# 2D Material Structures and Discrete Symmetries of Periodic Polygons

## TL;DR

This paper explores the geometric and symmetry properties of 2D materials with double lattice structures based on periodic polygons, proposing methods to engineer new materials by manipulating superstructures and symmetries.

## Contribution

It introduces a novel approach to designing 2D materials using dual superstructures and links these configurations to Lie symmetries for deeper understanding.

## Key findings

- Periodic polygons can generate complex 2D lattice structures.
- Engineered superstructures can double the atomic unit cell.
- Lie symmetries provide a theoretical framework for these geometries.

## Abstract

In this work, we reconsider the study of 2D materials involving double lattice structures associated with periodic polygons. In tessellated periodic representation, it appears two periodic polygons of $k$ sides of unequal side lengths at certain angles fixed by the underlying discrete symmetries. In this way, 2D materials could be engineered by using two superstructures on the same atomic sheet generated by two length parameters $a_{1}$ and $a_{2}$ and rotated by the angle $\phi _{n}^{k}=\frac{n\pi }{k}$, where $n$ is an arbitrary natural number. These geometrical configurations could be exploited to engineer 2D materials by doubling the number of the ordinary unit cell atoms. To support the present conjecture, we establish a link with Lie symmetries including finite and indefinite ones providing a room interpretation for $n$.