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Balanced triangulations. (English) Zbl 1285.52004
Summary: Motivated by applications in numerical analysis, we investigate balanced triangulations, i.e. triangulations where all angles are strictly larger than $$\pi /6$$ and strictly smaller than $$\pi /2$$, giving the optimal lower bound for the number of triangles in the case of the square. We also investigate platonic surfaces, where we find for each one its respective optimal bound. In particular, we settle (affirmatively) the open question whether there exist acute triangulations of the regular dodecahedral surface with 12 acute triangles [J.-I. Itoh and T. Zamfirescu, Eur. J. Comb. 28, No. 4, 1072–1086 (2007; Zbl 1115.52004)].
##### 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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