## 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.

### MSC:

 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry)

### Keywords:

triangulation; cover; acute triangulation; trapezoid

### Citations:

Zbl 1048.51501; Zbl 0998.52005
Full Text:

### References:

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