I want to throw a lot of copies of an object of nonzero volume, randomly into a large box. Ignoring boundary effects of the box, with which type of object will the expected packing density be the largest?

Is anything known about the 2D analogue, with any kind of random dropping of objects.

  • $\begingroup$ This is definitely not a topology question, and probably belongs on physics.SE. I'll wait a bit to allow any objections, then migrate the question if none arise. $\endgroup$ – Zev Chonoles Aug 7 '11 at 13:38
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    $\begingroup$ I dont think its a physics questions, its about randomly packing a box. No physical laws needed. We can ignore angular momentum and just throw objects into the box. $\endgroup$ – grok_it Aug 7 '11 at 13:40
  • $\begingroup$ Ok, I was concerned about the angular momentum aspect since you mentioned it in the question (that was why I assumed it was best for physics.SE), but if the question is how best to randomly pack a sphere of radius 100 with objects of volume 1, I think that's probably a fine question here on math.SE. Perhaps you could edit your question to clarify what setup you'd like to assume? $\endgroup$ – Zev Chonoles Aug 7 '11 at 13:42
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    $\begingroup$ What is the probability model? $\endgroup$ – Did Aug 7 '11 at 14:05
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    $\begingroup$ This seems relevant: en.wikipedia.org/wiki/Random_close_pack. It says "Random close packing does not have a precise geometric definition. It is defined statistically, and results are empirical." So coming up with a rigorous definition of what's being maximized might already be a useful contribution. $\endgroup$ – joriki Aug 7 '11 at 14:16

The beautiful paper by Torquato and Stillinger, "Jammed hard-particle packings: From Kepler to Bernal and beyond" (Rev. Mod. Phys. 82, 2633–2672 (2010)), will answer many of your questions. It focuses on spheres, but also touches upon ellipsoids, superballs, polyhedra, higher dimensions, and non-Euclidean geometries. It is an up-to-date survey with pages of references. You may recognize Torquato as a primary mover in the recent spectacular advances on tetrahedral packing.

I think that your question, made precise in some way, is likely open. But there are octahedral packings of density 0.947, considerably higher than the best tetrahedral or sphere packings. So a regular octahedron is a candidate answer.

Incidentally, this paper says that the notion of "random close packing" is now passé, being displaced by "maximally random jamming," a more precise concept.

Some specific information on random sphere packings can be found in my MO question "Average degree of contact graph for balls in a box."

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    $\begingroup$ I think "jamming" sounds much better than "packing"... :D $\endgroup$ – J. M. ain't a mathematician Aug 8 '11 at 4:00

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