  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.
