Taking powers of polytopes I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known. 
Imagine this situation where I want to think of such a thing : say one is given the unit square in the all-positive orthant of $\mathbb{R}^2$. Is there a ``simple" (hopefully piecewise linear!) map which will map this square to the square of side $2$ with its vertices at $(-1,-1),(-1,1),(1,1),(1,-1)$. 
Had one started with a quarter of a unit disk then one could have taken the map $z^4 : \mathbb{C} \rightarrow \mathbb{C}$ to get that to the unit disk about the origin. What is the closest analogue to this in the world of polytopes (and linear maps?)? 
 A: Given any $k$ and any polyheda $\newcommand{\bR}{\mathbb{R}}$ $P, Q\subset\bR^n$ with nonempty interiors  one can  easily   produce a $k:1$  covering $P\to Q$ branched over a codimension $2$-locus.
Here it goes. Fix  homeomorphisms $F_P: P\to C^{n-2}\times D^2$, $F_Q: Q\to C^{n-2}\times D^2$, where $C^m$ is  the cube $[-1,1]^m\subset\bR^n$ and $D^2$ is the unit disk in the plane.
Consider the canonical $k:1$ cover $S^1\to S^1$. Fix a triangulation $\newcommand{\eT}{\mathscr{T}}$ $\eT$ of $S^1$ and denote by $\tilde{\eT}$ its lift $p_k^{-1}(\eT)$. Then we can view $p_k$ as a $k:1$-simplicial covering. The triangulations $\eT$, $\tilde{\eT}$ of $S^1$ extend to triangulations of $D^2$, viewed as the cone over $S^1$.  The map $p_k$ then induces  a $k:1$   simplicial map $\beta_k:D^2\to D^2$ branched over the origin. Define
$$ \Psi_k: C^{n-2}\times D^2,\;\;\Psi_k(y,z)=(y,\beta_k(z)). $$
Then
$$ \Psi_k: F^{-1}_Q\circ\Phi_k\circ  F_P: P\to Q $$
is a $k:1$ map branched over $F_P^{-1}(C^{n-2}\times 0)$.
How  to  emulate this in the piecewise affine world?
Think of $\eT$ as the triangulation of the boundary $\newcommand{\pa}{\partial}$ $\pa \Pi_3$ of the regular $3$-gon $\Pi_3$ with vertices on $S^1$ and $\tilde{\eT}$ as the triangulation  defined by the  vertices of a regular $3k$-gon $\Pi_{3k}$.  Then $\beta_k$ can be viewed as a piecewise-affine $k:1$ covering map $\Pi_{3k}\to\Pi_3$, branched over the center of $\Pi_3$. Denote by $\Delta_m$  the $m$-simplex
$$  \{0\leq x_1\leq \cdots \leq x_m\leq 1\}\subset \bR^m. $$
Now  fix   piecewise-affine homeomorphisms
$$ F_P: P\to\Delta_{n-2}\times\Pi_{3k},\;\; F_Q: Q\to\Delta_{n-2}\times \Pi_{3}. $$
