Talk:Kissing number

message: "In 24 dimensions, the kissing packing places the spheres at the points of the Leech lattice and there is no space at all left over," but from Leech lattice, "It seems to be expected that this configuration also gives the densest packing of balls in 24-dimensional space, but this is still open."
These don't seem to agree. Correct me if I'm wrong but this page claims that the Leech latice is the best packing of spheres in 24 dimensions but the Leech latice page says that we aren't sure yet. Which is correct? I'd wager the Leech latice page is but I've never heard of this problem before so I'm not going to change it. Can a mathematician change one of these two pages please? -- Rory ☺ 23:01, May 31, 2004 (UTC)
:( It sucks that this has been around for so long without being fixed. There is only an upper bound proven how much better an irregular packing could be in the 24-d case, so the article was clearly wrong. You should have removed the suspect uncited claim as thats always the safe thing to do. --71.191.135.140 (talk) 04:44, 20 September 2008 (UTC)
replies: text: No, the two statements aren't in contradiction with each other. The Leech lattice (lambda-24) has been proven to have the highest possible _kissing number_ among all arrangements, whereas it has not been proven to have the highest _density_ among all arrangements. There is no evidence to suggest that there is a denser packing of 24-dimensional spheres, and, if there is such a packing, it can only be microscopically denser (by a factor of about 10^-30). --Calcyman (talk) 15:34, 11 April 2010 (UTC)
thread_title: Contradiction of [[Leech lattice]]

message: Could anybody give a ref to the statements in dim 8 and 24? Tosha 15:37, 8 Jul 2004 (UTC)
thread_title: Ref

message: Just a hunch. It rests better with the factors that be. :) --mwazzap 07:50, 29 October 2005 (UTC)
thread_title:  It's 24! It's 24! 

message: It was recently proven by Oleg Musin that in fact the kissing number in four dimensions is 24. See
http://arxiv.org/pdf/math.MG/0309430 for a preprint of the article, or
http://www.ams.org/notices/200408/fea-pfender.pdf to learn more about the topic.
* I think the kissing number K(n) of an n-D Euclidean space can be found from K(n–1) by constructing n-D equilateral "triangles"; that is,
   K(n)=K(n-1)+(number of admissible new vertices).
Admissible vertices are those at least one diameter apart. The approach gives K(1)=2, K(2)=6, K(3)=12, K(4)=24, but K(5)=48. The last number is larger than the reported bound. Is K(5)=48 correct? The approach also gives an upper bound as K(n)/2n < or =3/2 for large n. 209.167.89.139 16:22, 15 September 2006 (UTC)
thread_title:  It is 24 

message: This preprint claims better upper bounds for 5, 6, 7, 9, and 10. Melchoir 05:32, 18 August 2006 (UTC)
thread_title:  math.MG/0608426 

message: You can't draw circles in one dimension, right ?--George (talk) 11:03, 11 March 2008 (UTC)
replies: text: You can have a mathematical equivalent of a circle on 1-dimensional line - it is just the two points that are the same distance either side of some central point. Algebraically, a circle on a 1-dimensional line with centre x and radius y is the two points x-y and x+y, which bound a (closed) 1-dimensional disk which is just the interval or line segment [x-y, x+y]. This is what the very first diagram in the "Known kissing numbers" section is trying to illustrate. Gandalf61 (talk) 12:06, 11 March 2008 (UTC)
thread_title:  One dimension ? 

message: The link to Oleg Musin is to an article about a football player. Same person? There's no mention of a mathematical career in the article.
:Yes, link definitely points to wrong Oleg Musin, so I have removed it. Gandalf61 (talk) 08:41, 25 September 2009 (UTC)
thread_title:  Musin? 

message: The "Known kissing numbers" section currently has the text Oleg Musin proved the kissing number for n = 4 to be 24, using a subtle trick, followed by a reference. This doesn't seem quite right. If we're going to tell our readers it was a subtle trick, shouldn't we also tell them at least something about what the trick was?
It would be sifferent if subtle trick was in quotes, but that phrase doesn't appear in the JAMS (Pfender & Ziegler) article. They seem to be actually discussing Musin's ideas from a preprint, unrefereed form of the paper, and include the opinion "The bad news about Musin’s approach is that it
forces him to compute, at least approximately...", so it's not clear exactly how clever the whole thing is, since it seems to rely on approximations (and I realize that a lot of higher math proofs latterly use that approach). But the subtle trick bit of the article looks like a paraphrase of two people's opinion, which may or may not be confirmed over time and in any case is not properly explained within the text body.
So what I'm suggesting is: add a rudimentary explanation of what the trick actually was; quotationize the phrase and make it true to the reference, so any of "clever modification", "beautiful idea", "clever way" would work fine; or just drop the subtle trick synthesis. I'm not even close to competent at interpreting the actual maths, so I suppose my own response would be to pull the phrase. I'll leave it in the hands of editors better able to explicate this stuff. Regards! Franamax (talk) 07:40, 6 October 2009 (UTC)
thread_title:  Musin's subtle trick 

message: Based on my observation, I suspect the upper bound for 22 is no greater than 75600...--Billymac00 (talk) 23:38, 6 November 2009 (UTC)
thread_title:  Questionable bounds 

message: The intro definition seems to imply the kissing number is the maximum for a given dimension, while other contexts talk about maximum known kissing number . Is it wrong to say a kissing number of a given lattice/packing is X, even if X<maximum? 75.146.178.58 (talk) 21:30, 10 July 2010 (UTC)
replies: text: The kissing numbers in other dimensions. More generally we may define the kissing number (usually denoted by τ) of a sphere packing in any dimension to be the number of spheres that touch one sphere. For a lattice packing τ is the same for every sphere, but for an arbitrary packing τ may vary from one sphere to another. Other names for τ that have been used are Newton number (after the originator of the problem), contact number, coordination number of ligancy (the last being chemist's terms.)
So anyway, it looks like kissing number isn't the maximum itself, but a property of a given packing. This ought to be written in to the article. I'd try but not really my domain of confidence. :) 75.146.178.58 (talk) 21:41, 10 July 2010 (UTC)

text: I also notice that Kissing number redirects here which actually is called Kissing number problem, which is the content of the article, but I'll try adding the more general definition. 75.146.178.58 (talk) 21:48, 10 July 2010 (UTC)
thread_title:  Maximum number? 

message: It would be good for the article to say which ones are tight - in the sense that there is no slack.  The two-dimensional case is tight, but three and four are not.  IIRC, 8 and 24 are tight, but I'm not sure.  Bubba73 You talkin' to me? 04:32, 21 March 2016 (UTC)
thread_title:  Which ones are "tight"? 

message: The intro paragraph of this page is currently inconsistent:
"In geometry, a kissing number is defined as the greatest number of non-overlapping unit spheres that can be arranged such that they each touch a common unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another."
The first sentence describes a property of a space while the second sentence describes a property of a particular sphere within a particular sphere packing (within a particular space).  I am going to make both possible notions clear and distinct, but if it only is supposed to be one, feel free to correct.Jess_Riedel (talk) 15:43, 30 May 2020 (UTC)
thread_title:  Inconsistency 

message: if anyone is interested, they can update the known bounds with these two sources: https://www.semanticscholar.org/paper/Solving-clustered-low-rank-semidefinite-programs-Laat-Leijenhorst/b5e34dc768b72620da731874ed2bd1ea2c25cac6 and https://arxiv.org/abs/2207.08266#  — Preceding unsigned comment added by Dintre (talk • contribs) 20:27, 30 March 2023 (UTC)
replies: text: These appear not to have been peer-reviewed and published, so they do not yet meet Wikipedia's standards for reliable sources. —David Eppstein (talk) 22:14, 30 March 2023 (UTC)

text: Here's an arXiv link to the first one. Henry Cohn's page has already been updated with these sources, FWIW. Double sharp (talk) 06:54, 14 December 2023 (UTC)
thread_title:  Update