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Kissing number in dimension 5

Description of constant

In geometry, the kissing number problem asks for the maximum number $\tau_n$ of unit spheres that can simultaneously touch the unit sphere in $\mathbb{R}^n$ without pairwise overlapping. The value of $\tau_n$ is only known for $n=1,2,3,4,8,24$. [BV2008-def-known-dims]

We define

$$ C_{29}\ :=\ \tau_5, $$

the kissing number in dimension $5$.

Dimension $5$ is the first dimension in which the kissing number is not known; currently the rigorous range is

$$ 40\ \le\ \tau_5\ \le\ 44. $$

[MV2009-range-40-44]

One standard reformulation is in terms of spherical codes. Let $A(n,\theta)$ be the maximal size of a code on the unit sphere $S^{n-1}$ with minimal angular distance at least $\theta$. Then the kissing number problem is equivalent to computing $A(n,\pi/3)$. [BV2008-A-pi3]

Equivalently (inner-product form), if $A(n,s)$ is the maximal size of a spherical code $C\subset S^{n-1}$ with $\langle x,y\rangle\le s$ for all distinct $x,y\in C$, then

$$ A(n,1/2)\ =\ \tau_n. $$

[BDM2012-A-n-s-and-tau]

Known upper bounds

Bound Reference Comments
$48$ [Cox1963] Coxeter’s strongest bound in dimension $5$ (as summarized in the survey literature). [BDM2012-ub-story-d5]
$46.345$ [OS1979] Improvement attributed to Odlyzko–Sloane (as summarized in the survey literature). [BDM2012-ub-story-d5]
$45$ [BV2008], [MV2009] Semidefinite-programming upper bound recorded as the best “known upper bound” prior to the higher-accuracy SDP computations.
$44$ [MV2009] [MV2009-range-40-44]

Known lower bounds

Bound Reference Comments
$10$ Trivial construction via the cross polytope $\{\pm e_i\}_{i=1}^5\subset S^4$.
$40$ [KZ1873] Achieved by the $D_5$ root system (40 points), giving a kissing configuration of size $40$ in $\mathbb{R}^5$. [CR2024-lb-40-D5]

Additional comments and links

References

  • [BDM2012] Boyvalenkov, Petko; Dodunekov, Damyan; Musin, Oleg R. A survey on the kissing numbers. Serdica Mathematical Journal 38 (2012), 507–522. Preprint: arXiv:1507.03631. Google Scholar

    • [BDM2012-A-n-s-and-tau] loc: arXiv v1 PDF p.1, L18–L31 quote: “A spherical code is a non-empty finite subset of $S^{n-1}$. Important parameters of a spherical code $C \subset S^{n-1}$ are its cardinality $|C|$, the dimension $n$ (it is convenient to assume that the vectors of $C$ span $\mathbb{R}^n$) and the maximal inner product $s(C) = \max\{\langle x, y\rangle : x, y \in C, x \ne y\}$. The function $A(n, s) = \max\{|C| : \exists C \subset S^{n-1} \text{ with } s(C) \le s\}$ extends $\tau_n$ and it is easy to see that $A(n, 1/2) = \tau_n$.”
    • [BDM2012-ub-story-d5] loc: arXiv v1 PDF p.6, L32–L35 quote: “Now the first open case is in dimension five, where it is known that $40 \le \tau_5 \le 44$ (the story of the upper bounds is: $\tau_5 \le L_5(5,1/2) = 48$, $\tau_5 \le 46.345$ from [38], $\tau_5 \le 45$ from [5] and $\tau_5 \le 44.998$ from [33]).”

  • [BV2008] Bachoc, Christine; Vallentin, Frank. New upper bounds for kissing numbers from semidefinite programming. (2008). PDF: https://ir.cwi.nl/pub/12655/12655D.pdf. Google Scholar

    • [BV2008-def-known-dims] loc: JAMS PDF p.1, L1–L5 quote: “In geometry, the kissing number problem asks for the maximum number $\tau_n$ of unit spheres that can simultaneously touch the unit sphere in $n$-dimensional Euclidean space without pairwise overlapping. The value of $\tau_n$ is only known for $n = 1, 2, 3, 4, 8, 24$.”
    • [BV2008-A-pi3] loc: JAMS PDF p.2, L7–L11 quote: “$A(n,\theta) = \max\{\mathrm{card}(C): C \subset S^{n-1} \text{ with } c \cdot c' \le \cos\theta \text{ for } c,c' \in C, c \ne c'\}$....The kissing number problem is equivalent to the problem of finding $A(n,\pi/3)$.”

  • [Cox1963] Coxeter, Harold Scott Macdonald. An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. In: Proc. Sympos. Pure Math. 7 (1963), 53–71. Google Scholar

  • [CR2024] Cohn, Henry; Rajagopal, Sidhant. A modular design for optimal five-dimensional kissing configurations. Preprint (2024). arXiv:2412.00937

    • [CR2024-appears-40-ub-44] loc: arXiv v2 PDF p.3, L4–L6 quote: “The kissing number in five dimensions appears to be $40$, although the best upper bound that has been proved is $44$ (from [15]).”
    • [CR2024-lb-40-D5] loc: arXiv v2 PDF p.3, L5–L10 quote: “The first construction achieving $40$ is implicit in Korkine and Zolotareff’s 1873 paper [9], where they constructed the $D_5$ root lattice. Its root system achieves a kissing number of $40$ as the permutations of the points $(\pm 1, \pm 1, 0, 0, 0)$; these points form a kissing configuration because they each have squared norm $2$ and the inner product between distinct points is always at most $1$.”
    • [CR2024-leech-1967] loc: arXiv v2 PDF p.3, L11–L12 quote: “In 1967, Leech [11] constructed a different kissing configuration of the same size, not isometric to the $D_5$ root system.”

  • [KZ1873] Korkine, Aleksandr; Zolotareff, G. Sur les formes quadratiques. Mathematische Annalen 6 (1873), no. 3, 366–389. Publisher: Springer-Verlag Berlin/Heidelberg. Google Scholar

  • [Leech1967] Leech, John. Five dimensional non-lattice sphere packings. Canadian Mathematical Bulletin 10 (1967), no. 3, 387–393. Publisher: Cambridge University Press. Google Scholar

  • [MV2009] Mittelmann, Hans D.; Vallentin, Frank. High-accuracy semidefinite programming bounds for kissing numbers. Experimental Mathematics 19 (2010), no. 2, 175–179. Publisher: Taylor & Francis. Preprint: arXiv:0902.1105

    • [MV2009-range-40-44] loc: arXiv v3 PDF p.1, L16–L18 quote: “Dimension 5 is the first dimension in which the kissing number is not known. With our computation we could limit the range of possible values from $\{40,\dots,45\}$ to $\{40,\dots,44\}$.”

  • [OS1979] Odlyzko, A. M.; Sloane, N. J. A. 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. Google Scholar

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.