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:
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.
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.)
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 .
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 Origami 3 , edited by Tom Hull. You can download my slides here:
The book Origami 3 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. (now part of CRC Press).
Carlos Furuti has developed VRML models of several of the polypolyhedra; you can find his work here.
If you’d like to compute your own polypolyhedra, you can download the latest version of my Mathematica notebook for analyzing 1-uniform-edge polypolyhedra here.
