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Acute triangulations of the regular dodecahedral surface. (English) Zbl 1115.52004

The authors study acute (non-obtuse) geodesic triangulations of two-dimensional spaces, i.e. triangulations by geodesic triangles whose angles are smaller (not larger) than \(\pi/2\). The question is to find an acute (non-obtuse) triangulation of a given space \(M^2\) with as few triangles as possible. The problem was solved for the surfaces of four Platonic solids – tetrahedron, cube, octahedron and icosahedron [see the authors, Discrete Comput. Geom. 31, No. 2, 197–206 (2004; Zbl 1062.51014), J.-I. Itoh, Josai Math. Monogr. 3, 53–62 (2001; Zbl 0998.52015), T. Hangan, J.-I. Itoh and T. Zamfirescu, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43(91), No. 3–4, 279–285 (2000; Zbl 1048.51501)]. Actually the authors propose a solution for the dodecahedron: the main statement is that there exists a non-obtuse geodesic triangulation of the dodecahedron with only 10 triangles, and that this is the best possible.
Besides, it is demonstrated that there exist an acute geodesic triangulation of the dodecahedron with only 14 triangles, however it is still unclear whether an acute geodesic triangulation with less that 14 triangles does exist. Another more general open problem is the following: does a number \(N\) exist such that every compact convex surface in \(R^3\) admits an acute triangulation with at most \(N\) triangles.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B10 Three-dimensional polytopes
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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