In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres 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 n-dimensional spheres in (n + 1)-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.^[1][2] Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others, investigations have determined upper and lower bounds, but not exact solutions.^[3]

Known greatest kissing numbers

One dimension

In one dimension,^[4] the kissing number is 2:

Two dimensions

In two dimensions, the kissing number is 6:

Proof: Consider a circle with center C that is touched by circles with centers C₁, C₂, .... Consider the rays C C_i. These rays all emanate from the same center C, 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 C C₁ and C C₂, are separated by an angle of less than 60°. The segments C C_i have the same length – 2r – for all i. Therefore, the triangle C C₁ C₂ is isosceles, and its third side – C₁ C₂ – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction.^[5]

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 0.1 of the radius between two nearby spheres.

Some known bounds

The following table lists some known bounds on the kissing number in various dimensions.^[12] The dimensions in which the kissing number is known are listed in boldface.

Rough volume estimates show that kissing number in n dimensions grows exponentially in n. The base of exponential growth is not known. The grey area in the above plot represents the possible values between known upper and lower bounds. Circles represent values that are known exactly.

Generalization

The kissing number problem can be generalized to the problem of finding the maximum number of non-overlapping congruent copies of any convex body that touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body, translates of the original body, or translated by a lattice. For the regular tetrahedron, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.^[14]

Algorithms

There are several approximation algorithms on intersection graphs where the approximation ratio depends on the kissing number.^[15] 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 ${\displaystyle x_{n}}$ be a set of N D-dimensional position vectors of the centres of the spheres. The condition that this set of spheres can lie round the centre sphere without overlapping is:^[16]

$${\displaystyle \exists x\ \left\{\forall _{n}\{x_{n}^{\textsf {T}}x_{n}=1\}\land \forall _{m,n:m\neq n}\{(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})\geq 1\}\right\}}$$

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 which is why this problem has only been solved up to four dimensions. By adding additional variables, ${\displaystyle y_{nm}}$ this can be converted to a single quartic equation in N(N − 1)/2 + DN variables:^[17]

$${\displaystyle \exists xy\ \left\{\sum _{n}\left(x_{n}^{\textsf {T}}x_{n}-1\right)^{2}+\sum _{m,n:m<n}{\Big (}(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})-1-(y_{nm})^{2}{\Big )}^{2}=0\right\}}$$

Therefore, to solve the case in D = 5 dimensions and N = 40 + 1 vectors would be equivalent to determining the existence of real solutions to a quartic polynomial in 1025 variables. For the D = 24 dimensions and N = 196560 + 1, the quartic would have 19,322,732,544 variables. An alternative statement in terms of distance geometry is given by the distances squared ${\displaystyle R_{mn}}$ between the m^th and n^th sphere:

$${\displaystyle \exists R\ \{\forall _{n}\{R_{0n}=1\}\land \forall _{m,n:m<n}\{R_{mn}\geq 1\}\}}$$

This must be supplemented with the condition that the Cayley–Menger determinant is zero for any set of points which forms a (D + 1) simplex in D dimensions, since that volume must be zero. Setting ${\displaystyle R_{mn}=1+{y_{mn}}^{2}}$ gives a set of simultaneous polynomial equations in just y 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 y by small amounts to try to minimise the polynomial in terms of the y.

See also

• Equilateral dimension
• Spherical code
• Soddy's hexlet
• Cylinder sphere packing

Notes

References

• T. Aste and D. Weaire The Pursuit of Perfect Packing (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3
• Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower bounds)
• Bachoc, Christine; Vallentin, Frank (2008). "New upper bounds for kissing numbers from semidefinite programming". Journal of the American Mathematical Society. 21 (3): 909–924. arXiv:math.MG/0608426. Bibcode:2008JAMS...21..909B. doi:10.1090/S0894-0347-07-00589-9. MR 2393433. S2CID 204096.

External links

• Grime, James. "Kissing Numbers" (video). youtube. Brady Haran. Archived from the original on 2021-12-12. Retrieved 11 October 2018.