Kissing number

In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres (i.e., of radius 1) that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing, the kissing number is the same for every sphere; but for an arbitrary sphere packing, the kissing number may vary from one sphere to another.

Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number.

In general, the kissing number problem seeks the maximum possible kissing number for -dimensional spheres in -dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.

Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century.

Known greatest kissing numbers

One dimension

In one dimension, the kissing number is 2:

center

Two dimensions

In two dimensions, the kissing number is 6:

center

Proof: Consider a circle with center that is touched by circles with centers Consider the rays . These rays all emanate from the same center , so the sum of angles between adjacent rays is 360°.

Assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, say and , are separated by an angle of less than 60°. The segments have the same length – – for all . Therefore, the triangle is isosceles, and its third side – – has a side length of less than . Therefore, the circles 1 and 2 intersect – a contradiction.

Three dimensions

thumb|A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a [[regular icosahedron. This leaves slightly more than 10% of the radius between two nearby spheres.]]

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, with 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 19th century, most notably one by Reinhold Hoppe, but the first correct proof (according to Brass, Moser, and Pach) 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.

Larger dimensions

In four dimensions, the kissing number is 24. This was proven in 2003 by Oleg Musin. Previously, the answer was thought to be either 24 or 25: it is straightforward 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), but, as in the three-dimensional case, there is a lot of space left over — even more, in fact, than for — so the situation was even less clear.

The existence of the highly symmetrical lattice and Leech lattice has allowed to determine the kissing number for (namely, 240) and for (namely, 196,560). The kissing number in dimensions is unknown for other values of .

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

Some known bounds

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

thumb|right|upright=1.5|Rough volume estimates show that kissing number in dimensions [[Exponential growth|grows exponentially in . The base of exponential growth is not known. The gray area in the above plot represents the possible values between known upper and lower bounds. Circles represent values that are known exactly.]]

{| class="wikitable" style="text-align: center; margin-left: 40pt;"

!Dimension

!Lower<br />bound

!Upper<br />bound

|colspan=2 | 2

|colspan=2 | 6

|colspan=2 | 12

|colspan=2 | 24

|868

|12

|840

|1,355

|13

|1,154

|2,064

|14

|1,932

|3,174

|15

|2,564

|4,853

|16

|4,320

|7,320

|17

|5,730

|10,978

|18

|7,654

|16,406

|19

|11,948

|24,417

|20

|19,448

|36,195

|21

|29,768

|53,524

|22

|49,896

|80,810

|23

|93,150

|122,351

|24

|colspan=2 | 196,560

|25

|197,056

|265,006

|26

|198,550

Algorithms

There are several approximation algorithms on intersection graphs where the approximation ratio depends on the kissing number. For example, there is a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares.

Mathematical statement

The kissing number problem can be stated as the existence of a solution to a set of inequalities. Let be a set of -dimensional position vectors of sphere centres. The condition that this set of spheres can lie round the central sphere without overlapping is:

<math display=block>\exist x\ \left\{ \forall_n \{x_n^\textsf{T} x_n = 1\} \land \forall_{m,n: m \neq n} \{ (x_n - x_m)^\textsf{T}(x_n - x_m) \geq 1\} \right\}.</math>

Thus the problem for each dimension can be expressed in the existential theory of the reals. However, general methods of solving problems in this form take at least exponential time; that is why this problem has only been solved up to four dimensions. By using additional variables , this can be converted to a single quartic equation in <math>N(N-1)/2 + DN</math> variables:

<math display=block>\exist xy\ \left\{ \sum_n \left(x_n^\textsf{T} x_n - 1\right)^2 + \sum_{m,n: m<n}\Big( (x_n - x_m)^\textsf{T} (x_n - x_m) - 1 - (y_{nm})^2 \Big)^2 = 0 \right\}.</math>

Therefore, to solve the case in dimensions and vectors would be equivalent to determining the existence of real solutions to a quartic polynomial in 1025 variables. For the dimensions and , the quartic would have 19,322,732,544 variables. An alternative statement in terms of distance geometry is given by the squared distances between the -th and -th sphere centers:

<math display=block>\exist R\ \{ \forall_n \{R_{0n} = 1 \} \land \forall_{m,n: m<n} \{ R_{mn} \geq 1\} \}.</math>

This must be supplemented with the condition that the Cayley–Menger determinant is zero for any set of points which forms a -simplex in dimensions, since that volume must be zero. Setting <math>R_{mn} = 1 + y_{mn}^2</math> gives a set of simultaneous polynomial equations in just which must be solved for real values only. The two methods, being entirely equivalent, have various different uses. For example, in the second case, one can randomly alter the values of the by small amounts to try to minimise the polynomial in terms of the .

Notes

References

T. Aste and D. Weaire The Pursuit of Perfect Packing (Institute Of Physics Publishing London 2000)
Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower bounds)