Vol. 126 (2014)
ACTA PHYSICA POLONICA A
No. 2
Proceedings of the 12th International Conference on Quasicrystals (ICQ12)
An Icosahedral Quasicrystal
as a Packing of Regular Tetrahedra
F. Fang∗, J. Kovacs, G. Sadler and K. Irwin
Quantum Gravity Research
We present the construction of a dense, quasicrystalline packing of regular tetrahedra with icosahedral symme-
try. This quasicrystalline packing was achieved through two independent approaches. The first approach originates
in the Elser–Sloane 4D quasicrystal. A 3D slice of the quasicrystal contains a few types of prototiles. An initial
structure is obtained by decorating these prototiles with tetrahedra. This initial structure is then modified using
the Elser–Sloane quasicrystal itself as a guide. The second approach proceeds by decorating the prolate and oblate
rhombohedra in a 3-dimensional Ammann tiling. The resulting quasicrystal has a packing density of 59.783%. We
also show a variant of the quasicrystal that has just 10 “plane classes” (compared with the 190 of the original),
defined as the total number of distinct orientations of the planes in which the faces of the tetrahedra are contained.
This small number of plane classes was achieved by a certain “golden rotation” of the tetrahedra.
DOI: 10.12693/APhysPolA.126.458
PACS/topics: 61.44.Br, 61.46.–w
1. Introduction
The emergence of ordered structure as a result of self-
assembly of building blocks is far from being well under-
stood, and has received a great deal of attention [1–3].
Understanding this underlying scheme (an example of
“hidden order”) may open up the doors for explaining
phenomena that have thus far remained elusive (e.g.,
anomalies in water [4]). Studies have shown that order in
structure is directly related to its fundamental building
blocks [3]. Recently, 2D quasicrystalline order has been
achieved in a tetrahedral packing, when thermodynamic
conditions are applied to an ensemble of tetrahedra [5].
This discovery has stimulated further investigations into
quasicrystalline packings of tetrahedra.
Fig. 1. Distribution of bond lengths relative to the
tetrahedral edge length in the network.
This paper presents an icosahedrally symmetric qua-
sicrystal, as a packing of regular tetrahedra. We obtain
this by using two approaches, with very different ratio-
nales, but ultimately obtain the same quasicrystalline
packing. This structure might also help in the search
for a perfect 4-coordinated quasicrystal [6]: even though
the tetrahedra in our quasicrystal are packed in such a
∗corresponding author; e-mail:
fang@quantumgravityresearch.org
way that faces of neighboring tetrahedra can be paired
up (yielding a 4-connected network between tetrahedra
centers), the structure cannot be considered physically as
4-coordinated, due to the excessively high ratio between
the longest and shortest bonds stemming from each tetra-
hedron center (Fig. 1).
2. First approach
This is a guided decoration of the 3D slice of the
Elser–Sloane quasicrystal [7] (Fig. 2a), which contains
four types of prototiles: icosidodecahedron (IDHD), do-
decahedron (DHD), and icosahedron (IHD) — each as a
section (boundary of a cap) of a 600-cell — and a golden
tetrahedron. The way each of these polyhedra (Fig. 2c
— IHD, e — DHD, IDHD (not shown)) is decorated is
guided by the arrangement of the tetrahedra in the 600-
cells of the Elser–Sloane quasicrystal. For example, the
IDHD is a slice through the equator of the 600-cell and
is the boundary of a cap of 300 tetrahedra. Projecting
these 300 tetrahedra into the IDHD hyperplane results in
distorted tetrahedra, and gaps appear when tetrahedra
are restored (while avoiding collisions) back to a regular
shape. To maintain a higher packing density and a better
tetrahedral coordination [6], extra tetrahedra can be in-
troduced to fill in such gaps or, alternatively, some tetra-
hedra may be removed from each shell before regular-
ization to avoid conflicts resulting from collisions. Both
methods result in exactly the same quasicrystalline struc-
ture, as shown in Fig. 2f. This method can be thought
as a decoration of the 3D slice of the Elser–Sloane qua-
sicrystal, guided by the quasicrystal itself.
3. Second approach
This is an direct decoration of the 3D Ammann
tiling [8]. A 3D Ammann tiling can be generated by
cut-and-project from the Z6 lattice, and contains two
prototiles: a prolate rhombohedron and an oblate rhom-
bohedron.
The decoration consists in placing a 20-tetrahedron
“ball” (Fig. 2c) at each vertex, and a 10-tetrahedron
(458)
An Icosahedral Quasicrystal as a Packing of Regular Tetrahedra
459
Fig. 2.
(a) 3D slice of the ElserSloane quasicrystal.
(b) 10-tetrahedron ring.
(c) 20-tetrahedron ball
(icosahedral cluster), obtained by capping the ring with
two 5-tetrahedron groups. (d) 40-tetrahedron cluster,
obtained by placing a tetrahedron on top of each face
of the icosahedron in (c). (e) 70-tetrahedron dodecahe-
dral cluster, obtained by adding 30 more tetrahedra in
the crevices in (d). (f) Patch of the resulting quasicrys-
tal, which contains the clusters shown in (c), (d) and
(e) around its center.
ring (Fig. 2b) around each edge of each of the rhom-
bohedra, and then removing all those tetrahedra that do
not intersect the rhombohedron. This process creates
some clashes inside the oblate rhombohedron, and af-
ter excluding the appropriate tetrahedra, the packing re-
mains face-to-face and therefore the resulting network of
tetrahedron centers is 4-connected. (However, the bond-
length distribution has a rather long tail, Fig. 1b.) This
process yields some pairs of tetrahedra with large over-
laps shared between the balls and the rings, providing
degrees of freedom to choose either tetrahedron of each
pair, which in turn translates into the ability to ip 3
tetrahedra in each face of the prolate rhombohedra, sug-
gesting a novel phason mechanism for this type of qua-
sicrystal (Fig. 3b,c).
The oblate rhombohedron, due to its atness, does not
have these degrees of freedom, there being only one way
to choose the tetrahedra so that the resulting decoration
will have a 3-fold axis of symmetry (Fig. 4).
4. Twisting and plane-class reduction
The quasicrystal can also be obtained by placing a
40-tetrahedron cluster (Fig. 5a) at each vertex of the
3D Ammann tiling. Applying a golden rotation [9] of
arccos(τ2/2
√
2) ≈ 22.2388◦ (where τ = 12 (1 +
√
5) is the
golden ratio) to each of the tetrahedra in the cluster,
around an axis running through the tetrahedral center
and the center of the cluster, yields a twisted quasicrys-
tal (Fig. 5b  twisted 40-tetrahedron cluster, c  the
twisted quasicrystal). This golden twist reduces the total
number of plane classes from 190 to 10. The resulting rel-
ative face rotation at each face junction between adja-
cent tetrahedra is arccos( 14 (3τ − 1)) =
1
3 arccos(11/16) ≈
15.5225◦.
5. Analysis
Diraction patterns of the non-twisted quasicrystal
(Fig. 2f) reveal 2-, 3-, and 5-fold symmetry planes
(Fig. 6), conrming the icosahedral symmetry of this
quasicrystal.
Its packing density is
65
16464 (208800
√
2 +
64215
√
5 − 45499
√
10 − 294845) ≈ 0.59783. The deriva-
Fig. 3. Decoration of the prolate rhombohedron (a).
(b), (c) The two conformations that 3 of tetrahedra as-
sociated to each face can have. There are 52 tetrahe-
dra decorating the whole prolate rhombohedron: 16 of
them lie completely inside it, while the remaining 36 are
shared 50% with corresponding tetrahedra in a face of
an adjacent rhombohedron in the packing. The 3 tetra-
hedra (in each face) that can ip consist of 2 shared
ones and one internal one.
Fig. 4. Decoration of the oblate rhombohedron (a).
(b) The decoration consists of 36 tetrahedra, each
shared 50% with corresponding tetrahedra in an adja-
cent rhombohedron. (c) Here only the 24 tetrahedra
coming from the 20-tetrahedron balls (each centered
at a vertex of the rhombohedron) are displayed, in order
to get a better feel of the arrangement. The other 12
tetrahedra shown in (b) come from the 10-tetrahedron
rings (each centered at the midpoint of a rhombohe-
dron edge).
Fig. 5.
(a) Non-twisted and (b) twisted 40-tetrahedron
clusters. (c) Patch of the twisted quasicrystal.
460
F. Fang, J. Kovacs, G. Sadler, K. Irwin
tion of this expression is too lengthy to be included here,
and was done using theMathematica software. (AMath-
ematica notebook containing code for this calculation
is available upon request.) Basically, the calculation is
done for each rhombohedron, by solving equations that
minimize the rhombohedron's edge length relative to the
tetrahedron's, in such a way that the various tetrahedra
just touch each other. The degrees of freedom allowed
in this process are shifts in the directions of the rhom-
bohedron's edges and radially from them. Finally, the
density values for both rhombohedra are combined using
the fact that the relative frequencies of occurrence of the
prolate and oblate rhombohedra in the Ammann tiling is
the golden ratio.
The network of tetrahedral centers (Fig. 1a) is
4-connected, although it cannot be considered what the
chemistry community would call 4-coordinated, due to
the rather wide range of bond lengths (Fig. 1b), with a
ratio of 1.714 between the longest and shortest bonds.
Fig. 6. Diraction patterns on the 2-fold (a), 3-fold
(b), and 5-fold (c) planes of the quasicrystal dened
by the tetrahedral centers.
Fig. 7. Stereo pair of the network of tetrahedral cen-
ters.
6. Summary and outlook
Using two seemingly unrelated approaches, we have
surprisingly obtained the same quasicrystalline packing
of regular tetrahedra with global icosahedral symmetry.
After the fact, this convergence turned out not to be
by chance. The reason is that a 3-dimensional slice (in
the appropriate orientation) of the 4-dimensional Elser
Sloane quasicrystal can be obtained directly by cut-and-
project from the D6 lattice [10], D6 being a sublattice of
Z6 (and Z6 being a sublattice of 12D6, the 6-dimensional
face-centered cubic lattice).
To our knowledge, an icosahedrally symmetric packing
of tetrahedra with this relatively high density of almost
0.6 has not been shown before. Moreover, this pack-
ing provides a 4-connected network with bond lengths
ranging from 0.437 and 0.749 (in units of the tetrahe-
dron's edge length), a 1:1.714 ratio. These features are
non-trivial among icosahedral arrangements of tetrahe-
dra. For applications, an important step would be to
shrink the range of bond lengths to what can be consid-
ered as a realistic 4-coordinated network.
We also considered the number of plane classes and
ways to reduce it to the minimum possible. The above
packing has 190 plane classes. By applying what we call
the golden twist to each tetrahedron, the 190 plane
classes of the original quasicrystal coalesce to only 10.
(This is easily seen to be the minimum possible for an
icosahedral arrangement of tetrahedra.)
This quasicrystal also suggests interesting alternatives
to the classical phason ips, as shown in Fig. 3. We
are investigating the dynamics of this and other types of
phasons and their potential physical applications.
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