I recently watched 3Blue1Brown's YouTube video "The most unexpected answer to a counting puzzle", and the video suggested (at time 2:56) that an inefficient and elegant way to express $\pi$ is:
Implement a physics engine, where:
Create 2 blocks, and a wall on the left side, while the right side leads towards infinity
The second block is bigger, starts out at the right of the smaller one(which has a mass of 1).
Set the number of digits you want to compute to a variable N
Set the mass of the heavier block to 100 to the power of N-1
Give the second block some initial push. (Doesn't really matter speed)
The rules are:
All collisions are elastic (no energy is lost, thus the momentum is completely preserved and transferred)
There are no friction to affect the sliding of the blocks
Thus:
- $\pi$ with $N$ digits is equivalent to the number of collisions of both blocks and the wall.
Is there any more ways to prove the fact while being both easy-to-understand and able to be graphed out, apart from 3Blue1Brown's ways to prove it?