The moving sofa problem (2016)

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:

Hammersley’s idea works for every value between 0 and 1 of the radius of the semicircular hole at the bottom. By extending the techniques used by Gerver in his 1992 paper, I found such an “ambidextrous sofa” shape with an area of approximately 1.64495, which may be the largest possible area.

Source: www.math.ucdavis.edu