Tilings of Right Trapezoids with Similar Triangles

Abstract

We say that a triangle A of angles a, /?, and 7 tiles a polygon V, if V can be decomposed into finitely many non-overlapping triangles similar to A. A tiling is called regular if there are two angles of the triangles, say a and such that at each vertex V of the tiling the number of triangles having V as a vertex and having angle a at V is the same as the number of triangles having angle £ at V. Otherwise the tiling is called irregular. Denote by 71(6) a right trapezoid with acute angle 8. In this paper we first consider the regular tilings of convex polygons with similar triangles and prove that 72(5) has not any regular tiling with similar right triangles. Then we prove that if 11(8) has an irregular tiling with similar right triangles of angles (a,/3, n/2) then (a, 0) = (<5, tt/2 - 6) or (a, /?) = (6/2, tt/2 - 6/2), and prove that if Tl(6) has an irregular tiling with similar non-right triangles T then the angles of T are given by one of the four triples (7t/6, tt/6, 27t/3), (7r/8,7r/4,57r/8), (tt/4, tt/3, 5tt/12), and (tt/12, tt/4, 2tt/3). © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.