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   Jacek Wesołowski (jzillw@mastodon.gamedev.place)'s status on Tuesday, 14-May-2024 16:40:10 JST Jacek Wesołowski

   in reply to

   • Greg Egan
   • John Carlos Baez

   @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.

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

     • John Carlos Baez

     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