Kissing numbers $\tau_n$

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Numbers

$n$

$\tau_n$

comment: $\tau_1=2$, attained by the integer lattice $\mathbb{Z}$.

comment: $\tau_2=6$, attained by the hexagonal lattice $A_2$; a seventh circle cannot touch, since two of seven rays from the centre would meet at an angle smaller than $60^{\circ}$.

comment: $\tau_3=12$, attained by the face-centred cubic lattice $A_3$ and by many other arrangements, among them the vertices of a regular icosahedron; whether a thirteenth sphere fits was disputed by Newton and Gregory in 1694, and the first complete proof that it does not is by Schütte and van der Waerden [2].

comment: $\tau_4=24$, attained by $D_4$, whose minimal vectors are the vertices of the 24-cell; proved by Musin [3], and the arrangement is unique up to isometry [11].

[40, 44] †

comment: $40$ is the kissing number of $D_5$ [1]; the upper bound is the semidefinite programming bound of Mittelmann and Vallentin [8].

[72, 77] †

comment: $72$ is the kissing number of $E_6$ [1]; the upper bound is by de Laat, Leijenhorst and de Muinck Keizer [11].

[126, 134] †

comment: $126$ is the kissing number of $E_7$ [1]; the upper bound is by Mittelmann and Vallentin [8].

240

comment: $\tau_8=240$, attained by $E_8$; proved independently by Odlyzko and Sloane [4] and by Levenshtein [5], and the arrangement is unique up to isometry [6]. The theta series of $E_8$ is the Eisenstein series $E_4$, whose coefficient of $q$ is $240$.
equals: Q-expansion_of_the_Eisenstein_series_E4#1

[306, 363] †

comment: $306$ is attained by a nonlattice arrangement of Leech and Sloane [13]; the largest kissing number of a lattice in dimension $9$ is $272$, that of $\Lambda_9$ [12]; the upper bound is by Machado and de Oliveira Filho [9].

10:

[510, 553] †

comment: $510$ is attained by an arrangement of Ganzhinov [17]; the laminated lattice $\Lambda_{10}$ has $336$ minimal vectors; the upper bound is by Machado and de Oliveira Filho [9].

11:

[604, 868] †

comment: $604$ is attained by an arrangement found in 2026 by AI agents on the EinsteinArena platform [21]; $\Lambda_{11}$ has $438$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

12:

[841, 1355] †

comment: $841$ is attained by an arrangement of Takhanov, Assylbekov and Yun [20], one sphere more than the arrangements of size $840$ known before it; the Coxeter–Todd lattice $K_{12}$ has $756$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

13:

[1154, 2064] †

comment: $1154$ is attained by an arrangement of Zinoviev and Ericson [16]; $\Lambda_{13}$ has $906$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

14:

[1932, 3174] †

comment: $1932$ is attained by an arrangement of Ganzhinov [17]; $\Lambda_{14}$ has $1422$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

15:

[2564, 4853] †

comment: $2564$ is attained by a nonlattice arrangement of Leech and Sloane [13]; $\Lambda_{15}$ has $2340$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

16:

[4320, 7320] †

comment: $4320$ is the kissing number of the Barnes–Wall lattice $\Lambda_{16}$ [15]; the upper bound is by de Laat and Leijenhorst [10].

17:

[5730, 10978] †

comment: $5730$ is attained by an arrangement of Cohn and Li [18], obtained by changing signs in the minimal vectors of a lattice; $\Lambda_{17}$ has $5346$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

18:

[7654, 16406] †

comment: $7654$ is attained by an arrangement of Cohn and Li [18]; $\Lambda_{18}$ has $7398$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

19:

[11948, 24417] †

comment: $11948$ is attained by an arrangement of Ho [19], improving the $11692$ of Cohn and Li [18]; $\Lambda_{19}$ has $10668$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

20:

[19448, 36195] †

comment: $19448$ is attained by an arrangement of Cohn and Li [18]; $\Lambda_{20}$ has $17400$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

21:

[29768, 53524] †

comment: $29768$ is attained by an arrangement of Cohn and Li [18]; $\Lambda_{21}$ has $27720$ minimal vectors; the upper bound is by de Laat and Leijenhorst [10].

22:

[49896, 80810] †

comment: $49896$ is the kissing number of the laminated lattice $\Lambda_{22}$ [14]; the upper bound is by de Laat and Leijenhorst [10].

23:

[93150, 122351] †

comment: $93150$ is the kissing number of the laminated lattice $\Lambda_{23}$ [14]; the upper bound is by de Laat and Leijenhorst [10].

24:

196560

comment: $\tau_{24}=196560$, attained by the Leech lattice $\Lambda_{24}$ [14]; proved independently by Odlyzko and Sloane [4] and by Levenshtein [5], and the arrangement is unique up to isometry [6].

Definition

The kissing number $\tau_n$ [23] is the largest number of non-overlapping unit balls in $\mathbb{R}^n$ that can touch one unit ball; equivalently, the largest number of points on the unit sphere $S^{n-1}$ any two of which are at angular distance at least $60^{\circ}$.

Parameters

$n$

— dimension ($n\geq 1$)

Formulas

(1)

$\tau_n\geq\tau(L)$ for every lattice $L\subset\mathbb{R}^n$, where $\tau(L)$ is the number of minimal vectors of $L$, with equality for $L=\mathbb{Z}$, $A_2$, $A_3$, $D_4$, $E_8$ and the Leech lattice $\Lambda_{24}$ in dimensions $1$, $2$, $3$, $4$, $8$ and $24$.

(2)

$2^{0.2075\,n\,(1+o(1))}\leq\tau_n\leq 2^{0.401\,n\,(1+o(1))}$, the lower bound by Chabauty, Shannon and Wyner and the upper by Kabatiansky and Levenshtein [1]; more precisely $\tau_n\geq c\,n^{3/2}\,(2/\sqrt{3})^{n}$ for a constant $c>0$ [22], and $\log_2(2/\sqrt{3})=0.2075\ldots$.

Comments

(3)

$\tau_n$ is also called the Newton number or the contact number of $\mathbb{R}^n$ [25], after the question Newton and Gregory disputed in 1694, whether thirteen spheres can touch one. The maximum is over all arrangements; the lattice kissing number, the largest number of minimal vectors of a lattice in $\mathbb{R}^n$, is OEIS A001116 [27] and is known for $n\leq 9$ and $n=24$. Dimension $9$ is the smallest in which $\tau_n$ is known to exceed the lattice kissing number: there the lattice kissing number is $272$, that of $\Lambda_9$, and the largest arrangement known has $306$ spheres. In dimensions $16$, $22$, $23$ and $24$, as in dimensions $5$, $6$ and $7$, the largest arrangement known consists of the minimal vectors of a lattice; in dimensions $10$ to $15$ and $17$ to $21$ it is not a lattice arrangement, and has more spheres than any lattice known there has minimal vectors. The kissing number of each classical lattice is in the entry comments of the table of packing densities and Hermite numbers of the classical lattices, and those of the laminated lattices $\Lambda_n$ are OEIS A002336 [28].

(4)

$\tau_n$ is OEIS A257479 [26]; it is known for $n=1$, $2$, $3$, $4$, $8$ and $24$, and in no other dimension [24]. The values for $n=1$ and $2$ are elementary; $\tau_3=12$ is the theorem of Schütte and van der Waerden [2]; $\tau_4=24$ is Musin's [3]; $\tau_8=240$ and $\tau_{24}=196560$ were proved independently by Odlyzko and Sloane [4] and by Levenshtein [5] with Delsarte's linear programming bound, which gives exactly $240$ and $196560$ in those two dimensions. The arrangements attaining $\tau_4$, $\tau_8$ and $\tau_{24}$ are unique up to isometry, the minimal vectors of $D_4$ [11], of $E_8$ and of the Leech lattice [6]; the twelve spheres in dimension $3$ can be arranged in infinitely many ways.

(5)

Where $\tau_n$ is not known the entry is the interval $[a,b]$ in which it lies: $a$ is the number of spheres in the largest arrangement known and $b$ the smallest upper bound proven, as of 6 September 2026, both taken from Henry Cohn's table of kissing number bounds [24], which also lists dimensions $25$ to $48$ and $72$, and from the table in the Wikipedia article [23]; the two agreed in every dimension on that date. In dimensions $5$, $6$, $7$, $16$, $22$ and $23$ the lower endpoint is the kissing number of a lattice, $D_5$, $E_6$, $E_7$, $\Lambda_{16}$, $\Lambda_{22}$ and $\Lambda_{23}$; in every other dimension the largest arrangement known has more spheres than any lattice known there has minimal vectors. Every upper bound with $n\geq 5$ is a semidefinite programming bound in the line of Bachoc and Vallentin [7], computed by Mittelmann and Vallentin [8], Machado and de Oliveira Filho [9], de Laat and Leijenhorst [10] and de Laat, Leijenhorst and de Muinck Keizer [11].

Programs

(P1)

Sage

G = CartanMatrix(['E', 8])
pari(G).qfminim()[0]      # 240, the number of minimal vectors of E_8, which is tau_8

References

[1]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der mathematischen Wissenschaften 290, Springer, 1999, chapter 1.

[2]

K. Schütte and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Mathematische Annalen 125 (1953), 325–334.

[3]

O. R. Musin, The kissing number in four dimensions, Annals of Mathematics 168 (2008), 1–32.

[4]

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, Journal of Combinatorial Theory, Series A 26 (1979), 210–214.

[5]

V. I. Levenshtein, On bounds for packings in n-dimensional Euclidean space, Doklady Akademii Nauk SSSR 245 (1979), 1299–1303; Soviet Mathematics Doklady 20 (1979), 417–421.

[6]

E. Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canadian Journal of Mathematics 33 (1981), 437–449.

[7]

C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, Journal of the American Mathematical Society 21 (2008), 909–924.

[8]

H. D. Mittelmann and F. Vallentin, High-accuracy semidefinite programming bounds for kissing numbers, Experimental Mathematics 19 (2010), 175–179.

[9]

F. C. Machado and F. M. de Oliveira Filho, Improving the semidefinite programming bound for the kissing number by exploiting polynomial symmetry, Experimental Mathematics 27 (2018), 362–369.

[10]

D. de Laat and N. Leijenhorst, Solving clustered low-rank semidefinite programs arising from polynomial optimization, Mathematical Programming Computation 16 (2024), 503–534.

[11]

D. de Laat, N. Leijenhorst and W. H. H. de Muinck Keizer, Optimality and uniqueness of the D4 root system, preprint, 2024. (arXiv)

[12]

G. L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Mathematicae 84 (1971), 42 pp.

[13]

J. Leech and N. J. A. Sloane, Sphere packings and error-correcting codes, Canadian Journal of Mathematics 23 (1971), 718–745.

[14]

J. Leech, Notes on sphere packings, Canadian Journal of Mathematics 19 (1967), 251–267.

[15]

E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, Journal of the Australian Mathematical Society 1 (1959), 47–63.

[16]

V. A. Zinoviev and T. Ericson, New lower bounds for contact numbers in small dimensions, Problems of Information Transmission 35 (1999), 287–294.

[17]

M. Ganzhinov, Highly symmetric lines, Linear Algebra and its Applications 722 (2025), 12–37.

[18]

H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, preprint, 2024. (arXiv)

[19]

B. S. Ho, A new lower bound for the kissing number in 19 dimensions, preprint, 2026. (arXiv)

[20]

R. Takhanov, Z. Assylbekov and S. Yun, Structure of kissing arrangements in R^12 and a place for the 841st sphere, preprint, 2026. (arXiv)

[21]

F. Bianchi, Y. Kwon, A. Pappu and J. Zou, Harnessing the collective intelligence of AI agents in the wild for new discoveries, preprint, 2026. (arXiv)

[22]

M. Jenssen, F. Joos and W. Perkins, On kissing numbers and spherical codes in high dimensions, Advances in Mathematics 335 (2018), 307–321.

Links

[23]

Wikipedia: Kissing number

[24]

H. Cohn, Kissing numbers: a table of the best bounds known

[25]

MathWorld: Kissing Number

[26]

OEIS A257479: maximal kissing number in $n$ dimensions

[27]

OEIS A001116: maximal kissing number of an $n$-dimensional lattice

[28]

OEIS A002336: kissing number of the laminated lattice $\Lambda_n$

Similar tables

Packing densities and Hermite numbers of the classical lattices — the kissing number of each lattice, a lower bound for $\tau_n$, is in its entry comments

Hermite's constants $\gamma_n$ — known in dimensions $1$ to $8$ and $24$; in the six dimensions where $\tau_n$ is known, the same lattice attains both

$q$-expansion of the Eisenstein series $E_4$ — $E_4$ is the theta series of $E_8$, so its coefficient of $q$ is $\tau_8=240$

Diagonal Ramsey numbers — the same convention, an interval of integers where the value is not known

Data properties

Entries are of type: integer

How well the digits are known: exact

Sources of data: [24], [23]

Table is complete: no (every dimension $n\leq 24$ is here, the exact value in the six dimensions where $\tau_n$ is known and otherwise the interval between the largest arrangement known and the smallest upper bound proven, as of 6 September 2026)

How they were obtained:

Every entry is a fact from the literature, cited in its comment: a theorem for the six exact values, and for each interval a construction for the lower endpoint and a proof for the upper.

The endpoints were taken from Henry Cohn's table of kissing number bounds and from the table in the Wikipedia article, which agreed in every dimension listed on 6 September 2026, and were compared with the abstracts of the papers cited. Before an entry is returned, the generator recomputes the exact value or the lower endpoint for $n\leq 8$ as the number of minimal vectors of the lattice attaining it, from the identity matrix or the Cartan matrix with PARI's qfminim, and a disagreement is an error rather than an entry; outside the generator the same was done for $\Lambda_{16}$, $\Lambda_{22}$, $\Lambda_{23}$ and the Leech lattice from the Gram matrices of the table of the classical lattices, and the kissing numbers of the laminated lattices quoted in the comments were compared with OEIS A002336. No upper bound was recomputed: each is the optimal value of a semidefinite program reported in the paper cited. An interval remains true when either bound is improved.