Create sphere from diamond-like shape?

I am trying to recreate the green house glove from Disney World’s Epcot, and the shell is pretty much made of 2 triangles that meet in their own centers, forming a triangle. I have create a base shape that/is array’d, but I cannot figure out how to turn the giant array sheet (or single piece) into a sphere. I tried forming to sphere, but was not able to get bending to work properly (it would turn the object from no back into odd 3D transformation with LOADS of stretching).

I have searched around google alot and seen other odd methods that would not work- including hair/particle (just did not look right) and dupliface (both did not look right and had almost random rotations, even when using constraints)

Icosphere, poke, select, shrink/fatten

What method did you use to select just the inside/poke tips? (poke is a new tool to me)

Selected one, select similar: amount of connected edges.

Thank you. This helps a lot and I learned 2 new tools that are probably going to be highly useful.