Short··0:38·Dynamical systems and ergodic theory
Triangle Billiards: 1 Ball or 10,000?
Watch on YouTubeAbout the problem →
About this video
Shoot a ball inside a triangle where one angle is an irrational multiple of π. Does it eventually visit every spot, in every direction, evenly? That is ergodicity of the billiard flow. In 1986 Kerckhoff, Masur and Smillie proved it for a dense set of special triangles, but never for all of them.
OpenAI says its AI has now proved that the billiard flow in every triangle with at least one irrational angle is ergodic, and even weakly mixing. This is OpenAI's claim, not yet peer reviewed. The animation illustrates the idea; it is not the proof. The 60-60-60 table and the five other triangles are exact billiard paths.
Papers (OpenAI's manuscripts, result 150):
Ergodicity: github.com/openai/math/blob/main/preprints/Ergodicity-of-tria…
Weak mixing: github.com/openai/math/blob/main/preprints/Weak-mixing-of-tri…
Original synthesized audio: every note is a bounce. Voice (B): ElevenLabs.
About the problem
Dynamical systems and ergodic theory·OpenAI Math·Result 150
Weak mixing of triangular billiards with an irrational angle
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. (Lean)
Topics
triangle billiardsmathematical billiardsbilliards in a triangleergodic theoryergodicityweak mixingirrational angleKerckhoff Masur Smilliedynamical systemschaosOpenAI mathAI proofAI solves math problemmaths shortsGradient Damage
