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?
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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 amDan Asimov:
I would like to know what it means for a trajectory (which has measure zero) to “cover the entire surface”.
9 June 2026, 10:01 pmtanyakh:
Dan. It means the trajectory is dense: comes arbitrarily close to every point of the surface.
13 June 2026, 12:22 pmlucas:
The word covers seems crucial here: do you mean that the trajectory is dense in the table, rather than literally passing through every point? Making that distinction explicit would help keep the geometric interpretation of the puzzle clear.
19 October 2026, 8:06 amlucas:
The frictionless table with no pockets is a neat constraint. My first instinct is a stadium-like shape whose periodic orbits stay away from some open set, but I am not sure that stays finite and convex—curious whether a strictly convex oval already works.
31 October 2026, 6:08 am