Tiling the plane with equilateral triangles

TL;DR

This paper proves that tilings of the plane with equilateral triangles of bounded size are necessarily periodic and limited to at most three types of triangles, answering longstanding questions in geometric tiling theory.

Contribution

It establishes the periodicity and limited diversity of equilateral triangle tilings under size constraints, resolving open problems and confirming previous independent results.

Findings

Tilings are periodic if triangle sizes are bounded below.

Such tilings involve at most three different triangle types.

The results confirm and extend prior theorems in geometric tiling.

Abstract

Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar. The same result has been obtained independently by Richter and Wirth.