Über regulär pflasterbare konvexe Polygone. (On regular tilings of convex polygons)
The author considers tiling a convex n-gon edge-to-edge with regular polygons of edge length one in an Euclidean plane. If such a tiling is possible, the n-gon is called regular decomposible. \textit{J. Malkevitch} [Ann. N. Y. Acad. Sci. 440, 299-303 (1985; Zbl 0575.52007)] showed that for and only for every natural number n with \(3\leq n\leq 12\) there exists a convex n-gon which is regular decomposible into regular triangles and quadrangles. Now allowing all regular p-gons with edge length one the author proves two assertions: (1) For a regular decomposible convex n-gon (not regular with edge length 1, that means non-trivial regular decomposible) also holds \(3\leq n\leq 12\). (2) The tiles of such a tiling can only be regular p-gons with \(p=3,4,5,6\), or 12. - Because regular 6- gons and 12-gons can be dissected into regular triangles and quadrangles the set of regular triangles, quadrangles, and pentagons of edge length one is a generator set for all non-trivial regular decomposible polygons. - These results were elementary found by using the assumption that the polygons are convex and by using estimations for the angle sums at the vertices.
scientific article; zbMATH DE number 4013438