Regular hole

There’s 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. Good luck finding a copy for sale at an affordable price, but our university library has it. It’s a self-published book, about 11″ by 8 1/2″, 256+ pp., hand lettered in Chancery Script by the author. Copyright date 1970 for the first edition, 1980 for the second.

As you can tell from the title, it’s pretty much about regular-faced polyhedra with holes through them.

Here’s one:

Q3^2/S3S3

Q3^2/S3S3

Q3^2/S3S3

Crappy cell phone pictures but you get the idea. This is the one designated Q32/S3S3 and nicknamed “Tortuous Tunnel”. It is (in Norman W. Johnson’s nomenclature and notation) a triangular orthobicupola (Q32) from which has been removed two stacked triangular antiprisms (S3) to make a hole through the middle.

I’ve modelled it in six colors — I don’t think there’s a good, suitably symmetric way to do it in fewer than six and more than two. The construction technique is the “cardboard, rubber band” method recommended by Stewart. It has the obvious drawback of leaving extraneous tabs sticking out of the model. The advantages over the more familiar “network, tab, paste” method are that it’s quicker, it can be disassembled for storage, transport, or reuse of the parts, and, for some of Stewart’s more intricate models, it’s a lot easier to band together exterior tabs than to glue together interior tabs. And the advantage over computer modelling is you can pick it up and play with it — maybe our grandkids will have cheap high quality tactile-feedback virtual reality equipment, but I don’t. Some of Stewart’s models also have been done with Geomags. Nice looking, but pricey, and I imagine a lot of the models, like the one linked here, have problems with too many parts in too small a space.

Anyway, there it is. Cute, huh.

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