Short··0:21·Convex and metric geometry
Cylinder Covering: Bang's Half-Area Conjecture Is False
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About this video
Cover a regular tetrahedron with cylinders (infinite prisms) and add up their base areas. Bang's classic cover uses two cylinders along opposite edges and costs exactly half of the tetrahedron's smallest shadow, and the half-area conjecture said no finite cover could do better. OpenAI claims a counterexample: split the two cylinders into 16 million slightly tilted cylinders with triangular bases, and the total drops to about 0.49999999946 of the smallest shadow. The same construction disproves the stronger directionwise (1-codimensional) cylinder covering conjecture for every tetrahedron. These are OpenAI's preprints, not yet peer reviewed; the tilts in the animation are hugely exaggerated.
Papers (OpenAI, catalogue result 100):
- Finite angular cylinder covers below the half-area bound: github.com/openai/math/blob/main/preprints/Finite-angular-cyl…
- Slope-field perturbations of the two-cylinder covering: github.com/openai/math/blob/main/preprints/Slope-field-pertur…
- Finite cylinder approximation of ruled sets: github.com/openai/math/blob/main/preprints/Finite-cylinder-ap…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/100
About the problem
Convex and metric geometry·OpenAI Math·Result 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. (Lean)
Topics
cylinder coveringhalf-area conjectureBang's conjecturecylinder covering conjecture1-codimensional cylinder covering conjectureplank problemTarski plank problemBang plank theoremBezdek Litvakregular tetrahedronconvex geometrydiscrete geometrycovering problemsprojection areaOpenAIOpenAI mathAI mathematicsAI proofcounterexamplemathsmathematicsmath shorts3D animation
