Talk:Close-packing of equal spheres

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Only two regular lattices?[edit]

There are two regular lattices that achieve this highest average density.

There's actually a few other regular lattices that have just as high average density.

Let's start with a hexagonally-close-packed sheet of atoms (marbles, spheres, whatever): A. (See for some pretty illustrations). Pack a second sheet B on top of the first. The third layer is where something very interesting happens.

  • We *could* line up the atoms in the 3rd sheet directly over the atoms in A --

and keep repeating A-B-A-B-A-B... . This gives us "hexagonal close packed" hcp.

  • We *could* line up the atoms in the 3rd sheet in so they *don't* line up with the atoms in A. Then we could make the 4th sheet line up with A, and repeat: A-B-C-A-B-C-A-B-C. This is "cubic close packed" ccp.

Are these the *only* possibilities ?

I vaguely remember hearing that some real crystals form such a close-packed lattice, but one different from hcp or ccp. Perhaps it was something like

  • A-B-C-B-A-B-C-B-A
  • A-B-A-B-C-A-B-A-B-C.

Anyone remember exactly what it was that formed those crystals ? If I knew its name, I could google for more information.

--DavidCary 05:10, 8 Jan 2005 (UTC)

Those are certainly close packed, but they're not regular - not all spheres within them are identical under symmetries of the lattice.

I'm having a hard time visualizing the difference. Does anyone have a good image to show it off?

I added the 3D ray-tracings. I think they make it much easier to visualize what’s going on. Greg L (my talk) 14:11, 26 April 2007 (UTC)

Hexagonal lattice[edit]

The hexagonal lattice linked from here is not the packing intended; in terms of this article's notation it is AAA, less dense. --Tamfang 23:42, 15 January 2006 (UTC)

Lattice Formation Section[edit]

I originally came to this site hoping to find an easy way to make a lattice for close packing spheres. So, on the chance someone else might want that I added that section. It is somewhat unclear I realize and probably not in good wikipedia style. Thanks for the help. Mangledorf 18:37, 31 July 2007 (UTC)

Animated gif is misleading[edit]


The animated gif (Fig.6) shows how the hcp structure is built up. First, plane A is constructed showing red triangles with a magenta height h=a*sqrt(3)/2. Plane B is now positioned at a delta-y of h/3 which should be h/2.

wolf —Preceding unsigned comment added by (talk) 10:42, 30 October 2007 (UTC)

I am changing the caption of the animated gif. I am disappointed that such sloppy work has gone uncorrected for so long. Two layers of spheres do NOT differentiate between fcc and hcp structure. That is, the gif is wrong because it is incomplete. Apparently, since it is animated it has been uncritically accepted as being useful. My opinion is that it should be removed, it is mostly useless, especially without a third layer being added. But I will just edit the caption to say that it is showing the first two layers of either a fcc or a hcp structure. I also note here that not only are A-B-A-B... and A-B-C-A-B-C structures possible but so are structures with mixes of A-B-A and A-B-C ordering. (talk) 22:00, 1 May 2013 (UTC)

Too many graphics[edit]

I pruned the long list of graphics here. Some of them were redundant, and the large number made it hard to clearly see the difference between FCC and HCP. -- Beland (talk) 19:11, 24 April 2008 (UTC)

Random Packing[edit]

This article should mention why random packing could never exceed the FCC close-packing limit. This article talks about how physicists have very recently derived mathematics to prove this as it has never properly been proven until now.H0dges (talk) 20:30, 2 June 2008 (UTC)

Move Justification[edit]

Close-packing is too ambigous as a title and hence unsuitable, it leaves the shape and size distributions ambigous. The first move I performed resolved the shape ambiguity but there are in reality different classes of spherical packing problem, monodisperse (which the article discusses) and polydisperse (dealing with a distribution of particle sizes). Therefore, in my view, "Close-packing of monodisperse spheres" is the only suitable name which provides a "a reasonable minimum of ambiguity". "Close-packing of spheres" could be used, but the article would need content on the polydisperse and monodisperse cases, which only really have surface similarities.

Noodle snacks (talk) 10:23, 15 September 2008 (UTC)

I oppose this move because "monodisperse spheres" is not a term commonly used in relation to this topic, even in the mathematics community. I propose that the article should be moved back to close-packing of spheres, which is a sufficiently precise title that is still suitable for searching and linking. Gandalf61 (talk) 10:56, 15 September 2008 (UTC)
I agree with Gandalf: the article should be moved back to close-packing of spheres, for the reasons said. There is no reasonable expectation of ambiguity in the phrase "close packing of spheres", and the title of an article isn't necessarily supposed to resolve all ambiguity, just to be the most common variant. In mathematics, note that sphere packing nearly always refers to a monodisperse sphere packing. The extra word therefore adds little clarification. siℓℓy rabbit (talk) 11:30, 15 September 2008 (UTC)
I agree with the above two editors. Any ambiguity could be solved in the article, if necessary. I suggest we move the article back asap. Thenub314 (talk) 11:42, 15 September 2008 (UTC)
Agree with move back to close-packing of spheres for the reasons above. I'd remove the word "monodisperse" from the article text too as it's unnecessarily technical. It's defined at monodisperse as "[having the] same size, shape and mass". Spheres are all the same shape by definition, while mass is irrelevant to close-packing, so why not "equal-sized spheres"? Qwfp (talk) 12:09, 15 September 2008 (UTC)
The article sphere packing uses "indistinguishable", not that I think any such qualification is needed in the article title. siℓℓy rabbit (talk) 12:59, 15 September 2008 (UTC)
I also would like to see it at close packing of spheres (or even close packing). If there is information about close packing of spheres of varying sizes, I would prefer to see it in the same article. Also, the term is not widely in use among mathematicians. CRGreathouse (t | c) 17:51, 15 September 2008 (UTC)
What then, would you call an article for the "close-packing of spheres" if the spheres were not all the same size? Wikipedia doesn't have anything on that subject, but it is a significant mathematical problem in some areas. If that title was used then the ambigiuity would have to be resolved by treating both cases, which doesn't work well in many of the contexts that this page is linked to from. Noodle snacks (talk) 05:14, 16 September 2008 (UTC)
First of all, I think that the content from both cases should be covered in this article. But if it were necessary to split the pages, I would use uniform/nonuniform or identical/irregular.
CRGreathouse (t | c) 14:36, 16 September 2008 (UTC)
In geometry, uniform has a special meaning inappropriate here. Why not equal? —Tamfang (talk) 03:24, 17 September 2008 (UTC)
Equal would be fine. In any case I don't think the article needs to be split, so it's moot. CRGreathouse (t | c) 18:07, 21 September 2008 (UTC)
What then, would you call an article for the "close-packing of spheres" if the spheres were not all the same size? Wikipedia doesn't have anything on that subject
I would call it "unequal sphere packing".
Wikipedia now has some information on the subject, currently at sphere packing#Unequal sphere packing. Perhaps someday we will have enough information on unequal sphere packing to WP:SIZESPLIT that section out into a separate article. --DavidCary (talk) 13:35, 1 July 2013 (UTC)

Hermite constant[edit]

Far from the current controversy, I would like to make a modest suggestion: could someone include a mention of the relation of this to the constant above? Katzmik (talk) 16:07, 17 September 2008 (UTC)

Sphere packing application in Chemistry[edit]

I would like to provide a link to a very interesting web site that illustrates how close sphere packing can be used to explain the Periodic Table and the quantum numbers: Periodic Table explained on the basis of the close sphere packing..

O.K? Drova (talk) 00:38, 26 September 2008 (UTC)

Looks fine to me Noodle snacks (talk) 01:43, 26 September 2008 (UTC)
I rm this link as inappropriate. The topic of the link is the periodic table; chemistry -- not packing, which is geometry. Also, the page itself verges on a nonstandard interpretation of its topic. — Xiongtalk* 03:53, 28 October 2008 (UTC)
What is inappropriate about a link that shows how close sphere packing can be used to explain natural phenomena? This is extremely interesting link related to the close sphere packing theory and its application in science. Besides, there is another link in regard to crystallography. It is just another example of usefulness of the close sphere packing theory.

I feel strongly that the link in question has to stay.Drova (talk) 12:45, 28 October 2008 (UTC)

Illustrate, rather than explain, because it's clearly an incorrect explanation. Nonetheless, it may still be appropriate. I think it may not have sufficient credibility to meet WP:EL, though.Arthur Rubin (talk) 13:13, 28 October 2008 (UTC)
I agree that Illustrate fits better. Mnemonic diagrams are not new in Chemistry, the first one was intruduced by Charles Janet in 1930. However, they were always presented in 2D, as this one, for example. The link in questionis the same thing, only in 3D, thanks to the Close Packing of Spheres. It shows one extra quantum number ml. There is at least one recent book that discusses this web site "La Tabla periodica" by Osorio Giraldo and Maria Cano, PhD (ISBN: 958-655-530-5 and ISBN: 958-655-081-8) and at least one article by Dr. Philip Stewart of Oxford in "Foundations of Chemistry" (currently under review). That should be enough credibility for an external link.Drova (talk) 14:01, 28 October 2008 (UTC)
After careful study, it appears irrelevant to this article _and_ probably incorrect. — Arthur Rubin (talk) 13:00, 29 October 2008 (UTC)
would you, please, be more specific. What does "probably incorrect" mean?Drova (talk) 16:02, 29 October 2008 (UTC)
I see that in my last comment I was excessively tactful. Let me rephrase myself. My opinion of the link in question is that it is at worst crank stuff, pseudo-science; at best a highly flimsy pretext to novelty and without any acceptance within the scientific community. It bears all the hallmarks of pseudo-religious fanaticism -- bizarre connections, use of absolute adjectives, the revelation of universal truth, etc. As a link from any page, I think it inappropriate. See: WP:BULL. This content does not dwell within the Church of Reason.
Its relevance to this page is marginal at best, even if one were to suffer it on other grounds. In that way, it is no more appropriate than a link to the standard periodic table from Rectangle. This link's radical arrangement of the elements has nothing to do with spheres or packing; perhaps it has something vaguely to do with tetrahedronal numbers. — Xiongtalk* 23:15, 29 October 2008 (UTC)
It seems that you are not that familiar with the subject of the Periodic Law. I recommend you to study it little further. Before calling it bad names, like "pseudo-religious" How about trying to learn little more about it here. And, by the way, lets restore the link, so others can contribute.Drova (talk) 02:25, 30 October 2008 (UTC)
Any inclusion of such a link on this "topic" will require a reference to a peer reviewed article in a relatively prominent scientific journal. Otherwise I will support the accusation of this being possible WP:BULL. User A1 (talk)
See also discussion here [[1]]Drova (talk) 12:36, 30 October 2008 (UTC)
I have replied there. My claim is that your references are somewhat weak and more verifiable literature is required. User A1 (talk) 12:58, 30 October 2008 (UTC)
The references might be weak, because it is fairly new staff. I think that you are mistaken in regard to the requirements for the external links. Such links are recommended to be avoided, but not disallowed WP:ELNO Drova (talk) 15:25, 30 October 2008 (UTC)

I occasionally hear from someone who apparently thinks physics and chemistry are built from octahedra; let's put the octahedronists and the tetrahedronists in a pit and have them fight ;) —Tamfang (talk) 06:44, 26 December 2008 (UTC)

Another pooh-pooh statement form the critic. As usual, no specifics. Oh, by the way, earth happened to be spherical (not flat, as some still believe). Sphere, as you know, a three dimensional geometric shape. It describes nature fairly accurately, doesn't it. The tetrahedron describes the Periodic Table accurately too. I chalenge you to disprove it. :)Drova (talk) 13:58, 19 January 2009 (UTC)

Suggest closing this thread. Even if this were a published theorey, it's not an application to chemistry; at best, it's an application to an illustration of electron configuration; and it's not just "Close-packing of (equal) spheres", it's tetrahedral filling. (Must … resist … calling it … pyramid power.) — Arthur Rubin (talk) 14:41, 19 January 2009 (UTC)
Hm, John Baez's crackpot index doesn't seem to assign points for comparing one's detractors to flat-earthers. —Tamfang (talk) 08:11, 20 January 2009 (UTC)

It appears to me that the above suggestion to close this thread has been quickly ignored, so as my call for a serious discussion. Instead, more cheap shots were made by those who believe in flat periodic system. Therefore, I'd like to use this opportunity to point out to the above critics that the illustration of electron configurations is directly related to chemistry. Similarly, the cannonball stack (referred to as "tetrahedral filling" above) is directly related to the close-packing of (equal) spheres. Just look at the image presented in the article itself (Fig.2) and compare with this one. And, to Arthur: if you resist to call something in degrading manner, please do not call it. Drova (talk) 04:07, 21 January 2009 (UTC)

  • Thought you might want to know that Dr.Philip J. Stewart in his article entitled "Charles Janet: unrecognized genius of the periodic system." printed in Foundations of Chemistry, January, 2009 issue (ISSN 1386-4238 (Print) ISSN 1572-8463 (Online)) wrote following: "An interesting improvement to the Janet table has been made by Tsimmerman (2007), who shifts the p, d and f blocks so that each electron shell is represented by a single row (or column if turned through 90 degrees), displaying a symmetry that was not evident in the original." He also included ADOMAH PT web site in the references.

Drova (talk) 18:49, 29 January 2009 (UTC)

Close Packing Spheres ("on the head of a needle")[edit]

Ran into a german mathmattician on the street in SF. He said, pre-newton, there used to be an "Angel Theory", the basis of isaac disraeli's derision "lets not argue how many angels can fit on the head of a needle." Debate came up because angels are "perfect" hence they are "spheres" (cf ptolemaic), and "invisible" hence "infinitesmal". I can't find anything on the net or in history books. Does anyone know if this really from a historic debate regarding a packing problem of spheres, or was he pulling my leg? Tautologist (talk) 20:43, 27 September 2008 (UTC)


Can someone put up an additional section for simple fcc or ccp lattice??? They have included one for hcp but nor for fcc. So please add it up is someone can —Preceding unsigned comment added by Nradam (talkcontribs) 02:03, 15 February 2009 (UTC)

FCC and HCP are the same thing[edit]

Just been reading "Kepler's Conjecture" (Wiley 2003) by George G Szpiro. On page 18 he states "it turns out that the FCC and the HCP are the exact same packing, viewed from different angles!" citing an illustration in Barlow 1833. I am not an expert in close packing but what do you say to this? The book was well received by all the major journals [2] --Nick Green (talk) 16:46, 19 February 2009 (UTC)

If you consider only two adjacent layers, FCC and HCP look alike; so they have the same density, and if you're looking to prove or disprove the Kepler conjecture there may be no reason to distinguish them. But if you crush FCC you get rhombic dodecahedra, while if you crush HCP you get trapezo-rhombic dodecahedra. —Tamfang (talk) 01:26, 20 February 2009 (UTC)

No they are note same. Bravais lattices takes into consideration all elements of symmetry. The 14 lattices are in no way identical. —Preceding unsigned comment added by Nradam (talkcontribs) 09:41, 10 April 2009 (UTC)


It probably doesn't fit in the article but this picture is nice. Snowpyramids.jpg

JIMp talk·cont 01:00, 11 July 2009 (UTC)

Animated GIF Does Not Work[edit]

Doesn't function on a Mac running Mac OS 10.6.3 using Safari browser. —Preceding unsigned comment added by (talk) 07:36, 21 May 2010 (UTC)

Neither it works on Windows with Firefox. There are constant problems with displaying animated gifs on wikipedia, and while changing image size options sometimes solve this, wikimedia updates change things from time to time. For myself, I decided to wait until the developers fix this problem for good. Materialscientist (talk) 07:50, 21 May 2010 (UTC)

Pitch z[edit]

To whomever keeps reverting the correction.

You either

a) have to change your description of what the pitch parallel to z is ie. "The distance between sphere centers parallel to the z axis"


b) the equation that says the distance is

Unless you can explain to me how the centers of two spheres can be a distance of less than 1 diameter apart from each other. —Preceding unsigned comment added by (talk) 02:04, 17 October 2010 (UTC)

It is a projection on z axis, and thus is smaller than the diameter. Materialscientist (talk) 02:09, 17 October 2010 (UTC)

Ok, thats fine but the text doesn't say that it is the projection, it says that it is the distance between the spheres. Which should be twice the listed distance. —Preceding unsigned comment added by (talk) 02:46, 17 October 2010 (UTC)

Captions for figures 2 and 3[edit]

The end of the second line of the caption for Figure 2 runs into the beginning of the second line of the caption for Figure 3; it looks like "stack shown in Figure 3 only in stack,". Someone please fix, I suppose by moving the "only in" down to start the third line. I don't know how to do it.HowardJWilk (talk) 16:10, 14 May 2016 (UTC)