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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 packing conjectures are nuts (no pun intended)
@gregeganSF @johncarlosbaez Next maybe he can apply the same ideas to the moving sofa problem, https://en.wikipedia.org/wiki/Moving_sofa_problem
@11011110 @gregeganSF - Great! What's Mahler's Last Conjecture?
My guess is that Hales will not move on until he's finished proving Reinhardt's conjecture. He wrote an NSF proposal saying
In 1934, Reinhardt considered the problem of determining the shape of the centrally symmetric convex disk in the plane whose densest packing has the lowest density. In informal terms, if a contract requires a miser to make payment with a tray of identical gold coins filling the tray as densely as possible, and if the contract stipulates the coins to be convex and centrally symmetric, then what shape of coin should the miser choose in order to part with as little gold as possible? Reinhardt conjectured that the shape of the coin should be a smoothed octagon. The smoothed octagon is constructed by taking a regular octagon and clipping the corners with hyperbolic arcs. The density of the smoothed octagon is approximately 90 per cent. Work by previous researchers on this conjecture has tended to focus on special cases. Research of the PI gives a general analysis of the problem. It introduces a variational problem on the special linear group in two variables that captures the structure of the Reinhardt conjecture. An interesting feature of this problem is that the conjectured solution is not analytic, but only satisfies a Lipschitz condition. A second noteworthy feature of this problem is the presence of a nonlinear optimization problem in a finite number of variables, relating smoothed polygons to the conjecturally optimal smoothed octagon. The PI has previously completed many calculations related to the proof of the Reinhardt conjecture and proposes to complete the proof of the Reinhardt conjecture.
@johncarlosbaez That's a really fun description!
I was just about to post "I'm surprised a heptagon isn't worse", but I RTFA, and now I know what "centrally symmetric" means.
@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.
@gregeganSF @johncarlosbaez So, he's done best possible packings and worst-possible packings. Next up: most mediocre possible packings!