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

Question

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?

Answer 1

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