Cluster Models of Decagonal Tilings and Quasicrystals

Two different relaxed versions of Gummelt’s aperiodic cluster covering rules are considered. These relaxed covering rules produce certain random tiling type structures, which are precisely characterized, along with their relationships to various other random tiling ensembles. One variant of the relaxed covering rules allows for a natural realization in terms of a vertex cluster. It is shown with Monte Carlo simulations that the structures with maximal density of this cluster are the same as those produced by the corresponding covering rules. The entropy density of this covering ensemble is determined by Monte Carlo simulations, using entropic sampling techniques. Perfectly ordered structures, like those produced by Gummelt’s perfect covering rules, can be obtained in our model if a coupling between neighboring clusters is introduced. This coupling can order the random tiling type structures to perfectly ordered quasicrystals.