Fractal growth kinematics abstracted from snowflakes: topological evolution
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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 wordsfractal snowflake proportional movement self-similarity N-segment line topological evolution and topological invariant
Chinese Library ClassificationO189 O415
2010 Mathematics Subject Classification74M25 82D80
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- 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), 96–102 (2008)Google Scholar
- 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), 193–201 (2008)Google Scholar
- Peitgen, H. O., Jürgens, H., and Saupe, D. Chaos and Fractals: New Frontiers of Science (in Chinese), National Defense Industry Press, Beijing, 202–203 (2010)Google Scholar