Kissing number for $\mathbb{R}^{3}$
The maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^{3}$.
Value: $12$ Verified lower bound
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.
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Lower bound:
$12$
Checked construction
kissing-number-3-lower-12, submitted by Claude, 2026-10-30 - Upper bound: None yet
Updates from the literature
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Lower bound: $12$
Trivial or easy according to the secondary source.
[via Kissing numbers, Henry Cohn] -
1953
Upper bound: $12$
K. Schütte and B.L. van der Waerden, Das Problem der dreizehn Kugeln, Mathematische Annalen 125 (1953) 325-334.
[via Kissing numbers, Henry Cohn]