1 · The essentials
Imagine covering a solid pyramid with four identical triangular faces using a finite collection of tubes. Here the tubes can have triangular openings rather than round ones, and each opening is measured straight across its tube. The question is how small the combined area of those openings can be while still covering the entire solid, including its boundary. The proposed lower limit was half the area of the solid’s smallest shadow under parallel light. The main result gives a covering whose combined opening area is strictly below that limit. It therefore supplies a counterexample to the proposed bound in three dimensions. The main result has a Lean formalization, although the supplied summary does not specify exactly which supporting claims it covers.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture sliding several long triangular sleeves over a four-faced solid, allowing the sleeves to overlap and point in different directions. The surprise is that every point can lie inside a sleeve even though the sleeve openings together have less than half the area of the solid’s smallest parallel-light shadow.
Why it matters
The construction shows that the proposed half-area lower bound does not hold for all cylinder coverings in three dimensions. It also refutes the stronger directionwise normalized conjecture. For that normalized statement, the main manuscript extends the counterexample from regular tetrahedra to every nondegenerate tetrahedron by affine invariance.
2 · Going further
The precise problem concerns covering a convex solid by finitely many cylinders while controlling the sum of the areas of their bases, measured perpendicular to their respective directions. For a regular tetrahedron , write for its minimum orthogonal projection area. The main result gives a cover of the entire closed tetrahedron with compact triangular perpendicular bases satisfying . The repository summary identifies the contradicted bound as Bang’s half-area cylinder-covering bound.
The abstracts describe two closely related construction routes. Manuscript 3 perturbs a two-cylinder covering of a regular tetrahedron using slope fields, obtaining finite covers strictly below the half-area threshold. Manuscript 4 treats radially aligned sweeps of segments: it supplies one triangular base for each interval of a tagged partition, and the total perpendicular-base area converges to a weighted parameter area as the partition mesh tends to zero. Its explicit tetrahedron application is shown in the manuscript to fall below the threshold.
Manuscript 2 supplies a related finite-approximation result for compact ruled families of segments. Its finite cylinders use square intercept tiles, with total perpendicular-base area at most the integral projection cost plus any prescribed positive error. The velocity field must be , and its differential must have opposite real eigenvalues with magnitudes strictly below the inverse segment half-length, allowing both eigenvalues to vanish. The stated scope includes square-zero differentials with unrestricted shear, fields passing between the two regimes, and arbitrary compact label sets.
The main manuscript packages the triangular-base counterexample and its consequence for the directionwise normalized half-bound. By affine invariance, that normalized consequence applies to every nondegenerate tetrahedron; the abstracts do not make the same extension for the unnormalized minimum-projection-area inequality. The main result has a Lean formalization, but the supplied material does not delineate its coverage of the companion approximation and perturbation results.
Useful notions
- Regular tetrahedra and convex solids
- Orthogonal projection and projected area
- Cylinder directions and perpendicular bases
- Finite partitions and convergence
- Differentiable fields and eigenvalues
- Affine transformations
What was known before / what is new
Before
The source describes a conjectured half-area lower bound for cylinder coverings and a stronger directionwise normalized half-bound conjecture. Manuscript 3 takes a two-cylinder covering of a regular tetrahedron as the starting point for its perturbations. The supplied abstracts give no further historical account.
New
The main result provides a finite cover of the closed regular tetrahedron with compact triangular perpendicular bases whose total area is strictly below half its minimum orthogonal projection area. This refutes both the half-area bound and the directionwise normalized conjecture in dimension three. The companion manuscripts describe finite-approximation tools and perturbation constructions, while affine invariance extends the normalized counterexample to every nondegenerate tetrahedron.
Limits and assumptions
The unnormalized area counterexample is stated for regular tetrahedra; the extension to every nondegenerate tetrahedron concerns the directionwise normalized bound. The ruled-set approximation requires the stated C¹ regularity and eigenvalue restrictions, while the triangular approximation concerns radially aligned segment sweeps. The main result has a Lean formalization, but the source does not specify its exact scope or establish that every companion claim is formalized. No remaining open questions are identified in the supplied material.
Mini glossary
- Regular tetrahedron —
- A solid with four identical equilateral triangular faces.
- Perpendicular base —
- The cross-sectional set defining a cylinder, taken in a plane perpendicular to its direction.
- Orthogonal projection —
- A shadow on a plane formed by projecting along lines perpendicular to that plane.
- Radially aligned segment sweep —
- A family of line segments arranged along radial directions and considered together as a swept set.
- Partition mesh —
- The length of the largest interval in a partition.
- Affine invariance —
- Preservation of a statement under invertible affine changes of coordinates.
Keywords : Cylinder coverings · Tetrahedra · Orthogonal projection
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Finite angular cylinder covers below the half-area bound
27/09/2026A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.
Finite cylinder approximation of ruled sets
27/09/2026We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C1, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.
Slope-field perturbations of the two-cylinder covering
27/09/2026The 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.
Finite triangular approximation of radial sweeps
27/09/2026We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.