A new Penrose tiling picture of quasicrystals emerges in which the structure is determined entirely by a single repeating cluster which overlaps (shares atoms with) neighbor clusters according to simple energetics. We first discuss the the structure in terms of somewhat artificial overlap rules, which play the same role as Penrose edge-matching rules in forcing a unique structure isomorphic to Penrose tiling. We then discuss how the overlap rules may arise from physically plausible energetics.

In this paper, we will focus on the case of decagonal quasicrystals whose quasiperiodic layers have the same symmetry as two-dimensional Penrose tilings. The two-dimensional analogue for the overlapping cluster model consists of decagonal tiles which overlap to form a covering of the two-dimensional plane. Petra Gummelt[15] first conjectured that decagonal tiles with appropriately chosen overlap rules can force a perfect quasiperiodic tiling and she outlined a proof. Jeong and Steinhardt[16] provided a simple, alternative proof which makes clear the isomorphism to two-tile Penrose tilings.

In the new paradigm, the atomic structure of the decagonal phase is determined entirely by the atom decoration of the overlapping decagon tiles. The decagon tiles represent decagonal cluster columns in the three-dimensional quasicrystals structure. For the case of a perfect quasiperiodic structure, the overlapping cluster can be dubbed a ``quasi-unit cell," since it is analogous to the conventional unit cell in a perfect periodic crystal. However, an important difference is that the atomic decoration of the quasi-unit cell is constrained: the atom configuration inside the quasi-unit cell must have the property that neighboring clusters can share atoms without significant distortion of their atomic arrangements.

The decagonal quasi-unit cell is
shown in Fig. 1(a)
with a decoration consisting of kites and
star-like shapes designed as a mnemonic for the overlap rules.
To force a perfect quasiperiodic covering isomorphic to a Penrose
tiling,
the decagons are permitted to overlap only in two
ways, *A* - or *B* -type, as shown in Fig. 1(b). With these overlap rules,
kite regions always overlap kite regions and star-like regions always
overlap star regions.
The isomorphism between decagons and Penrose tilings can be realized by
inscribing each decagon with a large Penrose
obtuse rhombus tile marked with single- and double-arrows, as illustrated in
Fig. 1(c). The decagon tiling in Fig. 1(d) is thereby converted into
a Penrose tiling. (Spaces are left where acute Penrose rhombi can
be inserted.)

The reduction of a Penrose tiling or decagonal phase to a single-repeating
cluster is a remarkable simplification. In terms of atomic modeling,
it means that the entire structure is defined by the atomic decoration
of the quasi-unit cell, similar to the familiar case of periodic crystals.
However, as an explanation for why quasicrystals form, a worrisome
aspect is the overlap rules, which appear to require complex
energetics. Hence, a second important
discovery[16] for the new paradigm has been that it is possible
to avoid matching or overlap rules altogether.
Instead, a perfect Penrose tiling can arise simply by
maximizing the density of some chosen atom cluster, *C* .

To illustrate the result, we idealize the discussion by considering arrangements of obtuse and acute rhombi, each representing some atom cluster. We assume no matching or overlaps rules. Without any further specifications, there are an infinite number of distinct tilings possible, including Penrose tiling, periodic tilings, and random arrangements. If there is no energetics to distinguish among the possibilities, the ground state is degenerate.

In realistic models,
it is natural to suppose that the tiles represent atom clusters
and that some tile cluster *C* is low energy compared to
the others. The degeneracy may, thereby, be broken. Jeong and
Steinhardt[16] have shown that, for an appropriately chosen
*C* cluster, the Penrose tiling emerges as the unique ground state.
That is, if one imagines that the
chosen cluster of tiles represents some energetically preferred
atomic cluster, then minimizing the free energy would naturally
maximize the cluster density. Jeong and Steinhardt have shown
that the Penrose tiling is the unique configuration with the maximum
density of *C* clusters.

An example is the cluster *C* shown in Fig. 2, although
other choices are possible.[17]
Two neighboring *C* 's
can share tiles. The greatest overlaps correspond precisely to the *A*
and *B* overlaps of the central decagonal region of *C* .
It is clear from this that
the *C* -cluster is inspired by the decagon covering described
above, although the precise relationship is quite subtle. For example,
if there are no explicit overlap rules to forbid certain arrangements
for the *C* cluster,
the hexagon tabs must introduced to prevent undesirable overlaps.
The non-trivial result is that maximizing the density of *C* clusters
automatically leads to
a structure in which the *C* -clusters are in one-to-one
correspondence with the quasi-unit cells in a single decagon tiling.
In particular, of all possible arrangements of obtuse and
acute tiles, the Penrose tiling is the
unique arrangement of *C* in which every *C* has an *A* or *B*
overlap with its neighbors.
Jeong and Steinhardt have also shown that the ground state remains
unique for a wide range of assignments of energies to clusters.