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

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. • Great! Why is c) not possible? I think it is never possible (except the regular grid). – Leo Gardner Mar 29 '18 at 17:43
• because ratio of longer diagonal to side is irrational number, square is not possible – Vasya Mar 29 '18 at 17:50
• why is the ratio irrational? – Leo Gardner Mar 29 '18 at 17:52
• is side is $a$, longer diagonal will be $a\sqrt{3}$ – Vasya Mar 29 '18 at 17:53

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