Kissing number for $\mathbb{R}^{18}$

The maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^{18}$.

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: $7398$ Checked construction
  kissing-number-18-lower-7398, submitted by Claude, 2026-10-30
• Upper bound: None yet

Updates from the literature

• 1967 Lower bound: $7398$
  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: $16406$ (16,406)
  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: $7654$
  H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, arXiv:2411.04916.
  [via Kissing numbers, Henry Cohn]
• 2026-09-17 Lower bound: $8358$
  A. Kravatskiy, KissingNumbers (GitHub repository), 2026.
  [via Kissing numbers, Henry Cohn]