Mathematicians Develop New Method For Describing Extremely Complicated Shapes

Topology is a powerful branch of mathematics that looks at qualitative geometric properties such as the number of holes a geometric shape contains, while fractals are extremely complicatedgeometric shapes that appear similarly complicated even when viewed under a microscope of highmagnification. Bridging the topology and fractals, as described in the American Institute of Physics' Journal of Mathematical Physics (JMP) , reliesupon a recently developed mathematical theory, known as"persistent homology," which takes into account the sizes and number of holes in a geometric shape. The work described in JMP is a proof of concept based on fractals that have already been studied by other methods -- such as the shapes assumed by large polymer molecules as they twist orbend under random thermal fluctuation. Many geometric structures with fractal-like complexity arise in nature, such as the configuration ofdefects in a metal or the froth of a breaking wave. Their geometry has important physical effects too, but until now we haven't had a vocabulary rich enough to adequately describe these and other complicated shapes. The mathematicians plan to use the vocabulary provided by persistent homology methods to investigate and describe complicated shapes in a whole new way.