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Some determinants and relations in Heronian friezes

Anja Šneperger

TL;DR

The paper studies the algebraic structure of Heronian friezes arising from cyclic polygons, formalizing polygonal and plane polygonal friezes via squared distances $x_{ij}$ and four-times signed areas $S_{ijk}$. It derives determinant-vanishing relations for both single and adjacent diamonds, and shows these extend to plane friezes when $n$ is divisible by $4$, connecting geometric chord-length identities to frieze entries. The central result is an explicit alternating-sum identity $ extstyle\sum_{m=1}^{n}(-1)^{m+1}x(m)S(m)=0$ for even $n$, with $x(m)$ and $S(m)$ defined in terms of $x_{ij}$ and $S_{ijk}$, which translates into concrete polynomial relations (e.g., for $n=8$) among the frieze entries. By tying Heronian measurements to frieze patterns, the work bridges classical geometry with algebraic-frieze frameworks and yields new invariants for cyclic configurations, with potential implications for computational geometry and cluster-algebraic perspectives on friezes.

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 $4$.

Some determinants and relations in Heronian friezes

TL;DR

The paper studies the algebraic structure of Heronian friezes arising from cyclic polygons, formalizing polygonal and plane polygonal friezes via squared distances and four-times signed areas . It derives determinant-vanishing relations for both single and adjacent diamonds, and shows these extend to plane friezes when is divisible by , connecting geometric chord-length identities to frieze entries. The central result is an explicit alternating-sum identity for even , with and defined in terms of and , which translates into concrete polynomial relations (e.g., for ) among the frieze entries. By tying Heronian measurements to frieze patterns, the work bridges classical geometry with algebraic-frieze frameworks and yields new invariants for cyclic configurations, with potential implications for computational geometry and cluster-algebraic perspectives on friezes.

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 -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 -gon, where is a positive integer divisible by .
Paper Structure (3 sections, 20 theorems, 177 equations, 10 figures, 4 tables)

This paper contains 3 sections, 20 theorems, 177 equations, 10 figures, 4 tables.

Table of Contents

  1. Definition and some properties of Heronian friezes
  2. On Determinants in Heronian friezes
  3. Main theorem

Key Result

Proposition 1.6

fs Any $n$-gon $P$ in the complex plane gives rise to a Heronian frieze of order $n$, as given in the Figure friz, in the following way: where $i, j \in \{1,2,...,n\}$, addition is modulo $n$, and the $x_{ij}$, $S_{i,i+1,j}$ and $S_{i,j,j+1}$ are as in eqx and eqs. (Boundary conditions bound hold since the squared distance between a vertex and itself equals zero. Similarly, the signed area of a t

Figures (10)

  • Figure 1: A Heronian diamond. Here, $b$ and $d$ are associated to the dashed lines extending the bimedians of the diamond. The remaining eight numbers are placed at the vertices of the diamond and at the midpoints of its sides.
  • Figure 2: Two triangulations of a plane quadrilateral
  • Figure 3: A Heronian diamond for a quadruple of vertices $A_i, A_j, A_k, A_l$
  • Figure 4: Heronian frieze of order $n$
  • Figure 5: Polygonal Heronian frieze of order $n$
  • ...and 5 more figures

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6
  • Definition 1.7
  • Remark 1.8
  • Example 1.9
  • Remark 1.10
  • ...and 44 more