Kissing Number

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).

Consider the situation in 2 dimensions, a 2D sphere is just a circle and it is easy to verify that the kissing angle 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!

The one dimensional case is rather boring with a kissing number of 2.

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

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             at least     24  at most     25
   5             at least     40  at most     46
   6             at least     72  at most     82
   7             at least    126  at most    140
   8                         240
   9             at least    306  at most    380
  10             at least    500  at most    595
  11             at least    582  at most    915
  12             at least    840  at most   1416
  13             at least   1130  at most   2233
  14             at least   1582  at most   3492
  15             at least   2564  at most   5431
  16             at least   4320  at most   8313
  17             at least   5346  at most  12215
  18             at least   7398  at most  17877
  19             at least  10668  at most  25901
  20             at least  17400  at most  37974
  21             at least  27720  at most  56852
  22             at least  49896  at most  86537
  23             at least  93150  at most 128096
  24                      196560

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