# Parallelogram grid

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?

• if parallelograms are rhombuses, a, b and c is a possibility – Vasya Mar 29 '18 at 17:23
• all of them? can you explain it? – Leo Gardner Mar 29 '18 at 17:24
• Note that if b) is true then a) is true as well – Pet123 Mar 29 '18 at 17:37
• And if (a) is true then (b) is true as well. – Henning Makholm Mar 29 '18 at 17:46

• is side is $a$, longer diagonal will be $a\sqrt{3}$ – Vasya Mar 29 '18 at 17:53