The shape on top is not a built-in primitive. I made it from another shape, the rhombic dodecahedron. Starting with that shape, I bevelled the whole thing such that all the edges were the same length. A bit complicated, but really interesting. I don’t know what it’s proper name is, though. It has squares at the six cardinal points, rhombi between them, triangles, and more squares between the rhombi and the triangles.
I really like the shape you get when you decimate a UV Sphere, but unfortunately the facets are not proper quadrilaterals—the vertices are not co-planar. Whereas in this shape, all the facets are proper polygons with coplanar vertices.
