I hope people aren't too tired of these. I'm finding them to be really lovely.
This question is closely related to a question that I posted recently.
Another way of overlapping 3 circles looks like this:
If we want to size these circles such that they are as small as possible yet still large enough for a unit circle to fit in each of the 7 regions, then we get a diagram like this:
The question then, is what are the radii of the circles?
In contrast to my previous posts... I have a proposed answer! But the exact answer is sufficiently inelegant that it has me wondering whether I got it right. My explanation is shown below.
The smaller circle in the middle has radius A. The other two circles have radius B.
- EDIT FALSE: The red triangle must be a right triangle with legs B and hypotenuse $\sqrt{2}B$.
- We can also say that the red hypotenuse has length 2B - 2(A-2).
- The blue triangle gives us a right triangle using both A & B.
So we get these 2 equations:
- $\sqrt{2}B=2B-2(A-2)$
- $(B-1)^2=(A+1)^2+({\sqrt{2}\over2}B)^2$
Solving these equations gives exact answers that appear inelegant to me. So I question the result. But... the approximate answers are reasonably close to what I got just constructing the diagram. So perhaps the smaller circle, A, really is ${\sqrt{52\sqrt{2}+86}\over{2}}-{\sqrt{26\sqrt{2}+43}\over{2}}+{2\sqrt{2}}+{1\over2}≈5.178$
Confirmation that this seems valid or correction where it's not valid would be much appreciated.