Anna starts at the origin facing the positive x-axis. She can (A) move one unit in the direction she is facing or (B) stay in place while rotating counterclockwise by angle θ. At some point in a sequence of actions, Anna must take action (A) at least once. In other words, a sequence of BBB... is not of interest.
1)If θ = arccos(-1/3) ≈ 109.47°, how can Anna take a sequence of actions listed above and return to the origin facing the positive x-axis?
2)Which angles will allow Anna to return to the origin, not necessarily facing the positive x-axis?
3)Angles that will allow Anna to return to the origin after tracing a path that is not a triangle or a regular polygon?
4)Are there angles that will never allow Anna to return to the origin?
My friend made up this problem for us to discuss and attempt to answer. It looks like this is analyzing movement in the plane. I have found that if θ = arccos(-1/3) ≈ 109.47°, then we can return to the origin (not facing the positive x-axis) with the sequence of actions AAABAABAAA. Any thoughts on how to return to the origin facing the positive x-axis?
Also, for any angle θ that is the interior angle of a regular polygon, we can use the sequence of steps ABABA...to trace out the corresponding polygon in the plane and return to the origin facing the positive x-axis. I am not sure of what other angles allow for the return to the origin or how to prove such.