I would like to know if there is a way to do the following: calculate the maximal number of spheres of unit radius that can fit inside a sphere of radius 200 times the unit radius.
This is a generalisation of a question that was asked in a biology class. I was wondering if there exist some theorems on this, since I don't know how to start on it.
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There is also a packing arrangement known as Random Close Pack. RCP depends on the object shape - for spheres it is 0.64 meaning that the packing efficiency is 64% (as you can also see in Jack D'Aurizio's link). Therefore, if the balls are randomly distributed, then you can fit approximately $0.64 \cdot \frac{\frac{4}{3}\pi (20r)^3}{\frac{4}{3}\pi r^3} \approx 5120 $, which is far from the highest estimation but still pretty good.
A simple approach for producing reasonable lower bounds is to use a face-centered cubic packing or a hexagonal packing (both have the optimal density, $\frac{\pi}{2\sqrt{3}}\approx 74\%$, in the unconstrained space) and to count the number of spheres met by $x^2+y^2+z^2=(20)^2$. Recalling that the optimal packing density in the plane is $\frac{\pi\sqrt{3}}{6}$, in a sphere with radius $20$ it should be possible to pack around
$$ \frac{\pi}{2\sqrt{3}}\cdot 20^3 - \frac{\pi\sqrt{3}}{6}\cdot 4(20)^2 \approx\color{red}{5804}$$ spheres, but not many more. The estimated density is so $\approx 72.5\%$.