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?
Okay, it’s been pointed out to me that what I was thinking of in my update to yesterday’s post was a little more general than group theory. In the case of groups, preserving the composition is all that’s required. If I talk about semigroups later that extra condition will be needed.
So, I’ll leave it as a (relatively straightforward) exercise to show that if a function from one group to another preserves the composition that it also preserves identities. Oh, and inverses. May as well nail that down while we’re at it.
For now I’ll make this point, which I also make to all my calculus classes: it’s really not that bad to misremember something. That’s one of the nice things about mathematics. If you make a mistake it’s usually not hard to check. Also, when two people have different views on something it’s possible to check which one is right. There are real answers to be found. Despite the accusations of coldness and impersonality, it’s comforting to know that something has a real answer.