Acute triangulations of the regular dodecahedral surface

In this paper we consider geodesic triangulations of the surface of the regular dodecahedron. We are especially interested in triangulations with angles not larger than π/2π/2, with as few triangles as possible. The obvious triangulation obtained by taking the centres of all faces consists of 20 acute triangles.We show that there exists a geodesic triangulation with only 10 non-obtuse triangles, and that this is best possible.We also prove the existence of a geodesic triangulation with 14 acute triangles, and the non-existence of such triangulations with less than 12 triangles.