Acute Square Triangulation
References
M. Bern and D. Eppstein. Polynomial size non-obtuse triangulation of polygons. Proc. 7th ACM Symp. Comp. Geom., 1991, pp. 342-350; Int. J. Comp. Geom. & Appl., vol. 2, 1992, pp. 241-255. Shows that any simple polygon can be triangulated with O(n2) non-obtuse triangles.
M. Bern, D. Eppstein, and J. Gilbert. Provably good mesh generation. Proc. 31st IEEE Symp. Foundations of Comp. Sci., 1990, vol. I, pp. 231-241; J. Comp. Sys. Sci., vol. 48, 1994, pp. 384-409. Figure 5 of the journal version includes a tile (labeled bbbb) triangulating the square with all angles at most 80 degrees.
M. Bern, S. Mitchell, and J. Ruppert. Linear-size non-obtuse triangulation of polygons. Proc. 10th ACM Symp. Comp. Geom., 1994, pp. 221-230; Disc. Comp. Geom., vol. 14, 1995, pp. 411-428.
Charles Cassidy and Graham Lord. A square acutely triangulated. J. Rec. Math., vol. 13, no. 4, 1980/81, pp. 263-268.
Summary (from Math. Reviews 84j:51036): We examine the problem of triangulating the square into acute-angled triangles. By a proper triangulation we mean a subdivision of the square and its interior into nonoverlapping triangles in such a way that any two distinct triangles are either disjoint, have a single vertex in common, or have one entire edge in common; and by an interior vertex we mean a vertex (of a triangle) which lies inside the square but not on its boundary. We begin with a proof, alternative to that of Lindgren, of the minimality-uniqueness of eight. Then we show there is no triangulation into nine triangles! And finally we demonstrate that there is a triangulation of the square into n acute-angled triangles for all n greater than or equal to ten.
J. L. Gerver. The dissection of a polygon into nearly equilateral triangles. Geom. Dedicata, vol. 16, 1984, pp. 93--106. Shows how to compute a dissection of a polygon (that is, vertices embedded within sides of triangles are allowed) with no angles larger than 72 degrees, assuming all interior angles of the input measure at least 36 degrees.
H. Lindgren. Austral. Math. Teacher, vol. 18, 1962, pp. 14-15. Cited by Cassidy and Lord.