TITLE: Walking on a torus
QUESTION [19 upvotes]: Everyone knows that when you walk on a sphere along a straight line, you eventually get back to the point you started from.
I'm wondering about the same question for the torus. Obviously there are some directions which will bring you back from where you started.
By unfolding the torus as a rectangle, it seems that, for any starting point, some directions will generate closed trajectories and that others won't.
Are there results about what these 2 types of trajectories might look like ? (specially the "infinite" one)

REPLY [8 votes]: There is a nice web page by Mark Irons on geodesics on the torus: link.
Here is one image from that page, showing a period-2 geodesic that crosses itself three times: