A $p$-angulated generalisation of Conway and Coxeter's theorem on frieze patterns

TL;DR

This paper generalizes Conway and Coxeter's theorem on frieze patterns to include p-angulated and non-integral patterns, expanding the combinatorial connections to polygon dissections.

Contribution

It introduces a p-angulated generalization of frieze patterns and explores their relation to polygon dissections, extending prior integral cases.

Findings

Established a p-angulated generalization of frieze patterns.

Connected polygon dissections to non-integral frieze patterns.

Extended the bijection beyond triangulations to more general dissections.

Abstract

Coxeter defined the notion of frieze pattern, and Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$-angulated generalisation involving non-integral frieze patterns. We also show that polygon dissections give rise to even more general non-integral frieze patterns.