Heilbronn minimum tetrahedron volume
Place n points in the unit cube so that the smallest tetrahedron formed by any four of them is as large as possible.
Formal definition
- ContainerThe unit cube: the origin (0, 0, 0) at one corner, (1, 1, 1) at the opposite one
- SubmissionExactly n points, no four of them coplanar
- ConstraintsEvery point lies inside the cube or on its boundary
- ObjectiveMake the smallest tetrahedron over all quadruples as large as possible; compared internally by six times the volume, which is the determinant of the three edge vectors, exactly
Getting a feel for it
Why this is not the plane problem with a z
In the plane three points degenerate one way: they line up. In space four points degenerate two ways — they share a plane, or three of them already share a line. Tidy arrangements walk straight into the first: any neat layering puts three points of one layer nearly in a plane with a point of another.
Where the frontier is
The plane version is proved optimal for n = 5..9 and has long-maintained numerical tables beyond that. Space has no comparable table: even small n are open, published constructions are scattered, and this arena cites no external value for any row. Every score here is a site record.
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Data and citation
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/heilbronn-tetrahedron.json
Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.
GET https://minmaxarena.com/data/editions/2026-08/heilbronn-tetrahedron.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-tetrahedron-2026-08,
title = {{Heilbronn minimum tetrahedron volume} (P72)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/heilbronn-tetrahedron.json},
license = {CC BY 4.0}
}Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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