Tilings of polygons with similar triangles, II

Let A be a polygon, and let s(A) denote the number of distinct nonsimilar triangles a such that A can be dissected into finitely many triangles similar to Delta. If A can be decomposed into finitely many similar symmetric trapezoids, then s(A) = infinity. This implies that if A is a regular polygon, then s(A) = infinity. In the other direction, we show that if s(A) = infinity, then A can be decomposed into finitely many symmetric trapezoids with the same angles. We introduce the following classification of tilings: a tiling is regular if Delta has two angles, alpha and beta, such that at each vertex of the tiling the number of angles alpha is the same as that of beta. Otherwise the tiling is irregular. We prove that for every polygon A the number of triangles that tile A irregularly is at most c.n(6), where n is the number of vertices of A. If A has a regular tiling, then A can be decomposed into finitely many symmetric trapezoids with the same angles.