Not Your Usual Circle: Geometry on the Integer Grid

This talk was given by Rebecca Sheppard from the University of Manchester on 8th October 2025. Abstract: Integer geometry explores objects whose vertices lie on the integer lattice, with congruence defined by lattice-preserving affine transformations. In this project, I introduced remarkable geometric objects called integer circles: discrete analogues of the Euclidean circle. These objects challenge our geometric intuition regarding circles. Unlike their classical counterparts, integer circles are unbounded, exhibit nontrivial arithmetic structure, and possess positive density in the plane. In this talk, I will define integer circles, illustrate their unusual behaviour, and demonstrate how to rigorously compute their densities and intersection patterns. To find more about our seminar, check out our website: selected-topics-mathematics.github.io