A lozenge triangulation of the plane with integers

Abstract

We introduce and study a three-folded linear operator depending on three parameters that has associated a triangular number tilling of the plane. As a result the set of all triples of integers is decomposed in classes of equivalence organized in four towers of two-dimensional triangulations. We provide the full characterization of the represented integers belonging to each network as families of certain quadratic forms. We note that one of the towers is generated by a germ that produces a covering of the plane with {Löschian} numbers.

Figures (9)

• Figure 1: The geometrical representation of the core of $\mathcal{R}_H(0,0,0)$ and $\mathcal{R}_H(0,1,1)$ as the sets of nodes in the tiling of the plane with triples in $\mathcal{T}_H(0,0,0)$ and $\mathcal{T}_H(0,1,1)$.
• Figure 2: A cut-off representation of the triangular networks generated by $(9,2,6)$ (left) and $(1,8,3)$ (right). The weights are taken modulo $23$ on the left and modulo $37$ on the right. In order to distinguish them, the residue classes are represented in distinct colors in each of the two cases.
• Figure 3: The triples and how they appear oriented in the triangular networks generated by $(0,0,0)$ (left) and $(0,1,1)$ (right). Once can check that every triple that occurs appears only once in the left image and exactly six times in the right image.
• Figure 4: Comparison between local symmetries of the two types of triangular networks. On the top row is the network generated by $(0,0,0)$, and on the bottom row is the one generated by $(0,1,1)$. In each of these, the weights are taken modulo $2, 3, 5$, and $7$, and the residue classes are represented in distinct colors, so that the cases from Theorem \ref{['TheoremC']} can be verified.
• Figure 5: The lozenge expansion of $d$ from $a$ across the short diagonal $bc$.

Theorems & Definitions (19)

• Theorem 1
• Remark 1.1
• Theorem 2
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Theorem 3
• proof
• Remark 3.1