Triangle Tiling I: the tile is similar to ABC or has a right angle

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T , overlapping only at their boundaries. The triangle T is the “tile”. The tile may or may not be similar to ABC. This paper is the first of four papers, which together seek a complete characterization of the triples (ABC,N, T ) such that ABC can be N-tiled by T . In this paper, we consider the case in which the tile is similar to ABC, the case in which the tile is a right triangle, and the case when ABC is equilateral. We use (only) techniques from linear algebra and elementary field theory, as well as elementary geometry and trigonometry. Our results (in this paper) are as follows: When the tile is similar to ABC, we always have “quadratic tilings” when N is a square. If the tile is similar to ABC and is not a right triangle, then N is a square. If N is a sum of two squares, N = e + f, then a right triangle with legs e and f can be N-tiled by a tile similar to ABC; these tilings are called “biquadratic”. If the tile and ABC are 30-60-90 triangles, then N can also be three times a square. If T is similar to ABC, these are all the possible triples (ABC,T,N). If the tile is a right triangle, of course it can tile a certain isosceles triangle when N is twice a square, and in some cases when N is six times a square. Equilateral triangles can be 3-tiled and 6-tiled and hence they can also be 3n and 6n tiled for any n. We also discovered a family of tilings when N is 3 times a square, which we call the “hexagonal tilings.” These tilings exhaust all the possible triples (ABC,T,N) in case T is a right triangle or is similar to ABC. Other cases are treated in subsequent papers. 1 Examples of Tilings We consider the problem of cutting a triangle into N congruent triangles. Figures 1 through 4 show that, at least for certain triangles, this can be done with N = 3, 4, 5, 6, 9, and 16. Such a configuration is called an N-tiling. Figure 1: Two 3-tilings