Consider the ring $R=\mathbb{Z}[\sqrt{-2}]$. It is a lattice in the complex plane: the set of points with integer coordinates with respect to the basis: $1,\sqrt{2}i$. Each mesh of the lattice is a rectangle, with base parallel to the real axis of length $1$ and height of length $\sqrt{2}$.
Now consider $R=\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$. My purpose is to figure out what is the shape of the lattice in this case, as i did in the previous case. I have $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]=\mathbb{Z}+\mathbb{Z}\frac{1+\sqrt{-3}}{2}$, hence the generic element has the form $$\displaystyle\frac{a+b\sqrt{-3}}{2}$$ with $a,b\in\mathbb{Z}$ and $a=b\pmod{2}$. But i cannot see what kind of meshes i have, are they squares, rectangles, parallelograms or what?
Note that $\dfrac{1+\sqrt{-3}}{2}$ is a cube root of $-1$. The fundamental parallelogram is a rhombus which is the union of two equilateral triangles, so you have a "triangular lattice".