User:Tomruen/Kissing number

In geometry, the kissing number is the maximum number of spheres of radius 1 that can simultaneously touch the unit sphere in n-dimensional Euclidean space. The kissing number problem seeks the kissing number as a function of n.

n	1	2	3	4	5	6	7	8
Kissing number	2	6	12	24	40	72	126	240
Image
Isogonal polyhedron		{6}	t₁{3,3}	{3,4,3}	r{3,3,4}	1₂₂	2₃₁	4₂₁
Isogonal tessellation
Isotopic polyhedron		{3}	Rhombic dodecahedron	{3,4,3}
Isotopic tessellation

Some known bounds

The following table lists some known bounds on the kissing number in various dimensions.^[1] The dimensions in which the kissing number is known are listed in boldface.

For 2..8, the best reflective tessellation geometries are given, and a few suboptimal ones.

Dimension	Lower bound KNOWN	Upper bound UNKNOWN	Ball-centered vertex arrangement	Vertex transitive tessellation Centered on ball centers	Facet transitive tessellation Voronoi tessellation of ball centers
1	2		Segment	Apeirogon	Apeirogon
2	4 (not best)		Square	Square tiling	Square tiling
2	6		Hexagon	Triangular tiling	Hexagonal tiling
2	6		Hexagon	Triangular tiling	Hexagonal tiling
3	6 (not best)		Octahedron	Cubic honeycomb	Cubic honeycomb
3	12		Cuboctahedron	tet-oct	Rhombic dodecahedral honeycomb
3	12		Rectified octahedron	tet-oct	Rhombic dodecahedral honeycomb
4	24	24
4	8 (not best)		16-cell	Tesseractic honeycomb	Tesseractic honeycomb
4	20 (not best)		Runcinated 5-cell
4	24		Rectified 16-cell	Demitesseractic honeycomb	Icositetrachoric_honeycomb
5	40	44
5	10 (not best)		5-orthoplex	Penteractic honeycomb	Penteractic honeycomb
5	30 (not best)		Stericated 5-simplex
5	(40)		Rectified pentacross	Demipenteractic honeycomb
6	72	78
6	12 (not best)		6-orthoplex	Hexeractic honeycomb	Hexeractic honeycomb
6	42 (not best)		Pentellated 6-simplex
6	60 (not best)		Rectified hexacross	Demihexeractic honeycomb
6	(72)		1₂₂	2₂₂
7	126	134
7	14 (not best)		7-orthoplex	Hepteractic honeycomb	Hepteractic honeycomb
7	56 (not best)		Hexicated 7-simplex
7	70 (not best)		0₃₃	1₃₃
7	84 (not best)		Rectified heptacross	Demihepteractic honeycomb
7	(126)		2₃₁	3₃₁
8	240	240
8	16 (not best)		8-orthoplex	Octeractic honeycomb	Octeractic honeycomb
8	72 (not best)		Heptellated 8-simplex
8	84 (not best)		0₅₂	1₅₂
8	112 (not best)		Rectified octacross	Demiocteractic honeycomb
8	128 (not best)		1₅₁	2₅₁
8	240		4₂₁	5₂₁
9	306	364
10	500	554
11	582	870
12	840	1,357
13	1,154^[2]	2,069
14	1,606^[2]	3,183
15	2,564	4,866
16	4,320	7,355
17	5,346	11,072
18	7,398	16,572
19	10,688	24,812
20	17,400	36,764
21	27,720	54,584
22	49,896	82,340
23	93,150	124,416
24	196,560

Notes

^ Mittelmann, Hans D.; Vallentin, Frank (2009). "High accuracy semidefinite programming bounds for kissing numbers". arXiv:0902.1105. {{cite arXiv}}: Unknown parameter |accessdate= ignored (help)
^ ^a ^b В. А. Зиновьев, Т. Эриксон (1999). "Новые нижние оценки на контактное число для небольших размерностей". Пробл. передачи информ. (in Russian). 35 (4): 3–11. {{cite journal}}: Unknown parameter |ulr= ignored (help) English translation: V. A. Zinov'ev, T. Ericson (1999). "New Lower Bounds for Contact Numbers in Small Dimensions". Problems of Information Transmission. 35 (4): 287–294. MR 1737742.

GoogleBook: Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai

[1] Mittelmann, Hans D.; Vallentin, Frank (2009). "High accuracy semidefinite programming bounds for kissing numbers". arXiv:0902.1105. {{cite arXiv}}: Unknown parameter |accessdate= ignored (help)

[Zinoviev99-2] В. А. Зиновьев, Т. Эриксон (1999). "Новые нижние оценки на контактное число для небольших размерностей". Пробл. передачи информ. (in Russian). 35 (4): 3–11. {{cite journal}}: Unknown parameter |ulr= ignored (help) English translation: V. A. Zinov'ev, T. Ericson (1999). "New Lower Bounds for Contact Numbers in Small Dimensions". Problems of Information Transmission. 35 (4): 287–294. MR 1737742.

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