Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common Fundamental Domains

We provide a complete characterization of pairs of full-rank lattices in $\mathbb{R}^{d}$ which admit common connected fundamental domains of the type $N\left[ 0,1\right) ^{d}$ where $N$ is an invertible matrix of order $d.$ Using our characterization, we construct several pairs of lattices of the type $\left( M\mathbb{Z}^{d},\mathbb{Z}^{d}\right) $ which admit a common fundamental domain of the type $N\left[ 0,1\right) ^{d}.$ Moreover, we show that for $d=2,$ there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type $N\left[ 0,1\right) ^{2}.$