A polygon is a simple object: it consists of some number of points connected (in order) by line segments. Despite that simplicity, many real-world objects like robot arms, polymers, or images can be modeled by polygons. In applications it´s usually not enough to handle just one polygon at a time, though: we don´t only want to consider one rigid robot arm or one single polymer, we want to understand the possible motions of the robot arm or to say how likely this particular polymer´s configuration is among all possible polymer configurations.

To get a handle on these questions, we need to understand the space of all possible polygons with a given number of edges and ideally find nice coordinates on this space so that we know how to move around. This turns out to be fairly easy in the case of a robot arm that doesn´t form a loop, but surprisingly tricky in the case of "ring" polymers which do close up.

This talk will describe how to take the square root of a polygon and why this is an amazingly useful thing to do if you want to understand the geometry of polygon space. We will then apply that understanding to find optimal reconfigurations, to recognize contours, and to produce surprisingly mesmerizing GIFs.