# The Lumber Room

"Consign them to dust and damp by way of preserving them"

## Colliding balls approximate pi

Found via G+, a new physical experiment that approximates $\pi$, like Buffon’s needle problem: The Pi Machine.

Roughly, the amazing discovery of Gregory Galperin is this: When a ball of mass $M$ collides with one of ball $m$, propelling it towards a wall, the number of collisions (assuming standard physics idealisms) is $\pi \lfloor\sqrt{M/m}\rfloor$, so by taking $M/m = 10^{2n}$, we can get the first $n+1$ digits of $\pi$. Note that this number of collisions is an entirely determinstic quantity; there’s no probability is involved!

Here’s a video demonstrating the fact for $M/m = 100$ (the blue ball is the heavier one):

The NYT post says how this discovery came about:

Dr. Galperin’s approach was also geometric but very different (using an unfolding geodesic), building on prior related insights. Dr. Galperin, who studied under well-known Russian mathematician Andrei Kolmogorov, had recently written (with Yakov Sinai) extensively on ball collisions, realized just before a talk in 1995 that a plot of the ball positions of a pair of colliding balls could be used to determine pi. (When he mentioned this insight in the talk, no one in the audience believed him.) This finding was ultimately published as “Playing Pool With Pi” in a 2003 issue of Regular and Chaotic Dynamics.

The paper, Playing Pool With π (The number π from a billiard point of view) is very readable. The post has, despite a “solution” section, essentially no explanation, but the two comments by Dave in the comments section explain it clearly. And a reader sent in a cleaned-up version of that too: here, by Benjamin Wearn who teaches physics at Fieldston School.

Now someone needs to make a simulation / animation graphing the two balls in phase space of momentum. :-)

I’d done something a while ago, to illustrate The Orbit of the Moon around the Sun is Convex!, here. Probably need to re-learn all that JavaScript stuff, to make one for this. Leaving this post here as a placeholder.

Or maybe someone has done it already?

Written by S

Mon, 2014-06-23 at 23:03:18

Posted in mathematics, unfinished

### 2 Responses

1. I’ve made an example using JS canvas! But instead of balls I’m using squares.

Chris Scomersi

Sun, 2019-01-20 at 02:22:00

• That’s great! Thank you. I also saw a video by Grant Sanderson (3Blue1Brown) last week: https://www.youtube.com/watch?v=HEfHFsfGXjs

(Minor suggestion: IMO it’s a bit confusing to see “collisions: 3.14” instead of “collisions: 314”, as the number of collisions is not a fractional number.)

S

Sun, 2019-01-20 at 02:30:29

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