A Single Cluster Covering for Dodecagonal Quasiperiodic Ship Tiling

Abstract

Single cluster covering approach provides a plausible mechanism for the formation and stability of octagonal and decagonal quasiperiodic structures. For dodecagonal quasiperiodic pattern such a single cluster covering scheme is still unavailable. Here we demonstrated that the ship tiling, one of the dodecagonal quasiperioidic structures, can be constructed from one single prototile with matching rules. A deflation procedure is devised by assigning proper orientations to the tiles present in the ship tiling including regular triangle, 30°-rhombus and square, and fourteen types of vertical configurations have been identified in the deflated pattern, which fulfill the closure condition under deflation and all result in a T-cluster centered at vertex. This result can facilitate the study of physical properties of dodecagonal quasicrystals.

Interpretation of some Yb-based valence-fluctuating crystals as approximants to a dodecagonal quasicrystal

“…5a. A closely related covering by a single type of dodecagonal patch has been recently proposed by Liao with coworkers [27]; our shield tile always resolves into a square, a pair of triangles and a thin rhombus tile in their model. The possibility of single-cluster covering using the dodecagon patch in Fig.…”

Directional Scaling Symmetry of High-symmetry Two-dimensional Lattices

“…This fact has been noticed and extensively studied by Strang 14 15 and Richert 16 . In studying the 1D incommensurate structures, we found that the function y = sin(2 πμ n), where is the platinum number which is related to the dodecagonal quasiperiodic structure 8 11 12 13 , reveals an interesting picture as illustrated in Fig. 2a .…”

“…   is the platinum number which is related to the dodecagonal quasiperiodic structure [8,[11][12][13], reveals an interesting picture as illustrated in Fig.2a. In the boundary regions defined by y 1   , the graph seems folding together, reminding us of Escher's paintings based on the concept of Poincaré disc.…”

“…is the platinum number which is related to the dodecagonal quasiperiodic structure [8,[11][12][13], reveals an interesting picture as illustrated in Fig. 2a.…”

“…In the effort of investigating the 1D incommensurate systems such as specified by the function cos(2 πqn ) 1 2 3 4 , where n is integer and the parameter q is an irrational number such as the golden ratio, and 2D quasiperiodic structures 5 6 7 8 , we came across to the question whether there is any directional scaling symmetry for the square lattice and the equilateral triangular lattice (hence also the honeycomb lattice). We found that the square lattice exhibits directional scaling symmetry along a direction at 22.5° with respect to the side of a unit square, with the drag center of scaling transformation falling on the lattice point, and the scale factor is , k = 1,2,3….…”