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 complicated geometric shapes that appear similarly complicated even when viewed under a microscope of high magnification.
Bridging the topology and fractals, as described in the American Institute of Physics' Journal of Mathematical Physics (JMP), relies upon 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 or bend under random thermal fluctuation.
Many geometric structures with fractal-like complexity arise in nature, such as the configuration of defects 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.
Articles featured in AIP press releases will be freely accessible online for a minimum of 30 days following publication.
Article: “Measuring shape with topology,” is published in the Journal of Mathematical Physics.
Link: http://jmp.aip.org/resource/1/jmapaq/v53/i7/p073516_s1
Authors: Robert MacPherson (1), Benjamin Schweinhart (1)
(1) Institute for Advanced Study, Princeton, NJ
Media Contact
More Information:
http://www.aip.orgAll latest news from the category: Physics and Astronomy
This area deals with the fundamental laws and building blocks of nature and how they interact, the properties and the behavior of matter, and research into space and time and their structures.
innovations-report provides in-depth reports and articles on subjects such as astrophysics, laser technologies, nuclear, quantum, particle and solid-state physics, nanotechnologies, planetary research and findings (Mars, Venus) and developments related to the Hubble Telescope.
Newest articles
High-energy-density aqueous battery based on halogen multi-electron transfer
Traditional non-aqueous lithium-ion batteries have a high energy density, but their safety is compromised due to the flammable organic electrolytes they utilize. Aqueous batteries use water as the solvent for…
First-ever combined heart pump and pig kidney transplant
…gives new hope to patient with terminal illness. Surgeons at NYU Langone Health performed the first-ever combined mechanical heart pump and gene-edited pig kidney transplant surgery in a 54-year-old woman…
Biophysics: Testing how well biomarkers work
LMU researchers have developed a method to determine how reliably target proteins can be labeled using super-resolution fluorescence microscopy. Modern microscopy techniques make it possible to examine the inner workings…