Talk:Sphere packing

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This page conflicts with kissing number problem where it claims that the densest packings are known up to 8-D. They are known for 1, 2, 3, 8, and 24 dimensions, but not others. -- Taral 21:33, 16 Aug 2004 (UTC)

I don't think there is a conflict here. Kissing number problem is a different problem from sphere/hypersphere packing. The kissing number problem depends on local packing properties whereas sphere/hypersphere packing concerns the global property of average density. Also note that the sphere packing article says "the densest regular packings of hyperspheres are known up to 8 dimensions" - regular packings are sometimes called lattice packings. It is possible that denser irregular packings may exist in some dimensions - indeed the major difficulty in solving the 3D case is in ruling out the existence of an irregular packing with an average density that is higher than that of the cubic/hexagonal close packing (see Kepler conjecture). See MathWorld for more details. Gandalf61 09:15, Aug 17, 2004 (UTC)

I have to question the sanity of saying that spheres in the corners of a hypercube have their size determined by hamming distance. At the very least, the article on hamming distance seems to be the wrong topic. I'm very familiar with digital signal processing, but I don't know much about 4D geometry, so I am unwilling to make an edit. Still, someone should check this - and if they are related somehow, I would love to know! --Ignignot 20:23, Nov 19, 2004 (UTC)

i think the picture of the oranges is a bad example because it barely shows sphere packing.

Contents

[edit] and on a sphere?

Is there an article on arrangements of nodes on a sphere, including packing, covering, and the Thomson problem? —Tamfang 00:33, 26 June 2006 (UTC)

[edit] Current known packing inefficient?

The current known packing for three dimensions has density of about .74 and it was suggested that there might be a denser packing allowing a thirteenth sphere to be added. Can someone confirm this? —Preceding unsigned comment added by 24.149.204.116 (talk • contribs) 16:29, 30 July 2006

When two spheres of the same radius are tangent, each, as seen from the centre of the other, fills less than 1/13 of the "sky"; for this reason it was long believed by some eminent mathematicians that a "kissing" arrangement of 13 around 1 must be possible, but 12 was eventually proven to be the limit. —Tamfang 01:03, 31 July 2006 (UTC)
For more details see our article on the Kepler conjecture. Gandalf61 09:56, 16 June 2007 (UTC)
Or not. That article says little about the substance of the proof, and nothing about whether 13 spheres can kiss another, which is a question independent of the Kepler Conjecture. —Tamfang 00:36, 17 June 2007 (UTC)
Indeed. The question of how many spheres can be arranged to touch a central sphere is the kissing number problem, but the kissing number in 3 dimensions was shown to be 12 and not 13 in 1874, so I am not sure why the original questioner would be unclear about the status of the 3D kissing number problem. Possibly the original questioner had got the two problems mixed up. Gandalf61 14:16, 17 June 2007 (UTC)

[edit] Animated Image

The animated image was removed in a bold edit. However, the removal was reverted. An editor should not revert a revert until consensus has been established to remove the image. I think it's fine, and there is no absolute law against allegedly "distracting" images - it's still very illustrative and relevant to this article. I would ask that the user who is removing this image not engage in edit warring and instead discuss his changes instead of forcing them onto the article. This runs contrary to how Wikipedia is meant to function - collaboratively and through consensus-building. --Cheeser1 16:55, 21 September 2007 (UTC)

I didn't follow the revert history etc., but I think the image is distracting. The rotation does not show anything particular which would not be clear when giving a static picture. The effect of the pseudo-transparency is also questionable. Plus, the background of the image is just distracting, as well. Keith Devlin's book "Pattern in mathematics" (or similar) has some good illustrations on this topic. If a gifted editor is able to somehow reproduce this kind of style, I would prefer it over the rotating image given there right now. Jakob.scholbach 19:01, 21 September 2007 (UTC)
I also find it distracting and uninformative. I agree with Jakob.scholbach. -- Dominus 19:37, 21 September 2007 (UTC)
I would then propose that we attempt to find a suitable alternate image. Until then, I see no reason to have no such image in the article. There's nothing requiring us to immediately remove it, so let's leave it in until we find a replacement? I think that sounds reasonable. --Cheeser1 20:35, 21 September 2007 (UTC)
I've now replaced it with a single frame from the animation, which I think is enough to show the geometry. --Salix alba (talk) 20:46, 21 September 2007 (UTC)
Yes, that's better. However, now, that the animation is no longer distracting, it is even more obvious (to me, personally) that the image itself is ugly and doesn't convey the geometric message clearly. After a quick search, I ended up at [1], which has some nicer images. (don't know if they are free). Jakob.scholbach 21:54, 21 September 2007 (UTC)
I don't think those are illustrating the same packing principles. I also don't think it's "ugly" - that is subjective, certainly, but I believe the still image does a good job of conveying the idea (Salix, thanks for fixing it up). If there's no objective or agreed-upon reason to remove it, I'd say the still is fine. If you find a free alternative, I'd love to see it, and perhaps we can all agree on it as a better substitute. --Cheeser1 23:28, 21 September 2007 (UTC)

[edit] curved space

H. S. M. Coxeter remarks that there are arrangements of equal spheres in both positively and negatively curved space that exceed the Kepler density. I think it's in The Beauty of Geometry; will look for it later. —Tamfang (talk) 01:45, 29 February 2008 (UTC)

[edit] Sphere packing and Close-packing

I'm not a mathematician, but these two seem to overlap to a fair extent. Shouldn't they be merged? dorftrottel (talk) 20:53, 30 May 2008 (UTC)

No. The close-packing problem is a special case of sphere packing. Close-packing deals only with regular arrangements in 3 dimensions, whereas sphere packing deals with the more general problem of finding dense packings of any type (regular or irregular) in any number of dimensions. Gauss solved the close-packing problem in 1831, but the general sphere packing problem is much harder, and it was not solved in 3 dimensions until 1998, by Thomas Hales. The Sphere packing artcile links to the Close-packing article at the top of its Regular packing section, and it would become unbalanced and too long if we merged the two articles. I oppose the merge proposal, and I have amended the lead of the Close-packing article to clarify the difference between the articles. Gandalf61 (talk) 08:27, 31 May 2008 (UTC)
Oh, ok. Thanks for the explanation and for clarifying the article intro. It wasn't obvious to me from reading the article intros. Removing the merge tags btw. dorftrottel (talk) 13:31, 31 May 2008 (UTC)
If factually correct, a simple statement in the very first sentence of Close-packing might be a good idea. Speaking as the non-expert, the difference between the concepts (or rather: the relationship between the article topics) is still not ovbious. Maybe something as simple as "Close-packing is a special case of sphere packing"? dorftrottel (talk) 13:37, 31 May 2008 (UTC)