Close packing of small spheres around a large one It is well known that, given a sphere, the maximum number of identical spheres that we can pack around it is exactly 12, corresponding to a face centered cubic or hexagonal close packed lattice.

My question is: given a sphere of radius $R$, how many spheres of radius $r&lt;R$ can we closely pack around it?

With disks, the problem is rather easy to solve. Indeed, with reference to the picture at the bottom, we can see that we must have

$$\theta = \frac{2 \pi} n = 2 \arctan \left( \frac r {\sqrt{R^2+2 R r}} \right)$$

from which 

$$n = \left \lfloor \frac \pi {\arctan \left( \frac r {\sqrt{R^2+2 R r}} \right)}\right \rfloor$$

The last expression gives the correct result for $R=r$, namely $n=6$ (hexagonal lattice). Moreover, when $R \gg r$, we get 

$$n \simeq \left \lfloor \frac {\pi R}  {r}\right \rfloor$$

which is completely reasonable.

How can I tackle the same problem in the 3D case (spheres)?

It is clear that for  $R \gg r$ we must get 

$$n \simeq \left \lfloor \frac {4 \pi R^2}  {\pi r^2}\right \rfloor$$

and also that we must have $n(R=r)=12$.

Any hint/suggestion is appreciated.
My question was how many spheres within the large ball given that the spheres have a radius of smallest known atom and the number of dimensions of packing  is equal to the radius of smallest known atom
My question was how many spheres within the large ball given that the spheres have a radius of smallest known atom and the number of dimensions of packing  is equal to the radius of smallest known atom