What app are you using? I think in Max there's a primitive that does something like this. If there isn't then I might try what you said with angling and snapping verts on 5 sided planes. I'm not sure what the angle you'd have to rotate it would be though.
I'm a max user and I would make that shape by simply creating a hedra and adjusting the parameters, shapes like these are built right in max. Then I would adjust the shape by using FFD Modifiers or lattice I believe it is for maya user's.
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example image from google of one hedra - http://www.evermotion.org/tutorials_old/modeling/ball/2.jpg
And that looks like a Dodecahedron
No need to snap vertices=)
[edit] too slow.
I tried it and it works for me
I smoothed it before hand and it wont let me convert after my division levels are 4+
was just screwing around with them, seeing how the make the shape etc future ref