TILINGS OF POLYGONS WITH SIMILAR TRIANGLES, II

Let A be a polygon, and let s(A) denote the number of distinct nonsimi lar triangles a such that A can be dissected into finitely many triang les similar to Delta. If A can be decomposed into finitely many simila r symmetric trapezoids, then s(A) = infinity. This implies that if A i s 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 m any symmetric trapezoids with the same angles. We introduce the follow ing classification of tilings: a tiling is regular if Delta has two an gles, alpha and beta, such that at each vertex of the tiling the numbe r 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 ve rtices of A. If A has a regular tiling, then A can be decomposed into finitely many symmetric trapezoids with the same angles.