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A frictionless, perfectly elastic collision puzzle produces counts that match digits of π for mass ratios that are powers of 100. This introduction poses the puzzle and leaves the full explanation for a later video.
- 1Equal massesThe system produces three collisions.
- 2Powers of 100Larger mass ratios produce counts beginning 31, 314 and 3,141.
- 3A hidden circleThe explanation is deferred to the next video.
ポイント
- The setup assumes no friction and perfectly elastic collisions.
- Mass ratios of powers of 100 produce collision counts related to digits of π.
- The resulting algorithm is wildly impractical.
- The complete proof is left for the next video.
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A collision puzzle
Imagine two blocks sliding toward a wall on a frictionless surface. Collisions are perfectly elastic, so energy is conserved; the clicking sounds in the animation are only a counting aid.
Imagine two blocks sliding toward a wall on a frictionless surface. Collisions are perfectly elastic, so energy is conserved; the clicking sounds in the animation are only a counting aid.
With equal masses, the incoming block hits the stationary block, which hits the wall and then the first block. There are three collisions.
When the incoming block has 100 times the other block’s mass, the smaller block repeatedly bounces between it and the wall. The total is 31 collisions.
Digits of π
At a mass ratio of 10,000, there are 314 collisions; at one million, there are 3,141. These counts begin to look like digits of π.
At a mass ratio of 10,000, there are 314 collisions; at one million, there are 3,141. These counts begin to look like digits of π.
For mass ratios that are powers of 100, the collision count yields successive digits of π under the idealized setup. The video credits mathematician Gregory Galperin for the result.
An impractical algorithm
As a thought experiment, to compute d digits, choose a mass ratio of 100^(d−1), launch the larger block and count all collisions. The method is beautiful but impractically inefficient.
As a thought experiment, to compute d digits, choose a mass ratio of 100^(d−1), launch the larger block and count all collisions. The method is beautiful but impractically inefficient.
Even 20 digits would require an enormous mass ratio, an immense number of impacts and numerical precision far beyond a practical physical experiment. The example quickly departs from real-world physics.
Why a circle?
Why does π appear in a discrete collision count? The speaker hints that conservation of energy creates a hidden circle, but reserves the full explanation for the next video.
Why does π appear in a discrete collision count? The speaker hints that conservation of energy creates a hidden circle, but reserves the full explanation for the next video.
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