A short solution of the kissing number problem in dimension three
aa r X i v : . [ m a t h . M G ] D ec A SHORT SOLUTION OF THE KISSING NUMBER PROBLEM INDIMENSION THREE
ALEXEY GLAZYRIN
Abstract.
In this note, we give a short solution of the kissing number problem in dimension three. Introduction
The problem of finding the maximum number of non-overlapping unit spheres tangent to agiven unit sphere is known as the kissing number problem . Sch¨utte and van der Waerden [13]settled the thirteen spheres problem (the kissing number problem for dimension three) that wasthe subject of the famous discussion between Isaac Newton and David Gregory in 1694. A sketchof an elegant proof was given by Leech [6]. The thirteen spheres problem continues to be of interestto mathematicians, and new proofs have been published in recent years [8, 2, 1, 9]. In otherdimensions, the kissing number problem is solved only for d = 8 ,
24 [7, 11], and for d = 4 [10]. Theorem 1. [13]
The kissing number in dimension three is 12.
For our proof, we use the linear programming approach. The method was discovered by Delsarte[3] for the Hamming space, then extended to the spherical case [4] and generalized by Kabatyanskyand Levenshtein [5]. For the linear programming approach, we use the properties of Gegenbauerpolynomials defined recursively as follows. G ( d )0 ( t ) = 1 , G ( d )1 ( t ) = t, G ( d ) k ( t ) = ( d + 2 k − t G ( d ) k − ( t ) − ( k − G ( d ) k − ( t ) d + k − . In particular, the Delsarte method in the spherical case is based on the following proposition.
Proposition 1. [4, 5]
For any finite set X = { x , . . . , x N } ⊂ S d − and any k ≥ , X ≤ i,j ≤ N G ( d ) k ( h x i , x j i ) ≥ . A short proof of Theorem 1
Let f ( t ) = 0 . . G (3)1 ( t ) + 0 . G (3)2 ( t ) + 0 . G (3)3 ( t )+0 . G (3)4 ( t ) + 0 . G (3)5 ( t ) + 0 . G (3)9 ( t ) (see Figure 1 for the plot of f ( t )).Assume we have N non-overlapping unit spheres tangent to a given unit sphere S . Then allpairwise angular distances between points of tangency x , . . . , x N in S are at least π/
3. If weshow that for each i , P Nj =1 f ( h x i , x j i ) ≤ .
23 then we can conclude the statement of the theo-rem. Indeed, on the one hand P Ni,j =1 f ( h x i , x j i ) ≤ . N . On the other hand, Proposition 1 im-plies P Ni,j =1 f ( h x i , x j i ) ≥ P Ni,j =1 . . N . Therefore, N ≤ . / . ≈ . x = x i . The polynomial f is negative on [ − / √ , /
2] so the positive contribution to thesum P Nj =1 f ( h x, x j i ) can be made only by points x j in the open spherical cap C with the center Date : January 1, 2021. Figure 1.
Plot of f ( t ) for t ∈ [ − , / − x and the angular radius π/
4. No more than 3 points with pairwise angular distances at least π/ C . Indeed, if there are at least 4 points y , y , y , y in C then at least one angle ∠ ( y i , − x, y j ) is no greater than π/
2. By the spherical law of cosines, the angular distance between y i and y j is less than π/ y in C , then f (1) + f ( h x, y i ) ≤ f (1) + max t ∈ [ − , − / √ f ( t ) ≤ . . For two points y, z in C , the angular distance between y and − x is at least π by the triangleinequality for y, z, − x . Hence if h x, y i = t then t cannot be less than − cos π . By the triangleinequality, h x, z i ≥ α ( t ) = t − √ √ − t . Since f is decreasing on I = [ − cos π , − / √ f (1) + f ( h x, y i ) + f ( h x, z i ) ≤ f (1) + max t ∈ I ( f ( t ) + f ( α ( t ))) ≤ . . For three points y, z, w in C , we use the monotonicity of f on I and move them as close aspossible to − x . This way we get at least two of the three pairwise angular distances equal to π/
3. Assume h y, z i = h z, w i = 1 /
2. Note that z, w, x cannot belong to the same large circlebecause otherwise y does not fit in C . This means we can always move w keeping h w, z i = 1 / h x, w i . The process stops in two possible cases: w reaches the boundary of C or h y, w i becomes 1/2. In the former case we are left with the case of two points in C coveredabove. Now we can assume that h y, z i = h z, w i = h y, w i = 1 /
2. Without loss of generality, h x, y i ≤ h x, z ) ≤ h x, w i . We keep the point y intact and rotate the regular triangle yzw so that h x, z i decreases. Since h x, y i ≥ h x, z i , h x, z i ≤ − √ − . Note that h x, z i + h x, w i decreases inthis case as well and, due to convexity and monotonicity of f on the interval [ − √ − , − √ ], f ( h x, z i ) + f ( h x, w i ) increases. This process will stop either when w reaches the boundary of C orwhen h x, z i becomes equal to h x, y i . In the former case, we are left with two points in C . In the lattercase, if h x, y i = h x, z i = t then h x, w i = β ( t ) = t − q − t . Given that t ≤ h x, w i ≤ − / √ t must belong to J = [ − √ − , − q ]. Then f (1) + f ( h x, y i ) + f ( h x, z i ) + f ( h x, w i ) ≤ f (1) + max t ∈ J (2 f ( t ) + f ( β ( t ))) ≤ . . emark 1. This proof is similar to the proof in [9] and the solution of the kissing problem indimension four [10] (see also [12] ) but the function is chosen more carefully so the case analysis ismuch simpler.
Remark 2.
The function f ( t ) was found by using a fixed value of 1.23 and maximizing the constantterm in the Gegenbauer expansion while imposing required conditions. All inequalities are easilyverifiable. For convenience, their explicit forms are available in a separate file attached to the arXivsubmission of the paper. References
1. K. M. Anstreicher. The thirteen spheres: a new proof.
Discrete Comput. Geom. , 31(4):613–625, 2004.2. K. B¨or¨oczky. The Newton-Gregory problem revisited. In
Discrete geometry , volume 253 of
Monogr. TextbooksPure Appl. Math. , pages 103–110. Dekker, New York, 2003.3. P. Delsarte. An algebraic approach to the association schemes of coding theory.
Philips Res. Rep. Suppl. ,(10):vi+97, 1973.4. P. Delsarte, J. M. Goethals, and J. J. Seidel. Spherical codes and designs.
Geometriae Dedicata , 6(3):363–388,1977.5. G. A. Kabatyansky and V. I. Levenshtein. Bounds for packings on the sphere and in space.
Problems of Infor-mation Transmission , 14(1):3–25, 1978.6. J. Leech. The problem of the thirteen spheres.
Math. Gaz. , 40:22–23, 1956.7. V. I. Levenshtein. Boundaries for packings in n -dimensional Euclidean space. Dokl. Akad. Nauk SSSR ,245(6):1299–1303, 1979.8. H. Maehara. The problem of thirteen spheres—a proof for undergraduates.
European J. Combin. , 28(6):1770–1778, 2007.9. O. R. Musin. The kissing problem in three dimensions.
Discrete Comput. Geom. , 35(3):375–384, 2006.10. O. R. Musin. The kissing number in four dimensions.
Ann. of Math. (2) , 168(1):1–32, 2008.11. A. M. Odlyzko and N. J. A. Sloane. New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. J. Combin. Theory Ser. A , 26(2):210–214, 1979.12. F. Pfender. Improved Delsarte bounds for spherical codes in small dimensions.
J. Combin. Theory Ser. A ,114(6):1133–1147, 2007.13. K. Sch¨utte and B. L. van der Waerden. Das Problem der dreizehn Kugeln.
Math. Ann. , 125:325–334, 1953.
School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley,Brownsville, TX 78520
Email address : [email protected]@utrgv.edu