Cell Elements, Growth Modes and Topology Evolutions of Fractal Supper Fibers

摘要

This paper deals with the cell elements, growth modes and topology evolutions of fractal super fibers. Three types of cell elements, i.e., circles, hexagons and equilateral triangles, are involved. The closed boundary (or the datum line) of the cell element is found to divide the growth district of the fractal pattern into two parts: the inside part and the outside one. Related to the datum lines, the growth modes and topology evolutions of the super fractal sets show diversities: from different cell elements, identical fractal patterns with identical geometric shapes and topologies may be generated through the same growth modes; while from the same cell elements, identical fractal patterns with identical geometric shapes and topologies may be created through different growth modes. As far as the growth modes are concerned, there are inside growth and outside one. As far as the topologies are concerned, there are connected topology and disconnected one. As far as the topology evolutions are concerned, there are anamorphosis, mutation and anamorphosis-mutation. Such results may not only enrich our knowledge about the growth and evolution phenomena in nonlinear dynamics, but also stimulate new ideas for designing novel fractal structures.