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Polypolyhedra

Background

Makalu
Folded model of the 6x1x5 polypolyhedron named Makalu.

In 1999, I became interested in a family of origami modulars composed of interwoven polygonal frames and/or polyhedral skeletons, the most famous of which was devised by Tom Hull, which he titled Five Intersecting Tetrahedra, now commonly known by its abbreviation, FIT. (See here for a picture and directions.) The underlying polyhedron—a compound of five tetrahedra — has been known for hundreds of years, but Tom's implementation as an origami skeleton had an uncanny beauty that made it an instant classic of the origami repertoire.

Naturally, of course, I wished I'd thought of it. But it also got me to wondering if there were any other origami modulars that were similar. The first thing to do is to define a bit more precisely what one means by "similar". I decided that the salient features of FIT that I wanted to replicate were the following:

I resolved to find all such structure that satisfied these criteria. Because these structures were compounds of polyhedra, I coined the name polypolyhedra to describe them.

K2
Folded model of the 20x1x3 polypolyhedron named K2.

Before I could construct origami versions of any polypolyhedra, I first needed to identify the underlying abstract geometric shapes: the underlying polyhedra. Now, the field of polyhedra is vast. Innumerable classes, families, and variations of polyhedra and their derivatives have been identified and seemed that these things I called polypolyhedra were likely already to be found within the mathematical literature. For one example, the compound of five intersecting tetrahedra — the basis of Hull's FIT — is one of the 59 known stellations of the icosahedron enumerated by H. S. M. Coxeter. I eventually learned that many years ago a Bell Labs mathematician, Alan Holden, had examined a wide range of linked skeletal structures, which he dubbed polylinks, and described them in his book, Orderly Tangles). Nevertheless, I found no complete enumeration of what was possible. So, I set out to enumerate them all.

I realized that symmetry played a key role in identifying what shapes were possible, and eventually found that it was possible to enumerate all possible interwoven shapes foldable from a single type of unit by examining the possible relevant symmetry groups and sorting through all the possible combinations of symmetry groups and orbits using Mathematica. Eventually, I was able to show that there are exactly 54 topologically distinct varieties of polypolyhedra, not including mirror images (most are enantiomorphic). Quite a few of them have a geometric structure suitable for rendition in origami, and once I'd found the geometric structures, I developed origami folding sequences for several of them (including the ones on this page).

Once I'd folded a particular polypolyhedron in origami, there came the question: what to call it? Tom named his design for the underlying geometric solid: "Five Intersecting Tetrahedra." But a purely geometric description of some of these new structures could be a mouthful: "Five Intersecting Tetrahedrally Distorted Skew Rhombic Hexahedra", for example. So I decided that instead of a descriptive name, I'd just give them short, easily-memorable names, and began naming them after Himalayan peaks: Gasherbrum, Annapurna, and Makalu, for starters. The most interesting I dubbed Chomolungma, which is one of the local names for Mount Everest, and when I built one with the same number of pieces as Chomolungma but that was far harder to assemble, what else could I call it but K2?

I've now designed several of these structures in origami; some you'll find in the galleries, some have been published in various origami society publications. There are 54 distinct varieties, however, so the vast majority await rendition in origami. I've also written several articles that describe the mathematics of polypolyhedra and how I analyzed them, and you can download these below.

Links and Downloads

This page displays POV-Ray renderings of all 54 polyhedra rendered as origami models.

This page displays POV-Ray renderings of all 54 polyhedra rendered from cylindrical dowels.

These three articles describe the process I went through to analyze the various polypolyhedra. The files are big and there are plenty of digressions, but you'll find diagrams for a few of the polypolyhedra within and design sketches for several more.

I gave a presentation on the polypolyhedra at the 3rd International Conference on Origami in Mathematics, Science, and Education, which I subsequently wrote up as a contribution to the book Origami3, edited by Tom Hull. You can download my slides here:

The book Origami3 is one of the best collections of origami, math, and science available, and I highly recommend it. You can buy it from OrigamiUSA (OrigamiUSA members will get a discount) or directly from the publisher, A K Peters, Ltd.

Carlos Furuti has developed VRML models of several of the polypolyhedra; you can find his work here.