Heronian friezes

TL;DR

This paper introduces Heronian friezes, Euclidean analogues of Coxeter's frieze patterns, proving their periodicity and Laurent phenomenon, and explores algebraic conditions for related Cayley-Menger friezes.

Contribution

It defines Heronian friezes, proves their symmetry and periodicity, and establishes algebraic coherence conditions for Cayley-Menger friezes of geometric origin.

Findings

Heronian friezes are generically glide symmetric and periodic.

A Laurent phenomenon analogue is established for Heronian friezes.

Coherence conditions enable propagation and periodicity in Cayley-Menger friezes.

Abstract

Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter's frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic), and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley-Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley-Menger friezes, as well as the corresponding periodicity results.