---
title: Kissing Number Problem
url: https://www.emergentmind.com/topics/kissing-number-problem
type: topic
---

# Kissing Number Problem

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 study 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 $s$ [1507.03631].

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 [2003.11832].

## 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 [1209.6043])
- $\tau(4)=24$ (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 [1507.03631, 2301.08272, 2411.04916].

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 [2411.04916].

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., $\tau_{31}\geq232,874$ [1608.07270, 2511.13391].

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 [2301.08272].

## 3. Upper Bounds: Analytic and Optimization-Based Approaches

The classical approach for upper bounds is via Delsarte’s linear programming (LP) bound [1507.03631]. 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 [1609.05167]. 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 $6\leq n\leq9$ [2003.11832].

In high dimensions, asymptotic analytic upper bounds of Kabatianskii–Levenshtein type give
\[
\tau_d \le 2^{0.401\,d(1+o(1))} \text{ as } d\to\infty.
\]

### Table: Numerical SDP Upper Bounds for the Average Kissing Number (n=3 to 9) [2003.11832]

|   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 [2111.01255].

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 $k_8=240$ [2303.03515]. 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 $K$ [2407.17340]. 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 [2501.06792].

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 $\alpha = 2\sqrt{2}-2$ [2407.17340].

## 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 [2511.13391].
- 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 [1608.07270].
- New 5-dimensional arrangements were found by clique search in compatibility graphs generated from parametrized clouds of points [2301.08272].

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) [2205.15949]. 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 [2501.06792, 2407.17340]. These investigations deepen the interplay between lattice theory, combinatorial optimization, and the geometry of numbers.

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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 [1507.03631, 1609.05167, 2511.13391, 1608.07270, 2111.01255]. 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 $n$.

Source: https://www.emergentmind.com/topics/kissing-number-problem