The quotient space $\mathbb{R}/\mathbb{Z}$ be defined by the set of all equivalence classes with respect to the equivalence relation $x \sim y \iff x-y \in \mathbb{Z}$. This quotient space is then proved to be the set $\mathbb{S}^{1} := \{[x]: x \in [0,1)\}$. Analogously, $\mathbb{R}^{d}/\mathbb{Z}^{d}$ is isomorphic to $(\mathbb{R}/\mathbb{Z})^{d}$.
Now, let $L \in \mathbb{R}$ be an even number and consider the set $I = [-L/2,L/2]^{d}$. In an analogous way, the set $I\subset \mathbb{R}^{d}$ is, by definition this is the set of all equivalence classes $x \sim y \iff x-y \in \mathbb{Z}^{3}$ for $x,y \in I$.
What is the characterization of $I/\mathbb{Z}^{3}$? I mean,what set does it represent as a "subset of" $\mathbb{R}^{3}$?