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.
What is the least dense rigid congruent sphere packing?
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0Both 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
2 Answers
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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).