A 1-parameter family of packings of smoothed octagons.

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   Greg Egan (gregegansf@mathstodon.xyz)'s status on Tuesday, 14-May-2024 16:40:11 JST Greg Egan

   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