Rationality of certain 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 $π$), 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).

Figures (4)

• Figure 1: A link $PQ$ in $\Gamma_b$ gives rise to another link through $QR$.
• Figure 2: $N=1215$. The tile is $(3,5,7)$ and $\gamma = 2\pi/3$.
• Figure 3: Before and after adding sawtooth tiles
• Figure 4: Lemma 7.1

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof