TITLE: Is it possible to give the unit square a smooth structure?
QUESTION [8 upvotes]: At the beginning of Lee's "Introduction to Smooth Manifolds", Lee gives the example of a the square and the circle being homeomorphic as an intuitive motivation for smoothness not being invariant under homeomorphisms. But I read somewhere that the unit square does admit a smooth structure and now I'm confused. I can't come up with a structure for the square, but I can't figure out how to prove that there isn't one. Help?

REPLY [4 votes]: For the sake of having an answer: 

Since the unit square is homeomorphic to $S^1$, it has a smooth structure given by transporting a smooth structure along any such homeomorphism, but
No such smooth structure will make the unit square an embedded smooth submanifold of $\mathbb{R}^2$ smooth (because the derivatives of such an embedding can't be nonzero at the corners).