Trajectory on a Polyhedron from a Vertex to Itself
Given a net for a polyhedron, find a straight line from some vertex to itself that lies inside the net. For the regular dodecahedron, such a line exists that does not pass through any other vertex . In this Demonstration, a random net for a polyhedron is given. Drag the locators at the two ends of the red line to try to join the vertex to itself.
 J. S. Athreya and D. Aulicino, "A Trajectory from a Vertex to Itself on the Dodecahedron," The American Mathematical Monthly, 126(2), 2019 pp. 161–162. doi:10.1080/00029890.2019.1538475.