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Heronian friezes and Plücker relations

Anja Sneperger

TL;DR

The paper links polygon-based geometry encoded in Heronian friezes to Grassmannian geometry by leveraging Plücker relations in $Gr(3,n)$. It introduces Heronian minors (minors of the coordinate matrix corresponding to triangle-area data) and defines Heronian minor relations as Plücker relations that involve only these minors, translating them into quadratic relations among $S$-entries. It further connects substructures of Heronian friezes to Plücker friezes $P(3,n)$ and proves vanishing determinants for certain $(k+1)\times(k+1)$ diamonds, revealing deep algebraic constraints on the frieze entries. Collectively, the results deepen the interplay between polygon geometry, frieze patterns, cluster algebra phenomena, and Grassmannian geometry, yielding new invariants and consistency relations across these domains.

Abstract

In this article, we use Plücker relations in the Grassmannian $Gr(3,n)$ to give relations that hold amongst some of the entries of the Heronian frieze of order $n$. Furthermore, we make a connection between certain subfriezes of a Heronian frieze and Plücker friezes $P(3,n)$, and then show that some determinants of the matrices whose elements lie in those subfriezes are vanishing.

Heronian friezes and Plücker relations

TL;DR

The paper links polygon-based geometry encoded in Heronian friezes to Grassmannian geometry by leveraging Plücker relations in . It introduces Heronian minors (minors of the coordinate matrix corresponding to triangle-area data) and defines Heronian minor relations as Plücker relations that involve only these minors, translating them into quadratic relations among -entries. It further connects substructures of Heronian friezes to Plücker friezes and proves vanishing determinants for certain diamonds, revealing deep algebraic constraints on the frieze entries. Collectively, the results deepen the interplay between polygon geometry, frieze patterns, cluster algebra phenomena, and Grassmannian geometry, yielding new invariants and consistency relations across these domains.

Abstract

In this article, we use Plücker relations in the Grassmannian to give relations that hold amongst some of the entries of the Heronian frieze of order . Furthermore, we make a connection between certain subfriezes of a Heronian frieze and Plücker friezes , and then show that some determinants of the matrices whose elements lie in those subfriezes are vanishing.

Paper Structure

This paper contains 4 sections, 10 theorems, 63 equations, 15 figures, 3 tables.

Table of Contents

  1. Heronian friezes
  2. Exterior powers
  3. The Grassmannian
  4. Heronian minor relations

Key Result

Proposition 1.7

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. (Note that the boundary conditions bound hold since the squared distance between a vertex and itself equals zero. Similarly, the sign

Figures (15)

  • 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: The combinatorial pattern underlying a Heronian frieze of order $n=4$
  • Figure 3: Two triangulations of a plane quadrilateral
  • Figure 4: Heronian frieze of order $n$
  • Figure 5: A Heronian diamond for a quadruple of vertices $A_i, A_j, A_k, A_l$
  • ...and 10 more figures

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Proposition 1.7
  • Definition 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 36 more