Dissection of a triangle into similar triangles

Abstract

We prove that the number of non-similar triangles T which can be dissected into two, three or five similar non-right triangles is equal to zero, one and nine, respectively. We find all these triangles. Moreover, every triangle can be dissected into n similar triangles whenever n = 4 or n ≥ 6. In the last section we allow dissections into right-triangles but we add another restriction. We prove that in any perfect, prime and simplicial dissection into at least three tiles, the tiles must have one of only three possible shapes. © Springer 2005.