Suppose we have the following scenario:
You are standing in a room that is in the shape of a regular n sided polygon with mirrors for walls. You shine a light, a single ray of light, in a random direction. Will the light ever return to its original position (the single point where the light originated from)? If so, will it return to its position an infinite amount of times or a definite amount of times? Will it ever return to its original position in the original direction?
This question is similar to my previous question concerning a circular room, but I want to focus on the harder part of regular polygons.
This is like a single point inside of a regular polygon, say a triangle, being the endpoint of a ray. The ray goes away from the point in a random direction and reflects off the sides of the triangle (staying inside the triangle). Will it ever return to its original point? If so, will it return infinitely or definitely? Will it return to the original point in the original direction?
When looking at my previous question, the answers were seemingly simple only because it was a circle. However, I imagine a regular polygon scenario is much harder, but still possible to solve.
One way of thinking about this is, when the ray of light hits a wall, construct a copy of the room reflected in that wall, completely with a reflected copy of the starting point. Then continue drawing the ray in the original direction (without reflection) so that it passes into the reflected room. The path of the (unreflected) ray in that room will simply be a reflection of the path of the reflected ray in the original room.
When the (unreflected) ray reaches a wall of the reflected room, make a reflected copy of the reflected room and continue drawing the ray in a straight line. Continuing this process, whenever the ray reaches a wall of the last copy of the reflected room, make a reflected copy of that room and continue the ray.
The ray returns to its starting point in the original room if and only if it reaches the reflected copy of the starting point in one of the reflected rooms that are constructed by the process above.
Clearly there are directions you can shine the ray such that it will return to the starting point. (Shining it perpendicularly onto one of the walls is one way, but it is not hard to construct others for any given polygon.)
On the other hand, since there are an uncountably infinite number of directions you can take out of the starting point, but a countably infinite number of images of the starting point that can be constructed by reflecting the room repeatedly, there are directions the ray can travel so that it will never return to its starting point.
This does not answer all parts of your question but I hope it provides a helpful way of thinking about it.