Kissing Number Problem - Known Greatest Kissing Numbers

Known Greatest Kissing Numbers

In one dimension, the kissing number is 2:

It is easy to see (and to prove) that in two dimensions the kissing number is 6.

In three dimensions the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, but the first correct proof did not appear until 1953.

The twelve neighbors of the central sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size (as in a chemical element). A coordination number of 12 is found in a cubic close-packed or a hexagonal close-packed structure.

In four dimensions, it was known for some time that the answer is either 24 or 25. It is easy to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled 24-cell centered at the origin). As in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was even less clear. In 2003, Oleg Musin proved the kissing number for n = 4 to be 24, using a subtle trick.

The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560). The results in these dimensions stem from the existence of highly symmetrical lattices: the E₈ lattice and the Leech lattice.

If arrangements are restricted to regular arrangements, in which the centres of the spheres all lie on points in a lattice, then this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions. For 5, 6 and 7 dimensions the arrangement with the highest known kissing number is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.