How do you parameterize a sphere so that there are "6 faces"? I'm trying to parameterize a sphere so it has 6 faces of equal area, like this:

But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice").

I can't seem to get the $\theta$ elevation parameter correct.  Help!

The following doesn't have much to do with spherical coordinates, but it might be worth noting that these 6 regions can be seen as the projections of the 6 faces of an enclosing cube.
In other words, each of the 6 regions can be parametrized as the region of the sphere
$$S=\{\{x,y,z\}\in\mathbb R^3\mid x^2+y^2+z^2=1\}$$
for which, respectively:


*

*$x > \max(\lvert y\rvert,\lvert z\rvert)$

*$x < -\max(\lvert y\rvert,\lvert z\rvert)$

*$y > \max(\lvert z\rvert,\lvert x\rvert)$

*$y < -\max(\lvert z\rvert,\lvert x\rvert)$

*$z > \max(\lvert x\rvert,\lvert y\rvert)$

*$z < -\max(\lvert x\rvert,\lvert y\rvert)$


For example, the top region can be parametrized by $z =\sqrt{1-x^2-y^2}$, with region
\begin{align*}
\lvert x\rvert&<\frac{1}{\sqrt{2}}\\
\lvert y\rvert&<\min\left(\sqrt{\frac{1-x^2}{2}},\sqrt{1-2x^2}\right)
\end{align*}
