Tiling an Equilateral Triangle

TL;DR

This paper investigates the tiling of an equilateral triangle using smaller tiles, proving that for more than three tiles, the number of tiles cannot be prime, and explores specific cases where the tile has a  angle.

Contribution

It establishes new constraints on the number of tiles needed for tiling an equilateral triangle and analyzes cases with  angles in the tile, advancing understanding of geometric tilings.

Findings

For N > 3, N cannot be prime.

Certain shapes of tiles are only possible for specific N.

Detailed analysis of tiles with  angles in tiling configurations.

Abstract

Let $ABC$ be an equilateral triangle. For certain triangles $T$ (the "tile") and certain $N$, it is possible to cut $ABC$ into $N$ copies of $T$. It is known that only certain shapes of $T$ are possible, but until now very little was known about the possible values of $N$. Here we prove that for $N>3$, $N$ cannot be prime, and study more closely the possible tilings when the tile has a $\pi/3$ angle.