", and "spherical" locally dense aggregates. Additionally, the aperiodic

Boerdijk-Coxeter helix [23, 24] (tetrahelix) is transformed into a structure of 3- or 5-fold

periodicity—depending on the relative chiralities of the helix and rotation—herein

referred to as the "philix". Further, using this same rotation, we build (1) a shell structure

which resembles a Penrose tiling upon projection into two dimensions, and (2) a

"tetragrid" structure assembled of golden rhombohedral unit cells. Our results indicate

that this rotation is closely associated with Fuller's "jitterbug transformation" [21] and

that the total number of face-plane classes (defined below) is significantly reduced in

comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of

tetrahedra which is both dynamic and dense. The golden rotation that we report presents a

novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.

Packings of spheres and the platonic

solids have been of great interest to

mathematicians since ancient times [1-7]. With in

the past few years, rapid progress has been made

in the problem of dense packings of tetrahedra [9-

20]. Previous studies have mainly focus on

crystalline packings [8, 10, 15, 17, 19] and have

since appeared to reach a plateau. Recently, the

possibility of quasicrystals displaying tetrahedral

phases has been examined [27], and a subsequent

report [14] has interestingly and unexpectedly

produced a quasicrystalline packing of tetrahedra,

motivating the present work. In fact, the close

relationship between the tetrahedron,

icosahedron, and pentagonal bipyramid

(discussed below) strongly suggests the existence

of a dense, quasicrystalline packing of tetrahedra,

as pentagonal and icosahedral symmetry are

forbidden in crystals but are common in

quasicrystals (QC) [22].

In our analysis of tetrahedral packings and

assemblages, we have found utility in a geometric

object's "plane classes". Planes are said to belong

to the same plane class if and only if their normal

vectors are parallel. The number of plane classes

for a collection of polytopes is defined as the

number of distinct plane classes comprising the

collection's two-dimensional faces. A

characteristic feature of crystals and QCs is that

they have a finite number of plane classes. For

example, a three-dimensional cubic lattice has a

total of three plane classes. A three-dimensional

a These authors contributed equally to this work.

b Group director. Correspondence should be directed to: klee@quantumgravityresearch.org

1

generalization of the Penrose tiling [28, 29] may

be put into correspondence with rhombic

triacontahedron, and consequently comprises 15

plane classes. Finally, the tetrahedral,

quasicrystalline approximant produced by Haji-

Akbari et al. [14], in principle, comprises an

infinite number of plane classes due to the

underlying stochastic nature of that structure's

effecting Monte Carlo simulation.

The irrationally valued dihedral angle of

the tetrahedron presents a difficulty towards the

production of tetrahedral packings with long-

range order and/or a finite number of plane

classes. As an extreme example, consider the

Boerdijk-Coxeter helix [23, 24] depicted in

Figure 1e. This structure is well-known to be

aperiodic and, if extended to comprise an infinite

number of tetrahedra, produces infinitely many

plane classes.

In the pursuit of a tetrahedral structure

2

Figure 1 | Tetrahedral structures subjected to the golden rotation and associated junction types. Groups

of tetrahedra uniformly distributed about a common edge (a–c), and vertex (d). A linear arrangement of

tetrahedra forms the Boerdijk-Coxeter helix (tetrahelix) (e). "Twisting" these structures eliminates inter-

tetrahedral gaps and reduces the number of plane classes (f–j). The corresponding projections of facial junctions

are shown in (k–o). The relative angular relationship between faces in k, m, n, and o is related to the golden

ratio. Hence, this rotational relationship is referred to as the golden rotation. In its canonical form, the Boerdijk-

Coxeter helix is aperiodic, however, following application of the golden rotation, this structure forms the "philix" (j)

with a 3- or 5-tetrahedron periodicity, depending on the relative chiralities of the helix and rotation. Axial

projections of this helix show the 5-period (p, q) and 3-period (r, s) cases. The significant reduction of plane

classes obtained using these face junctions vs. a randomly generated set is shown in (t).

with long-range order and a small number of

plane classes, we have found useful five

categories of face junctions—defined as

orientational facial relationships between

coincident tetrahedra. These face junctions are

shown in Figures 1k–o and are referred to here as

3G, 4G, 5G, 20G, and FC (face centered),

respectively. The 3G, 4G, and 5G face junctions

may be obtained by "twisting" the n tetrahedra of

a collection (n < 6) arranged about a common

edge (Figures 1a–c) by an angle αn, expressed as

α

n=tan−1

(

√

cos2

(

θ

2

)

−cos2

(

θ

n

2

)

sin

(

θ

2

)

cos

(

θ

n

2

)

)

(1)

where θ = arccos(1/3) is the tetrahedral dihedral

angle and θn = 2π/n, about an axis passing

between the midpoints of the central and

peripheral edge. (In the case of the 20G,

tetrahedra are rotated by an angle of

arccos((√5+3)/(4√2)) about an axis that passes

from the icosahedral center through each

tetrahedron's exposed face.) This procedure is

equivalent to twisting the tetrahedra of Figures

1a–d uniformly until all gaps have been closed

(Figures 1f–i). The resulting angular relationship

between faces in a junction pair is determined by

β

n=2tan−1

(

√

cos2

(

θ

2

)

−cos2

(

θ

n

2

)

cos

(

θ

n

2

)

)

(2)

Assembling tetrahedral structures with these

facial junctions has the effect of reducing the

total number of plane classes on certain

3

Type Twisting angle, αnFacial rotation angle, βnFacial center offset Symmetry Num.

plane

classes

3G

arccos

(

1

√

6

)

≈65.91̊

arccos

(

−1

4

)

≈104.48̊

3

τ

+1

2

√

6

τ

2≈0.4564

3-fold 9

4G

arccos

(

1

√

2

)

=45 ̊

arccos

(

1

2

)

=60 ̊

1

2

√

3≈0.29

4-fold 4

5G

arccos

(

√

1

2+1

√

5

)

=arccos

(

τ

2

√

2

(

τ

+2

)

)

≈13.28 ̊

β

g≡arccos

(

1+3

√

5

8

)

=arccos

(

3

τ

−1

4

)

≈15.52̊

1

√

6

(

√

5+3

)

=1

2

√

6

τ

2≈0.078

5-fold 10

20G

arccos

(

√

5+3

4

√

2

)

=arccos

(

τ

2

2

√

2

)

≈22.24̊

β

g≡arccos

(

1+3

√

5

8

)

=arccos

(

3

τ

−1

4

)

≈15.52̊

2

√

6

(

√

5+3

)

=1

√

6

τ

2≈0.15

A510

FC n/a

β

g≡arccos

(

1+3

√

5

8

)

=arccos

(

3

τ

−1

4

)

≈15.52̊

0

3-fold 9

5-fold 10

Table 1 | Important parameters for facial junction types. "Twisting" angles are given by αn. The relative

angular relationship between face pairs in a junction are given by βn. The golden rotation angle is denoted by βn.

Values for facial center offsets represent the distance between facial centers of a junction pair. Here, A5 denotes

the orientation-preserving subgroup of the icosahedral symmetry group, the alternating group on 5 letters.

arrangements of tetrahedra (see Figure 1t).

A list of relevant parameters for these

"closed-twisted" structures is provided in Table 1.

Of note is the fact that the relative face-to-face

rotations for the 5G, 20G, and FC junctions are

all equal (βg = arccos((3τ – 1)/4) ≈ 15.5225°,

where τ is the golden proportion). Hence, this

value is referred to as the golden rotation, βg. In

these structures—as for the twisted 4G

arrangement—the number of plane classes is

reduced compared with the uniform

arrangements. Additionally, the twisting angle of

the 20G is equal to the rotational angle of Fuller's

"jitterbug transformation" which transmutes the

icosahedron to the cuboctahedron [21]. Taken

together, these two features strongly suggest a

"transformable" quasicrystalline packing of

tetrahedra. (See Figure 2.)

In attempting to construct a dense

quasicrystalline packing of tetrahedra, we have

employed both bottom-up and top-down

approaches. The former consists of using the

various face junctions of Figures 1k–o either (1)

to construct a skeleton of the quasicrystal,

subsequently filling gaps with appropriately

shaped tiles (e.g., "the shell model", Figure 3a),

or (2) to build construct a primitive cell of the

quasicrystal (e.g., the "tetragrid model", Figures

3d,e). These models may be assembled using the

following procedures.

Shell model construction

1. To an initial tetrahedron, four tetrahedra

are appended such that the facial centers

of their incident faces coincide. Each

appended tetrahedron is then rotated such

that the relative angle between the faces is

the golden rotation angle, βg.

2. Tetrahedra are then iteratively appended

to each "exposed" face (i.e., not already

affixed to another tetrahedron), and

oriented in the same manner as above.

3. Following each iteration, tetrahedra are

scaled down (with centers fixed) until all

collisions are eliminated.

Tetragrid model construction

1. Apply the golden rotation to the

constituent tetrahedra of a Boerdijk-

Coxeter helix to obtain a "philix" with

linear periods of 3 or 5 tetrahedra.

2. The periodic nature of the philix along its

axis allows for the formation of three-

dimensional grids of philices, by spacing

their points of intersection every 3 or 5

4

Figure 2 | Jitterbug transformation associated with handedness of the golden rotation. 8 selected faces

(red) on the icosahedron with octahedral symmetry (a). When rotated through an angle of α20 these faces

coincide with 8 outer faces of the "twisted" 20G (b). When the handedness of rotation is alternated, these faces

coincide with 8 faces of the cuboctohedron (c).

tetrahedra.

3. At each crossing point, there are three

directions along which to place philices,

each given by one of the opposing faces

in a dipyramid (two consecutive

tetrahedra sharing a face)

4. Since these grid lines form three families

of parallel lines, rhombohedra may be

formed. In the case of 3-periodic philices,

golden rhombohedra are produced,

whereas rhombohedra with a diagonal to

edge ratio of √2 are formed for 5-periodic

philices.

These models are shown in Figures 3a, d,

and e. The two-dimensional projection of the

shell model very closely resembles a Penrose

tiling (Figures 3b, c). In the tetragrid case

employing 3-periodic philices, a golden

rhombohedral primitive cell is obtained,

reminiscent of the golden rhombohedron prototile

of the three-dimensional Penrose tiling [28, 29].

These are encouraging indications for the

feasibility of generating quasicrystalline

packings using the face junctions described

above.

Our top-down approach involves the

production of a four-dimensional quasicrystal via

5

Figure 3 | The shell and tetragrid models. 3D shell structure with 5 iterations and a total of 284 tetrahedra (a)

and its 2D projection down a 5-fold symmetry axis (b). Coloring this projection reveals similarities to the Penrose

tiling (c). Tetragrid structure comprised of 5-periodic (d) and 3-periodic (e) philices. A layer of the 3D projection of

V', viewed from an axis of 10-fold symmetry, is shown in (f).

the cut-and-project method [22, 26] from the E8

lattice. Selected points of this structure are

subsequently projected into three-dimensional

space. The details of this procedure are as

follows:

1. Determine the set of edge of minimal

length l among all points of the projected

(8D to 4D) set. Call this set E.

2. From this set, select those vertices that

belong to 12 or more of these edges (i.e.,

vertices forming connections with at least

12 others at a distance of l). Let the set of

these vertices be denoted by V.

3. Select those edges of E whose end points

are contained in V. Denote this set by E',

and let V' denote the set of endpoints of

edges in E'.

4. Finally, project the vertices of V' from

four-dimensions to three-dimensions.

We expect this procedure to produce a

three-dimensional quasicrystal of irregular

tetrahedra. Through the application of our golden

and 4G face junctions, we anticipate the ability to

regularize tetrahedral cells while introducing

inter-tetrahedral gaps. We have achieved some

preliminary results in this direction. Figure 3f

shows a layer of the 3D projection from V',

viewed along a five-fold axis of symmetry. This

layer is clearly a cartwheel-type quasicrystal,

similar to the two-dimensional projection of the

shell model. An interesting fact about this

projection method is its relationship with the

"Sum of Squares" law [25], which states that for

any edge-transitive polytope projected

orthographically from m dimensions to n

dimensions, the ratio of the sum of squared edge

lengths is m/n (m > n > 0).

In conclusion, the golden rotation

described in this writing produces plane-class

reduction in tetrahedral packings, and is closely

related to Fuller's jitterbug transformation.

Consequently, we suspect that a tetrahedral

quasicrystal may be constructed using this golden

rotation, and that a dynamic quasicrystal may be

achieved using jitterbug-like transformation

between consecutive quasicrystal frames.

Towards this goal, both bottom-up and top-down

approaches are currently under investigation.

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