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Equable Parallelograms on the Eisenstein Lattice

Christian Aebi, Grant Cairns

Abstract

This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors' previous study of equable parallelograms whose vertices lie on the integer lattice.

Equable Parallelograms on the Eisenstein Lattice

Abstract

This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors' previous study of equable parallelograms whose vertices lie on the integer lattice.
Paper Structure (9 sections, 16 theorems, 76 equations, 7 figures, 4 tables)

This paper contains 9 sections, 16 theorems, 76 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['T:suff']}
  3. Proof of Theorem \ref{['T:345']}
  4. Comments on Theorem \ref{['T:345']}
  5. The tree of ELEPs
  6. Diagonals, heights and altitudes
  7. ELEPs with horizontal diagonal
  8. ELEPs with a vertical diagonal
  9. ELEPs with a vertical side

Key Result

Theorem 1

Given positive integers $a,b$, an Eisenstein lattice equable parallelogram with sides $a\sqrt3,b\sqrt3$ exists if and only if $9a^2 b^2 -12(a+b)^2$ is a square.

Figures (7)

  • Figure 1: The root ELEP, $(a,b)=(2,4)$
  • Figure 2: Contraction of the $\phi_3$ edges
  • Figure 3: The tree of ELEPs; the elements $(a,b)$ are shown above the corresponding pairs $(s,t)$.
  • Figure 4: Diagonals, heights and altitudes
  • Figure 5: Two ELEPs with horizontal diagonal
  • ...and 2 more figures

Theorems & Definitions (38)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • proof : Proof of Theorem \ref{['T:suff']}
  • ...and 28 more