Parallelogram grid

Question

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,

c) square,

such that the vertices would lie on the grid points, i.e. the vertices of the parallelograms?

Answer 1

Here is a picture that shows that equilateral triangle and regular hexagon is a possibility is we use rhombus with 60 degree angle for our grid. Square is not possible in this configuration but rectangle is possible. Each rhombus consists of two equilateral triangles and each regular hexagon has six.

Answer 2

c) is possible too, but asks for a different setting than a) and b).

E.g. when having angle 45° and sides ratio of parallelogram of sqrt(2) to 1. Then clearly a square grid can be superimposed on a subset of nodes, then having a side length of sqrt(2) too.

--- rk