  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 $n$.
