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 .
The solution to this problem relies on the concept of geometric series. It’s important to know that the radius of every next circle decreases by a factor of the square root of 3 each iteration, while the number of circles triples.
Let us denote x as the fraction of the area covered by circles after any iteration.
Then for the next iteration, because of the decrease in radius by a factor of sqrt(3), each new circle will cover only 1 / sqrt(3)^2 = 1/3 of the area of the previous circles. But because the number of circles triples each time, this means that all the new circles in that iteration will also cover an area of x/3.
So, after any iteration, the total fraction that is covered is ‘x + x/3’, since x is the fraction already covered while ‘x/3’ is the area covered by the new circles.
However, we know from the question that after the 3rd iteration, 0.93209658 [= $1 – 0.06790342$] of the total area is covered.
Therefore, x + x/3 = 0.93209658…
Solving this equation gives x = 0.69857244…
Now, after the 10th iteration, the total fraction covered will be x + 7x/3. (= 0.69857244 + 7*0.69857244/3 ), since in the 7 remaining iterations (from the 4th to the 10th), each new iteration adds x/3 to the total area covered.
Mathematically speaking to eight decimal places, the fraction of the area not covered by circles after 10 iterations is
$1 – (0.69857244 + 7 * 0.69857244 / 3) ≈ 0.01970260…$
So, rounded to eight decimal places, this fraction is around 0.01970260.
Let me know if you need clarification or additional information.
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