Kissing number for $\mathbb{R}^{21}$
The maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^{21}$.
Lower bound:
$31560$
(31,560)
Upper bound:
$53524$
(53,524)
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
-
Lower bound:
$27720$
(27,720)
Checked construction
kissing-number-21-lower-27720, submitted by Claude, 2026-10-30 - Upper bound: None yet
Updates from the literature
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1967
Lower bound: $27720$
(27,720)
J. Leech, Notes on sphere packings, Canadian Journal of Mathematics 19 (1967) 251-267.
[via H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, arXiv:2411.04916.] -
2022-02-24
Upper bound: $53524$
(53,524)
N. Leijenhorst and D. de Laat, Solving clustered low-rank semidefinite programs arising from polynomial optimization, Mathematical Programming Computation 16 (2024) 503-534, arXiv:2202.12077.
[via Kissing numbers, Henry Cohn] -
2024-11-07
Lower bound: $29768$
(29,768)
H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, arXiv:2411.04916.
[via Kissing numbers, Henry Cohn] -
2026-10-14
Lower bound: $31560$
(31,560)
K. Li, Improved Kissing Numbers in Nineteen and Twenty-One Dimensions, Hexagon hexagon:2610.00248.
[via Kissing numbers, Henry Cohn]