Packings of Smoothed Polygons

Some shapes can be packed more easily into the plane than others. A square tiles the plane, completely filling the plane, but a circle packing can fill no more than about 90% of the plane. Motivated by Minkowski’s theory of numbers, researchers in the 1920s investigated the problem of finding the worst shape for the packing problem. This lecture will describe our recent proof that the worst shape must be a smoothed polygon – a polygon with rounded corners. This was conjectured by Mahler in 1947. This work is joint research with Koundinya Vajjha.