Geodesics on the regular tetrahedron and the cube
Given a convex polyhedron \(S\) in \({\mathbb{R}}^3\), a \textit{geodesic} on \(S\) is a curve \(\gamma\) on \(S\) such that for any sufficiently close points \(p\) and \(q\) on \(\gamma\), the shortest path on \(S\) from \(p\) to \(q\) coincides with \(\gamma\). A natural way how to study geodesics on polyhedra is by unfolding the polyhedron into the plane and, if possible, tiling the plane with faces of the polyhedron. In this way, geodesics on \(S\) correspond to straight lines in the plane.NEWLINENEWLINEIn the present paper, the regular tetrahedron and the cube are studied, because their faces give tiling of the plane. For the tetrahedron, geodesics from an arbitrary point to an arbitrary vertex are characterized in terms of corresponding directions. In particular, vertex-to-vertex geodesics are characterized this way. For the cube, vertex-to-vertex geodesics are studied. Depending on the pair of vertices chosen, the number of geodesic differs. Surprisingly, for both the tetrahedron and the cube, there are no geodesics which start and end in the same vertex.
scientific article; zbMATH DE number 6642332