Tiling triangle ABC with congruent triangles similar to ABC

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), each of which is similar to ABC. The more general problem when the tile is not similar to ABC is not treated in this paper; see [1]. We give a complete characterization of the numbers N for which some triangle ABC can be tiled by N tiles similar to ABC, and also a complete characterization of the numbers N for which a particular triangle ABC can so tiled. It has long been known that there is always a “quadratic tiling” when N is a square. We show that unless ABC is a right triangle, N must be a square. On the other hand, if ABC is a right triangle, there are two more possibilities: N can be a sum of two squares e + f if the tangent of one of the angles is the rational number e/f , or in case ABC is a 30-60-90 triangle, N can be three times a square. The key idea is that the similarity factor √ N is an eigenvalue of a certain matrix. The proofs we give involve only undergraduate level linear algebra. 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 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 1 Figure 2: A 4-tiling, a 9-tiling, and a 16-tiling Figure 3: Three 4-tilings 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. Let N be of the form n +m. Let triangle T be a right triangle with perpendicular sides n and m, say with n ≥ m. Let ABD be a right triangle with base AD of length m, the right angle at D and altitude mn, so side BD has length mn. Then ABD can be decomposed into m triangles congruent to T , arranged with their short sides (of length m) parallel to the base AD. Now, extend AD to point C, located n past D. Triangle ADC can be tiled with n copies of T , arranged with their long sides parallel to the base. The result is a tiling of triangle ABC by n + m copies of T . The first 5-tiling exhibited in Fig. 3 is the simplest example, where n = 2 and m = 1. The case N = 13 = 3 + 2 is illustrated in Fig. 5. We call these tilings “biquadratic.” More generally, a biquadratic tiling of triangle ABC is one in which ABC has a right angle at C, and can be divided by an altitude from C to AB into two triangles, each similar to ABC, which can be tiled respectively by n and m copies of a triangle similar to ABC. The second 5-tiling shows that this can be sometimes be done more generally than by combining two quadratic tilings. Figure 4: Two 5-tilings If the original triangle ABC is chosen to be isosceles, then each of the n triangles can be divided in half by an altitude; hence any isosceles triangle can be decomposed into 2n congruent triangles. If the original triangle is equilateral, then it can be first decomposed into n equilateral triangles, and then these triangles can be decomposed into 3 or 6 triangles each, showing that any equilateral triangle can be decomposed into 3n or 6n congruent triangles. These tilings are neither quadratic nor biquadratic. For example we can 12-tile an equilateral triangle in two different ways, starting with a 3-tiling and then subdividing each triangle into 4