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An explicit finite cylinder cover of the regular tetrahedron saving 3.86×10−43.86\times10^{-4} below the half-area bound

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometrymath.CO — Combinatorics

Contributed by Ricardo Pinho

Version 2 / Oct 17, 2026 / CC BY 4.0

Abstract

We give an explicit cover of a regular tetrahedron by 40964096 cylinders with compact triangular perpendicular bases. The sum of their base areas, divided by the tetrahedron's minimum orthogonal projection area, lies in [0.499613830783714148, 0.499613830783714149)[0.499613830783714148,\,0.499613830783714149). Thus the saving below 1/21/2 exceeds 0.0003861692162858510.000386169216285851. The construction uses a finite tilt in the angular-sector framework of OpenAI's September 2026 preprint and rationally specified radial cutoffs. Coverage is certified by exact linear-programming dual witnesses, and independently verified by exhaustive rational polyhedron clipping. We describe the geometric reduction, the certificate obligations, and reproducible verification of the full closed tetrahedron and the perpendicular base areas. No optimality is asserted.

Provenance statement

The search, exact certificates and checkers were developed with AI coding assistants (OpenAI Codex and Claude) under the author's direction, and the text was drafted with AI assistance. The author directed the work and reviewed the construction and results but has not verified every step of the argument by hand. Correctness rests on exact rational certificates (linear-programming dual witnesses), checked independently by exhaustive rational polyhedron clipping; certificates and verification code: https://github.com/rpinho/tetrahedron-cylinder-cover. Also submitted to arXiv (math.MG); identifier to be added on announcement.

References

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Version history

  1. v1Submitted by Ricardo PinhoInitial depositSupersededOct 17, 2026
  2. v2Submitted by Ricardo PinhoSubmitted as a new versionCurrentOct 18, 2026