Given a lattice $L$ defined by its generators $w_1$ and $w_2$, how can we tell if the Dirichlet region of $L$ is a rectangle?
For example, the Dirichlet region for $L = (w_1, w_2) = \left(1, \frac 1 2 e^{i \pi / 3}\right)$ looks like a rectangle:
If instead $L = \left(1, \frac 1 2 e^{i \pi / 4}\right)$, then the Dirichlet region looks like a hexagon:
Given the origin point $O$ of a cell, the boundary is made up of line segments of the perpendicular midpoint lines between $O$ and $w_1$, $O$ and $w_2$, $O$ and $w_1+w_2$, etc.
Let $p(a,b)$ represent the perpendicular midpoint line between $a$ and $b$. Conjecture: You have a rectangle iff the distance of the point of intersection of $p(O, w_1)$ and $p(O, w_2)$ to the origin is less than the distance from $(w_1+w_2)/2$ to the origin.
I'll leave it up to you to prove it. You can also explicitly compute the lines and the intersected point in terms of $w_1$ and $w_2$.
