## Kissing NumberWho said geometry wasn't romantic?
Written by Paul Bourke
The so called "kissing number" is the maximum number of times a sphere in N dimensional space can touch a central sphere (all spheres are the same size and cannot intersect another sphere). 2D
Consider the situation in 2 dimensions, a 2D sphere is just a circle and it is easy to verify that the kissing number is 6. That is, at most 6 circles of equal radius can be packed around a central circle of the same radius....try it with 7 coins all of the same denomination!
1D
The one dimensional case is rather boring with a kissing number of 2. 3D
In 3 dimensions the kissing number is 12, this can be verified with pingpong balls and bits of masking tape to hold them together. There is more than one way to pack the 12 spheres, the example below is a very symmetric solution. Another method is to arrange the spheres so their centers lie along at the vertices of an icosahedron. There does seem to be lots of "empty" space but there isn't enough for another sphere!
For a slightly more non-symmetric example see the following coordinates (center of each kissing sphere) and corresponding image. x y z 0.25531102 0.89156330 -0.37407374 -0.13044368 -0.77593450 -0.61717914 0.12484695 0.78152529 0.61125401 0.79480098 -0.47827054 -0.37356217 0.45181161 -0.14070529 0.88094738 0.91933526 0.36212485 0.15390992 0.21657532 -0.92622152 0.30855928 -0.91836971 -0.36091967 -0.16227776 -0.62695983 0.48432758 -0.61020338 -0.79681573 0.47329850 0.37559716 0.22672985 0.08894927 -0.96988742 -0.51617898 -0.41001744 0.75196074
Higher dimensions
The kissing number is known for certain for many higher dimensions and suspected for others. The table below gives the values for a range of dimensions. In the cases where the maximum hasn't been proved the number below has generally be determined by exhaustive computer searches. The exact value for 24 dimensions was found in 1979 by A.M. Odlyzko and N.J.A. Sloane. Dimension Kissing Number 1 2 2 6 3 12 4 24 5 at least 40 at most 44 6 at least 72 at most 78 7 at least 126 at most 134 8 240 9 at least 306 at most 364 10 at least 500 at most 554 11 at least 582 at most 870 12 at least 840 at most 1357 13 at least 1130 at most 2069 14 at least 1582 at most 3183 15 at least 2564 at most 4866 16 at least 4320 at most 7355 17 at least 5346 at most 11072 18 at least 7398 at most 16572 19 at least 10688 at most 24812 20 at least 17400 at most 36764 21 at least 27720 at most 54584 22 at least 49896 at most 82340 23 at least 93150 at most 124416 24 196560 4D
For those with an interest in 4D geometry, here are the coordinates for a solution with 24 kissing hyperspheres. x y z w 0.75380927 -0.28878253 -0.17107694 -0.56489726 -0.75380927 0.28878253 0.17107694 0.56489726 0.00158908 0.23980955 -0.40088438 -0.88418356 0.88625386 0.32965258 -0.25236023 0.20542051 0.33787744 -0.01190412 -0.92962202 -0.14662890 0.41593183 -0.27687841 0.75854507 -0.41826836 0.13403367 0.85824466 -0.48216766 -0.11386579 0.33628835 -0.25171367 -0.52873764 0.73755466 0.54837642 0.34155670 0.67726179 0.35204941 -0.33628835 0.25171367 0.52873764 -0.73755466 0.20384377 -0.87014878 -0.44745436 -0.03276311 0.13244459 0.61843511 -0.08128328 0.77031777 -0.41593183 0.27687841 -0.75854507 0.41826836 0.75222019 -0.52859208 0.22980744 0.31928630 -0.13244459 -0.61843511 0.08128328 -0.77031777 -0.88625386 -0.32965258 0.25236023 -0.20542051 -0.54837642 -0.34155670 -0.67726179 -0.35204941 -0.20384377 0.87014878 0.44745436 0.03276311 -0.54996550 -0.58136625 -0.27637741 0.53213415 -0.13403367 -0.85824466 0.48216766 0.11386579 0.54996550 0.58136625 0.27637741 -0.53213415 -0.00158908 -0.23980955 0.40088438 0.88418356 -0.75222019 0.52859208 -0.22980744 -0.31928630 -0.33787744 0.01190412 0.92962202 0.14662890 |