Abstract
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 / 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.
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