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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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