Trapping Light Rays with Segment Mirrors

Definitions and mathematical statement

Choose finitely many line segments in $\mathbb R^2$ whose closures are pairwise disjoint. Their relative interiors are two-sided mirrors; their endpoints are not reflecting. Let $s$ be a point off the mirrors, and let $H$ be the convex hull of all mirror closures.

A ray from $s$ travels along straight lines. At a transverse hit in the interior of a mirror it obeys specular reflection: the tangential velocity component is preserved and the normal component changes sign. Passing through a missing endpoint gives no reflection. A ray is trapped when no point of its forward trajectory lies strictly outside $H$; a ray escaping after a very large but finite number of reflections is not trapped.

Can the mirrors and source be chosen so that every initial direction from $s$ gives a trapped ray? The conjectured answer is no: for every finite arrangement and source there is an escaping direction. It suffices to consider $s\in H$, since otherwise trapping already fails at the source.

The quantifier over all directions is essential. Trapping selected periodic rays, or even a finite collection of nonperiodic rays, does not give a trap for a point source emitting in every direction.

Short English statement

Can finitely many separated straight mirrors trap every ray emitted by one point, or must some direction eventually escape?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500031 (AMR-054-0031), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 31. Original source indexed by AMR-054-0031. https://topp.openproblem.net/p31.