I have some problems with the proof(you can see the paper here) in section 3: Honeycombs on a Torus. As is stated in the article on page 8
Before moving the cluster to a torus, we first move it to a cylinder. Pick a diameter to the cluster (a segment between maximally separated points p1 and p2 on $\Gamma$). Then move the cluster to the cylinder $\mathbb{R}^2/\mathbb{Z}v$, where v is the translation along the length of the diameter . The map of the cluster to the cylinder is injective, except at the points p1 and p2, which become identified. Since p1 and p2 are maximally separated, the cluster fits inside a square of edge length $|v|$ with a pair of sides parallel to $v$.
Here, I can't understand why $\mathbb{R}^2/\mathbb{Z}v$ is a cylinder. Besides, I don't understand how and why we move the cluster to the cylinder before we move it to the torus.
Hopefully, if someone have read this article, I really want to discuss with you.