keggin and keggin-ish structures

    I'm on a 2-week break, since the store that my husband and I own is closed for the holidays.  Just when I had lots of free time, the beaded molecules folks (thebeadedmolecules.blogspot.com) posted a set of pictures that set me off on another structural journey.
The structure is called a keggin structure and wikipedia says it's the form of certain acid molecules.  I confess to not being much interested in the chemistry of it, but only in the geometry.  Geometrically. the structure is made of groups of 3 octahedrons that serve as the triangles in a larger structure.  The top  picture shows one such unit.  You can see on one edge (the top in this pic) there's a big triangle, and on the other edge (bottom) there are 3 small triangles that meet at a point in the center.   4 of these units, with the 1-triangle side on the outside and the 3-triangle side to the cente, form a  tetrahedron-ish thing.
    In thebeaded molecules site they showed 5 versions of this structure, identified by Greek letters that I don't have the font to write.  Picture 2 is the alpha version.  Till I made it, I couldn't tell how the others differed.  Turns out that in the alpha version, the triangles on the outside all line up with edges parallel to one another.  In the beta version one of them is point-to-edge.  Each of the others has 1 more triangle arranged point-to-edge till all 4 are that way.  But I had to see it in front of me before I understood it.  It's really hard to show a 3 dimensional structure like this well in a flat picture.  Even more so for me because I don't have 2 colors of bugle beads in the same length just now.
  Anyway, that's it for the Keggin structure.  But for me, since I'm dealing with it just as a geometric structure, not as a chemical thing, it seemed that the next logical step would be to take 8 of these triangle-ish units and make an octahedral version of the structure. So that's picture 3.  Then I took the 8 triangle version and made all the triangles point-to-edge instead of edge-to-edge.  That's the last picture.  I also improved things in that one by making the seed bead accents on the outside a different color from those on the inside of the structure, so that it's easier to read what's going on.
     A person just a little crazier than I  could also take 20 of the 3-octahedron units and make an icosahedron, but I think I've just about exhausted my interest in this.