# What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner. I know that this is a famous problem, but what is it called?

• Manhole problem ? – Hachino Apr 1 '15 at 14:19
• @Hachino Actually not... The right ans is from Ian Farrell. – Yijun Yuan Apr 1 '15 at 14:22
• A 13-minute-long video featuring Dan Romik on the Moving Sofa Problem. – Rodrigo de Azevedo Jun 5 '17 at 13:59

The Moving sofa problem, I believe.

• From personal experience ... sofas move in 3 dimensions. Many years ago another mathematician was helping me move. We had to carry a sofa up a flight of stairs, then turn a corner at the top. It didn't fit. But he says, "Just rotate it this way..." and it did fit. I still don't know what he did. – Gerald Edgar Apr 1 '15 at 16:03
• An instance where transseries wouldn't have helped, I suppose. Gerhard "Guess Geometry's Good For Something" Paseman, 2015.04.01 – Gerhard Paseman Apr 1 '15 at 18:22

A supplement to Ian's answer: Here is the largest-area sofa known, due to Gerver:

Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata 42 (3): 267–283. (Springer link.)

Added (triggered by @GeraldEdgar's remark). The computational complexity of algorithms grows exponentially in the dimension, about $n^5$ for polyhedral objects with $n$ vertices moving in $\mathbb{R}^3$. Here is an algorithm moving an $n{=}4500$-triangle piano through a challenging apartment requiring several tricky maneuvers:

Kuffner, James J., and Steven M. LaValle. "RRT-connect: An efficient approach to single-query path planning." Robotics and Automation, 2000. Proceedings. ICRA'00. IEEE International Conference on. Vol. 2. IEEE, 2000. (IEEE link.)

Not surprisingly, the problem is also called The Piano Mover's Problem.

• Manifestly, to find the largest solid that would pass through a unit cube corner, the solution would be obtained by endowing the largest-area sofa known with height 1. – Bob Spaghetti Apr 1 '15 at 19:53
• New (Jun 2017) paper by Yoav Kallus and Dan Romik: "Improved upper bounds in the moving sofa problem," arXiv abstract. – Joseph O'Rourke Jun 22 '17 at 19:25