Based on a comment here, the following 4-d parametrization represents a Torus (it also happens to lie completely on $S^3$):
$$ x=3/5 \cos(\theta)\\ y=3/5 \sin(\theta)\\ z=4/5 \cos(\phi)\\ w=4/5 \sin(\phi) $$
I'm trying to represent these as the standard equations of a Torus:
$$ x=(R+r \cos(\alpha))\cos(\beta)\\ y=(R+r \cos(\alpha))\sin(\beta)\\ z = r \sin(\alpha) $$
That will tell me what the $R$ and $r$ parameters of the 4-d Torus are.
My attempt:
From the first set of equations, $w$ is already in the correct form (if we assume $\alpha = \phi$). We can express $z$ as a function of $w$. Now I don't know how to infiltrate the $x$ and $y$ which don't depend on $z$ and $w$ in any way.