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The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.
To understand what makes this question tricky, let's think what kind of "sofa" shapes we can construct that can move around a corner. How about a unit square?
Well, a unit square only has area 1; surely we can do better? For example, a semicircle with radius 1 is another simple example that works.
The semicircular sofa has a larger area than the square one, ᴨ/2 (approximately 1.57). It is also more interesting, because in order to move around the corner it rotates, whereas the square sofa merely translates. Now, if only we could combine rotation and translation, maybe we could construct an even bigger sofa shape? Indeed, the mathematician John Hammersley noticed that if the semicircle is cut into two quarter-circles, which are pulled apart and the gap between them filled with a rectangular block, we get a larger sofa shape, which could be moved around the corner if only a smaller semicircular hole is also removed from the rectangular block. Here is the resulting shape, that is starting to look a bit more like an actual sofa.
The shapes involved are interesting. Not really sofa shaped, but then theory is rarely like reality.
-- submitted from IRC
(Score: 2) by DECbot on Wednesday January 04 2017, @08:40PM
Don't be upset that you didn't catch the loose wording of the problem. Where does it explicitly state in the summary that the 90° hallway with the width of 1 is in the same plane as the shape with the largest area? It is your assumption that they are. The problem states there is an object of undetermined size and shape on a 2-d plane. It then states there is a 2-d hallway with a width of 1. The shape passes through the hallway--there is no indication of the orientation of the plane containing the shape and the hallway. Just because the examples given assume the two are on the same plane, that doesn't mean that is the only solution.
Anyway, this is really a problem for the physicists to tell us how much energy is needed to change the phase of the dimensional plane of the object to make it perpendicular to the plane of the hallway and give us hints on how to produce that much energy if it is beyond our conventional means and then a problem for the engineers to figure how to actually produce and deliver that much energy. So, yes, it is still an engineering problem. Given that we live and think in 3-dimensions only makes the problem look trivial. If you really wanted to pass an infinitely large area of 0 mass through that corner, you will still need infinite amounts of energy to pass it through the hallway regardless of your means of transporting the plane.
cats~$ sudo chown -R us /home/base