A Billiard Table
Puzzle. A ball rolls forever on a frictionless billiard table with no pockets. Can you find a finite convex shape of the table for which no trajectory of the ball ever covers the entire surface?
Share:
Share:
Tanya Khovanova's Math Blog
Mathematics, applications of mathematics to life in general, and my life as a mathematician.
Share:
Follow Me
Izzy Grosof:
A circle has this property, right? Whenever the ball bounces off the circular wall, its path reflects a ross a diameter of the circle. So every line segment has the same minimum distance from the center. If this distance is nonzero, a circle with that radius is never covered. If the line segment goes through the center, it is reflected along the same line segment, and nothing else is ever covered.
18 April 2026, 11:13 pmLilac:
Circle.
Some internet commenter analysts suggest it might be an open question for polygons.
There also exists the Birkhoff-Poritsky Conjecture and results on caustics.
19 April 2026, 6:36 am