A new paradigm emerges from recent mathematical discoveries about Penrose tilings.[15,16,17] The structure of perfect quasiperiodic tilings can be interpreted in terms of a single quasi-unit cell and matching rules can be replaced by simple energetics which favor the formation of some specific atom cluster.

In the new paradigm, the atomic structure of a quasicrystal can be totally characterized by the decoration of a single cluster, rather than two clusters as the Penrose tiles would suggest. The result simplifies the problem of specifying and of determining the atomic structure since the only degrees of freedom are the atom types and the atom positions within the quasi-unit cell. Jeong and Steinhardt have shown that every atomic decoration of the conventional Penrose tiling can be reinterpreted in terms of an atomic decoration of the quasi-unit cell, although the converse is not true.[23] Some decorations of the quasi-unit cell are not equivalent to decorating each obtuse tile identically and each acute tile identically. In this sense, the quasi-unit cell picture encompasses more possibilities.

In the new paradigm, the atomic decoration of the quasi-unit cell encodes the symmetry of the structure. In the past, the symmetry of the structure has been determined by appealing to reciprocal space (see Chapter 6 by Mermin in this volume) or to perp-space (see Chapter 3 by Janot and de Boissieu in this volume). These indirect techniques can be substituted by a a real-space description. That is, there is a well-defined correspondence between the atomic decoration of the quasi-unit cell in real space and the space group symmetry of the structure.[23]

The new paradigm implies a closer physical relationship between quasicrystals and crystals. Now it appears that both can be described in terms of the close-packing of a single cluster or unit cell. In a crystal, the unit cell packs edge-to-edge with its neighbors. Quasicrystals correspond to a generalization in which the quasi-unit cells overlap. In both cases, the formation of the particular structure appears to be explained by a low-energy atomic cluster, although the atomic arrangement in the case of quasicrystals is constrained to allow overlap. Hence, the new paradigm makes plausible why many materials form quasicrystals and, at the same time, explains why quasicrystals are less common than crystals.

The new paradigm requires a mechanism to explain how quasicrystals
grow. If the quasicrystals are grown slowly, then thermodynamic relaxation
to the ground state is possible. However, some of the most perfect
quasicrystal samples,
including *AlNiCo* , are formed by rapid quenching.
In this volume, Socolar has described a scheme for solids
equivalent to Penrose patterns based on obtuse and acute rhombi
using
vertex rules and stochastic growth similar to diffusion limited
aggregation. This approach can be adapted to overlapping clusters.
(Janot[13] has already suggested a similar mechanism for
overlapping clusters, although his vertex rules allow random tilings
as well as perfect quasiperiodic tilings.)
If quasicrystals form due to a particular cluster being energetically
favored, a simpler
kinematic
mechanism may be through local atomic
rearrangement that increases the local density of the given atomic
cluster.

The overlapping cluster picture may also account for other physical properties of quasicrystals. Janot[14] has suggested that the cluster picture can naturally explain the inelastic neutron scattering properties, and the electrical and thermal conductivity behavior. Finally, the new paradigm suggests a natural explanation of why quasicrystals form, shedding new light on an old mystery. In the new picture, the problem reduces to the behavior of small atom clusters. Perhaps total energy calculations based on a modest number of atoms may be used to understand why quasicrystals form and to predict new ones.

From future structural
and kinematical studies of known quasicrystals, such as *AlNiCo* ,
these principles may be
established providing a new understanding of and new control
over the formation and structure of quasicrystals.