Self-similar transformation and quasi-unit cell construction of quasi-periodic structure with twelve-fold rotational symmetry

Abstract

The structural properties of a quasicrystal model with twelve-fold rotational symmetry are studied. We correct the errors in the self-similar transformation of the square-rhombus-hexagon tiling model proposed by Socolar. Based on the Stampfli-Ghler square-rhombus-triangle tiling model， the quasi-unit cell is successfully constructed， which can describe the dodecagonal quasiperiodic structure by the covering theory.

Construction Rules of P3 Penrose Tiling and Its MATLAB Implementation

“…It is important to note that when drawing a figure, the rule of symmetry and the coordinates of the symmetry point can be considered. For instance, if you need to locate a point p3 in the tiling (assume p3 is a point of symmetry about the line connected by p1 and p2), you can first locate the symmetry point about the line connected by p1 and p2 to determine the center position, and then locate the coordinates of point p3 [13]. Before executing the preceding procedure, we must define the expressions of two variables.…”

Covering Rules and Nearest Neighbour Configurations of the Dodecagonal Quasiperiodic Structure

“…If one could find a simpler quasi-unit cell, the coverings and configurations would be simplified. However, our previous work [13] showed this seems almost impossible although no rigorous proof was provided. The present work gives the fundamental properties of the dodecagonal quasiperiodic structure following the covering theory and it is expected that more progress can be made based on it.…”

“…After a careful investigation, a quasi-unit cell named T-cluster is chosen from many candidates. [13] A T-cluster consists of 7 squares, 20 regular triangles, and 2 rhombi, as shown in Fig. 2.…”

A Single Cluster Covering for Dodecagonal Quasiperiodic Ship Tiling

“…[12] In Ref. [13], one of the current authors, in studying the structural properties of the dodecagonal quasiperiodic ship tiling, [12] noticed the existence of a turtle-like cluster, which is dubbed the T-cluster and comprises seven squares, 20 regular triangles and two 30 ∘ -rhombuses, as highlighted in dark gray in Fig. 1.…”