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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.00440 |
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| _version_ | 1866916867474456576 |
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| author | Banaian, Esther Farrell, Libby Tao, Amy Wright, Kayla Zhang, Joy Zhichun |
| author_facet | Banaian, Esther Farrell, Libby Tao, Amy Wright, Kayla Zhang, Joy Zhichun |
| contents | A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00440 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Friezes over $\mathbb Z[\sqrt{2}]$ Banaian, Esther Farrell, Libby Tao, Amy Wright, Kayla Zhang, Joy Zhichun Combinatorics A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze. |
| title | Friezes over $\mathbb Z[\sqrt{2}]$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.00440 |