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

Abstract

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 $ABC$ not similar to $T$: 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 $(2\alpha ,\alpha ,2\beta )$. We prove that the tile is rational (the ratios of its sides are rational). We then exhibit several families of previously unknown tilings. We also exhibit, for each shape, a Diophantine equation in $N$ and the parameters of the tile that must have a solution whenever there is a tiling of some triangle of that shape. By means of these Diophantine equations, some conclusions about the possible values of $N$ are drawn. We prove, for example, that there is no $N$-tiling with $N$ prime when $3\alpha +2\beta =\pi$, for three of the five shapes; that question remains open for the shape $(2\alpha ,\beta ,\alpha +\beta )$ and for isosceles triangles with base angles $\beta$. 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.) For each of the five shapes, we give a construction method that works under certain sufficient conditions, but there remain values of $N$ that satisfy our necessary conditions and for which we do not know whether a tiling exists.

103 pages, 30 figures. This version corrects errors in the previous version. It supplies a correct proof of the rationality of the tile. It retracts the claim that $K|M^2$ is a necessary as well as sufficient condition for tiling the shape $2\alpha ,\beta ,\alpha +\beta )$. It retracts the claim that $N$ cannot be prime for two of the five shapes; it stands for the other three shapes