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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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[1] Alexandrov, A. D., On partitions and tessellations of the plane, Mat. Sb., 2, 307-318, (1937), (in Russian)
[2] Baker, B. S.; Grosse, E.; Rafferty, C. S., Nonobtuse triangulations of polygons, Discrete Comput. Geom., 3, 147-168, (1988) · Zbl 0634.57012
[3] Bern, M.; Mitchell, S.; Ruppert, J., Linear-size nonobtuse triangulations of polygons, Discrete Comput. Geom., 14, 411-428, (1995) · Zbl 0841.68118
[4] Cassidy, Ch.; Lord, G., A square acutely triangulated, J. Recreat. Math., 13, 263-268, (1980-1981)
[5] Eppstein, D.
[6] Fulton, C., Tessellations, Amer. Math. Monthly, 99, 442-445, (1992) · Zbl 0790.52009
[7] Gerver, J. L., The dissection of a polygon into nearly equilateral triangles, Geom. Dedicata, 16, 93-106, (1984) · Zbl 0547.05026
[8] Hangan, T.; Itoh, J.; Zamfirescu, T., Acute triangulations, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 43, 91, 279-285, (2000) · Zbl 1048.51501
[9] Itoh, J., Acute triangulations of sphere and icosahedron, Josai Math. Monogr., 3, 53-62, (2001) · Zbl 0998.52015
[10] Itoh, J.; Zamfirescu, T., Acute triangulations of the regular icosahedral surface, Discrete Comput. Geom., 31, 197-206, (2004) · Zbl 1062.51014
[11] Itoh, J.; Zamfirescu, T., Acute triangulations of the regular dodecahedral surface, European J. Combin., 28, 1072-1086, (2007) · Zbl 1115.52004
[12] Niven, I., Convex polygons that cannot tile the plane, Amer. Math. Monthly, 85, 785-792, (1978) · Zbl 0403.52001
[13] Yuan, L., Acute triangulations of polygons, Discrete Comput. Geom., 34, 697-706, (2005) · Zbl 1112.52002
[14] Zamfirescu, C. T., Survey of two-dimensional acute triangulations, Discrete Math., 313, 35-49, (2013) · Zbl 1261.52012
[15] T. Zamfirescu, Acute triangulations: a short survey, in: Proc. 6th Ann. Conf. Roum. Soc. Math. Sci., Vol. I, 2002, pp. 10-18.
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