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

TL;DR

This paper extends the study of friezes from integers to the ring 2, exploring their connection to polygon dissections and providing a classification of those that produce unitary friezes.

Contribution

It introduces the concept of friezes over 2, analyzes their relation to polygon dissections, and proposes a conjecture for classifying dissections that admit unitary friezes.

Findings

Identified a family of dissections that produce unitary friezes.

Extended the theory of friezes from 1 to 2.

Conjectured a complete classification of dissections with unitary friezes.

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 $\mathbb{Z}$ 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.