Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.
Bibliographical noteFunding Information:
Our interest in this problem began at the American Mathematical Society’s Mathematics Research Communities workshop on Discrete and Computational Geometry in June 2012 at Snowbird. We thank Satyan Devadoss, Vida Dujmovic, Joseph O’Rourke, and Yusu Wang for organizing this workshop, and especially Joseph O’Rourke for introducing us to this problem. O’Rourke also made Fig. 5.6 . Travel funding for additional collaborations, during which we continued our work on this problem, was provided by the American Mathematical Society. We thank anonymous referees for suggesting additional references. Victor Dods partially supported by the 2011–2012 ARCS Fellowship. Jed Yang partially supported by NSF GRFP grant DGE-0707424 and NSF RTG grant DMS-1148634 .
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- Regular tetrahedron
- Stern–Brocot tree