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Wow, Thomas Hales and Koundinya Vajjha have proved Mahler’s First Conjecture! (That’s Kurt Mahler, not Gustave.)
https://arxiv.org/abs/2405.04331
Mahler’s First Conjecture says that the centrally symmetric convex shape with the *worst possible* packing ratio is made up of straight lines and arcs of hyperbolas, like the smoothed octagons shown in the animation below (demonstrating a 1-parameter family of slightly different packings all with the same density).
Whether these smoothed octagons are, as Karl Reinhardt conjectured, the actual worst case is still an open problem.
A bit more detail on Reinhardt’s conjecture in this article by @johncarlosbaez :
https://blogs.ams.org/visualinsight/2014/11/01/packing-smoothed-octagons/
and this thread by Koundinya Vajjha on Twitter:
@gregeganSF @johncarlosbaez Some people will say this is the worst possible packing *ratio*, but I can see where the handles for these containers can go, and rounded edges are usually safer than sharp ones, so in terms of efficiency and safety this is a pretty good packing, actually.