Irregular tilings of regular polygons with similar triangles

Abstract

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