Kissing Number Problem

Updated 24 November 2025

Kissing number problem is defined as determining the maximal count of non-overlapping unit spheres touching a central sphere in n-dimensional space.
It connects combinatorics, coding theory, lattice optimization, and spherical codes, with exact values known in select highly symmetric dimensions.
Recent computational and analytic methods, including semidefinite programming and simulated annealing, have improved bounds while significant gaps remain in higher dimensions.

The kissing number problem asks for the maximal number of non-overlapping unit spheres in $\mathbb{R}^n$ that can be arranged to simultaneously touch a central unit sphere of the same radius. It is a singular instance of a wider family of extremal geometry questions and has deep connections to combinatorics, coding theory, lattice theory, optimization, and high-dimensional discrete geometry. Determining the kissing number, usually denoted $\tau(n)$ or $K(n)$ , remains a challenging open problem in all but a finite list of dimensions, despite over three centuries of paper since its formulation in the Newton–Gregory correspondence of 1694.

1. Formal Definitions and Equivalent Formulations

Given $n\in\mathbb{N}$ , the $n$ -dimensional kissing number $\tau(n)$ is the maximum cardinality $k$ for which there exists a set $\{x_1,\dots,x_k\}\subset S^{n-1}$ of unit vectors such that

$\langle x_i, x_j\rangle \leq \frac{1}{2}, \quad \forall\, i\neq j.$

Equivalently, it is the maximal number of pairwise non-overlapping unit balls in $\mathbb{R}^n$ that are all tangent to a central unit ball. The problem can be recast as a spherical code packing problem, with $\tau(n)=A(n,1/2)$ , where $A(n,s)$ is the maximal cardinality of a code in $S^{n-1}$ with pairwise inner products at most $s$ (Boyvalenkov et al., 2015).

Central to the analysis of sphere packings and their contacts is the notion of the contact graph: for a packing $P$ of spheres, the contact graph $G=(P,E)$ joins two spheres whenever they are tangent, and the average degree $\overline{\delta}(G)$ is relevant for "average kissing number" relaxations (Dostert et al., 2020).

2. Known Exact Values and Lower Bounds

Exact values of $\tau(n)$ are known only in certain highly symmetric dimensions:

$\tau(1)=2$ (interval endpoints)
$\tau(2)=6$ (vertices of a regular hexagon)
$\tau(3)=12$ (proved by Schütte–van der Waerden; see also classification via hexagonal layers (Hales, 2012))
$\tau(4)=24$ (Musin 2003)
$\tau(8)=240$ (root system $E_8$ )
$\tau(24)=196560$ (Leech lattice minimal vectors)

For $n=5,6,7,9,\dots,23$ only lower and upper bounds are known (Boyvalenkov et al., 2015, Szöllősi, 2023, Cohn et al., 7 Nov 2024).

Recently, new combinatorial and probabilistic constructions have yielded improved lower bounds in many dimensions. Cohn–Li improved kissing number lower bounds in dimensions 17 to 21 substantially, showing, for instance, in dimension 19: $\tau_{19}\geq 11692,$ outperforming Leech's cross-section construction (Cohn et al., 7 Nov 2024).

In high dimensions, classical constructions using the Leech lattice $\Lambda_{24}$ and probabilistic methods, such as the automorphism method and simulated annealing, produce record lower bounds in dimensions $25\leq n\leq31$ ; e.g., $\tau_{31}\geq232,874$ (Kallal et al., 2016, Ma et al., 17 Nov 2025).

For $d=5$ , the lower bound $\tau(5)\ge 40$ was previously believed to be realized only by the $D_5$ and Leech-type arrangements, but a third, non-isometric arrangement was recently discovered by Szöllősi (Szöllősi, 2023).

3. Upper Bounds: Analytic and Optimization-Based Approaches

The classical approach for upper bounds is via Delsarte’s linear programming (LP) bound (Boyvalenkov et al., 2015). Given a family of positive definite polynomials $f$ on the sphere satisfying sign and degree conditions, one obtains

$\tau(n) \leq \frac{f(1)}{f_0},$

where $f_0$ is the lowest-degree Gegenbauer coefficient. The Levenshtein universal bound and extensions optimize over higher-degree auxiliary polynomials, in some cases exactly attaining the kissing number (notably in dimensions $8$ and $24$).

Semidefinite programming (SDP) methods generalize LP to three-point (or higher) correlations. Initiated by Bachoc–Vallentin and advanced via symmetry reduction and sum-of-squares techniques, these produce the sharpest known rigorous upper bounds in dimensions up to 23 (Machado et al., 2016). For example, improved SDP bounds in $n=9,10,11$ are strictly better than previous LP or SDP results.

For average kissing numbers, further SDP relaxations (with variable ball radii, harmonic analysis, and envelope functions) have yielded first nontrivial upper bounds below $2\tau(n)$ in dimensions $6\leq n\leq9$ (Dostert et al., 2020).

In high dimensions, asymptotic analytic upper bounds of Kabatianskii–Levenshtein type give

$\tau_d \le 2^{0.401\,d(1+o(1))} \text{ as } d\to\infty.$

n	Lower Bound	Previous Upper Bound	New SDP Upper Bound
3	12.612	13.955	13.606
4	24	34.681	27.439
5	40	77.757	64.022
6	72	156	121.105
7	126	268	223.144
8	240	480	408.386
9	272	726	722.629

4. High-Dimensional and Asymptotic Results

In large dimensions, both upper and lower bounds are exponential in $n$ , but with a persistent gap. Recent progress uses probabilistic methods, the hard sphere model, and intricate spatial Markov analysis. The best known lower bound for $K(d)$ is now: $K(d)\;\geq\; (1+o(1))\,\frac{\sqrt{3\pi}}{4\sqrt{2}}\;\log\frac{3}{2}\;\,d^{3/2}\left( \frac{2}{\sqrt{3}} \right)^d,$ which constitutes a $3.442\dots$ fold improvement over the previous bound (Fernández et al., 2021).

Further, sum-product methods have been generalized to nonassociative algebras (octonions and 16-ons). Mendelsohn gives an abstract inequality for the 16-dimensional kissing number, bootstrapping from $k_8=240$ (Mendelsohn, 2023). However, concrete numeric improvements from this approach in $n=16$ remain open.

5. Extensions: Generalized and Convex Kissing Numbers

The classical notion of the kissing number for Euclidean balls extends naturally to arbitrary convex bodies $K\subset\mathbb{R}^d$ , defining the (translative) kissing number $\tau(K,d)$ as the maximal number of pairwise non-overlapping translates of $K$ touching $K$ (Li et al., 24 Jul 2024). Exact results are established for the Euclidean ball in $n=3,4,8$ , with values $12,24,240$ respectively—the sphere, 24-cell, and $E_8$ root lattice.

The generalized lattice kissing number $\kappa^*_\alpha(K)$ counts all translates of $K$ with centers in a lattice $\Lambda$ lying within a thickened shell $2\leq \|v\|_K \leq 2+\alpha$ . For $K=B^n$ , the exact values in $3\leq n\leq8$ and a canonical $\alpha=2\sqrt{3}-2$ have recently been determined (Li et al., 12 Jan 2025).

Open conjectures include the uniqueness of optimal configurations, bounds for general convex bodies, and the behavior of generalized kissing numbers for nonzero $\alpha$ . For instance, in $n=8$ the second-shell problem conjectures $\kappa_\alpha(B^8) = 2400$ for $\alpha = 2\sqrt{2}-2$ (Li et al., 24 Jul 2024).

6. Computational and Algorithmic Approaches

Algorithmic enumeration, code-based constructions, multiangular cloud methods, and AI-driven matrix-completion games have been critical in advancing kissing number research:

PackingStar, a two-player cooperative RL/game-theoretic matrix-completion system, surpassed previous record lower bounds for $n=25$ –$31$ and broke a half-century-old barrier in $n=13$ by constructing fully rational configurations (Ma et al., 17 Nov 2025).
Simulated annealing and group-automorphism-based approaches allowed the discovery of larger mutually compatible subsets in Leech lattice cross-sections, increasing lower bounds in high dimensions (Kallal et al., 2016).
New 5-dimensional arrangements were found by clique search in compatibility graphs generated from parametrized clouds of points (Szöllősi, 2023).

These computational advances not only raise current records but also amplify structural diversity—refuting previously held beliefs that only a finite set of configurations realize maximal arrangements in specific dimensions.

7. Variants and Generalizations

The kissing number problem admits several combinatorial and geometric generalizations. One such variant asks for the maximum cardinality of a packing with prescribed "kissing distance," that is, allowing spheres to be separated via a given number of touching intermediaries (kissing radius) (Golovanov, 2022). For instance, in the disk $k=3$ case, the maximum is exactly $37$, confirming conjectures on higher order "kissing layers".

Extensions to convex bodies, cross-polytopes, and the inclusion of shells or covering radii further diversify the field (Li et al., 12 Jan 2025, Li et al., 24 Jul 2024). These investigations deepen the interplay between lattice theory, combinatorial optimization, and the geometry of numbers.

In summary, the kissing number problem is a central and deeply connected challenge in high-dimensional discrete geometry, with growing evidence that advances in semidefinite optimization, combinatorial construction, and high-performance computational techniques are required for further progress (Boyvalenkov et al., 2015, Machado et al., 2016, Ma et al., 17 Nov 2025, Kallal et al., 2016, Fernández et al., 2021). Open problems include the determination of exact values in moderate dimensions, uniqueness of optimal configurations in $n=4$ and beyond, and narrowing the exponential gap between best upper and lower bounds in large $n$ .