Tiling the plane with billiards (Billiards part 2)

(This is part 2 of a series on dynamical billiards. Go to part one)

There's another way to picture billiards that is often useful. Instead of holding the polygon still and drawing many reflected rays over it, we can draw the ray's path in a single straight line, and reflect the polygon itself outward when the ray hits it:

This still draws rays in all the same places in the triangle, they're just spread out over different copies of it instead of being overlaid on the same one. Like the previous demos, use the arrows or WASD to move, and hold down shift to move more slowly. Uncheck "reflect polygon" to see what it looks like as a billiard path.

Although billiards are hard to analyze in general, we can find all solutions for the equilateral triangle this way: since it can be drawn on an infinitely reflected grid, and billiard paths on it follow a straight line on that grid, the cyclic paths all come from straight lines in the plane, going between a point and itself on a different copy of the triangle.

You might suspect then that other tiled figures are also going to be nice to analyze, and you'd be right:

But of course with the triangle, square and hexagon we've exhausted all the regular polygons that are easy to tile. Some variations also work, like rectangles and some triangles. But take a look at the irregular triangle from part 1:

It skitters around and reshapes itself almost too fast to see. Clearly we can't get the answer from simple distance ratios like with tiled shapes.

But there are some cases that are in between. For example, even though most regular polygons don't tile the plane, they're still very symmetric. Look at the pentagon:

But if you try reflecting the pentagon through to tile the plane, it doesn't seem very stable when you turn. But the pentagon does have a tiling of a different sort:


If we reflect the pentagon around this surface, it will mesh together on the dodecahedron, just like the triangle did in the plane. In fact, all regular polygons can be tiled nicely on a surface in 3 dimensions. Unfortunately, that's still pretty tricky to analyze -- we usually try to find ways to treat the surface as flat. But that will have to wait for another post.