Parallel covering a rhombus with equilateral triangles

Abstract

Suppose that ${R}^α$ is a rhombus with side length $1$ and with an interior angle $α$, where $0<α\leq \fracπ{2}$. Let $\triangle$ be an equilateral triangle with a side parallel to a side of ${R}^α$ and let $\{\triangle_{n}\}$ be a collection of homothetic copies of $\triangle$. In this paper, we show the following two results: if $0<α\leq\fracπ{3}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\cosα+\frac{\sqrt{3}}{3}\sinα)^{2}$, then these equilateral triangles can parallel cover the rhombus ${R}^α$; if $\fracπ{3}<α\leq\fracπ{2}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\frac{2\sqrt{3}}{3}\sinα)^{2}$, then they can parallel cover the rhombus ${R}^α$. Furthermore, these bounds are optimal on their respective intervals.