Parallel covering a rhombus with equilateral triangles

Abstract

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