So I wondered, what would happen if I took rhombic dodecahedra…

…and then affixed them to the sixty wider faces of a rhombic enneacontahedron?

Well, it turns out that this is what you get:

It’s at time like these — urgent situations in recreational mathematics — that I am most glad I bought Stella 4d, the program with which I made these images (and which you can try, for free, at http://www.software3d.com/stella.php). This would have taken months to figure out without the proper software! The next thing that occurred to me was to take the convex hull of the last polyhedron. That’s like draping a sheet around it and then pulling it tight. Here’s the result:

Stella owes its name, in part, to a complex operation involving extensions of edges into lines, or faces into planes, called stellation. Stellating the above figure gave me something I didn’t like, but stellating it again gave me this:

And then, after six more stellations, I arrived at the end of this particular polyhedral journey.