Roskilde plywood dome  

I'm sure
that Bucky Fuller would have found it, but -- is there any geometry in which identical 'petals' can be joined to form a sphere ? I stress "identical" . . .

Well...
For starters, there are the five Platonic solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). Those are the only ones that use regular (i.e., equilateral) polygons.
If you don't mind using other types of polygons, like isosceles triangles or kite-shaped quadrilaterals, there are lots more. Search the web for something like:
tile sphere congruent
to see.