TITLE: Diffeomorphism between $\Bbb{R}^{4}$ and the cube
QUESTION [3 upvotes]: I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple looking derivatives (I need to induce a metric on the cube from the space, and I'd prefer it didn't look terribly complicated at the end). Is anybody able to help me with this?

REPLY [4 votes]: We can use the result proved in this question to build one.  There it is shown that $f : (-\frac{\pi}{2}, \frac{\pi}{2}) \to \mathbb{R}$ defined by
  $$f(x) = \tan(x)$$
is a diffeomorphism.  To obtain the function you are looking for, use four copies of that one:
$f : (-\frac{\pi}{2}, \frac{\pi}{2})^4 \to \mathbb{R}^4$
$$f(x_1, x_2, x_3, x_4) = (\tan(x_1),\tan(x_2),\tan(x_3),\tan(x_4))$$
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