A ball rolls on a frictionless table shaped like a regular n-gon and bounces off the edges, leaving at the same angle it arrived. Choose where on the first edge (the one running clockwise from the top corner) it starts and which way it heads, then watch the path it traces. Some starts close up into a repeating orbit; most never do.

Unfolding turns every bounce into a straight line. Instead of reflecting the ball at an edge, reflect the whole table in that edge and let the ball carry straight on into the mirror copy. The bouncing path is the straight line folded back into the original table. For triangles, squares and hexagons the mirror copies fit together into a tiling of the plane, so a closed orbit is a straight line that reaches an identical copy of its starting point and direction.