Some determinants and relations in Heronian friezes

TL;DR

This paper investigates algebraic relations and determinant conditions in Heronian friezes derived from cyclic polygons, extending to plane friezes and establishing new equalities for their entries.

Contribution

It introduces new algebraic relations and determinant vanishing equalities for Heronian friezes, including those from multiple adjacent diamonds and plane friezes from cyclic polygons.

Findings

Derived algebraic relations for Heronian diamonds

Established determinant vanishing equalities for friezes

Extended results to plane Heronian friezes from cyclic polygons

Abstract

In this article, we give algebraic relations and determinant vanishing equalities that hold for the entries of a single Heronian diamond of a Heronian frieze arising from a cyclic $n$-gon. We also give algebraic relations that hold between entries of multiple adjacent diamonds of such a frieze. Furthermore, we define a plane Heronian frieze, and establish some more determinant vanishing equalities for the entries of a plane Heronian frieze arising from a cyclic $n$-gon, where $n$ is a positive integer divisible by $4$.