Frieze patterns of integers

TL;DR

This paper reviews the classical theorem of Conway and Coxeter on integer frieze patterns, explores its developments over the past decade, and highlights its connection to cluster combinatorics.

Contribution

It provides a comprehensive review of frieze patterns, their geometric interpretation, and recent advances linking them to cluster algebra theory.

Findings

Revisits Conway-Coxeter's geometric interpretation of frieze patterns

Summarizes recent developments in frieze pattern research

Highlights the connection between friezes and cluster combinatorics

Abstract

The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics.