2020/02/27 by Laczkovich, M.
#52C20 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2002.12013
We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if N>42, then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of π and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, \al and \be, such that at each vertex of the tiling the number of angles \al is the same as that of \be. Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if N>10, then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of π. Therefore, the numbers of triangles tiling the regular N-gon irregularly is at most three for every N>42.