Talia Blum

The winding number of a closed curve counts the number of rotations about a point. A geometric interpretation of Rouché's theorem relates winding numbers of the paths of a person walking a dog—or, in this case, a pig—on a leash. If a person walks a pig on a leash around a lamppost, such that the leash is always shorter than the distance between the person and the lamppost, then the winding numbers of the paths of the person and the pig about the lamppost will always be equal. In other words, the leash will never get tangled around the lamppost.

Polygonal billiards studies trajectories of pointlike balls moving in straight lines and reflecting along edges of a polygon. An equivalent formulation is to think of rays of light bouncing around a polygon with mirrors as edges. Standing inside the polygon and looking directly into a mirror, the trajectory of a ray of light appears to follow a straight line off to infinity. This is called an unfolded billiards trajectory, produced by reflecting the polygon repeatedly along its edges. The triangle in the image is reflecting on its unfolded self.