Cluster Approach for Quasicrystals

We consider a "cluster picture" to explain the structure and formation of quasicrystals. Quasicrystal ordering is attributed to a small set of low-energy atomic clusters which determine the state of minimum free energy. The nature of the ground state as $T\to 0$, depending upon the clusters and cluster energies, ranges from energetically stable and unique (as in Penrose tilings), to entropically stable and degenerate (as in random tilings).