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 [1]. 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.

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DETAILS

Reference
[1] 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.
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