Fractal growth kinematics abstracted from snowflakes: topological evolution

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• 作者：Fan Yang (1)
  Yajun Yin (1)
  Bin He (2)
  Qinshan Fan (1)
  
• 关键词：fractal snowflake ; proportional movement ; self ; similarity ; N ; segment line ; topological evolution and topological invariant ; O189 ; O415 ; 74M25 ; 82D80
• 刊名：Applied Mathematics and Mechanics
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：36
• 期：2
• 页码：243-264
• 全文大小：1,076 KB
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  2. Prozorov, T., Palo, P., Wang, L., Nilsen-Hamilton, M., Jones, D., Orr, D., Mallapragada, S. K., Narasimhan, B., Canfield, P. C., and Prozorov, R. Cobalt ferrite nanocrystals: out-performing magnetotactic bacteria. / ACS NANO, 1, 228鈥?33 (2007) CrossRef
  3. Du, J., Niu, H., Dong, J. Y., Dong, X., and Han, C. C. Self-similar growth of polyolefin alloy particles in a single granule multi-catalyst reactor. / Advanced Materials, 20, 2914鈥?917 (2008) CrossRef
  4. Yin, Y., Zhang, T., Yang, F., and Qiu, X. M. Geometric conditions for fractal super carbon nanotubes with strict self-similarities. / Chaos, Solitons and Fractals, 37, 1257鈥?266 (2008) CrossRef
  5. Yin, Y., Yang, F., Zhang, T., and Fan, Q. S. Growth condition and growth limit for fractal super fibers and fractal super tubes. / International Journal of Nonlinear Sciences and Numerical Simulations, 9(1), 96鈥?02 (2008)
  6. Yin, Y., Yang, F., Fan, Q. S., and Zhang, T. Cell elements, growth modes and topology evolutions of fractal supper fibers. / International Journal of Nonlinear Sciences and Numerical Simulation, 10(1), 1鈥?2 (2009) CrossRef
  7. Yin, Y. J., Yang, F., Li, Y., and Fan, Q. S. Fractal geometry and topology abstracted from hair fibers. / Applied Mathematics and Mechanics ( / English Edition), 30(8), 983鈥?90 (2009) DOI 10.1007/s10483-009-0804-5 CrossRef
  8. Yin, Y., He, B., Yang, F., and Fan, Q. S. Centroid evolution theorem induced from fractal super fiber or fractal super snowflakes. / International Journal of Nonlinear Sciences and Numerical Simulation, 10(6), 805鈥?09 (2009) CrossRef
  9. Yin, Y., Yang, F., and Fan, Q. S. Growth kinematics of fractal super snowflakes. / Chinese Science Bulletin, 55(7), 573鈥?80 (2010) CrossRef
  10. Yin, Y. J., Li, Y., Yang, F., and Fan, Q. S. Multiple-cell elements and regular multifractals. / Applied Mathematics and Mechanics ( / English Edition), 31(1), 55鈥?5 (2010) DOI 10.1007/s10483-010-0106-2 CrossRef
  11. Yin, Y. J., Yang, F., and Fan, Q. S. Isologous fractal super fibers or fractal super lattices. / International Journal of Electrospun Nanofibers and Applications, 2(3), 193鈥?01 (2008)
  12. Peitgen, H. O., J眉rgens, H., and Saupe, D. / Chaos and Fractals: New Frontiers of Science (in Chinese), National Defense Industry Press, Beijing, 202鈥?03 (2010)
  
• 作者单位：Fan Yang (1)
  Yajun Yin (1)
  Bin He (2)
  Qinshan Fan (1)
  
  1. Department of Engineering Mechanics, School of Aerospace, Key Laboratory of Applied Mechanics, Tsinghua University, Beijing, 100084, China
  2. Division of Mechanics, Nanjing University of Technology, Nanjing, 211816, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Applications of Mathematics
  Mechanics
  Mathematical Modeling and IndustrialMathematics
  Chinese Library of Science
  
• 出版者：Shanghai University, in co-publication with Springer
• ISSN：1573-2754

文摘

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.