Configurations And Transformations Of Penrose Tiling

In this thesis, we systematically study configurations and transformations of Penrose tiling including the eight nearest neighbor(NN) and the next nearest neighbor(NNN) configurations.First, we briefly introduce the structural and geometrical properties of quasicrystals, mainly the one-dimensional and two-dimensional theoretical models, including the Fibonacci model, the Penrose tiling, the eight-fold and twelve-fold symmetric quasicrystals.Secondly, we focus on the structure and properties of two-dimensional quasiperiodic Penrose tilings. The self-similar transformation is used to generate Penrose tiling. In the structural properties of Penrose tilings, we study the self-similar transformation, distribution and correlation of the eight types of vertex configurations in the whole Penrose tiling pattern. Further, we get the next stage configurations of the eight vertex types, namely the twenty-three types of NNN vertex configurations. We also study the self-similar transformation of the twenty-three types of NNN vertex configurations. Based on the results on the NNN configurations, the next stage structures are explored.Finally, we analyze the NN and NNN concentrations in an infinite Penrose tiling and their relationship with the golden mean t, which equals( 5 -1) / 2. It is believed that in an infinite Penrose tiling, the self-similar transformation does not change. the proportionality of the thin and fat rhombi. This also holds for the configuration concentration of the eight types of NN vertex and twenty-three types of NNN vertex. So we can establish the related equations and further calculate the proportional values through Matlab. Then we get the relationship with the golden mean. At the end of the work, we calculate the number of the vertex configurations by numerical simulation and check the relationship of the eight types of NN vertex configurations and the twenty-three types of NNN vertex configurations. The numerical results are in good agreement with the analytical ones.