Configurations Of Octagonal Ammann-Beenker Tiling And Growth Rules For Quasicrystal Model

There are two main models to describe the growth of quasicrystals: an energy-driven perfect quasiperiodic tiling model and an entropy-driven random-tiling model.For the growth of two-dimensional quasicrystals,research on the basis of quasiperiodic tiling model is an effective theoretical method.There are many geometrical models of two-dimensional octagonal quasicrystals,among which the Ammann-Beenker(hereinafter referred to as AB)tiling is widely adopted.In this thesis,through the study of configurations and geometric properties of AB tiling,its local growth rules are explored.By combining configurations properties with local growth rules,the perfect infinite two-dimensional eight-fold symmetric quasiperiodic tiling can be obtained.Firstly,several aspects of quasicrystals are briefly introduced,including the development history of quasicrystals,methods of studying quasicrystals,and applications of quasicrystals,etc.Secondly,three representative kinds of tiling models for two-dimensional quasicrystals are presented,including the two-dimensional five-fold symmetric Penrose tiling model,the twodimensional eight-fold symmetric AB tiling model and the two-dimensional twelve-fold symmetric Stampfli-G?hler tiling model.Next,a further research on the self-similar transformation and properties of AB tiling are made.A large AB tiling is obtained by self-similar transformation method and vertex configurations of AB tiling are determined.By analyzing the self-similar transformation of vertex configurations,their relationship during the transformation process and concentrations in the whole AB tiling are obtained.Further study on next-nearest configurations of AB tiling,a preliminary scheme for the growth of perfect eight-fold symmetric quasiperiodic tiling is put forward.Finally,based on the previous research results,we further refine growth rules for twodimensional eight-fold symmetric quasiperiodic tiling.The problem with the boundary vertices is solved at first.After that,a set of growth rules for the selection of the growth configurations of the boundary vertices is proposed.A special phenomenon is encountered when simulating growth according to the set of rules.In the study of this phenomenon,we find the growth essence of the two-dimensional eight-fold symmetric quasiperiodic tiling.Through our study,we find that when the continuity of Ammann lines and the correctness of configurations growth are guaranteed,it is possible to generate a perfect infinite AB tiling.