I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra dimensions to work out its kinks and orient itself.
Is this true? I can't find any discussion of it online.
This is not true. Obviously, we have an embedding of the Möbius strip into $\mathbb{R}^3$, and there is a rather obvious embedding of $\mathbb{R}^3$ into $\mathbb{R}^n$ for any $n \geq 3$ (namely, the map which sends $(x, y, z)$ to $(x, y, z, 0, \ldots, 0)$). The composition of these maps is an embedding of the Möbius strip in $\mathbb{R}^n$.
The Möbius strip cannot "work out its kinks" in any space, since it is not homeomorphic to a cylinder.