## The Sofa ProblemMay 1, 2017 My elementary school mathematics courses included light introductions to trigonometry, geometry, and number theory. One problem that I remember had an illustration of a practical problem with a mathematical solution. A boy wanted to bring his model airplane through a door, but he was having problems. The plane, which would fit inside a perfect square, wouldn't pass through in the common configurations of wings parallel to the ground, and wings perpendicular to the ground. It was our task to see whether the boy could ever get the model airplane through the door. As can be seen in the figure, rotating the containing square would work; provided, however that the width of the doorway was compatible with the airplane length, a diagonal of the square. Happily, the length of the airplane was slightly less than√2 times the width of the doorway. Left unsaid was why the boy didn't just rotate the airplane to take advantage of its thinner profile in the other direction. Perhaps his smarter sister would have done that.
π/2 (1.57) will traverse a right-angled corner in a corridor of unit width, but there are shapes that will do that with a larger area.
One such shape with an area of 2.2195 was devised by Gerver in 1992, and it is conjectured to be the solution of largest area (see figure).[7] Gerver's shape is formed from eighteen curves, each defined by an equation. The sofa problem has been revisited by Dan Romik, a professor in the Department of Mathematics, University of California Davis, as the "ambidextrous sofa problem" in which the object must traverse both a right- and left-angled corner. Surprisingly, Romik's shape is formed, also, from eighteen curves.
X + arctan(Y), where X and Y are solutions of the cubic equations, x and ^{2}(x+3) = 8x(4x, respectively.[2-3]
^{2} + 3) = 1
## References:- Evelyn Lamb, "The Serenity of Kakeya," Roots of Unity blog on Scientific American. February 27, 2017.
- Dan Romik, "Differential Equations and Exact Solutions in the Moving Sofa Problem," Experimental Mathematics, Advanced Online Publication (15 pages), January 19, 2017, http://dx.doi.org/10.1080/10586458.2016.1270858.
- Dan Romik, "Differential equations and exact solutions in the moving sofa problem," arXiv, July 11, 2016. A PDF file can be found here.
- Becky Oskin, "New Twist on Sofa Problem That Stumped Mathematicians and Furniture Movers," University of California Blogs, March 20, 2017.
- Dan Romik's page on the moving sofa problem (with animations and 3-D printer files).
- Leo Moser, "Moving furniture through a hallway," SIAM Rev., vol. 8, no. 3 (1966), p. 381, DOI:10.1137/1008074.
- Joseph L. Gerver, "On moving a sofa around a corner," Geometriae Dedicata, vol. 42, no. 3 (June, 1992), pp. 267-283, DOI: 10.1007/BF02414066.
- Animation of Romik's ambidextrous sofa moving through a corridor.
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