We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good idea of the exact maximum density. In particular, we get new intervals for $r$ which does not allow any packing more dense that the hexagonal packing of equal discs.

中文翻译：

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二元圆盘填料的密度：下限和上限

我们为任何 $r\in (0,1)$ 提供了半径为 $1$ 和 $r$ 的圆盘的欧几里得平面中最大密度的下限和上限。下限大多是民间的，但上限改进了任何 $r\in[0.11,0.74]$ 之前已知的最好的。对于 $r$ 的许多值，这给出了准确的最大密度的相当好的概念。特别是，我们得到了 $r$ 的新间隔，它不允许任何比相等圆盘的六边形堆积更密集的堆积。