题目199：迭代圆形包装

欧拉计划

Iterative Circle Packing

Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles.

At each iteration, a maximally sized circle is placed in each gap, which creates more gaps for the next iteration. After 3 iterations (pictured), there are 108 gaps and the fraction of the area which is not covered by circles is 0.06790342, rounded to eight decimal places.

What fraction of the area is not covered by circles after 10 iterations?
Give your answer rounded to eight decimal places using the format x.xxxxxxxx .

题目：

三个半径相同的圆放入一个更大的圆中，相互之间相切且不重叠。这样，就出现了四个未覆盖的区域，我们可以用更多相切的圆来填充这些区域。

每次迭代之后，空白区域中会放入一个最大圆，因此会形成更多下次迭代要处理的空白区域。三次迭代之后（如上图），总共有 108 个零散的空白区域，它们的总面积与整个大圆的面积之比为 0.06790342 （保留小数点后面八位）。

请问，10 次迭代之后，这个面积比（空白区域与整个大圆的面积比）为多少？

答案精确到小数点后面 8 位，形如 x.xxxxxxxx。