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Friezes over $\mathbb{Z}[\sqrt{2}]$

Esther Banaian, Libby Farrell, Amy Tao, Kayla Wright and Joy Zhichun Zhang

Vol. 18 (2025), No. 4, 683–705
Abstract

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 [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 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
Mathematical Subject Classification
Primary: 05E16, 51M04
Secondary: 11A55
Supplementary material

Basic triangles with tower arcs

Milestones
Received: 14 September 2023
Revised: 29 March 2024
Accepted: 16 April 2024
Published: 30 July 2025

Communicated by Kenneth S. Berenhaut
Authors
Esther Banaian
Aarhus University
Aarhus
Denmark
Libby Farrell
University of Minnesota
Minneapolis, MN
United States
Amy Tao
University of Wisconsin
Madison, WI
United States
Kayla Wright
University of Minnesota
Minneapolis, MN
United States
Joy Zhichun Zhang
Cornell University
Ithaca, NY
United States