I was recently looking at random walks in 2 dimensions, and was able to simplify a problem involving the integer lattice $\mathbb{Z}^2$ by rotating it by $\frac{\pi}{4}$ radians, as below:
Each point on the new grid is represented with $(x,y)$, and the grid being diagonal also ensures that $x+y=2n$ for some $n\in\mathbb{Z}$.
How would I describe this space in such a way as to be able to do mathematics with it?