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
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$\begingroup$This is an unsolved problem. Here is a description for $X\le 12$.
Also, follow this other thread for more information on the subject
$\endgroup$ 4 $\begingroup$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.
$\endgroup$ 3 $\begingroup$The maximum density of an infinite number of congruent identical spheres packed within a larger sphere is 0.74048... or Pi/(sqrt(18)), as stated in the Kepler Conjecture, partially proven by Gauss and not formally proven by Thomas C. Hale until 2014.
The Kepler Conjecture states that “no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.”
I actually worked this out myself (but didn’t prove it) by calculating how many spheres could fit into a regular tetrahedral container and a square pyramidal container.
When the number of spheres got well into the millions, the values converged at .074048...
I still have the Excel spreadsheets that I used to do the calculations.
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