The Moving Sofa Problem is an unsolved mathematical problem and it only addresses two-dimensional objects. The problem was first posed by Leo Moser in 1966, probably because he had some super exciting furniture moving to do. Essentially, the question posed by this problem is this, “What is the maximum area that a sofa-shaped object can be and still pass through a constant L-shaped corridor?” This is a visualization of the Moving Sofa Problem, showing the successful maneuver of the object through an L-shaped region:
One unit of area is the width of the corridor and so far the largest area for a sofa that has been proven to slip through the corridors is 2.2195 units, also known as “Gerver’s Sofa” because it was found by Joseph Gerver in 1992. Gerver’s Sofa is also shown in the animation above.
But before Gerver’s Sofa, mathematicians experimented with different objects attempting to find the solution to this problem. At first a simple semi-circle was used:
What’s the area of the semicircle? πr2/2, where r is the radius. Clearly, the radius of this semicircle is 1 unit, so using some crazy math we get π/2 or ≈ 1.5708 units. But mathematicians knew they could come up with something more complex. Just two years after the problem was posed, a man named John Hammersley created the “Hammersley Sofa” (I know, very creative names here but bear with me) which is shaped like this:
It is the shape of a semicircle cut in half, with a rectangle stuck in between. Then a smaller semicircle is cut out of this square. Hammersley realized that a sofa like this would have a greater area than just a semicircle while also being able to move about the L-shaped corridor. The calculation of the area is rather simple. The area of half a semicircle is πr2/4, and again, each quarter semicircle has the radius of 1 unit, meaning each has an area of π/4 or ≈ 0.7854 units. Since there are two, we can just add the areas, and look at that, we get 2(π/4) or ≈ 1.5708 units.
Then, middle rectangle’s area has to be taken into account. Now, the rectangle cannot just be any size. The height of the rectangle is 1 unit, while, the width of the rectangle is 4/π. The area of the rectangle is 1 * 4/π which of course, is 4/π ≈ 1.2732 units.
But, this shape cannot go around the L-shaped corridor, so Hammersley figured out that a semicircle with a radius of 2/π units would be the largest possible to make the whole “sofa” go around the corridor. So, 2/π units are removed from the rectangle, to get 4/π - 2/π = 2/π ≈ 0.6366 units.
To get the total area of the object, you do π/4 + π/4 + 2/π = π/2 + 2/π = (4 + π2) / 2π ≈ 2.2074 units. This is Hammersley's sofa in action:
Yay! 2.2074 units is clearly greater than the measly 1.5708 units of a semicircle! But as shown by Gerver’s Sofa, we can do better. Only 0.0124 units better...but still better! So what did Gerver do to make his sofa 0.0124 units greater? Well, just look at this image of Gerver’s Sofa overlayed on top of Hammersley's Sofa.
It may be difficult to see, but the red is Gerver’s Sofa, and the white (which you can see popping though on top of the inner semicircle) is Hammerly Sofa. Essentially Gerver, changed the semicircle in the middle so that it was not a perfect semicircle, as the corners are rounded off at the very bottom. This creates a loss in area, however, the two quarter semicircles were modified slightly to be taller, increasing the area. The net area difference turns out to be 0.0124 units greater for the Gerver Sofa than the Hammersley Sofa. The Gerver Sofa is also much more complicated than Hammersley’s. Hammersley’s is only made up of 4 different geometrical shapes, two quarter circles, a rectangle, and a cutout of a semicircle. But Gerver’s Sofa has 18 distinct geometrical shapes.
But, as I mentioned earlier, this is still an unsolved mathematical problem. Gerver never was able to fully determine that this was the greatest are possible and there have not been many advancements since 1992. Recently, a study by Philip Gibbs came out, in which Gibbs created a software program that would create the ideal shape to get through an L-shaped corridor. His software spit out a shape whose area is exactly the same as Gerver to 8 significant digits. Thus, it is widely believed that Gerver’s sofa most likely is the most optimal shape. However, there have technically been no mathematical proofs that solidify Gerver’s Sofa as the most optimal, and therefore, the problem is still unsolved.
Who would have thought that moving furniture would be so interesting? These kind of mathematical problems could actually help people when it comes to moving real-life furniture...though I have never seen a couch that is shaped like a semicircle or a Gerver’s sofa...this was the best I could find:
Regardless, the Moving Sofa Problem is just one of those fun yet pointless problems in mathematics that could possibly contribute to new discoveries in math. Who knows, the way in which someone proves the largest possible area could be used for another unsolved geometric problem, like the “Einstein Problem” or the “Circle Packing in an Equilateral Triangle Problem”. If you are interested in learning more about the Moving Sofa Problem, Numberphile has an excellent video that you can find here.