Triangle Tiling IV: A non-isosceles tile with a 120 degree angle

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. We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N-tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T; or given N, determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is one of four papers on this subject. In this paper, we take up the last remaining case: when ABC is not similar to T, and T has a 120 degree angle, and T is not isosceles (although ABC can be isosceles or even equilateral). Here is our result: If there is such an N-tiling, then the smallest angle of the tile is not a rational multiple of \pi. In total there are six tiles with vertices at the vertices of ABC. If the sides of the tile are (a,b,c), then there must be at least one edge relation of the form jb = ua + vc or ja = ub + vc, with j, u, and v all positive. The ratios a/c and b/c are rational, so that after rescaling we can assume the tile has integer sides, which by virtue of the law of cosines satisfy c^2 = a^2 + b^2 + ab. A simple unsolved specific case is when ABC is equilateral and (a,b,c) = (3,5,7). The techniques used in this paper, for the reduction to the integer-sides case, involve linear algebra, elementary field theory and algebraic number theory, as well as geometrical arguments. Quite different methods are required when the sides of the tile are all integers.