I was reading about random close packing of spheres on Wikipedia and Wolfram Mathworld, and if I did not interpret both incorrectly, the conclusion is that if I pack a volume V randomly with spheres, the spheres will occupy a volume of approximately 0.6V.

I feel like I am missing something because I would expect the volume fraction occupied by the spheres to be larger if the spheres were smaller (in the extreme case, if each sphere were the size of a water molecule, packing with spheres would be the same as filling the volume with a fluid). Is there something I'm missing in the definition of packing density that gives a size dependence?

  • $\begingroup$ Another problem with using physical intuition in mathematics is that one certainly should be careful to get the physics "right". In the case of thinking of filling space up with a fluid, one should come from the understanding that most physical fluids are mainly empty space. $\endgroup$ – Ron Kaminsky Apr 19 '18 at 13:58

The volume fraction constant is not dependent on the size of the spheres because the scaling transformation which transforms larger spheres into smaller ones scales the volume of the spheres with exactly the same constant that it scales the volume of the spaces between the spheres (the "voids" of the packing) --- so the ratio is unaffected.

Your intuition is only correct when the volume being packed is finite and and not very large compared to the volume of an individual sphere. The references you cite discuss instead the case of the limit as V becomes all of $\mathbb R^3$, which has infinite volume.

  • $\begingroup$ Are there any general results for finite V? $\endgroup$ – The Hagen Apr 18 '18 at 22:30
  • $\begingroup$ Most of the results I am familiar with concern optimal packings rather than random ones. For example, if we consider the volume fraction of $n$ spheres versus the volume of their convex hull, it is known that the densest arrangements of up to 55 spheres is a "sausage" where the centers of the spheres are colinear. It is known that for $n=56$ and $n>=58$ there exist cluster arrangements which are denser than sausages, a phenomenon called the "sausage catastrophe" by J. M. Wills. $\endgroup$ – Ron Kaminsky Apr 19 '18 at 7:53
  • $\begingroup$ Based on the comments on math.stackexchange.com/a/186316/195155 the sausage of 57 spheres is also not the optimal arrangement. $\endgroup$ – Ron Kaminsky Apr 19 '18 at 8:42

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