Also a math question - I was unable to figure out how to calculate the radius of the outer ring of circles and approximated it by eye to 3.21725 times the radius of the inner ring. Anyone better at geometry than me who could give me a precise equation?
I did some trig. If the radius of each small circle is 1, then the radius of the next larger circles solves r^2 - 2( 1 / cos 36 + tan^2 36 ) r + 1 = 0, or approximately r=3.2170. If the center of the small pentagon of circles is the origin, then the coordinates of the center of the small circle above the origin is (0, 1/tan 36) = (0, 1.3764). The center of the medium circle above and to the right is (r, r/tan 36) = (3.2170, 4.4278). The other centers are rotated by an angle of 72 degrees about the origin.
I got something that I think is trigonometrically equivalent (after more time and some math errors), deriving it as a law-of-cosines problem relating the distances from the center and the sum of the two radii. The quadratic has two roots which are reciprocals, because it doesn't distinguish as to which is the inner circle. An interesting problem; it almost looks like a Descartes' theorem problem, but the circle positioning makes that not apply.
I didn't think about the interpretation of the other root, just that it was obviously less than 1 so not correct for this problem. Your reading makes a lot of sense.
When I got this solution, I poked at it with trig identities. You can write it in other forms, but I didn't find any which are obviously better.
Y'all are awesome. I'll update the code later but I don't expect any visible change to the image :)