Given a three-dimensional object, such as a bone or a large molecule, how might we determine if it has a mirror plane or a rotational axis of symmetry? What if this symmetry is only approximate? Can we make a quantitative assessment that one object is closer to having a mirror symmetry than another?  

A recent honours project established a method doing exactly this for two-dimensional objects. We used a computational tool from topological data analysis called the persistent homology transform. 

The three-dimensional case is more challenging, but of much greater relevance to applications. The first step will be to extend the two-dimensional approach and develop a method to measure how close an object is to its mirror image when reflected in various planes.  

References 
“Planar Symmetry Detection and Quantification using the Extended Persistent Homology Transform” Bermingham, Robins, Turner (2023) DOI: 10.1109/TOPOINVIS60193.2023.00007

“The extended persistent homology transform of manifolds with boundary” Turner, Robins, Morgan (2024) DOI: 10.1007/s41468-024-00175-8

Helpful mathematical background: metric spaces, linear algebra, elementary group theory

Helpful computational background: data analysis using python and numpy, basic knowledge of data structures and algorithms for graphs and geometry