How to make a three-way grid along a hemisphere

The image that you shown is NOT a good thing for real-life structures (for more than obvious reasons). That said I’m in the AEC market sector and thus - for me - everything is rated/evaluated accordingly

I would strongly advise to read a thing or two about Geodesic Structures/Domes. See this:

Recognizing same/similar shapes in different orientations?

That said there’s tons of Dome related material in the Internet Search this and the good Old (dead) Forum as well.