Applied Mathematics and Mechanics

, Volume 36, Issue 2, pp 243–264 | Cite as

Fractal growth kinematics abstracted from snowflakes: topological evolution

  • Fan Yang
  • Yajun YinEmail author
  • Bin He
  • Qinshan Fan


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.

Key words

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

Chinese Library Classification

O189 O415 

2010 Mathematics Subject Classification

74M25 82D80 


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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, School of Aerospace, Key Laboratory of Applied MechanicsTsinghua UniversityBeijingChina
  2. 2.Division of MechanicsNanjing University of TechnologyNanjingChina

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