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.

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:

=-=-=-=-=-=-=-=-=

!enter image description here

=-=-=-=-=-=-=-=-=

Hilbert and Cohn-Vossen, pages 50-51:

=-=-=-=-=-=-=-=-=

!enter image description here

=-=-=-=-=-=-=-=-=