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Periodic billiard paths: http://web.colby.edu/thegeometricviewpoint/2014/04/25/periodic-billiard-paths/

If the boundary of a given polygon is made of mirrors, these are paths that a laser beam could take that would eventually reflect back to the starting point and angle and then repeat infinitely. It remains a heavily-studied open question whether such paths exist in every triangle. This blog post from 2014 provides a proof that they do exist in polygons whose vertex angles are all rational multiples of π.