Geodesic Domes

Icosahedrons are the best possible approximations to spheres with congruent polygons and identical vertices:

 

Consider dividing the surface into smaller congruent triangles:

Also, consider moving the new vertices farther from the center to better approximate a sphere:

Geodesic domes are built from portions of such figures.  The American Institute of Architects calls geodesic domes the "strongest, lightest and most efficient means of enclosing space known to man".