The Unapologetic Mathematician

Billiards course

This semester I’m sitting in on Jayadev Athreya’s course on billiards. “Billiards?” I hear you cry, “The game like pool but with all the red balls?” No, that’s snooker. Besides, the course is on (not surprisingly) a mathematical model inspired by balls bouncing around on a table.

When playing mathematical billiards, we place a ball down on a polygonal table and send it off in some direction. When it hits the edge of the table it rebounds, making the same angle as it leaves as it did when it came in. We simplify a bit by assuming that the ball isn’t spinning, has no friction with the table, and so travels at a constant speed in a straight line between bounces.

So let’s start with a square table. How do we determine how the path behaves as time goes on? Imagine the ball approaching a side of the table. Instead of the ball reflecting off the edge, let’s reflect the whole table through the edge and let the ball continue on its straight-line path. Whenever the ball’s about to hit an edge, we have a reflection of the table ready for it.

But we really don’t need all that many reflections. In fact, four will do it: the first table, a horizontal reflection, a vertical one, and one reflected in both directions. We can put these four squares together into one bigger square. Instead of reflecting the ball at an edge, we can just let it wrap around to the other side of the table travelling in the same direction, just like a game of Asteroids. That means we’re really just rolling a ball across the surface of a torus.

Now the whole picture is pretty clear: by Weyl’s criterion, if the tangent of the angle between the path and an edge of the table is rational the path eventually closes back up and runs over itself again. If not, the path covers every point on the table equally. By this I mean that if the whole table has area $T$, then given a section of the table of area $A$, the ball spends $\frac{A}{T}$ of its time in that section.

There’s a lot of interesting material here, and a lot of it can be broken down to bite-sized chunks, tying into all sorts of other areas of mathematics. I’m starting a category (in the WordPress sense) so if you’re interested, there will be plenty more.

January 30, 2007 - Posted by | Billiards

1. Would you please add some hotlinks to pictures of other shaped billiards tables, and the trajectories on them? I’ve seen diagrams in papers and lectures of circular, triangular, elliptical, and recently “mushroom” shaped. The formal math will be beyond the reach of your less-math-educated readers, but a picture is worth a kiloword.

Great start, though, for this blog!

Comment by Jonathan Vos Post | January 30, 2007 | Reply

2. I looked for some good ones, even maybe with the unfolding shown, that I could use for free. No luck.

I’m asking Jay if he has some I can use, so when he’s back in town in a couple days (NSF gives a lot of travel money) I might update this a bit.

Comment by John Armstrong | January 30, 2007 | Reply

3. “. . . just like a game of Asteroids.”

You have no idea how many years I’ve waited for somebody to say that!

Comment by Blake Stacey | January 30, 2007 | Reply

4. Fascinating! I’ll certainly keep an eye on your blog and I look forward more entries in this category. đź™‚ Thanks!

Comment by hroswith | January 31, 2007 | Reply

5. Hummm I’m trying to figure out how to combine the four ‘sides’ to form a torus……

ot I’m I being to literial

Comment by Michael D. Cassidy | July 12, 2007 | Reply

6. Michael, think of a piece of flexible “paper” (a fruit roll-up perhaps) that you first wrap into a tube by pasting the two vertical sides together, and then fold into a torus by pasting the ends of the tube together.

Comment by Todd Trimble | July 15, 2007 | Reply

7. Well, I find it extraordinarily interesting.Good luck to

all of you. And Iâ€™m sure youâ€™ll do fine. Really. Just fine.

Comment by Paul | August 15, 2007 | Reply