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So Gerver’s 1992 curved “sofa” (area ≈ 2.2195) is not just a good guess but actually optimal.

The problem asks for the largest 2D shape that can be slid around a right-angle corner in a unit-width hallway.

Here is the perfect fitting sofa: https://en.wikipedia.org/wiki/File:Gerver.svg



I think it would be cool to have a sofa in that shape as a joke. It could go in a room tiled with a single shape non-repeatable pattern.


A very functional sofa joke as it's the optimal shape for moving.


Not really, because it's only optimal in 2d.


Not a problem, we can just shape it like the space covered by rotating the whole thing and make the sofa kind of "bone shaped", then we should be set for 3 dimensions. The only remaining issue would be, where to put the actual sitting area on the bone, but that problem I leave for homework.


3dimensions still allows for more freedom than that though since the couch can stand on end.

I would contend that it's still useful since you'd be able to turn the corner without over-complicating it by getting it into some weird tilt position.


> Not a problem, we can just shape it like the space covered by rotating the whole thing and make the sofa kind of "bone shaped", then we should be set for 3 dimensions.

That might give you a feasible solution, but I doubt it's optimal.


Most of us don't live in Escheresque labyrinths


If you've ever moved any furniture at all, you'll notice that it's often much easier to get around corners (or through doorways), if you can turn them sideways.

That's especially easy to imagine with tables, but sofas also count.

There are also sofas that can be easily taken apart. Eg one of our sofas at home, an L-shaped sofa, comes apart into two pieces.


You don’t have to move house many times to realise that yes, we do.




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