1. Geodesics on the regular tetrahedron and the cube
- Author
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Victor Dods, Jed Yang, Diana Davis, and Cynthia M. Traub
- Subjects
Discrete mathematics ,Geodesic ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Disphenoid ,Vertex (geometry) ,Combinatorics ,Polyhedron ,010201 computation theory & mathematics ,Stern–Brocot tree ,Tetrahedron ,Mathematics::Metric Geometry ,Discrete Mathematics and Combinatorics ,Mathematics::Differential Geometry ,0101 mathematics ,Cube ,Trirectangular tetrahedron ,Mathematics - Abstract
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 SternBrocot 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.
- Published
- 2017
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