Visualizing A Flattened $2v/3v$ Geodesic Dome

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I had a really nifty mathematic/art project idea and was wondering how flattening a partial sphere made of triangles could look. From what I can just visually see it seems like I can take the face triangle and "make a star" out of them with one common vertex at the top, but are there any other ways I could layout the triangles so they could assemble in this way, i.e. more of a spiral or something "organic" that doesn't appear to be made from 1-2 components? I'm leaning toward no but if there is, what resource could help me figure out what 2d "box designs" I could come up with? I'm about $10$ years removed from my math theory BS so I'm a bit rusty, but hopefully not totally inept at figuring things out.

Cut me in '3/8' and make a box design out of me