In the plane consider a system of lines given by equation $x=m, y=n$, where $m$ and $n$ are integers. These lines form a lattice of squares or an integer lattice. The vertices of the squares, i.e. the points with integer coordinates, called the notes of the integer lattice.
First of all, for $n\neq 4$, a regular $n$-gon is impossible to place so that the vertices would lie on the nodes of an integer lattice. I can prove it.
But here is a problem. Using congruent parallelograms we can cover completely the plane as shown in the figure. That is called a parallelogram grid. For what type of parallelograms, i.e. what type of parallelogram grids is it possible to place a
a) regular triangle,
b) regular hexagon,
such that the vertices would lie on the grid points, i.e. the vertices of the parallelograms?