Turns out you can do the same thing with 47 icosahedrons. The rhombic dodecahedron is clearer here.
[Edit: Better GIF]
Here’s a nice one, from Stewart’s Chapter VII ‘Exploration of (R)(A) toroids’. Take an octahedron and attach a pair of octahedrons to each face:(That’s 17 octahedra, one in the center and two in each of the eight arms.) That makes something roughly cube shaped. Now make a kind of pyramid of five octahedra:and you can attach one of these to each face of the cube. That’s 30 more octahedra for a total of 47, and it looks like this:If you stare at it long enough you can see each of the resulting twelve holes is in the center of a rhombus — there’s a rhombic dodecahedron underlying all this:Spin her up.
So I now have created all of the examples from Chapter V, ‘Simplest (R)(A)(Q)(T) toroids of genus p=1’, in Antiprism, and they’re in this Google Drive folder. The relation between the file names and Stewart’s designations should be fairly clear. In addition to the OFF files there are shell scripts which generate them. The tor.sh script is something I threw together to automate some of the process.
And because why not, here are six Z4 surrounding a cube:
off_align -F z4.off,0,1,0 cube | off_align -F z4.off,4,1,2 | \ off_align -F z4.off,1,1,1 | off_align -F z4.off,2,1,3 | \ off_align -F z4.off,0,1,2 | off_align -F z4.off,0,1,3 | antiview
And then that’s subtracted from six J91 surrounding a cube (essentially) because, yes, six J91 will fit around a cube:
off_align -F J91,1,1,0 z4_6.off | off_util -M b | off_align -F J91,21,1,0 | \ off_util -M b | off_align -F J91,83,1,0 | \ off_util -M b | off_align -F J91,103,1,0 | \ off_util -M b | off_align -F J91,41,1,0 | \ off_util -M b | off_align -F J91,61,1,0 | \ off_util -M b | antiview
So, yeah, a regular faced polyhedron with three mutually perpendicular tunnels passing through its center.
And here’s J91/Z4.
off_align -F z4.off,1,1,4 j91 | off_util -M b | antiview
Okay, so if I’m not doing J91/Z4 in Antiprism anytime soon, how come I’ve gone ahead and done Z4?
off_align -F J2,5,0,5 J63 > piece1.off off_align -F piece1.off,5,6,9 J91 | off_util -M b -M a -x V | \ off_align -F piece1.off,8,6,6 | off_util -M b -M a -x V | antiview
I checked out Adventures Among the Toroids again. Not sure when the library stopped stamping due dates in books but the last and (I think) only stamped date after the book’s acquisition in 1982 was June 2, 2008, which presumably was me. It’s mine again for the next year if it’s not recalled.