Cluster coverings as an ordering principle for quasicrystals

Cluster density maximization and (maximal) cluster covering have emerged as ordering principles for quasicrystalline structures. The concepts behind these ordering principles are reviewed and illustrated with several examples. For two examples, Gummelt’s aperiodic decagon model and a cluster model for octagonal Mn–Si–Al quasicrystals, these ordering principles can enforce perfectly ordered, quasiperiodic structures. For a further example, the Tübingen triangle tiling (TTT), the cluster covering principle fails to enforce quasiperiodicity, which sheds some light on the limitations of this approach.