Tilings of Parallelograms with Similar Right Triangles

摘要

We say that a triangle (T) tiles the polygon (mathcal A ) if (mathcal A ) can be decomposed into finitely many non-overlapping triangles similar to (T). A tiling is called regular if there are two angles of the triangles, say (alpha ) and (beta ), such that at each vertex (V) of the tiling the number of triangles having (V) as a vertex and having angle (alpha ) at (V) is the same as the number of triangles having angle (beta ) at (V). Otherwise the tiling is called irregular. Let (mathcal P (delta )) be a parallelogram with acute angle (delta ). In this paper we prove that if the parallelogram (mathcal P (delta )) is tiled with similar triangles of angles ((alpha , beta , pi /2)), then ((alpha , beta )=(delta , pi /2-delta )) or ((alpha , beta )=(delta /2, pi /2-delta /2)), and if the tiling is regular, then only the first case can occur.