Tiling a Triangle with Congruent Triangles

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), which may or may not be similar to ABC. We wish to characterize the numbers N for which some triangle ABC can be tiled by N tiles, or more generally to characterize the triples (N,T ) such that ABC can be N-tiled using tile T . In the first part of the paper we exhibit certain families of tilings which contain all known tilings. We conjecture that the exhibited tilings are the only possible tilings. If that is so, then for there to exist an N-tiling of any triangle ABC, N must be a square, or 2, 3, or 6 times a square, or a sum of two squares. We were able to reduce this conjecture to a special case. The case we could not solve is when tile has angles α, β, and γ with 3α = 2β, and sin(α/2) is rational. Some number-theoretic properties of N are also necessary. The triangle ABC must have angles 2α, β, and β + γ and α is not a rational multiple of π. The simplest unsolved case is N = 28, with a tile whose sides are 2, 3, and 4, and triangle ABC has sides 12, 14, and 16. In particular, there are no N-tilings for N = 7. We have earlier given a (rather long) traditional Euclid-style proof of the impossibility of a 7-tiling, but could not handle evenN = 11, let alone N = 19. Now we know there are no N-tilings for N = 11, 19, and 23. The proof in this paper goes beyond Euclidean geometry, by bringing the tools of linear algebra and field theory to bear on the problem. The key idea for the case when the tile is similar to ABC is that the similarity factor √ N is an eigenvalue of a certain matrix. In the case when T is not similar to ABC, various geometrical considerations deal with all but a few special cases. When the tile has angles π/11, 3π/11, and 7π/11, or π/14, 4π/14, and 9π/14, we use the theory of cyclotomic fields, and when the tile has one angle of 120◦ and one angle of 2π/15 = 24◦, or π/9 = 20◦, or π/12 = 15◦, we also use some algebraic number theory about the primes in cyclotomic fields. 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. The method illustrated for N = 4 ,9, and 16 clearly generalizes to any perfect square N . While the exhibited 3-tiling, 6-tiling, and 5-tiling clearly depend on the exactly angles of the triangle, any triangle can be decomposed into n congruent triangles by drawing n − 1 lines, parallel to each edge and dividing the other two edges into n equal parts. Moreover, the large (tiled) triangle is similar to the small triangle (the “tile”). We call such a tiling a quadradtic tiling. It follows that if we have a tiling of a triangle ABC into N congruent triangles, and m is any integer, we can tile ABC into Nm triangles by subdividing the first tiling, replacing each of the N triangles by m smaller ones. Hence the set of N for which an N-tiling of some triangle exists is closed under multiplication by squares.