Survey of two-dimensional acute triangulations
Discrete Math. 313, Iss. 1 (2013) 35–49.
http://czamfirescu.tricube.de/CTZamfirescu-08.pdf


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In the same busy year of 1960, Lindgren [54] described an acute triangulation of
the square (see Fig. 6), proving that it can be done with 8 triangles and that this is
optimal – Federico solved this independently in the same year. In 1966, Gardner also
gave a construction, which he reports in one of his mathematical columns (reprinted
in [35]), saying: “For days I was convinced that nine was the answer; then suddenly I
saw how to reduce it to eight”. We remark that, in fact, Gardner was trying to find
dissections of the square, and it happened that his configuration on eight triangles is a
triangulation. If he were looking for triangulations, this would have contradicted the
following matter.
Cassidy and Lord [15] continued the investigations of acutely triangulating the
square, publishing their results in 1980. They gave an alternative proof of the min-
imality (and combinatorial uniqueness) of the Federico-Lindgren construction on 8
triangles. They also showed that there is no triangulation consisting of exactly 9 tri-
angles, and proved that there exist acute triangulations of the square with k triangles
whenever k ≥ 10.


[54] H. Lindgren. A quadrilateral dissection. Austral. Math. Teacher 16 (1960) 64–65.
[15] Ch. Cassidy and G. Lord. A square acutely triangulated. J. Recr. Math. 13 (1980–
81) 263–268.