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How much volume do spheres take up when filling a rectangular prism of a shape?

I assume it's somewhere in between $\frac{3}{4}\pi r^3$ and $r^3$, but I don't know where.

This might be better if I broke it into two questions:

First: How many spheres can fit into a given space? Like, packed optimally.

Second: Given a random packing of spheres into space, how much volume does each sphere account for?

I think that's just about as clear as I can make the question, sorry for anything confusing.

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2 Answers

up vote 3 down vote accepted

You can read a lot about these questions on Wikipedia.

Concerning the random version, there are some links at Density of randomly packing a box. The accepted answer links to a paper that "focuses on spheres".

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Note that the Kepler conjecture claims the optimum density (ratio of spheres' total volume to space filled) can be attained in either of two packings. This optimum density (approaching $\pi/\sqrt{18}$ for increasingly large "rectangular prism" to fill, or increasingly small spheres) does not depend on radius(?) $r$ in the way OP's question suggests. –  hardmath Aug 24 '11 at 9:31
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There are some links here (Rusin's known-math site). Also here (Eppstein's Geometry Junkyard).

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