John Carlos Baez on Nostr: A big geometry problem may have been cracked! You’ve probably tried to move a sofa ...

A big geometry problem may have been cracked!

You’ve probably tried to move a sofa around a bend in a hallway. It’s annoying. But it leads to some interesting math puzzles. Let’s keep things simple and work in 2 dimensions. Then the 'moving sofa problem' asks:

"What’s the largest possible area of a 2-dimensional region that can be maneuvered using rigid motions through an L-shaped hallway of width 1?"

This question was first published by Leo Moser in 1966. The animation here, made by Claudio Rocchini, shows one attempt to solve this problem. It was discovered by John Michael Hammersley in 1968. Its area has this charming value:

π/2 + 2/π = 2.2074…

But it’s not the best possible sofa! In 1992, Joseph Gerver found a more sophisticated shape of area

2.219531668871…

that also works. Basically Gerver rounded off some of the corners of Hammersley’s sofa. The resulting shape has a boundary consisting of 3 straight line segments and 15 curved segments, each described by its own formula.

For decades nobody could prove Gerver's sofa was the best. Now Jineon Baek claims he's done it!

• Jineon Baek, Optimality of Gerver's Sofa, https://arxiv.org/abs/2411.19826

This paper is 119 pages long, and I certainly haven't read it, but a brief peek suggests it's serious. We'll have to wait for dedicated people to study it.

I thank nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqg7va94ywu9fjkh2y6sz90vcqwv00shpyfgy5ndy8ltq59sy6szpsr4pmdx (nprofile…pmdx) for pointing this out! For an animation of Gerver's sofa, see my next post.

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