Balanced triangulations

Motivated by applications in numerical analysis, we investigate balanced triangulations, i.e. triangulations where all angles are strictly larger than π/6π/6 and strictly smaller than π/2π/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 1212 acute triangles [J. Itoh, T. Zamfirescu, Acute triangulations of the regular dodecahedral surface, European J. Combin. 28 (2007) 1072–1086].