If you have 100 spheres packed into a sphere shape, how many will be on the surface?

Question

My question is more about ratios. I'm wondering is there a calculator or formula I can input an X number of Spheres. If the spheres are packed in a spherical shape what is the ratio of The interior spheres compared to the spheres on the surface? Thanks

Answer 1

This is an unsolved problem. Here is a description for $X\le 12$.

Also, follow this other thread for more information on the subject

Answer 2

Suppose you have $N$ spheres, with radius 1.

Suppose you pack them together in a sphere with radius $R$. Random sphere packings have a density of about 64%, so you can use this to approximate what $R$ would be.

$$0.64 \frac43 \pi R^3 = N \frac43 \pi 1^3\\ 0.64 R^3 = N\\ R = \left( \frac{N}{0.64} \right)^{\frac13}$$

To find the number of spheres on the surface, we can consider it a circle packing of the area $4\pi (R-1)^2$. I subtracted $1$, the radius of the small spheres, because the centres of the surface spheres are located on a sphere of that radius, and that is where the packing takes place.

Random circle packings have a density of about 82%, so packing an area of $4\pi (R-1)^2$ with circles of area $\pi 1^2=\pi$ we get:

$$N_s = 0.82 \frac{4\pi (R-1)^2}{\pi} = 3.28 (R-1)^2$$

Substituting $R$ gives the formula: $$N_s = 3.28 \left(\left( \frac{N}{0.64} \right)^{\frac13}-1 \right)^2$$

This is just a very rough approximation that is not applicable to relatively small numbers of spheres, but then again the problem is somewhat loosely defined.