Talk:Sphere packing

Kissing Number
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)

Hamming distance
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)

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

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


 * It appears so. There is this: Circle packing and this: Thomson problem 130.66.206.111 (talk) 14:56, 17 March 2014 (UTC)

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. &mdash;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.  &mdash;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)

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, 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)

The image displayed in the still frame and in the animation is a tetrahedron, not a pyramid. A pyramid has a four-sided, square base and four three-sided walls (like the pyramids in Egypt). A tetrahedron has a equilateral triangle for a base and three equilateral triangles for walls, giving the shape four equal sides (i.e. tetra- (four) hedron). I suggest renaming it. I'd do it myself, but the process looks complicated. Tdbostick (talk) 13:00, 5 May 2011 (UTC)Tim Bostick


 * The Egyptian pyramids are square pyramids. A general pyramid's base can be any polygon. —Tamfang (talk) 06:56, 6 May 2011 (UTC)

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. &mdash;Tamfang (talk) 01:45, 29 February 2008 (UTC)


 * Ah, here we go:
 * Elliptic 3-space presents a different state of affairs: sixty spheres of radius &pi;/10, each touching twelve others, form a packing of density 0.774, and somewhat larger spheres (having the same centres) form a covering of density 1.439, whereas in Euclidean space the maximum packing-density and minimum covering density are almost certainly 0.740 and 1.464.
 * In hyperbolic 3-space, no superior arrangements of finite spheres are known, but horospheres provide a packing of density 0.853 and a covering of density 1.280.
 * —"Arrangements of Equal Spheres in Non-Euclidean Spaces", Acta Mathematica Acad. Sci. Hungaricae, Tomus V (1954); reprinted in Twelve Geometric Essays (1968), retitled The Beauty of Geometry (1999).
 * —Tamfang (talk) 06:38, 14 November 2008 (UTC)

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)

Sphere packing in Chemistry
It was recently found that sphere packing has direct, although still unexplained, connection to the arrangement of the elements in the periodic table, as shown here. It is thought provoking and fascinating. My view is that it would be benefitial for atracting more attention to the field of Sphere Packing if this web site is listed among the external links. What do you think? Drova (talk) 13:29, 24 December 2008 (UTC)


 * I don't think that belongs in this article. The point of the article is address how spheres stack, not geometric consequences relating to the periodic table. If anything it belongs on that page or with something related to atomic chemistry. IncidentalPoint (talk) 16:52, 24 December 2008 (UTC)


 * Drova, you made a similar suggestion a couple of months ago at Talk:Close-packing of spheres. This link was rejected as inappropriate for Wikipedia. It is not within Wikipedia's remit to promote non-notable unpublished theories, no matter how thought provoking and fascinating they might appear to be. Sorry, but I don't think your link belongs here. Gandalf61 (talk) 17:14, 24 December 2008 (UTC)


 * Just to let you know that this web site was recently refered to by Oxford professor Philip Stewart in his January 2009 article in Foundations of Chemistry magazine entitled "Charles Janet: unrecognized genius of the periodic system" where he noted ADOMAH PT as the improved version of Janets LSPT.Drova (talk) 14:25, 19 May 2009 (UTC)

Density in 24-dimensions
If it is proven that the 24-dimensional regular sphere packing has the highest density, is the actual (numerical) density known? If it is, that certainly should be in the article. Eebster the Great (talk) 07:17, 17 February 2009 (UTC)

Merger proposal
I propose to merge the page Packing problem into Sphere packing. This shouldn't be too difficult as the second article is written beautifully whereas the first one contains lots of numbers (without citations) that don't tell you anything. And the style in the first one is quite relaxed (eg., "Packing problems are one area where mathematics meets puzzles (recreational mathematics)"). There is a second reason I want to cleanup Packing problem: In computer science, packing problems are combinatorial optimization problems (eg., the set packing problem) and they are LP-dual to covering problems. Of course, the combinatorial meaning is related to sphere packing, and there will be a disambiguation at the top of the page. I want to extend the article packing problem to the combinatorial and computational aspects and before I can do that, it needs a cleanup of experts in sphere packing. ylloh (talk) 22:49, 11 March 2009 (UTC)
 * Only problem I have is that sphere packing is often not a puzzle but a real-world problem of atomic packing, packed beds, etc. The two could be combined if the focus was changed a little. IncidentalPoint (talk) 00:26, 12 March 2009 (UTC)


 * I think the word "puzzle" is more meant as a funny description of what is going on. The article Packing problem addresses different versions of circle packing but also square packing. At least the first one is the two-dimensional case of Sphere packing. The "puzzle" metaphor is quite good, but really shouldn't be the introductory explanation of this kind of packing problems. Now that I looked at Packing problem a little more closely, it seems to be just a list of optimal packing constant in different situations. This definitely does not belong into a main article but should rather be something like list of optimal packing constants and needs good references. ylloh (talk) 12:15, 12 March 2009 (UTC)


 * Oppose merge. Packing problems involve the efficient packing of objects (typically spheres) into various finite shapes and regions; sphere packing involves efficient packing of spheres and hyperspheres in infinite spaces. Very different types of problem. Gandalf61 (talk) 11:51, 5 June 2009 (UTC)


 * I also oppose. Not only are the problems typically different, but the Packing problem page is growing large.  CRGreathouse (t | c) 00:13, 3 July 2009 (UTC)


 * I also oppose, packing is clearly about more than spheres, and there is plenty of content here, it just needs to be cleaned up. 99of9 (talk) 00:50, 1 September 2009 (UTC)

small spheres on large spheres
It would be nice to have a simple discussion and formula for how many small spheres of diam D1 can pack on the surface of a larger sphere diam D265.220.64.105 (talk) 18:06, 12 May 2009 (UTC)


 * Lots of people would be pleased to have such a formula. —Tamfang (talk) 00:31, 13 May 2009 (UTC)

Tangled edit links
At the top of section 3 Hypersphere Packing, there are two edit links. The second is for the previous subsection 2.2 Irregular Packing. Obviously this is incorrect, but I don't know how it was caused or how to fix it. Any ideas? Anywhere else that this could be asked? Elroch (talk) 09:02, 18 November 2010 (UTC)


 * It happens because your browser (and mine) doesn't cope well with multiple 'floating' objects on the right margin. —Tamfang (talk) 18:00, 18 November 2010 (UTC)

Merger proposal
Shouldn't Close-packing of spheres be merged into this article? Toshio Yamaguchi (talk) 14:21, 8 May 2011 (UTC)


 * Or vice versa. —Tamfang (talk) 21:20, 8 May 2011 (UTC)


 * Since according the article Close-packing of spheres, a Close-packing is a regular arrangement of spheres and according to this article a sphere packing can also be irregular, I think it would make more sense to merge the way I suggested above, because the close packings seem to be a subset of the sphere packings. Toshio Yamaguchi (talk) 21:46, 8 May 2011 (UTC)


 * I'd prefer not to. Close-packed structures are one very particular solution to one very particular sphere packing problem (equal spheres in 3-d Euclidean space).   I agree it's a subset of this page, but this page should cover much more, and will get very long if we explain the details of every solution.  That particular solution is so important that it deserves a page of it's own, and since it's already got sufficient content, I think we should leave it like that. --99of9 (talk) 06:23, 11 May 2011 (UTC)

Can we bring this to GA quality?
Is anyone interested in collaborating with me for a drive to get this article to GA quality? --99of9 (talk) 11:47, 24 November 2011 (UTC)

Irregular Packing Density
In the section on irregular packing (Sphere packing), it says, "This irregular packing will generally have a density of about 64%." But then in the next sentence it says, "Recent research predicts analytically that it cannot exceed a density limit of 63.4%". This seems contradictory. Shouldn't the first number be at most 63% (if we are rounding)? Even then, it looks odd to me that the general packing density is so close to the upper bound. 130.66.206.111 (talk) 15:03, 17 March 2014 (UTC)

Number of regular close-packs?
I once saw a table providing the number of different regular close-packs in various dimensions. If I recall correctly, it was 1 in most dimensions, except its 2 in 3D (fcc and hcp), some number in 6 and a bigger number in 10 dimensions, and an explosion in 24 due to leech lattice, and then back to 1 for the rest. Is my memory faulty? Where can this table be found? 67.198.37.16 (talk) 17:13, 22 September 2015 (UTC)

Optimal packings for dimensions n>3 not yet shown
The article states that "In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions.[8] ". But the optimal (i.e. densest) packings for dimensions higher than 3 are only postulated and not known, as [8] clearly states. — Preceding unsigned comment added by 2A02:168:7406:0:A544:DB60:A8BB:3AEF (talk) 15:20, 27 June 2018 (UTC)
 * The densest regular packings are known for all those dimensions, not the densest packings in general (save for dimensions 8 and 24, in which we now know that the densest regular packing is actually the best overall). XOR&#39;easter (talk) 15:26, 27 June 2018 (UTC)

Ok thanks — Preceding unsigned comment added by 2A02:168:7406:0:A544:DB60:A8BB:3AEF (talk) 16:48, 27 June 2018 (UTC)

4D sphere packing = doughnut packing! ~  JasonCarswell   (talk)   21:29, 28 August 2018 (UTC)

What is this called?
1 center circle ringed by 6, then those now 7 center circles fenced in by 9 around the border, then those 16 ringed by 18, then those 34 ringed by 24, then those 58 ringed by ---,...  This 2D hexagonal circle packing ratio/equation must have a name as well as versions for 3D and higher. Please link to it here and/or try to include some of it on these "packing" pages. Thanks in advance. ~  JasonCarswell   (talk)   21:26, 28 August 2018 (UTC)

FCC and HCP are not both lattices?
The article currently states "Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or face-centred cubic, "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP")—where the layers are alternated in the ABAB... sequence." I have tagged that statement as dubious. says on page 127 "The f.c.c. and h.c.p. packings have the same density, but they are different: one is a lattice, the other is not." So the article currently seems to be wrong in stating both arrangements correspond to lattices. Toshio Yamaguchi (talk) 12:04, 13 April 2022 (UTC)


 * You're right. In a lattice, the translation that takes the first layer to the second layer should, when applied to the second layer, produce the third layer. This is the case for the FCC packing but not for the HCP packing. In the article, the packings are formed by stacking horizontal layers of balls, each layer a translate of a regular triangular lattice. Let $$A=\left\{m(1,0)+n\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right) \mid m,n\in\mathbf{Z}\right\},$$ be the points of this lattice, $$B=A+v,$$ with $$v=\left(\frac{1}{2},\frac{1}{2\sqrt{3}}\right),$$ the set of barycenters of the upward pointing triangles in this lattice, and $$C=B+v=A+2v$$ the set of barycenters of downward pointing triangles in this lattice. Note that $$3v\in A$$, so that $$A=C+v$$.


 * Then the FCC lattice is


 * $$\ldots\cup (A-3u)\cup (B-2u)\cup (C-u)\cup A\cup (B+u)\cup (C+2u)\cup (A+3u)\cup\ldots,$$


 * while the sphere centers in the HCP arrangement are the points of


 * $$\ldots\cup (B-3u)\cup (A-2u)\cup (B-u)\cup A\cup (B+u)\cup (A+2u)\cup (B+3u)\cup\ldots,$$


 * where $$u$$ is the vertical displacement between layers. The HCP arrangement contains the point $$v+u$$. If it were a lattice, it should also contain $$2v+2u,$$ which would have to lie in the $$A+2u$$ layer. One can easily see that this is not the case.


 * The subsequent section of the article mentions the "hexagonal lattice" and the "tetrahedral lattice". I'm not sure what these are, but it should be checked that they also really are lattices. Also, the definition of lattice, which is given in the preceding section of the article, could stand to be tightened up a little. Will Orrick (talk) 15:48, 13 April 2022 (UTC)