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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:

https://twitter.com/KodyVajjha/status/1790211313618362848