Triangle Tiling: The case $3\alpha + 2\beta = \pi$

An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This paper takes up the case when $3\alpha + 2\beta = \pi$. Then there are (as was already known) exactly five possible shapes of $ABC$: either $ABC$ is isosceles with base angles $\alpha$, $\beta$, or $\alpha+\beta$, or the angles of $ABC$ are $(2\alpha,\beta,\alpha+\beta)$, or the angles of $ABC$ are $(2\alpha, \alpha, 2\beta)$. In each of these cases, we have discovered, and here exhibit, a family of previously unknown tilings. These are tilings that, as far as we know, have never been seen before. We also discovered, in each of the cases, a Diophantine equation involving $N$ and the (necessarily rational) number $s = a/c$ that has solutions if there is a tiling using tile $T$ of some $ABC$ not similar to $T$. By means of these Diophantine equations, some conclusions about the possible values of $N$ are drawn; in particular there are no tilings possible for values of $N$ of certain forms. We prove, for example, that there is no $N$-tiling with $N$ prime when $3\alpha + 2\beta = \pi$. These equations also imply that for each $N$, there is a finite set of possibilities for the tile $(a,b,c)$ and the triangle $ABC$. (Usually, but not always, there is just one possible tile.) These equations provide necessary, and in three of the five cases sufficient, conditions for the existence of $N$-tilings.