The maximum number of cycles in a triangular-grid billiards system with a given perimeter
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910416308797440 |
|---|---|
| author | Zhu, Honglin |
| author_facet | Zhu, Honglin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The maximum number of cycles in a triangular-grid billiards system with a given perimeter Zhu, Honglin Combinatorics 05D99, 05E18, 37B20, 51M04, 52B60 G.2.1 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. |
| title | The maximum number of cycles in a triangular-grid billiards system with a given perimeter |
| topic | Combinatorics 05D99, 05E18, 37B20, 51M04, 52B60 G.2.1 |
| url | https://arxiv.org/abs/2309.00100 |