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

摘要

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a unionof N trianglescongruent to T, overlapping only at their boundaries. Thetriangle T is the "tile". The tile may or may not be similar to ABC. We wish tounderstand possible tilings by completely characterizing the triples (ABC, T,N) such that ABC can be N-tiled by T. In particular, this understanding shouldenable us to specify for which N there exists a tile T and a triangle ABC thatis N-tiled by T; or given N, determine which tiles and triangles can be usedfor N-tilings; or given ABC, to determine which tiles and N can be used toN-tile ABC. This is one of four papers on this subject. In this paper, we takeup the last remaining case: when ABC is not similar to T, and T has a 120degree angle, and T is not isosceles (although ABC can be isosceles or evenequilateral). Here is our result: If there is such an N-tiling, then the smallest angle ofthe tile is not a rational multiple of \pi. In total there are six tiles withvertices at the vertices of ABC. If the sides of the tile are (a,b,c), thenthere 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 thatafter rescaling we can assume the tile has integer sides, which by virtue ofthe law of cosines satisfy c^2 = a^2 + b^2 + ab. A simple unsolved specificcase is when ABC is equilateral and (a,b,c) = (3,5,7). The techniques used inthis paper, for the reduction to the integer-sides case, involve linearalgebra, elementary field theory and algebraic number theory, as well asgeometrical arguments. Quite different methods are required when the sides ofthe tile are all integers.