The Wrapped Displacement and the Flat Torus Distance

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\lVert\,\cdot\,\rVert$ the Euclidean norm of $\mathbb{R}^{d}$. By Euclidean Points as Tuples of Real Numbers a point of $\mathbb{R}^{d}$ is the $d$-tuple of its coordinates, so it is determined by prescribing them.

1. (Wrapped displacement)¶ For $z\in\mathbb{R}^{d}$, the wrapped displacement $\varpi(z)$ is the point of $\mathbb{R}^{d}$ with coordinates

$$\varpi(z)_{i}=z_{i}-\bigl\lfloor z_{i}+\tfrac12\bigr\rfloor\qquad(i\in[d]).$$

2. (Flat torus distance)¶ For $x,y\in\mathbb{R}^{d}$, the flat torus distance between $x$ and $y$ is the nonnegative real number

$$d_{\mathbb{T}}(x,y)=\lVert\varpi(y-x)\rVert .$$