Kissing number in dimension 21
- lower bound 31,259by Griffin Longgithub.com
- lower bound 31,272by Griffin Longgithub.com
- lower bound 31,288by Griffin Longgithub.com
- Last checked
Cite this frontier
DiscoverIA. Kissing number in dimension 21 [Frontier, checked on Oct 12, 2026]. DiscoverIA registry. https://discoveria.ai/frontiers/kissing-d21
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The exact value lies somewhere in between, within a factor of ×1.71. The lower bound has moved 5 times since 1967, most recently on Oct 10, 2026. The upper bound has not moved since 2022.
The maximum number of unit spheres with disjoint interiors that can touch a central unit sphere in R^21. A valid configuration of N spheres gives the lower bound N; a linear or semidefinite programming bound gives an upper bound.
Exact value unknown: gap 31288 versus 53524 (ratio 1.711 on Cohn's table consulted 2026-10-13). Lower bounds are explicit constructions (the recent ones announce an exact-arithmetic program check); no named third-party review appears in the sources read. The sources read declare AI use for these lower bounds: GPT-6-Astra Pro, Claude Opus 5.5 (30779, 2026-10-04); agent-afk (31259, 2026-10-09); agent-afk (31272, 2026-10-10); agent-afk (31288, 2026-10-10). The latest (31288, Griffin Long, 2026-10-10) has AI use declared by its source. The upper bound (53524) is a semidefinite programming bound (Cohn's table, credited to Leijenhorst-de Laat); no source read attributes it to an AI.
Analogy
Take a marble and surround it with identical marbles that all touch it without overlapping: how many can you fit? Flat on a table, with coins around a coin, a few tries give the answer. In space, with marbles, there is room for them to shift, and it took centuries to prove that no extra one can be squeezed in. The kissing number asks the same question in spaces with more than three dimensions.
Where the picture stops
Beyond three dimensions there are no marbles to move around: a marble becomes a list of numbers, "touching" becomes a distance to compute, and the exact answer is known in only a handful of dimensions.
In the glossary : Kissing number
Every step, as a table : Lower bound (6 points)
Best known configuration (constructions), from the Leech, Kallal-Kan-Wang or Cohn-Li records to the AI-assisted constructions of 2025-2026.
| Date | Exact value | Nature | Authors | Source |
|---|---|---|---|---|
| Oct 10, 2026 | 31288 Current record | published | Griffin Long | github.com |
| Oct 10, 2026 | 31272 Current record | published | Griffin Long | github.com |
| Oct 9, 2026 | 31259 Current record | published | Griffin Long | github.com |
| Oct 4, 2026 | 30779 Current record | published | Kevin Li | github.com |
| Nov 7, 2024 | 29768 Current record | published | Henry Cohn, Anqi Li | arxiv.org |
| 1967 | 27720 Current record | published | John Leech | doi.org |
Every step, as a table : Upper bound (1 points)
Best known upper bound (three-point semidefinite programming).
| Date | Exact value | Nature | Authors | Source |
|---|---|---|---|---|
| Feb 24, 2022 | 53524 Current record | published | David de Laat, Nando Leijenhorst | arxiv.org |
Kissing number (21)
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