# What is the least dense rigid congruent sphere packing?

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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There are rigid circle packings with arbitrarily low density: see mathworld.wolfram.com/RigidCirclePacking.html. – mjqxxxx Nov 24 '12 at 18:49
Both the Wikipedia and the Mathworld articles on sphere packing cite a book by Martin Gardner 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. – begeistzwerst Nov 24 '12 at 19:15

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).

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