×

The dissection of a polygon into nearly equilateral triangles. (English) Zbl 0547.05026

Every polygon can be dissected into acute triangles. In this paper it is proved that every polygon, such that all interior angles are at least \(\pi/5\), can be dissected into triangles with interior angles all less than or equal to \(2\pi/5\). Necessary conditions are found on the interior angles of the polygon in order to obtain a dissection into triangles with interior angles all \(\leq\alpha\) (where \(\pi/3< \alpha< 2\pi/5),\) and the conjecture is stated that these conditions are also sufficient. These results are proved first for regular polygons, using Euler’s formula, and extended to irregular polygons via the Riemann mapping theorem.

MSC:

05B45 Combinatorial aspects of tessellation and tiling problems
51M15 Geometric constructions in real or complex geometry
PDF BibTeX XML Cite
Full Text: DOI