Infinite friezes and triangulations of the strip

TL;DR

This paper establishes a bijection between infinite friezes of positive integers and certain triangulations of the infinite strip, extending the understanding of their geometric and combinatorial structures.

Contribution

It adapts existing constructions to create a bijection between infinite friezes and admissible triangulations without special points, linking them to the infinity-gon.

Findings

Infinite friezes correspond to admissible triangulations of the infinite strip.

Infinite friezes with enough ones are in bijection with triangulations of the infinity-gon.

The construction provides a new geometric interpretation of infinite friezes.

Abstract

The infinite friezes of positive integers were introduced by Tschabold as a variation of the classical Conway-Coxeter frieze patterns. These infinite friezes were further shown be to realizable via triangulations of the infinite strip by Baur, Parsons and Tschabold. In this paper, we show that the construction of Baur, Parsons and Tschabold can be slightly adapted in order to obtain a bijection between the infinite friezes and the so-called admissible triangulations of the infinite strip with no special marked points on the upper boundary. As a consequence, we obtain that the infinite friezes with enough ones are in bijection with the admissible triangulations of the infinity-gon.