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

TL;DR

This paper investigates triangle tilings where the tile's angles satisfy 3α + 2β = π, discovering new tilings, deriving Diophantine equations relating the number of tiles to triangle dimensions, and establishing conditions for tiling existence.

Contribution

It identifies all possible triangle shapes for tiling under the given angle condition, introduces new tilings, and formulates Diophantine equations to determine tiling feasibility.

Findings

Five possible triangle configurations for tilings are classified.

New families of tilings are constructed for each configuration.

Diophantine equations relate the number of tiles to triangle dimensions and restrict possible tilings.

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 $ABC$: 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.