3
$\begingroup$

If you go to images.google.com and search on "bubble rafts", you'll see various pictures of disk packings that in large patches look like the six-around-one dense packing of the plane by equal-sized disks but exhibit defects of various kinds. Is there a precise and rigorous mathematical theory of such defective packings, and more precisely, a model of defective versions of packings of the whole plane?

It is not essential to me that the mathematical theory connect the geometry with some underlying physical model (via notions of energy) or that the infinite packings be conjecturally optimal in some sense. It is however important to me that every picture be accompanied by a precise prescription for where the center of each and every disk is. Materials science has some nice ideas, like the Burgers vector, but it doesn't seem to traffic in such specific prescriptions. And physicists who take the work of materials science to high levels of abstraction to create models of the vacuum with "topological defects" don't seem to be interested in getting down to brass tacks about what a specific state would look like. Or maybe I just don't understand what they're saying.

At a very basic visual/mathematical level, when I see a picture of a bubble raft, I don't know whether those patches I see are really exact excerpts of a six-around-one packing, or tiny perturbations thereof, where the amount of perturbation falls off rapidly as one moves away from the visible defect (becoming invisible to the naked eye) but never goes all the way down to zero. A mathematical model of a bubble raft should be unambiguous (and indeed quantitative) about such things. It might not correspond to reality, but it would describe a very specific geometry for an infinite collection of unit disks with disjoint interiors.

$\endgroup$
  • 1
    $\begingroup$ When we moved to the SE 2.0 network, we lost the ability to make questions CW. You can flag your own questions for moderator attention; although I have done that for you this time. $\endgroup$ – Asaf Karagila Feb 18 '15 at 17:48
  • $\begingroup$ This is already community wiki, actually. $\endgroup$ – Asaf Karagila Mar 2 '15 at 5:02
  • 1
    $\begingroup$ In response to the earlier MO question, "Expanding disks lead to what packing of the plane?," Henry Cohn pointed to the Lubachevsky, Graham, Stillinger paper, "Spontaneous Patterns in Disk Packings." I included one figure from that paper, which resembles your bubble-raft images. $\endgroup$ – Joseph O'Rourke Mar 9 '15 at 9:52

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.