用等边三角形平行覆盖菱形
Parallel covering a rhombus with equilateral triangles
AI总结：
本文针对内角为$\alpha$（$0<\alpha\leq\frac{\pi}{2}$）的菱形${R}^{\alpha}$，证明了两类不同$\alpha$区间下，满足对应面积和下界的等边三角形位似拷贝可平行覆盖该菱形，且下界为最优值。
AI中文摘要：
设${R}^{\alpha}$为边长为1且内角为$\alpha$（其中$0<\alpha\leq \frac{\pi}{2}$）的菱形。令$\triangle$为一条边与${R}^{\alpha}$的一条边平行的等边三角形，且$\{\triangle_{n}\}$为$\triangle$的一系列位似拷贝。在本文中，我们证明了以下两个结果：若$0<\alpha\leq\frac{\pi}{3}$，且$\{\triangle_{n}\}$中等边三角形的面积和至少为$\frac{\sqrt{3}}{4}(1+\cos\alpha+\frac{\sqrt{3}}{3}\sin\alpha)^{2}$，则这些等边三角形可平行覆盖菱形${R}^{\alpha}$；若$\frac{\pi}{3}<\alpha\leq\frac{\pi}{2}$，且$\{\triangle_{n}\}$中等边三角形的面积和至少为$\frac{\sqrt{3}}{4}(1+\frac{2\sqrt{3}}{3}\sin\alpha)^{2}$，则它们可平行覆盖菱形${R}^{\alpha}$。此外，这些界在各自的区间上是最优的。
英文摘要：
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.
