Toroidal Circle Packing

This work was the focus of my summer research at Grand Valley State University as part of their 2019 NSF Research Experience for Undergraduates chort. I worked with Sarah Van (now a Ph.D. student at UNC), and we were advised by Professor William Dickinson. Building off of previous results, we determined the local and global maximally dense arrangements of three circles on any flat torus with ratio \(\sqrt{2}-1\). This ratio was chosen as a result of Heppes’ bound, a supremum on the density of circles with radii in this ratio. Using techniques from rigidity theory and topological graph theory, we constructed graphs representing possible circle packings, and then examined their embeddings on a flat torus in the fundamental domain of \(SL(2,\mathbb{Z})\).

Below are links to our final report containing our original results and to a GitHub repository with the Mathematica scripts we used for algebraic reduction. Additionally, within the repo the Embedding Results folder contains the data for all possible graph embeddings.