Tiling Triangles with $2π/3$ Angles

TL;DR

The paper tackles the tiling problem for triangles in the non-reptile, non-commensurable-angles regime, focusing on the case where the tile $R$ contains an angle of $2\pi/3$ and the outer triangle $T$ is one of six sporadic shapes. It introduces ideal trapezoids as a key intermediate construct and develops two main families of tilings based on equilateral and $\pi/3$-angled intermediate steps, yielding infinite, arithmetic-structured families of tile counts $N$ (notably of the form $N = m^2ab$ and $N = (a+2b)(b+2a)m^2$) and explicit constructions for several $(T,R)$ pairs. The authors conjecture that these constructions capture all large-$N$ tilings for the considered cases and propose further work to prove conciseness of the trapezoid criterion and to generalize the approach to all non-commensurable configurations. The work blends geometric decompositions with number-theoretic tools (Frobenius-type results) to map the feasible $N$-values and provides concrete examples (e.g., $(a,b,c)=(3,5,7)$ and $(5,8,7)$) to illustrate the method and its potential reach.

Abstract

Motivated by a question of Erdös 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 $π$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2π/3$ 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.

Figures (12)

• Figure 1: A List of the possible non-commensurable-angles cases, where $ABC$ is the triangle being tiled. This table is taken from Beeson beeson-seven, though it contains the same content as Theorem 4.1 of laczkovich1995tilings.
• Figure 2: The basic ideal trapezoid. The marked angles are equal to $\alpha$.
• Figure 3: How to tile different parallelograms. The bigger parallelograms $ABCD$ and $BCFE$ are tiled by the same small $(a,b,a,b)$ parallelogram, which in turn tiles into two $(a,b,c)$ triangles each.
• Figure 4: Tiling more complex ideal trapezoids.
• Figure 5: Tiling the equilateral triangle into $3$ ideal trapezoids.

Theorems & Definitions (35)

• Lemma 1
• proof
• Proposition 1: Frobenius number of $2$ Elements
• Lemma 2
• proof
• Proposition 2
• proof
• Theorem 3
• proof
• Lemma 3