Hexagon

math.MG — Metric Geometry

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

Contributed by Ricardo Pinho

We give an explicit cover of a regular tetrahedron by $4096$ cylinders with compact triangular perpendicular bases. The sum of their base areas, divided by the tetrahedron's minimum orthogonal projection area, lies in $[0.499613830783714148,\,0.499613830783714149)$ . Thus the saving below $1/2$ exceeds $0.000386169216285851$ . 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.

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Oct 17, 2026PDF ↗View work →

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

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