Which shape is worst of all for packing the plane? That is, which has the lowest maximal packing density? Suppose we demand that our shape be convex and also centrally symmetric: that is, a subset $S \subseteq \mathbb{R}^2$ such that $x \in S$ implies $-x \in S$. Then a certain ‘smoothed octagon’ is conjectured to be the worst. Amazingly, this shape has a 1-parameter family of maximally dense packings, shown in this image created by Greg Egan.