Tiling Triangles with $2\pi/3$ Angles

TL;DR

This paper investigates the tiling of triangles with angles of 2π/3, focusing on specific sporadic cases and proposing constructions and conjectures about possible tiling counts.

Contribution

It introduces new tiling constructions for six special triangles with a 2π/3 angle and conjectures these are the only solutions for these cases.

Findings

Constructed tilings for six specific triangles with 2π/3 angles.

Proposed conjectures that these are the only tiling counts for these triangles.

Extended understanding of tiling possibilities beyond known reptile and commensurable-angles cases.

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.