**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

#geometry

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- What you wrote didn't make sense to me, but I was in a rush when I read it, so I didn't have time to figure out who was confused!Oct 21, 2014
- +John Baez Sometimes I have trouble translating something that I "see" clearly into words that others will understand. Only after reading back through what I had written did my utter failure become apparent. (And this was after I had performed several edits trying to find the right words.) I am still thinking on this and may, or may not, have more to say after I spend more time thinking.Oct 21, 2014
- Mathematical methods themselves have homotopy and entropy which could be studied mathematically. Addition is simple. Multiplication has entropy one higher than addition. Exponentiation has entropy one higher again. 2+2=4 and 4+3=7 are two discontinuous things whereas 2+2+3=7 is one continuous thing. Find the method with the lowest entropy and you have the easiest method. Either I just invented a whole new field of maths or I'm in my own wonderful bubble of delusion!Oct 21, 2014
- +The Road Less Travelled I don't know whether you're "in your own wonderful bubble of delusion!" or I am just not capable of understanding your point. I have to admit that I have no idea what the phrase "Their complexity derives from the entropy of the discontinuity of the method required to solve them" even means ... although I have to admit it is a grand phrase. (But, then again, so are Chomsky's --
*"Colorless green ideas sleep furiously.*" -- and Carroll's --*"'Twas brillig, and the slithy toves did gyre and gimble in the wabe."*-- phrases) ;-)Oct 21, 2014 - I had some time today to try the computational approach. Still needs some work but here is the picture so far https://plus.google.com/109950732766352929935/posts/e4PqirU5F2SOct 22, 2014
- Or a phone receiver sneaking off a table and escaping down a narrow crevice.Nov 4, 2014

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