Balanced triangulations. (English) Zbl 1285.52004

Jia, Lijie; Yuan, Liping; Zamfirescu, Carol T.; Zamfirescu, Tudor I.

Discrete Math. 313, No. 20, 2178-2191 (2013).

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)

Keywords:

balanced triangulations; acute triangulations; platonic solids

Citations:

Zbl 1115.52004

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