A Short Solution of the Kissing Number Problem in Dimension Three

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12-30-2020

A short solution of the kissing number problem in dimension three

Alexey Glazyrin The University of Texas Rio Grande Valley

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Recommended Citation Glazyrin, A. (2020). A short solution of the kissing number problem in dimension three. ArXiv:2012.15058 [Math]. http://arxiv.org/abs/2012.15058

This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Mathematical and Statistical Sciences Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact [email protected], [email protected]. arXiv:2012.15058v1 [math.MG] 30 Dec 2020 12 plies e.Ide,o h n hand one the on Indeed, rem. hwta o each for that show arieaglrdsacsbtenpit ftangency of points between distances angular pairwise 0 npriua,teDlat ehdi h peia aei b is case spherical 1. the Proposition in method Delsarte the particular, In n eesti 5.Frtelna rgamn prah w approach, follows. programming as linear recursively cas the defined spherical For polynomials the to [5]. extended Levenshtein then and space, Hamming the for [3] 1. Theorem omteaiin,adnwpof aebe ulse nrec in for only published solved been is problem have number spher proofs kissing thirteen the new The dimensions, and [6]. an Leech by mathematicians, Newton given Isaac to was between pr proof discussion elegant number an famous kissing of the (the of problem subject spheres the as thirteen known the is settled sphere unit given sum . 18905584 . Fix Let suew have we Assume o u ro,w s h ierpormigapoc.Teme The approach. programming linear the use we proof, our For h rbe ffidn h aiu ubro non-overlapping of number maximum the finding of problem The Date 99405263. P P f aur ,2021. 1, January : x j N Abstract. HR OUINO H ISN UBRPOLMIN PROBLEM NUMBER KISSING THE OF SOLUTION SHORT A i,j N ( =1 t = =1 0 = ) G f G x 0 ( d ( f i 4 (3) ) h h polynomial The . ( ( ,x x, [13] h . t 9689+0 + 09465869 ( x 1 = ) t i 0 + ) 4 5] [4, x , j nti oe egv hr ouino h isn ubrpr number kissing the of solution short a give we note, this In i h isn ubri ieso he s12. is three dimension in number kissing The a emd nyb points by only made be can ) j N i G , i , ) . 00334265 P o n nt set finite any For o-vrapn ntshrstnett ie ntsphere unit given a to tangent spheres unit non-overlapping ≥ j N P 1 ( =1 d ) i,j N ( . f 17273741 t = ) ( =1 2. h P x h isn ubrproblem number kissing the G f 0 i i,j N hr ro fTerm1 Theorem of proof short A x , 5 (3) ,G t, . sngtv n[ on negative is 9689=0 = 09465869 =1 IESO THREE DIMENSION j ( 1 i t ≤ ) 0 + ) X f G i,j LXYGLAZYRIN ALEXEY ( k ( 1. ≤ 1 (3) d ≤ h X ) x N ( 1 Introduction i ( . t x , 03616728 t . G = = ) 3te ecncnld h ttmn ftetheo- the of statement the conclude can we then 23 0 + ) k ( j { d i ) x ) ( ( . 1 1 09465869 h d ≤ . x , . . . , x 33128438 − x 2 + i j 1 x , 1 G . / nteoe peia cap spherical open the in 23 k √ j 9 (3) i N − ) 2 N x ( , ≥ ⊂ } N nteohrhn,Pooiin1im- 1 Proposition hand, other the On . 1 t 4) 1 d seFgr o h ltof plot the for 1 Figure (see ) x , . . . , G / sdo h olwn proposition. following the on ased 0 c¨teadvndrWedn[13] Waerden der van Schütte and . 2 8 = ]s h oiiecnrbto othe to contribution positive the so 2] G t be o ieso he)ta was that three) dimension for oblem . Therefore, . 4 n eeaie yKabatyansky by generalized and [4] e 2 (3) spolmcniust eo interest of be to continues problem es S ai rgr n19.Asketch A 1694. in Gregory David d s h rpriso Gegenbauer of properties the use e d d k ( ( , − d t − + 4[,1] n for and 11], [7, 24 ) N 0 + ) 1 n er 8 ,1 ] nother In 9]. 1, 2, [8, years ent 1 hdwsdsoee yDelsarte by discovered was thod ( k n any and t in ) − be ndmninthree. dimension in oblem − . 17275228 S 3 ntshrstnett a to tangent spheres unit 2 ( k N r tleast at are − k ≤ 1) ≥ 1 G C G 0 . 23 k ( , 3 (3) d − ihtecenter the with ) / d 2 ( 0 ( S [10]. 4 = t . t )+ 09465869 2 ) π/ hnall Then . . f .I we If 3. ( t )). ≈ 0.2

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Figure 1. Plot of f(t) for t [ 1, 1/2]. ∈ −

x and the angular radius π/4. No more than 3 points with pairwise angular distances at least − π/3 can fit in C. Indeed, if there are at least 4 points y1,y2,y3,y4 in C then at least one angle ∠(y , x,y ) is no greater than π/2. By the spherical law of cosines, the angular distance between i − j yi and yj is less than π/3. If there is exactly one point y in C, then f(1) + f( x,y ) f(1) + max f(t) 1.23. h i ≤ t [ 1, 1/√2] ≤ ∈ − − π For two points y, z in C, the angular distance between y and x is at least 12 by the triangle inequality for y,z, x. Hence if x,y = t then t cannot be less− than cos π . By the triangle − h i − 12 inequality, x, z α(t)= 1 t √3 √1 t2. Since f is decreasing on I = [ cos π , 1/√2], h i≥ 2 − 2 − − 12 − f(1) + f( x,y )+ f( x, z ) f(1) + max(f(t)+ f(α(t))) 1.23. h i h i ≤ t I ≤ ∈ For three points y, z, w in C, we use the monotonicity of f on I and move them as close as possible to x. This way we get at least two of the three pairwise angular distances equal to π/3. Assume− y, z = z, w = 1/2. Note that z,w,x cannot belong to the same large circle because otherwiseh yi doesh noti fit in C. This means we can always move w keeping w, z = 1/2 and decreasing x, w . The process stops in two possible cases: w reaches the boundaryh i of C or y, w becomesh 1/2.i In the former case we are left with the case of two points in C covered above.h i Now we can assume that y, z = z, w = y, w = 1/2. Without loss of generality, x,y x, z) x, w . We keep theh pointi hy intacti andh rotatei the regular triangle yzw so that h i ≤ h ≤ h i x, z decreases. Since x,y x, z , x, z √2 1 . Note that x, z + x, w decreases in h i h i ≥ h i h i≤− 4 − 2 h i h i this case as well and, due to convexity and monotonicity of f on the interval [ √2 1 , 1 ], − 4 − 2 − √2 f( x, z )+ f( x, w ) increases. This process will stop either when w reaches the boundary of C or whenh ix, z becomesh i equal to x,y . In the former case, we are left with two points in C. In the latter h i h i case, if x,y = x, z = t then x, w = β(t)= 2 t 2 3 2t2. Given that t x, w 1/√2, t h i h i h i 3 − 3 q 2 − ≤ h i≤− must belong to J = [ √2 1 , 2 ]. Then − 4 − 2 −q 3 f(1) + f( x,y )+ f( x, z )+ f( x, w ) f(1) + max(2f(t)+ f(β(t))) 1.23. h i h i h i ≤ t J ≤ 2 ∈ Remark 1. This proof is similar to the proof in [9] and the solution of the kissing problem in dimension four [10] (see also [12]) but the function is chosen more carefully so the case analysis is much simpler. Remark 2. The function f(t) was found by using a fixed value of 1.23 and maximizing the constant term in the Gegenbauer expansion while imposing required conditions. All inequalities are easily verifiable. For convenience, their explicit forms are available in a separate file attached to the arXiv submission of the paper. References 1. K. M. Anstreicher. The thirteen spheres: a new proof. Discrete Comput. Geom., 31(4):613–625, 2004. 2. K. Böröczky. The Newton-Gregory problem revisited. In Discrete geometry, volume 253 of Monogr. Textbooks Pure 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. Schütte and B. L. van der Waerden. Das Problem der dreizehn Kugeln. Math. Ann., 125:325–334, 1953.

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