Spheres in Cubes

After pondering this for a while and coming up with the same f(k) for the volume ratios, I questioned why you would ever input k = 0. That situation wouldn't make physical sense, because here k must be limited to sqrt(3) - 1 < k < 1/[sqrt(3)-1]. If k is outside this range, you end up implying the spheres can intersect each other.

Hmm. I didn't think of it that way. When I first posed the question, I wanted k = 0 to look like this:



Thanks. Now I know.