A long-range hierarchical clustering model for constructing perfect quasicrystalline formations

Since their discovery, the unique structural signature of quasicrystals has posed a perplexing puzzle. Whereas significant progress has been made in determining their physical properties, a deeper understanding of the structure of quasicrystals remains an open question. Although available structural models provide important insights into understanding the quasiperiodic structural order, there is still significant information lacking concerning the determination of the long-range order of quasicrystals. A global multi-level hierarchical framework is presented that is able to describe the long-range translational and orientational order of the global empire pertaining to perfect Penrose tiling (PPT) models. In this order, the global empire is constructed of infinite levels of hierarchical clusters. Any cluster level, in this hierarchy, is composed of four smaller building clusters; two highly symmetrical clusters and their two fragments. Cluster configurations and the geometric arrangements of the multi-level clusters are determined entirely by a geometric progression of framework of nested decagrams. The proposed model suggests that the position of building units, locally and globally, is defined by one global framework, and not tiled based on local rules (matching or overlapping, etc). In this way, quasicrystal formations can grow rapidly ad infinitum without the need for any defects or mismatches.