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I was wondering what the least dense rigid uniform packing of congruent spheres was. The lowest density packing of circles is the truncated hexagonal packing.

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    Both the [Wikipedia](http://en.wikipedia.org/wiki/Sphere_packing#Jammed_packings_with_a_low_density) and the [Mathworld](http://mathworld.wolfram.com/SpherePacking.html) articles on sphere packing cite a [book by Martin Gardner](http://www.amazon.com/exec/obidos/ASIN/0883855178/ref=nosim/weisstein-20) as the reference for the loosest rigid sphere packing. Unfortunately, the articles don't tell us anything about this packing except that is has a density of 0.0555.2012-11-24

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It appears these very loose packings are not lattice packings. They are periodic, but given a fixed origin, if there are spheres centered at vectors $u,v$ there may not be a sphere centered at $u+v.$ Instead, a condition referred to as rigid or jammed is used.

Gardner, page 88:

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Hilbert and Cohn-Vossen, pages 50-51:

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The answer seems to depend on what the restrictions are; Fischer and Dorozinski & Fischer present sphere packings of arbitrary low density. See also Dorozinski's web page (in German; English translation by Google here).