# Diffeomorphism between $\Bbb{R}^{4}$ and the cube

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?

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: $f : (-\frac{\pi}{2}, \frac{\pi}{2})^4 \to \mathbb{R}^4$ $$f(x_1, x_2, x_3, x_4) = (\tan(x_1),\tan(x_2),\tan(x_3),\tan(x_4))$$