Ever since their discovery in 1984,[1] quasicrystals have posed a perplexing puzzle: Why do the atoms form a complex, quasiperiodic pattern rather than a regularly-repeating, crystal arrangement? Most explanations have been based on some analogy with tilings. For example, in the Penrose tiling picture, the notion has been that atoms arrange themselves into two types of clusters analogous to the obtuse and acute rhombic Penrose tiles and have interactions which force connections between clusters analogous to the Penrose matching rules for tiles.[2] In the original formulation of the random tiling picture,[3] two clusters are also needed corresponding to obtuse and acute rhombic tiles without matching rules.

The tiling models suggest that the conditions necessary to form quasicrystals are significantly more complex than the conditions for forming quasicrystals. For example, the requirement of two types of cluster appears to be necessary to obtain quasiperiodicity. Yet, it is difficult to imagine energetics that permit two clusters in the just the right proportion in density (and exclude any other clusters), especially considering that most known quasicrystals are composed of metallic elements with central force potentials rather than rigid covalent bonding. In the case of the Penrose picture, there is the additional problem of finding energetics that impose the matching rules.

An important development has been the emergence of a new paradigm for the atomic structure of quasicrystals which simplifies the description of quasicrystal structure and suggests a simple thermodynamic mechanism for quasicrystal formation. In the new paradigm, quasicrystals are described in terms of a single, repeating cluster. The repeating cluster is analogous to the unit cell in periodic crystals. The novel feature is that the neighboring clusters ``overlap." Atoms in the overlap region are shared by the two clusters enabling the hypothetical surfaces that bound the clusters to interpenetrate. However, the sharing means that there is no duplication or crowding of atoms. The new picture does not have a simple interpretation in terms of tiling; the term ``covering" is more apropos. We shall refer to the general class of models whether applied to the random tiling or Penrose tiling pictures as ``overlapping cluster models."

Early work on overlapping cluster models focused on the decagonal
phase. From high resolution lattice imaging
studies of decagonal *AlCuCo* and *AlNiCo* , it was realized
that the atomic structure contains overlapping
decagonal atomic cluster columns. Various theoretical models emerged in which
the atomic structure was described entirely in terms of a single, repeating
decagon cluster with overlap rules which constrain the way neighbor
clusters can join.[4,5,6,7,8]
These early models differed in detail, but they shared the common feature
that
the overlap rules did not force a unique structure. For example,
Burkov's model[4] produced a structure analogous to a binary
tiling in which infinitely many arrangements are possible.
Similar experimental evidence for overlapping clusters has been
found for icosahedral quasicrystals.[9]-[12]
Janot has proposed
an overlapping cluster model[13]
to explain the structure and growth of the icosahedral phase using
self-similar growth algorithm rather than overlap rules.
As with the Burkov's overlap rules, the self-similar growth
model permits periodic and random structure as well as a perfect quasiperiodic
structure.

A remarkable mathematical breakthrough has been the discovery that clusters and overlap rules can be chosen so as to obtain a unique arrangement isomorphic to a Penrose tiling.[15,16,17] The discovery overturns the strongly-held view based on Penrose's construction that two atom clusters are necessary to force perfect quasiperiodicity. Instead of two incommensurate length scales arising from two tile shapes, the two length scales arise from the overlap rules which permit two different nearest neighbor distances between clusters. The fact that the overlap rules are as powerful as matching rules in forcing a unique structure is a subtle surprise. The new paradigm for quasicrystals that arises from these mathematical results has the potential of revolutionizing our concept of the structure, growth, and physical properties of quasicrystals.