Fractal geometry and topology abstracted from hair fibers

Based on the concepts of fractal super fibers, the (3, 9+2)-circle and (9+2,3)-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the (3)-circle and the (9+2)-circle fractal sets as subsets. As far as the (9+2) topological patterns are concerned, the following propositions are proved: The (9+2) topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent (or fundamental). Besides, we demonstrate that the (3, 9+2)-circle and (9+2, 3)-circle fractal sets are golden ones with symmetry breaking.