Acute triangulations of sphere and icosahedron (Q2752886)

An acute triangulation is a decomposition into geodesic triangles such that each of the interior angles is strictly smaller than \(\pi/2\). For a smooth metric this implies that around each vertex one must have at least 5 triangles. Consequently, for the round 2-sphere one needs at least 20 triangles for any acute triangulation.NEWLINENEWLINENEWLINEOne of the main results of the paper states that precisely those numbers of triangles \(n\geq 20\) can occur which are even and distinct from 22. (In the paper there are still two undecided cases, \(n = 28\) and \(n = 34\), which were solved later, according to communication by the author.) There are also results on acute triangulations of the cube and the dodecahedron. Here it is admitted that the intrinsic geodesic triangles may contain original vertices in their interiors. The regular icosahedron provides a natural acute triangulation with 20 triangles. However, it is shown that the same metric can be realized with any even number \(n\geq 14\) of acute triangles except for \(n = 18\) and \(n =24\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00049].