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The maximum number of cycles in a triangular-grid billiards system with a given perimeter

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arxiv 2309.00100 v3 pith:IOT4FTW2 submitted 2023-08-31 math.CO

Honglin Zhu This is my paper · ORCID
classification math.CO
keywords operatornamebilliardssystemdefantgivengridjiradiloknumber
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abstract

Given a (simple) grid polygon $P$ in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside of $P$. We study the relationship between the perimeter $\operatorname{perim}(P)$ of $P$ and the number of different trajectories $\operatorname{cyc}(P)$ that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality $\operatorname{cyc}(P) \leq (\operatorname{perim}(P) + 2)/4$ and characterize the equality cases.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Homology in Combinatorial Refraction Billiards

    math.CO 2025-02 accept novelty 8.0 of 10

    A graph is expelling, sending every billiard loop around the torus non-contractibly, exactly when it is bipartite; ensnaring graphs have a partial structural theory.

  2. Random Subwords and Billiard Walks in Affine Weyl Groups

    math.PR 2025-01 conditional novelty 7.0 of 10

    For random subwords of b^K in an irreducible affine Weyl group, the normalized alcove position converges to a central spherical Gaussian, with an explicit variance formula.

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