If you want to compute the circumference of the observable universe to within, say, the width of a human hair, you’ll need to know about 35 digits of π, though this never seems to deter a certain sort of person from memorizing the first 100, 200 or 500 digits. But it turns out there’s no need to memorize anything at all! You can recover any number of digits you like from a simple little physics experiment that I just learned about, though it was invented over ten years ago by Professor Gregory Galperin of Eastern Illinois University. His lovely little paper is here.
To see how it works, start with two identical billiards lined up in front of a wall like so:
Now push Ball 2 toward Ball 1 and count the collisions: First Ball 2 collides with Ball 1 and pushes it toward the wall. (At this point Ball 2 has transferred all its momentum to Ball 1 and stops moving). Then Ball 1 collides with the wall and bounces back toward Ball 2. Then Ball 1 collides with Ball 2 and pushes it off to a far-away place. Three collisions. That tells you that π starts with a 3.
If you want more accuracy, make Ball 2 exactly 100 times as heavy as Ball 1. This time the sequence of events is a little more complicated, but it turns out there are exactly 31 collisons. That tells you that π starts with 3.1.
Or if you prefer, make Ball 2 exactly 10,000 times as heavy as Ball 1. You’ll get exactly 314 collisions. π starts with 3.14.
You want 100 digits of accuracy? No problem. Just make Ball 2 10200 times as heavy as Ball 1 and count the collisions. There should be exactly
of them.(Be sure not to lose count halfway through or you’ll have to start all over again!).
Granted, this is not a terribly practical way to remind yourself of the first 100 digits of π, but then there’s no reason a terribly practical person would ever care about the first 100 digits of π in the first place. Galperin’s result is way cool, and so is the argument that proves it (which, if you go for this sort of thing, you can read in his paper). I thought it was well worth sharing.