To get the edges equal AND the angles 90/120/150 is not possible in 2D.
So we have to bend the lines.
I didn't get it work on a round sphere; it just doesn't fit.
when I use an egg-shape it should be possible!
I don't have the means and capabilities to make the calculations for it, so I just give the idea in drawings:
We bend the Pi-, e- and Phi-lines and with the diagonal we can calculate the edges.
The three lines together would give something like this:
In a somewhat bigger picture you would get................