Irregular Tilings of Regular Polygons with Similar Triangles

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 $$N>42$$ N > 42 , then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of $$\pi $$ π and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $$\alpha $$ α and $$\beta $$ β , such that at each vertex of the tiling the number of angles $$\alpha $$ α is the same as that of $$\beta $$ β . 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 $$N>10$$ N > 10 , then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of $$\pi $$ π . Therefore, the number of triangles tiling the regular N-gon irregularly is at most three for every $$N>42$$ N > 42 .