Is it possible for an $n$-dimensional object to ever cover an $(n+1)$-dimensional object? For example, could a square ever cover a cube?
Note: Definition of "cover" here means to completely cover the surface area, while not breaking the dimensionality. For example, a piece of paper could be folded around a cube, but then you'd just have another cube.
Edit for clarification: By "cover," I mean something akin to "wrapping" a 2D object around a 3D object without simply creating a 3D object. One of my "mathy" friends phrased his answer this way:
"Form an $n$-dimensional basis using $n$ orthonormal vectors in $\mathbb{R}^n$; this is space $A$. Form an $(n-1)$-dimensional basis using $n-1$ orthonormal vectors in $\mathbb{R}^{n-1}$; this is space $B$. To prove that you can't do what you're saying, prove that there exists a surface in space $A$ that contains points that do not exist in space $B$ (which is trivial)."
By the way, this ISN'T a homework problem, I thought of it lying in bed last night and nobody seems to have an answer.
For $n = 2$, the answer is yes if one restrict to 3-d objects whose surfaces are formed by connected 2-d polygons.
This is proved in a paper Folding Flat Silhouettes and Wrapping polyhedral packages: New Results in Computational Origami by E. Demaine, M. Demaine and Mitchell around year 2000.
An online copy can be found here.