Fractal geometry and topology abstracted from hair fibers

The study explored the fractal geometry and topology of hair fibers, particularly wool fibers and human hairs, revealing that certain fractal patterns like the (9+2)-circle have broken symmetry and golden properties, with a fractal dimension close to the golden mean (approximately 1.6379). These patterns are capable of transferring topological information consistently due to their self-similarity. The research identified nine (9+2) topological patterns, with only two being fundamental, and demonstrated that the fractal sets exhibit symmetry breaking originating from the (9+2) pattern. This understanding of hair fiber structure could have implications for their functional properties, supported by previous findings that golden mean fractal dimensions enhance wool fibers' heat transformation capabilities. The study was funded by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangsu Province.