The Sofa Problem

May 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.

Thou shalt not pass. The model airplane, which fits inside a square, is too wide to pass through a doorway in configurations a or b, but it will pass in configuration c, provided that it's length is not larger that the width of the doorway times the square-root of two. (Inkscape.)

A similar problem, called the Kakeya needle problem, was mentioned in a recent article by Evelyn Lamb of the University of Utah in her Roots of Unity blog at the Scientific American web site.[1] This problem, posed in 1917 by Japanese mathematician, Sōichi Kakeya, concerns the minimum area in the plane in which a needle can be turned through 360°. One example of such turning is shown in the figure.

Kakeya needle example. (Wikimedia Commons graphic by Claudio Rocchini.)

The object of Lamb's article was a sailing ship, and not an airplane. The Kakeya needle problem relates to the ship, since the long mast of a sailing ship approximates a needle. The example she cites is the removal of a display ship from a Paris outdoor exhibit with the necessary avoidance of trees, sculptures, and other nearby objects. The crew that removed the ship were likely heartened by the idea that, if it was put where it was, then it could likewise be removed, and this was accomplished successfully with the proper rigging.

While most people are not likely to be moving sailing ships, they do move a lot of bulky furniture in their homes. In my own home, we have a massive sofa bed in the living room, and we still can't believe that the movers were somehow able to bring it through our front door, around a corner, and into the room. As it turns out, the "moving sofa problem" has been on mathematicians' minds for quite some time. Simply stated, the moving sofa problem asks what planar shape of maximal area can move around a right-angled corner in a hallway of unit width.[2-5]

The problem is simplified by limiting movement within a plane, so the tilting tricks of actual movers cannot be employed. The problem was posed by Leo Moser in 1966.[6] A semicircle of area 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.

Gerver's sofa, which has an area of 2.2915, expressed in units of the corridor width. This sofa will only navigate around right-angled corners. (Modified Wikimedia Commons image by "Weburbia.")

Romik was inspired to investigate the problem through his hobby of 3-D printing. Says Romik, "I'm excited by how 3-D technology can be used in math... Having something you can move around with your hands can really help your intuition."[4] As Romik translated Gerver's equations into code that a 3-D printer could understand, he became interested in the mathematics of the solution. This inspired Romik to try his hand at developing new equations and refining them through computer code.[4]

In April, 2016, Romik decided to shift his attention to the ambidextrous moving sofa problem, a problem previously investigated by John Horton Conway and others.[2-3,5] Assisted by software, Romik found a dumbbell-shaped object that worked (see figure). The final object has an area of 1.64495, which might be the largest possible area for an ambidextrous moving sofa.[5] Says Romik, "I remember sitting in a café when I saw this new shape for the first time... It was such a beautiful moment."[4]

Romik's ambidextrous moving sofa. This object has an area of 1.64495521, expressed in units of the corridor width. (Modified Wikimedia Commons image by "Weburbia.")

Gerver conjectured that his shape is the shape of maximal area that solves the right-angled corner sofa problem, and no one has discovered an object of larger area since Gerver's 1992 paper.[5] Romik points out, however, that there is no proof that this is the case, and there's always the possibility that an object of larger area will be found. "It's a surprisingly tough problem... It's so simple you can explain it to a child in five minutes, but no one has found a proof yet."[4]

Romik has one advantage over Gerver, since his ambidextrous sofa is defined by a closed form solution that's defined by equations that don't contain any arbitrary constants. It differs in that respect from previous ambidextrous sofas designed by Kiyoshi Maruyama in 1973 and Philip Gibbs in 2014.[5] The area of Romik's sofa is equal to X + arctan(Y), where X and Y are solutions of the cubic equations, x²(x+3) = 8 and x(4x² + 3) = 1, respectively.[2-3]

Nicely pivoted. The illustration shows successive images of a Romik ambidextrous sofa moving down a corridor with a right- and left-angled turn.[8] (Drawn using Inkscape.)

References:

1. Evelyn Lamb, "The Serenity of Kakeya," Roots of Unity blog on Scientific American. February 27, 2017.
2. 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.
3. Dan Romik, "Differential equations and exact solutions in the moving sofa problem," arXiv, July 11, 2016. A PDF file can be found here.
4. Becky Oskin, "New Twist on Sofa Problem That Stumped Mathematicians and Furniture Movers," University of California Blogs, March 20, 2017.
5. Dan Romik's page on the moving sofa problem (with animations and 3-D printer files).
6. Leo Moser, "Moving furniture through a hallway," SIAM Rev., vol. 8, no. 3 (1966), p. 381, DOI:10.1137/1008074.
7. 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.
8. Animation of Romik's ambidextrous sofa moving through a corridor.