For those of us who do the dishes and tidy up the kitchen after a meal, the problem of the sofa is a common one. When, having finished the chores, we get to the sofa with the intention of taking a nap, the best seats are already occupied and the TV remote control is already taken. The solution to this problem is simple: take a nap in bed.
But, in mathematics, the so-called “moving sofa problem” refers to something else, which also happens to be related to a household chore, heavier than washing up: what is the largest sofa that can be moved along a corridor with an “L” shaped corner? Of course, the corridor is the same width before and after the corner, and you are not allowed to lift or tilt the sofa. If the width of the corridor is \(1\), the area of the largest sofa that can turn the corner is called the “sofa constant”. And what is the shape of that sofa?
By the way, now that Holy Week is approaching, and the streets of Seville will be filled with processional pasos, we can also think about what shape we should use to build the largest paso that can turn the corner between two perpendicular streets of equal width. Such an idea is not original, and can already be read in a book by the well-known writer-mathematician from Cabra (Córdoba) who runs this blog [1, p. 65].
Although the sofa problem may have been posed before, the oldest reference dates from 1966, and is due to the Austrian-Canadian mathematician Leo Moser [2]. The fact is that the problem has turned out to be much more complicated than it seemed, and quite a few mathematicians have made small advances; here we will only mention a few.
Of course, if the sofa is a square of side \(1\), it will be able to turn the corner without difficulty (and, in fact, without turning). One might think that, for rectangular sofas, we would be able to construct a sofa of larger area that can turn the corner. But, surprisingly, this is not true; you can, for example, turn the corner with a rectangular sofa of length \(\sqrt{2}\) and width \(1/\sqrt{2}\), but the area is the same as with the square sofa.
In order to get improvements, we need sofas with other silhouettes. It is clear that a sofa with the shape of a semicircle of radius \(1\) will also be able to turn the corner, and the area of this sofa is \(\pi/2 \approx 1.5708\). We can see it (together with the square sofa) in the adjacent diagram.
In 1968, John M. Hammersley improved the design by separating the semicircle in two halves and adding, in the centre, a rectangle with a semicircular cut on one of its sides; if the length of the cut-out central rectangle is, at most, \(4/\pi\), that sofa will also manage to turn the corner. The area of this figure, which resembles the handset of a traditional telephone, is \(\pi/2 + 2/\pi \approx 2.2074\). We can see this sofa in the accompanying illustration. Hammersley also proved that no couch of area greater than \(2\sqrt{2} \approx 2.8284\) was going to be able to turn the corner; this imposes an upper bound for the sofa constant.
In 1992, Joseph Gerver proposed a sofa design visually similar to Hammersley’s with rounded inner corners that could also turn the corner and had a somewhat larger area. The boundary of Gerver’s sofa has a complicated description, and is composed of \(18\) arcs defined by analytical expressions dependent on four constants whose value is known only numerically (these constants are solutions of a non-linear system of equations). The approximate area of this sofa is \(2.2195\), but we do not have an algebraic expression for this number. This sofa is shown in the accompanying picture; note, in particular, the marks separating the different pieces of the boundary. To date, no one has been able to find a sofa of larger area, but it has not been proved that it does not exist.
A variant of the sofa problem is known as the “ambidextrous sofa problem” and was proposed, at least, by the mathematicians John H. Conway and Geoffrey C. Shepard (sometimes also called “Shepard’s piano problem”). The difference is that we now require that the sofa (or piano) can make turns of \(90\) degrees both to the right and to the left. Admittedly, our house has to be very big to have such a sophisticated corridor, but in a processional procession this is crucial, as we cannot expect that, in the course of the procession through the streets of a city, all the turns will be to the right or all to the left.
In 1973, Kiyoshi Maruyama proposed, based on polygonal approximations using numerical procedures, a silhouette of a sofa that could perform both types of rotation. In 2014, Philip Gibbs, using a different numerical technique, arrived at a very similar silhouette, of approximate area \(1.64495\). The shape of these sofas has been described as a bikini (top only) without straps; this writer fails to understand the utility of such a bra, unless it is designed to be worn in the absence of gravity.
Pretty recently, Dan Romik has made a significant breakthrough in the search for the best ambidextrous sofa. Extending Gerver’s techniques to the non-ambidextrous sofa problem, he has found a silhouette of an ambidextrous sofa whose boundary is also composed of \(18\) arcs, each defined with a different analytic expression. The shape of such a sofa is indistinguishable from the one drawn numerically by Gibbs, and so is its area. We can see Romik’s sofa in the accompanying figure; again the marks separating the different pieces of the boundary are shown.
The surprise, and the difference from the Gerver sofa case, is that the parameters involved in the Romik sofa boundary definitions do have a closed-form algebraic expression, and the boundary segments are algebraic curves. This makes it possible to explicitly calculate the area of such a sofa, which has a rather quaint expression:
$$
\sqrt[3]{3+2\sqrt{2}} + \sqrt[3]{3-2\sqrt{2}} – 1
+ \arctan\left( \frac{1}{2} \left( \sqrt[3]{\sqrt{2}+1} + \sqrt[3]{\sqrt{2}-1} \right) \right)
\approx 1.644955218425440.
$$
It is also possible to calculate the length of the sofa, which is
$$
\frac{2}{3} \sqrt{4 + \sqrt[3]{71+8\sqrt{2}} + \sqrt[3]{71-8\sqrt{2}}}
\approx 2.334099633100619.
$$
But, of course, Romik’s sofa does not solve the ambidextrous sofa problem, as it is not proven that a better one cannot exist.
Note. All images of sofas accompanying this post are reproduced from [3].
References
[1] A. J. Durán, Historia, con personajes, de los conceptos del cálculo,
Alianza Editorial, 1996.
[2] L. Moser, Problem 66-11: Moving furniture through a hallway,
SIAM Rev. 8 (1966), 381.
[3] D. Romik, Differential equations and exact solutions in the moving sofa problem,
Experimental Mathematics, en prensa.
Prepublicación disponible en https://arxiv.org/abs/1606.08111
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