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!

Following on FelixCQ's approach, take the cube to have corners $\pm 1$ and we want to project onto the unit sphere.  So one side is $(-1+2t,1,1)$  Projecting that onto the unit sphere gives $\frac{1}{\sqrt{3-4t+4t^2}}(-1+2t,1,1)$, $\theta=\arccos(\frac{1}{\sqrt{3-4t+4t^2}}, \phi=\arctan(-1+2t)$.  The other sides can be treated similarly, but you have to worry about the quadrants for $\phi$.
