TITLE: Parallelotope fundamental domains of the n-torus
QUESTION [6 upvotes]: The group $\mathbb Z^n$ acts on the topological space $\mathbb R^n$ by translation: if $z = (z_1, \cdots, z_n) \in \mathbb Z^n$ and $x = (x_1, \cdots, x_n) \in \mathbb R^n$, then $z\cdot x := z+x$. The quotient space of this action is the $n$-dimensional torus $\mathbb R^n/\mathbb Z^n$. In this setting, a fundamental domain is a convex set $Z \subset \mathbb R^n$ such that $Z$ surjects onto $ \mathbb R^n / \mathbb Z^n$ via the usual quotient map,  and such that this map is injective on the interior of $Z$. 
My question is: when is a parallelotope in $\mathbb R^n$ a fundamental domain of the $n$-dimensional torus?. By a parallelotope, I mean the $n$-dimensional analogue of a parallelogram. More precisely, a parallelotope is set of the form $\left\{ \sum_{i = 1}^n a_i v_i \mid 0 \leq a_i \leq 1 \right\}$ for some linearly independent set $\left\{ v_1, \cdots, v_n \right\}\subset \mathbb R^n$. 
I'm fine with assuming that our parallelotopes are rational, meaning $v_i \in \mathbb Q^n$ for all $i$. I'm also inerested more generally in which parallelitopes in $\mathbb R^n$ surject onto $\mathbb R^n / \mathbb Z^n$ via the usual quotient map. 
My initial guess was that any parallelotope with sufficiently large volume would at least surject onto the $n$-torus, but this dream was quickly crushed by the following example: if $Z$ is the rectangle $[0, 0.9999]\times [0,10000000]$ in $\mathbb R^2$,  then $Z$ doesn't surject onto $\mathbb R^2 / \mathbb Z^2$, but a small rotation of $Z$ does surject. 
On the other hand, it's easy to see that this property is preserved by the action of $SL_n(\mathbb Z)$ on $\mathbb R^n$. Thus we may assume that the matrix $[v_1 v_2 \cdots v_n]$ is in Hermite normal form (and in particular, upper-triangular). Using this trick, I found it's not too hard (though quite messy) to figure out the $n=2$ case by hand, but I'm not sure what to do in higher dimensions.

REPLY [5 votes]: Instead of asking which unit-volume n-parallelotopes ($TC$, where $T\in SL_n(\mathbb{R})$, and $C=[0,1]^n$) tile $\mathbb{R}^n$ when translated by $\mathbb{Z}^n$, we can equivalently ask, which unit-determinant lattices $T^{-1}\mathbb{Z}^n$ tile space when applied to the cube $C$ via translations. This is the subject of Hajós's theorem, previously a conjecture of Minkowki, which says that such a lattice has an upper triangular basis with ones on the diagonal.