Friezes over $\mathbb Z[\sqrt{2}]$
Combinatorics
2025-07-30 v2
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 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.
Cite
@article{arxiv.2307.00440,
title = {Friezes over $\mathbb Z[\sqrt{2}]$},
author = {Esther Banaian and Libby Farrell and Amy Tao and Kayla Wright and Joy Zhichun Zhang},
journal= {arXiv preprint arXiv:2307.00440},
year = {2025}
}