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Friezes over
$\mathbb{Z}[\sqrt{2}]$
Esther Banaian, Libby Farrell, Amy Tao, Kayla Wright and Joy Zhichun Zhang |
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Vol. 18 (2025), No. 4, 683–705
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Abstract |
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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 and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over 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 gives rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze. |
Keywords
friezes, triangulated polygons, Catalan, cluster algebras
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Mathematical Subject Classification
Primary: 05E16, 51M04
Secondary: 11A55
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Supplementary material |
Milestones
Received: 14 September 2023
Revised: 29 March 2024
Accepted: 16 April 2024
Published: 30 July 2025
Communicated by Kenneth S. Berenhaut
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Authors |
| Esther Banaian | |
| Aarhus University Aarhus Denmark |
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| Libby Farrell | |
| University of Minnesota Minneapolis, MN United States |
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| Amy Tao | |
| University of Wisconsin Madison, WI United States |
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| Kayla Wright | |
| University of Minnesota Minneapolis, MN United States |
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| Joy Zhichun Zhang | |
| Cornell University Ithaca, NY United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
