After looking for images of billiards I could use with no luck I just sat down and sketched an example.
The billiard table itself is the lower left square. I’ve drawn a path moving towards the upper right and bouncing around a few times. The other three squares are the reflections of the original table that I spoke of. We can imagine the path continuing into them in a straight line, and wrapping from one side of the big square to the other, and from the top to the bottom. The sides marked “A” are identified, as are the sides marked “B”. When we wrap up the sides as marked we get a donut-shaped surface called a “torus”.
Now we didn’t have to start with a square table. If we start with a rectangle we’ll get pretty much the same picture, except the torus we get will either be longer and thinner or shorter and fatter. There are a lot of different tori out there, but they all come from taking some parallelogram and identifying the opposite sides like we did here. So, what sorts of parallelograms are there?
Let’s fix one side to have length 1. Expanding or contracting the whole figure isn’t very interesting anyhow, so this isn’t a big restriction. Now we can put our parallelogram into the regular Cartesian plane so that this fixed side runs from to and the rest of the parallelogram lies in the upper half-plane. There’s a side running from to as well as one running to , and that determines the parallelogram entirely.
Now, different parallelograms might give us “the same” torus. It turns out there are essentially two ways this can happen. If we change to all the same points line up, so this doesn’t change anything.
A little weirder at first glance is the fact that replacing with also does nothing. It’s essentially like flipping the parallelogram over and rescaling it to make a different side the base. In terms of the point it first reflects the point through the y-axis, then “reflects” it through the unit circle. It takes a little while to get the hang of this one, but if you try plotting a few points and their images it should come to you.
So we have these two transformations throwing around points on the upper half-plane. One just slides points by one unit to the left and to the right, and one “reflects” the point in some sense. Each point in the half-plane determines a parallelogram, and two parallelograms wrap up to give “the same” torus only if they are related by a sequence of these transformations. We want some region of the plane that contains exactly one representative point for each different shape of torus. Here’s the picture.
Every point on the half-plane can be sent to a point in the grey area by some transformation, and no two points in the grey area can be sent into each other. We call this area the “fundamental domain” for these transformations of the plane. Just like the big square above, the edges get identified so walking out of the grey area on one side walks back in somewhere else. The long sides up the left and right get sewn together, and the arch at the bottom sews to itself (work out how!). The whole thing has its own sense of “geometry” — a notion of length and distance and angle and everything — one you may not be too familiar with, but which you’ve probably seen. I’ll go more into this another time.
What’s really amazing is that the behavior of straight paths on this “moduli space” of all tori can tell us a lot about how billiard paths behave on various tables. This space also ties into many other regions of mathematics. What seemed like a little toy problem of bouncing a ball around inside a rectangle has suddenly become very deep indeed…