I noticed that I could arrange eight great rhombcuboctahedra into a ring, but that ring, rather than being regular, resembled an ellipse.
I then made a ring of four of these elliptical rings.
After that, I added a few more great rhombcuboctahedra to make a meta-rhombcuboctahedron — that is, a great rhombcuboctahedron made of rhombcuboctahedra. However, it’s squashed. (I believe the official term for this is “oblate,” but “squashed” also works, at least for me.)
So now I’m wondering if I can make this more regular. In other words, can I “unsquash” it? I notice that even this squashed metapolyhedron has regular rings on two opposite sides, so I make such a ring, and start anew.
I then make a ring of those . . .
. . . And, with two more ring-additions, I complete the now-unsquashed meta-great-rhombcuboctahedron. Success!
To celebrate my victory, I make one more picture, in “rainbow color mode.”
[All images made using Stella 4d, available here: http://www.software3d.com/Stella.php.]
It’s like the snub dodecahedron’s big brother.
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
In this construction, I took the polyhedral cluster found in the last two posts, and augmented every pentagonal face with yet another great dodecahedron. I used software you can find at http://www.software3d.com/stella.php.