Talk:Kissing number problem

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[edit] Contradiction of Leech lattice

"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)

[edit] Ref

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

http://www.research.att.com/~njas/doc/interview.html says this is by Sloane/Odlyzko and V. Levenshtein, independently. Charles Matthews

[edit] It's 24! It's 24!

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

[edit] It is 24

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)

[edit] math.MG/0608426

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

[edit] One dimension ?

You can't draw circles in one dimension, right ?--George (talk) 11:03, 11 March 2008 (UTC)

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)

Right, thanks --George (talk) 21:08, 11 March 2008 (UTC)