This is a question inspired by the number of arrangements of n circles in the affine plane which was discussed on Numberphile.
One way of overlapping 3 circles looks like this:
My informal question is: what are the sizes of the 3 smallest circles for this arrangement subject to the limitation that each section is large enough to enclose a unit circle?
The answer perhaps looks like this:
That is approximate (and possibly wrong!). In my drawing the 2 lower circles have radius 2, which is clearly the smallest they can be and still enclose 2 distinct unit circles. The upper circle has a radius of… a bit more than 2.
I think that to be well-defined, we must specify that we're looking for the smallest total area. What are the radii of circles which form the arrangement of circles that cover the minimum total area, subject to the constraint that each of the 5 sections encloses a unit circle?
If anyone can answer the question, that would of course be great. But I'd also appreciate any useful partial answers, since I'm really not sure how to attack the problem.
Note: In the overlapping circles problem, circles cannot be tangent. I'm not worried about that detail in the question I'm posing here. If the smallest arrangement includes 2 circles that touch in a tangent point (as I suspect it does), then the answer should more accurately be thought of as the limit as the circles get arbitrarily close.
Best Answer
Since we are looking for the limit, we will solve for the situation where the two bottom circles are actually tangent.
Assuming the large middle circle has radius $R$, the lengths are calculated and labeled in the picture in terms of $R$. In the red triangle we have $$2^2+(\sqrt{5}-R+1)^2=R^2$$ which gives $R=\sqrt{5}$