Atom-generated planar lattices

TL;DR

This paper investigates the structure of planar lattices generated by atoms, establishing bounds on boundary sizes and demonstrating the potential for arbitrarily large lattices with a fixed number of atoms.

Contribution

It proves bounds on boundary sizes of atom-generated planar lattices and constructs examples of arbitrarily large lattices with a fixed number of atoms.

Findings

Bound on boundary elements: at most n+1 for n atoms.

Existence of arbitrarily large lattices with 4 atoms.

Construction method for large lattices with fixed atoms.

Abstract

In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can be arbitrarily large. For every $k > 1$, we construct a planar lattice $L$ generated by $4$ atoms such that $L$ has more than $k$ elements.