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**Acute triangulations of trapezoids.**
*(English)*
Zbl 1205.52006

A polygon is said to have an acute triangulation of size \(n\) (or can be triangulated by \(n\) acute triangles) if it can be covered by \(n\) acute triangles each two of which share one vertex or one edge at most.

It is proved by C. Cassidy and G. Lord in [J. Recr. Math. 13, 263–268 (1980/1981)] that every square can be triangulated by \(8\), and no less than 8, acute triangles. This result is extended to arbitrary rectangles by T. Hangan, J. Itoh, and T. Zamfirescu in [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43 (91), No. 3–4, 279–285 (2000; Zbl 1048.51501)]. The paper under review proves that a trapezoid that is not a rectangle can be triangulated by 7 or less acute triangles and that there is a trapezoid that cannot be triangulated by less than 7 acute triangles. Thus a rectangle can be defined to be a trapezoid that does not have an acute triangulation of size 7 or less.

The paper also talks about analogous known results pertaining to other figures such as quadrilaterals, pentagons, and Platonic surfaces. It mentions H. Maehara’s result in [Proc. JCDCG 2000, Lecture Notes in Comput. Sci. 2098, 237–243 (2001; Zbl 0998.52005)] that every quadrilateral can be triangulated by 10 or less acute triangles and that there is a quadrilateral, albeit non-convex, for which 10 is minimal, and raises the question whether every convex quadrilateral can be triangulated by 8 or less acute triangles.

It is proved by C. Cassidy and G. Lord in [J. Recr. Math. 13, 263–268 (1980/1981)] that every square can be triangulated by \(8\), and no less than 8, acute triangles. This result is extended to arbitrary rectangles by T. Hangan, J. Itoh, and T. Zamfirescu in [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43 (91), No. 3–4, 279–285 (2000; Zbl 1048.51501)]. The paper under review proves that a trapezoid that is not a rectangle can be triangulated by 7 or less acute triangles and that there is a trapezoid that cannot be triangulated by less than 7 acute triangles. Thus a rectangle can be defined to be a trapezoid that does not have an acute triangulation of size 7 or less.

The paper also talks about analogous known results pertaining to other figures such as quadrilaterals, pentagons, and Platonic surfaces. It mentions H. Maehara’s result in [Proc. JCDCG 2000, Lecture Notes in Comput. Sci. 2098, 237–243 (2001; Zbl 0998.52005)] that every quadrilateral can be triangulated by 10 or less acute triangles and that there is a quadrilateral, albeit non-convex, for which 10 is minimal, and raises the question whether every convex quadrilateral can be triangulated by 8 or less acute triangles.

Reviewer: Mowaffaq Hajja (Irbid)

### MSC:

52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |

52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |

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\textit{L. Yuan}, Discrete Appl. Math. 158, No. 10, 1121--1125 (2010; Zbl 1205.52006)

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### References:

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