From: 3blue1brown
Mathematician Gregory Galperin discovered a surprising relationship between the collisions of idealized blocks and the mathematical constant pi (π) [02:35:00]. This fact was originally discovered by Galperin in 1995 and subsequently published in 2003 [02:39:00]. The speaker was introduced to this concept by a viewer named Henry Cavill [02:31:00].
The Collision Experiment
The discovery revolves around a thought experiment involving two sliding blocks and a wall, under highly idealized conditions:
- No friction is assumed [00:19:00].
- All collisions are perfectly elastic, meaning no energy is lost [00:23:00].
The setup involves one block starting with a velocity from the right, while the second block begins stationary [00:14:00], [00:17:00]. The goal is to count the total number of collisions between the blocks and the wall [00:32:00].
Collision Examples and the Pi Connection
The number of collisions observed changes dramatically depending on the mass ratio of the blocks:
- Equal masses: If both blocks have the same mass, there are 3 total collisions [00:42:00], [00:55:00].
- Mass ratio 100:1: If the first block is 100 times the mass of the second, there are 31 total collisions [00:57:00], [01:23:00].
- Mass ratio 10,000:1: With the first block being 10,000 times the mass of the second, the total number of collisions is 314 [01:31:00], [01:40:00], [01:56:00].
- Mass ratio 1,000,000:1: If the first block’s mass is 1,000,000 times that of the second, the total number of collisions reaches 3,141 [01:59:00], [02:08:00].
The remarkable paradoxical pattern observed is that when the mass of the first block is a power of 100 times the mass of the second, the total number of collisions corresponds to the digits of pi [02:18:00], [02:23:00]. This observation “defies all expectation” [02:17:00] and is described as “mind-boggling” [04:09:00].
Implications and Mechanism
While elegant [02:49:00], this method is described as comically inefficient for computing pi [02:49:00]. For example, calculating just 20 digits of pi would require one block to be approximately 10 times the mass of the supermassive black hole at the center of the Milky Way, resulting in 31 billion billion collisions at frequencies of 100 billion billion billion billion clacks per second [03:16:00], [03:21:00], [03:28:00], [03:35:00], [03:38:00].
The appearance of pi in this seemingly discrete collision-counting problem, where pi usually describes continuous phenomena [04:17:00], [04:20:00], is explained by a “hidden circle” that arises from the conservation of energy within the system [04:26:00], [04:29:00]. The full explanation of this phenomenon is deferred to a subsequent discussion [04:40:00].