Moving Sofa Problem

The problem of “What is the largest area of a sofa that can turn a right angle in a corridor?” has finally been solved. On the other hand, the problem of “What is the smallest area of a box that can store a line that can bend freely?” is still unsolved. The mathematician Leo Moser was the one who considered both of these problems.

What is the maximum size of a sofa that can be carried down a corridor that bends at a right angle?

It’s been a month since we moved our laboratory to the new campus. The initial chaos of the move is finally starting to settle down. The thing that worries me most about moving is large pieces of furniture. There’s the question of whether they’ll fit through the door, and the question of whether they’ll be able to turn the corner in the corridor.

Mathematician Leo Moser also had to think hard about moving furniture. This is the *moving sofa problem.* The moving sofa problem is a question about how big a sofa can be if it has to be carried through a right-angled corner in a one-metre-wide corridor, i.e. an L-shaped bend.

This is a math problem, so don’t worry about where you lift or pull the sofa. For example, a sofa with a square base measuring 1 metre on each side can move through a corridor 1 metre wide and can turn corners.

Conventionally, it has been thought that a telephone-receiver-shaped sofa like the one in the following figure would have the largest surface area.

The telephone-receiver-shaped sofa can turn corners in a straight hallway.

And this month, Jineon Baek of Yonsei University in Korea gave a mathematical solution to the moving sofa problem. Problems like this in geometry are always full of fascination!

Thus, I would like to introduce another kind of puzzle-like geometry problem. That problem is Kakeya’s problem. And at the end of the newsletter, I will also introduce the unsolved problem left by Leo Moser.

The Matchbox

A matchbox containing a single matchstick

Suppose there is a single matchstick in a matchbox.

Because the matchbox is so cramped (for the matchstick inside), the matchstick cannot rotate 180 degrees inside the matchbox. If the matchbox were big enough for the matchstick to rotate freely, it would be inconvenient because you would have to check the direction of the head every time you took out a match. (Please don’t say that people don’t use matches these days.)

Now, if we were to make a box that a matchstick could rotate 180 degrees, or even 360 degrees, what shape would have the smallest area? Here, we’ll assume that the length of a matchstick is exactly 1. If you don’t like the fact that there are no units, for the time being, please think that there is a unit called 1 matchstick.

Square matchbox

If there is a square matchbox with one side measuring 1 matchstick (unit), it is of course possible to store matches in it, and it is also possible to rotate the matchstick 360 degrees. As one side measures 1 matchstick, the area of the box is 1 square matchstick.

Circular matchbox

The matchbox doesn’t have to be square. Even a circular box with a diameter of one matchstick can rotate 360 degrees. In this case, the area is π/4 square matchsticks, or about 0.785 square matchsticks. This is a 21% reduction compared to the square matchbox.

Reuleaux-triangle matchbox

The mathematician who came up with this matchbox problem was Soichi Kakeya. That’s why we call it the Kakeya Problem. He thought that the shape known as *Reuleaux triangle would have the smallest area. The area of a Reuleaux triangle that a unit-length matchstick can be rotated is approximately 0.705 square matchsticks.* This is 29% smaller than a unit square.

Triangular matchbox

Unfortunately, it seems that Dr. Kakeya was using his head too much at this time, and in fact, even a regular triangle with a height of 1 matchstick can make a matchstick rotate 360 degrees. It was Kakeya’s colleagues Matsusaburo Fujiwara and Tadahiko Kubota who made the discovery. The area of this regular triangle is approximately 0.5774 square matchsticks, which is 42% smaller than a unit square.

Looking for Even Smaller Matchbox

Deltoid matchbox

The triangle that Fujiwara and Kubota found can be further cut off. Kubota later discovered that matchstick can also be rotated using a shape called a *Deltoid. The area is approximately 0.3853 square matchstick.* This is 61% smaller than a unit square, and is now half the size of a perfect circle.

You can compare perfect circles, Reuleaux triangles, equilateral triangles and deltoids on this site.

Wow, mathematics is amazing…

But it doesn’t end there.

The Russian mathematician Abram Besicovitch discovered that if the width of a matchstick is zero — — a common assumption used by mathematicians — — then the area of a matchbox can be made infinitesimally small. When mathematicians say “infinitesimally small,” they don’t mean “zero,” but rather “a finite number that is infinitely small compared to zero.” It’s a bit complicated, but think of it as, say, “0.000001” rather than “0.” Well, when we engineers say “infinitesimally small” or more commonly “nearly zero,” we mean “we want to say zero, but we don’t want to take responsibility for it.”

What shape do you think this Besicovitch matchbox will be?

Beshkovich’s matchbox

So, what is this??? (The original image is here.)

There is a explanatory video on YouTube, so please take a look when you have time. In short, it’s like a car driver changing direction on a narrow road, and they just keep reversing. In particular, the matchsticks with zero width use the principle that if you reverse them over an incredibly long distance, you can change direction with almost zero area.

Math is interesting, isn’t it?

Now, Leo Moser also posed another problem. It is called *Moser’s worm problem. It asks for the region of smallest area that can accommodate* every plane curve of length 1. Here “accommodate” means that the curve may be rotated and translated to fit inside the region. And as of 2024, it is still unsolved.

I look forward to the future mathematicians who will solve it.

One More Thing

Thank you for reading this article. If you are interested in relationship between art and mathematics, please watch this video.

[embed]

This article is originally published in Japanese.