Jump to content

User:Tomruen/Kissing number

From Wikipedia, the free encyclopedia

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

[edit]

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

[edit]
  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)
  2. ^ 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.