Tilings of Convex Polygons with Congruent Triangles

Abstract

By the spectrum of a polygon A we mean the set of triples (α,β,γ) such that A can be dissected into congruent triangles of angles α,β,γ. We propose a technique for finding the spectrum of every convex polygon. Our method is based on the following classification. A tiling is called regular if there are two angles of the triangles, α and β such that at every vertex of the tiling the number of triangles having angle α equals the number of triangles having angle β. Otherwise the tiling is irregular. We list all pairs (A,T) such that A is a convex polygon and T is a triangle that tiles A regularly. The list of triangles tiling A irregularly is always finite, and can be obtained, at least in principle, by considering the system of equations satisfied by the angles, examining the conjugate tilings, and comparing the sides and the area of the triangles to those of A. Using this method we characterize the convex polygons with infinite spectrum, and determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough.