Tilings of an Isosceles Triangle

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N trianglescongruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, for N even, but leave open whether N can be odd. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or twice a prime.Comment: 34 pages, 18 figures. This version supplies corrections and simplification