# 3D version of couch problem - well THAT wasn't much fun!!

Discussion in 'Other Advanced Math' started by EmptySetBlues, Mar 31, 2022.

1. ### EmptySetBlues

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I recently read about the couch (or moving sofa) problem, involving the largest area shape that can be moved around the corner of an L shaped hallway of constant width. The Wikipedia article is a good intro. It's a 2D problem. If we consider it to be in the XY plane, the sofa's rotations occur around the sofa's Z axis. Call it C2 (couch 2).
I searched for any signs that anyone was working on the 3D version, with a hallway of rectangular cross section formed by stacking the L shaped hallway in the Z direction, and can't find any mention.
After a little doodling, I realized why. Just extend the 2D version into 3D by stacking the 2D hallway and sofa together, making the 3D sofa (C3) a very strange cylindroid. The Z direction height can be any nonzero value.
With C3 as tall as the hallway height, it can move around the corner using the same combined rotation/translation as the 2D version, also rotating around the Z axis. And, none of it can move at all in the Z direction, therefore it cannot be rotated around any axis other than Z. C3 is the largest volume shape that can make the passage with only XY translation and Z rotation, as is the case with C2.
Try to find another figure of larger volume. There is no larger figure that only undergoes XY translation and Z rotation, so any proposed different figure must be able to undergo other translations and rotations. There are only Z translations and non-Z axis rotations available to try.
The only way to make C3 undergo Z translations is to shorten it in the Z direction, reducing its volume. The only way to make it undergo non-Z axis rotations is to shave its edges down so that it is not locked into the corners of the hallway. Rotation angles up to 90 degrees demand shaving off more and more material, including perhaps from entire faces of the couch, and this again reduces volume. Minimal necessary shaving occurs if the hallway's Z height is equal to (or perhaps nearly equal to) its X width, and height values increasingly different from X result in greater proportional loss of volume. Some may even cut the couch in two or more pieces, invalidating the problem!
This is not a rigorous proof, but it certainly seems to me that C3 is the winner, and that any other figure that can pass around the corner is a subset of C3. Any thoughts?

EmptySetBlues, Mar 31, 2022