Problem 199

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 .

迭代放圆

在一个大圆内放入三个等大的圆，这三个圆以及大圆两两相切，而且相互不重叠。这样一来，大圆内就有四块未覆盖的“空隙”，我们将迭代地在里面放入更多相切的圆以填补它们。

每一轮迭代，我们都在每个空隙中放入尽可能大的圆，如此一来下一轮迭代时将会有更多的空隙。经过三轮迭代（如上图），大圆中一共有108个空隙，大圆内未被其它圆覆盖的面积占大圆面积的比例为0.06790342，四舍五入到八位小数。

经过10轮迭代后，未被覆盖的面积占大圆面积的比例是多少？
你的答案应当四舍五入到八位小数，也即x.xxxxxxxx的格式。