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arXiv · 2512.22696

Tiling Triangles with $2\pi/3$ Angles

Yan X Zhang
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Abstract

Motivated by a question of Erd\"{o}s and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $\pi$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2\pi/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles.

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Yan X Zhang. 2025-12-27. Tiling Triangles with $2\pi/3$ Angles. https://arxiv.org/abs/2512.22696

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