When a ball put inside a box has a longer diameter than the box (What happens when we put a ball inside a box in a high dimensional world?)

When a ball put inside a box has a longer diameter than the box

I had a chance to read Ferenc's interesting post about counter-intuitive things happening in a high dimensional space. This reminds me a very simple but another interesting example I met in a topology class.

I found this example in a series of YouTube lectures about topology and geometry by professor Tadashi Tokieda.  This example is very easy to understand with a simple logic though it triggers a lot of interesting thoughts and broadens my sight. I hope this would also help the readers to glimpse a deep and extraordinary world of a high dimensional space.

To start, let's begin with an easy example of 2D space. Think about the situation that you put every corner of the square (each side has a length of one, $I^2$ with a white disk. Then, try to fit a red disk in the middle.

2D square example

Then, what is the diameter of the red disk? You can solve this problem in various ways but let me introduce a very intuitive and simple way.

Solution

As you can see in the above picture, you would immediately notice that the length of the diagonal line is equal to the sum of two white disks' and two red disks' diameters. Therefore, you can derive the diameter of the red ball $d_{red}$ as below:
$$d_{red} = \frac{\sqrt{2}-1}{2}.$$
Since the value of $\sqrt{2}$ is approximately 1.414, the diameter of the red disk is approximately 0.2, which is definitely smaller than the length of each side of the square. This simply means that this disk in the middle is smaller than the square. Well... Of course! Because we put it in the square.

What would happen when we go to 3D box $I^3$?

3D box example

Again, it is not hard to find $d_{red}$. In an analogous way with the previous solution, we can simply find the fact that the length of the diagonal line of the box would be also same with the sum of two white balls' and red balls' diameters.

By the Pythagorean theorem, we can easily derive the diameter of the red ball by
$$d_{red} = \frac{\sqrt{3}-1}{2}\approx 0.35 <1.$$
It is still smaller than the box though it has increased a little.

Let's do the same in $I^m$. Still the length of the diagonal line across the high dimensional box would be found by the square root of the sum of the squares of all the sides it has.

Therefore, the general solution for $d_{red}$ in $m$-th dimension can be derived as follows:
$$d_{red} = \frac{\sqrt{m}-1}{2}.$$
You may already notice that this will give a very interesting result when $m\geq 9$. Whenever $m$ goes beyond 10, we would find the ball which is put inside the box would have longer diameter than the length of each side of the box!

"When a ball put inside a box has a longer diameter than the box's side!"

This simple but very counter-intuitive example shows how much our intuition can be wrong when it comes to a high dimensional space. As written in the front desk of my blog, I always try to think in pictures and visualize more when I meet a new concept. However, this example always rings me a warning that I have to be very careful whenever I cross the line above three-dimensional space.

I hope the readers would also find the fresh impression that I felt when I met this example for the first time. Thank you for your reading and please leave the comments if you liked it.