## Forty-seven

Here’s a nice one, $47S_3$ 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.

## Folder full of holes

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.

Here’s an animation of $Q_4\left (T_4\right )Q_4 / B_4 \left(P_4\right )B_4$:

## Virtual insane holes

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.

## Virtual insane inverse hole

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.

## Virtual regular hole

Way way back, before there was a Mathematrec, on my other blog I posted about a book titled Adventures Among the Toroids: a study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus having Regular Faces with Disjoint Interiors, being an elaborate Description and Instructions for the Construction of an enormous number of new and fascinating Mathematical Models of interest to Students of Euclidean Geometry and Topology, both Secondary and Collegiate, to Designers, Engineers and Architects, to the Scientific Audience concerned with Molecular and other Structural Problems, and to Mathematicians, both professional and dilettante with hundreds of Exercises and Search Projects many completely outlined for Self-Instruction (Revised Second Edition), by B. M. Stewart, and I showed a picture of a paper model I’d made of one of Stewart’s toroidal polyhedra, designated Q32/S3S3. Today after much trial and even more error, I figured out how to draw that same polyhedron using the open source Antiprism software. The command is

off_align -F oct,0,0,0 oct | off_align -F J27,2,0,3 | \
off_util -D f19,8 | antiview


(which stacks and merges two octahedra, stacks them inside a triangular orthobicupola and subtracts them from it, removes the two additional coincident faces which off_align doesn’t, and then displays the result). If you don’t have Antiprism but you have something that’ll display OFF files, here it is: https://drive.google.com/open?id=1M_a474kL95XMdTl4qHHfH6AfCrm2Tnfw. And if you have neither, here’s a static image of it:And no, I am not planning on doing the same for J91/Z4 anytime soon. Feel free to take that on yourself.