Rationality of certain triangle tilings

TL;DR

This paper investigates tilings of triangles with congruent triangles having a 120° angle and non-rational angles, proving that such tilings must have sides with rational ratios, except in special cases.

Contribution

It establishes that tilings with these specific triangles necessarily have sides with rational ratios, revealing a fundamental geometric constraint.

Findings

Tilings with a 120° angle triangle and non-commensurable angles have sides that are rationally related.

Most triangle tilings either have commensurable angles or sides, with few exceptions.

The results impose algebraic restrictions on possible triangle tilings.

Abstract

We consider tilings of a triangle $ABC$ by congruent copies of a triangle that has one angle equal to $120^\circ$, has non-commensurable angles (that is, not all angles are rational multiples of $\pi$), and is not similar to $ABC$. We prove that any such tiling has commensurable sides, meaning that the side lengths can be taken to be integers after scaling. As a consequence, we show that outside of a couple of special cases, a triangle (allowing all angles) tiling must either have commensurable angles or commensurable sides (that is, all sides have rational ratios).