The moving sofa problem

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

What is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? 

This movie shows one attempt to solve this problem.  It's called the Hammersley sofa, since it was discovered by John Hammersley.  It has an area of

π/2 + 2/pπ = 2.20741609916...

But it's not the best known solution!  In 1992, Joseph Gerver found a shape of area 2.2195 that works.

On the other hand, Hammersley showed that any solution has area at most

2 sqrt(2) = 2.82842712475...

So, the moving sofa problem remains unsolved.  Another easily stated but very hard geometry problem!

You can see Joseph Gerver's sofa on this page by my friend Steve Finch:

http://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm

Basically he rounded off some of the corners of Hammersley's sofa!

The movie here was made by Claudio Rocchini, and appears on the Wikipedia article:

https://en.wikipedia.org/wiki/Moving_sofa_problem

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