Ingeometry, thekissing number of amathematical space is defined as the greatest number of non-overlapping unitspheres that can be arranged in that space such that they each touch a common unit sphere. For a givensphere 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 alattice 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 areNewton number (after the originator of the problem), andcontact number.

In general, thekissing number problem seeks the maximum possible kissing number forn-dimensional spheres in (n + 1)-dimensionalEuclidean 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]

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

In two dimensions, the kissing number is 6:

Proof: Consider a circle with centerC that is touched by circles with centersC₁,C₂, .... Consider the raysCC_i. These rays all emanate from the same centerC, 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, sayCC₁ andCC₂, are separated by an angle of less than 60°. The segmentsC C_i have the same length – 2r – for alli. Therefore, the triangleCC₁C₂ isisosceles, and its third side –C₁C₂ – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction.^[5]

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 mathematiciansIsaac Newton andDavid 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, most notably one byReinhold Hoppe, but the first correct proof (according to Brass, Moser, and Pach) did not appear until 1953.^[1][2][6]

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

In four dimensions, the kissing number is 24. This was proven in 2003 by Oleg Musin.^[7][8] 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 scaled24-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 forn = 3 — so the situation was even less clear.

The existence of the highly symmetricalE₈ lattice andLeech lattice has allowed known results forn = 8 (where the kissing number is 240), andn = 24 (where it is 196,560).^[9][10]The kissing number inndimensions is unknown for other dimensions.

If arrangements are restricted tolattice arrangements, in which the centres of the spheres all lie on points in alattice, then this restricted kissing number is known forn = 1 to 9 andn = 24 dimensions.^[11] 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.

The following table lists some known bounds on the kissing number in various dimensions.^[12][13] The dimensions in which the kissing number is known are listed in boldface.

The kissing number problem can be generalized to the problem of finding the maximum number of non-overlappingcongruent copies of anyconvex 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 theregular 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.^[16]

There are severalapproximation algorithms onintersection graphs where the approximation ratio depends on the kissing number.^[17] For example, there is a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares.

The kissing number problem can be stated as the existence of a solution to a set ofinequalities. Let${\displaystyle x_n}$ be a set ofND-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:^[18]$${\displaystyle \mathrm{\exists }x\left\{{\mathrm{\forall }}_n\{x_n^{\mathsf{\text{T}}}{x}_n=1\}\wedge {\mathrm{\forall }}_{m,n:m\ne n}\{(x_n-x_m{)}^{\mathsf{\text{T}}}(x_n-x_m)\ge 1\}\right\}}$$

Thus the problem for each dimension can be expressed in theexistential theory of the reals. However, general methods of solving problems in this form take at leastexponential 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 singlequartic equation inN(N − 1)/2 +DN variables:^[19]

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

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

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

• Cohn, Henry; Li, Anqi (2024). "Improved kissing numbers in seventeen through twenty-one dimensions".arXiv:2411.04916 [math.MG].
• T. Aste andD. WeaireThe Pursuit of Perfect Packing (Institute Of Physics Publishing London 2000)ISBN 0-7503-0648-3
• Table of the Highest Kissing Numbers Presently Known maintained byGabriele Nebe andNeil 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.

• Grime, James (10 October 2018)."Kissing Numbers"(video).youtube.Brady Haran.Archived from the original on 2021-12-12. Retrieved11 October 2018.

• Discrete geometry
• Packing problems

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