Short··0:14·Differential geometry
Infinitely Many Closed Geodesics
Watch on YouTubeAbout the problem →
About this video
Follow one loop. Watch an entire linked world appear.
Infinitely many closed geodesic images on every Riemannian sphere, a reported result from OpenAI, the company behind ChatGPT.
The manuscript states that every smooth Riemannian metric on the standard sphere S^n, n at least 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; its claimed advance includes arbitrary, possibly degenerate metrics in higher dimensions. Repeated laps around one loop do not count as new geodesics.
The animation shows 36 distinct Hopf fibers: great circles on the unit round 3-sphere S3 in four real coordinates, stereographically projected into three dimensions. Each pair is linked. The circles are a known round-metric example; they do not visualize or prove the new arbitrary-metric theorem. The reveal, camera chase, colour choices and playback speed are editorial. No physical force or motion simulation is implied.
On screen: covariant acceleration is zero, and the path is periodic. Construction: gamma(t) = (cos(eta) exp(it), sin(eta) exp(i(t+phi))), eta = 0.64, on S3 in C2. The projection is (x1,x3,x2)/(1-x4), with a uniform display scale. Geodesic speed is constant in the original S3 parameter, not in its 3D projection.
Full manuscript, Theorem 1.1:
github.com/openai/math/blob/main/preprints/Infinitely-many-cl…
GitHub repository: github.com/openai/math
Corrections and withdrawals: github.com/openai/math/blob/main/history.md
Source-reported result from an internal research model. The proof has not been independently audited in this production. No corresponding top-line Lean formalization was linked in the inspected catalogue entry. This is not a claim that the public ChatGPT product proved the theorem.
'Aurora' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
scottbuckley.com.au/library/aurora/
Licence: creativecommons.org/licenses/by/4.0/
Music edits: excerpted, crossfaded for looping, and gain adjusted. Original Three.js visuals, native 4K at 60 fps.
About the problem
Differential geometry·OpenAI Math·Result 345
Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
Proves that every smooth Riemannian metric on Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.
Topics
closed geodesicsRiemannian geometryHopf fibration3 sphereOpenAIChatGPTmathematics3D math animationinfinite loops
