Year: 2026
Type: preprint
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).
Source: arXiv (Cornell University)
Authors: Beeson, Michael, Zhang, Yan X
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7148731506
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