Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
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
add comment

2 Answers

up vote 2 down vote accepted

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

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

share|improve this answer
add comment

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.