Fractal growth kinematics abstracted from snowflakes:topological evolution

作     者：Fan YANG Ya jun YIN Bin HE Qinshan FAN 

作者机构：Department of Engineering Mechanics School of Aerospace Key Laboratory of Applied Mechanics Tsinghua University Division of Mechanics Nanjing University of Technology 

出 版 物：《Applied Mathematics and Mechanics(English Edition)》 (应用数学和力学（英文版）)

年 卷 期：2015年第36卷第2期

页      面：243-264页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Project supported by the National Natural Science Foundation of China(Nos.10872114,11072125,and 11272175) the National Natural Science Foundation of Jiangsu Province(No.SBK201140044) the Fundation of Tutor for Doctor Degree of Higher Education of China(No.20130002110044) 

主　　题：fractal snowflake proportional movement self-similarity N-segment line topological evolution and topological invariant 

摘      要：Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.