Parametrization of hypertorus $S^1 \times S^1 \times S^1$

Question

I'm looking for a parametrization of $T^3$ but have not been able to find one online. Can somebody offer one, point me to where I can find one, or provide instructions on setting one up? Thanks!!

Answer 1

Here's a way to realize $T^3$ as a parametrized submanifold of $\mathbb R^6$:

$$f(u,v,w) = \begin{pmatrix}\cos u\\ \sin u\\ \cos v\\ \sin v\\ \cos w\\ \sin w\end{pmatrix}.$$

This generalizes easily to an embedding of $T^n$ in $\mathbb R^{2n}$.