Tilings of an Isosceles Triangle

TL;DR

This paper investigates tilings of an isosceles triangle using congruent tiles of specific angle types, providing characterizations, side ratio conditions, and restrictions on the number of tiles.

Contribution

It offers a complete characterization for right-angled tiles and establishes side ratio rationality and tiling existence conditions for other angle configurations.

Findings

N must be even for right-angled tiles.

Side ratios of tiles are rational in certain cases.

N cannot be prime or squarefree when one angle doubles another.

Abstract

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, and prove that N must be even. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or even squarefree.