# Polypolyhedra

## Background

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 description.) 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:

• The structure was composed of units that followed the edges of geometric polyhedra;
• The structure must be composed of a single type of origami unit;
• The structure must contain multiple polyhedra;
• The multiple polyhedra must be linked together;
• The structure must hold together by folded locks.

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.

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?

Left: Folded model of the 6x1x5 polypolyhedron named Makalu. Right: Folded model of the 20x1x3 polypolyhedron named 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 —wait, I mean 55 — wait, I mean $$55 + \infty$$ distinct varieties, however, so the vast majority await rendition in origami. (Why the changing numbers? See below.) I've also written several articles that describe the mathematics of polypolyhedra and how I analyzed them, and you can download these below.

## First Cook

I mentioned above that the number of polypolyhedra has changed—twice! The first happened in 2014, when Aaron Pfitzenmeyer, to whom I had given my enumeration code, came to me and said, "I think you missed one." And indeed I had. It turned out that when I was counting the topologically distinct versions, I had done so by counting the distinct lobes in a plot of inter-stick-distance versus scaling parameter in Mathematica plots. But the adaptive scaling in 2001-era Mathematica had smoothed over a tiny sliver of a lobe in the plot for 10x3x4 (ten rhombic trihedral dipyramids). In later versions of Mathematica, the lobe was picked up; and Aaron spotted it.

Left: The inter-stick distance plot by 2001-vintage Mathematica. Right: the 2014-era version of the equivalent plot. Note the small spike next to the tallest lobe.

So there is, in fact, a 55th polypolyhedron, which I would like to name "Pfitzenmeyer" in honor of its discoverer. (Who has gone on to construct and build amazing 2-uniform and higher polypolyhedra far more complex than anything in my collection; see here.)

## Second Cook

But it's even worse! Also in 2014, Tom Hull and sarah-marie belcastro pointed out that I'd erroneously overlooked the dihedral group (in this case, because my computer-aided search routine had erroroneously discarded them as being unlinked). In this case, there was an example of a dihedral polypolyhedron for each even rotational order. So, as I like to say, it was a small error in my counting; I was only off by infinity. You can read about Tom and sarah-marie's analysis, and some lovely related coloring problems, in their article, "Symmetric Colorings of Polypolyhedra", which appears in the book Origami^6.