The moving sofa problem is inspired by the real-life problem of moving furniture around. Probably, a mathematician was carrying his sofa down a corridor and he had to navigate some obstacles and he asked himself that question; what is the sofa of the largest area we can move around the corner?
If you have a corridor with 1-meter width, then you can make a semicircle with a radius of 1 meter. When you push your semicircle down your corridor you can easily rotate and push when it meets the opposite wall because it is a semi-circle. It works perfectly. Here the area of the semicircle will be pi/2.
In 1968, a mathematician John Hammersley made his own geometric shape with a greater area to answer this question. Again, you push Hammersley’s sofa and when you reach the wall you just smoothly rotate it and that’s it. It also works perfectly.
OK, but what is the difference between the first sofa and Hammersley’s sofa? Basically, Hammersley cut the semicircle sofa into two pieces and he got 2 quarter circles. Then he pulled them apart and filled the gap with a rectangle. Finally, he carved a hole in the middle of his shape to do the rotational part. He also optimized how far apart you want to push the two-quarter circles. The area of his sofa was pi/2 + 2/pi which is 2.2074…
However, Hammersley wasn’t sure if his sofa was optimal or not. After two decades mathematicians realized that Hammersley’s solution was not optimal. Joseph Gerver made a very similar sofa to Hammersley’s sofa. He made subtle differences. Gerver used the same sofa but he just polished off a little bit of the sharp inner edges and some other points. He got 18 different curves.
The area of Grover’s sofa was 2.2195 and it was 1% bigger than Hammersley’s sofa. Even it was a small improvement it was very interesting because of the way he thought to have a larger area. He also claimed that his sofa is the optimal one and it is still not proved or disproved. In particular, it would have to be optimal because of how it was derived. His approach is so logical and that’s a pretty good indication that it might be optimal. Or we haven’t been able to find something that works better because our imagination is limited. Anyway, it is still an open problem for mathematicians.
On the other hand, another mathematician Dan Romik found some new advances in sofa technology. Romik had two goals at the beginning. First, he wanted to find a better sofa than Gerver did. Second, he would try to prove that Gerver’s sofa is the best. However, he did something else.
Romik realized that we cannot move Gerver’s sofa in more complicated structures. We can make rotate to the right easily, but if we need to rotate to the left? Obviously, we get stuck because Gerver’s sofa can only rotate to the right. That’s why Romik considered an ambidextrous sofa shape that they can rotate in both directions and has the largest area. He ended up finding a new shape that satisfies those conditions. It might not seem a realistic sofa but mathematically, it was a well-defined shape. It is perfectly working because it is an asymmetric shape. It’s not a circle or a square. It’s something new.
Romik’s sofa has also of 18 different curves that are needed to glue together in a very precise way. The ends are also made a certain angle which is 16.6 degrees and more interestingly Romik found a precise formula for them.
In mathematics, if you claim something, you need to describe it and most of the time you need to write in closed form. For instance, the equation x²=2 is the closed form of the square root of 2. Romik discovered that his equations can be written in closed form.
We still have no clue that all those sofas are optimal or not. If somebody shows that they are not optimal that would not surprise any mathematician.