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Kissing number in dimension 21

31,288≤ value ≤53,524lower boundupper bound
40,000198020002020updated on Oct 12, 2026Lower bound 27,720, 1967, John Leech, publishedLower bound 29,768, Nov 7, 2024, Henry Cohn and Anqi Li, publishedLower bound 30,779, Oct 4, 2026, Kevin Li, publishedLower bound 31,259, Oct 9, 2026, Griffin Long, publishedLower bound 31,272, Oct 10, 2026, Griffin Long, publishedLower bound 31,288, Oct 10, 2026, Griffin Long, published31,288Upper bound 53,524, Feb 24, 2022, David de Laat and Nando Leijenhorst, published53,524lowerupper30,00040,00050,000198020002020updated on Oct 12, 2026Lower bound 27,720, 1967, John Leech, publishedLower bound 29,768, Nov 7, 2024, Henry Cohn and Anqi Li, publishedLower bound 30,779, Oct 4, 2026, Kevin Li, publishedLower bound 31,259, Oct 9, 2026, Griffin Long, publishedLower bound 31,272, Oct 10, 2026, Griffin Long, publishedLower bound 31,288, Oct 10, 2026, Griffin Long, published31,288Upper bound 53,524, Feb 24, 2022, David de Laat and Nando Leijenhorst, published53,524lower boundupper bound
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  1. lower bound 31,259by Griffin Longgithub.com
  2. lower bound 31,272by Griffin Longgithub.com
  3. lower bound 31,288by Griffin Longgithub.com
  4. Last checked
Lower bound · published · Oct 10, 2026
Exact value : 31288
Long (agent-afk) : 31288 en dimension 21 (échange d'orbites) · github.com
by Griffin Long
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Cite this frontier

DiscoverIA. Kissing number in dimension 21 [Frontier, checked on Oct 12, 2026]. DiscoverIA registry. https://discoveria.ai/en/frontieres/kissing-d21/

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.

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.

Points : Lower bound
DateExact valueNatureAuthorsSource
Oct 10, 202631288 Current recordpublishedGriffin Longgithub.com
Oct 10, 202631272 Current recordpublishedGriffin Longgithub.com
Oct 9, 202631259 Current recordpublishedGriffin Longgithub.com
Oct 4, 202630779 Current recordpublishedKevin Ligithub.com
Nov 7, 202429768 Current recordpublishedHenry Cohn, Anqi Liarxiv.org
196727720 Current recordpublishedJohn Leechdoi.org
Every step, as a table : Upper bound (1 points)

Best known upper bound (three-point semidefinite programming).

Points : Upper bound
DateExact valueNatureAuthorsSource
Feb 24, 202253524 Current recordpublishedDavid de Laat, Nando Leijenhorstarxiv.org
Kissing number (21)

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