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Slope-field perturbations of the two-cylinder covering
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Abstract
The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Ricardo Pinho, An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (hexagon:2610.00304v2, v1 17 Oct 2026, v2 18 Oct 2026). Gives 4,096 triangular cylinders with normalized cost approximately 0.499613830783714 and exact rational certificates; it cites this paper among the four family preprints but builds mainly on the angular-sector construction of the companion paper. [AI-generated per source] [family-level]
Formalization
- OpenAI math release, Lean scope notes for family 100: Cylinder coverings below the half-area bound (lean/docs/100.md, 2026). States that the formalization proves that every regular tetrahedron has finite covers with parallelogram bases below both the absolute and the directionwise half bounds. The affine extension to every nondegenerate tetrahedron (this paper's corollary) is outside the selected statements. [family-level]
Possibly relevant literature (found by the agent, not checked)
- K. Bezdek, A. E. Litvak, Packing convex bodies by cylinders (arXiv:1507.05115; Discrete Comput. Geom. 55 (2016) 725–738). Packing counterpart of the Bezdek–Litvak relative-cost bounds ($\mathcal R\ge1/3$, and $\ge1$ for ellipsoids) that the paper uses to place the tetrahedral infimum in $[1/3,1/2)$.
Digests and expositions
- Latent Seminar, Finite cylinder coverings below the tetrahedral half-area bound (6 October 2026). Explains the four manuscripts' constructions, approximation hypotheses and two area normalizations, while expressly disclaiming independent verification of the complete proofs. [family-level]
Elsewhere
- Hexagon: An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (AI-generated; human understanding: some)
- Emergent Mind: family 100
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026,
author = {{OpenAI}},
title = {{Slope-field perturbations of the two-cylinder covering}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf}{OAI:Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/100.md. The Lean statements have not been checked by people against the paper's claims.
Family 100: Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Other papers in this family:
- Finite angular cylinder covers below the half-area bound · September 27, 2026
- Finite cylinder approximation of ruled sets · September 27, 2026
- Finite triangular approximation of radial sweeps · September 27, 2026