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.