Six Intersecting Pentagrams: Diagrams and the Puzzle of Its Genealogy

Click on the image to download the diagrams

One of the challenges that I ran into while documenting this model was getting clear on its attribution. I'd assumed it was Meenakshi Mukerji, but upon further investigation I found that the attribution was not only not straight-forward, it was downright fuzzy.

Mukerji had introduced me to the model initially, when she challenged me to come up with a simple, easily repeatable algorithm for assembly. But she'd sent me links to images of the model on flickr that had been folded 4 years prior, as well as a link to a computer generated image depicting it on Robert Lang's website, which he had posted almost a decade earlier. On the other hand, Francesco Mancini (the owner of the flickr images) had credited Leong Cheng Chit and Jared Needle as having independently discovered it, as well as Tom Hull and Francis Ow as being the co-designers of the unit and its use. So confusing…

I checked in with everyone mentioned and found that this model has a long history and has had many creators/collaborators and emissaries over the years. It's fascinating how there are so many examples and occurrences of models being designed simultaneously by different people who were neither working together nor even aware of each other.

In this case, it appears that while the original "sketch" of the model was posted several years prior, four different creators had each completed their own, unique versions of the work – all inspired by Robert Lang's original illustration.

Correspondence

Rather than try to convey the exact history of each creator as they designed their own version, I think presenting each person's account from their own point of view makes for an intriguing juxtaposition.

Jared Needle

Leong Cheng Chit

Francesco Mancini

(and in a later correspondence)

Each Creator's Personal Touch

Another thing that I found intriguing is the fact that each creator put their own, personal signature on the model. Each one implemented the internal module with a slightly different angle, bringing a different esthetic to their work. The model, as it was presented by Leong Cheng Chit, bends the Ow unit at an angle of 30°, while Mancini utilized an angle of 22.5° (one half of 45°). While working with the model myself, I used the CrOw 30 unit as taught to me by Mukerji (I've nicknamed the bent units CrOw 30, 27 and 22.5 respectively for “Crimped-Ow 30” etc.).

I found the fact that the same design worked with two different angle fascinating and in correspondence with Mancini he reiterated that he felt that the “proper” angle was probably 27° and that he wanted to work on figuring that out at some point in the future. Coincidentally, I soon heard back from Needle who informed me that he had used the 27° in his original design.

I was suspicious that this was no coincidence at all and that it was probably possible to create the model with any angle all the way to 0° and checked with Lang around that prediction. He congratulated me on my intuition and sent me a link to his implementation of that exact configuration from all the way back in 2002, which he'd published in the OUSA Annual Collection that year.

History and Attribution

It turns out that the SIP is one of 54 forms discovered by Robert Lang during a series of investigations he made in the late 90s. Lang was originally inspired by the idea of creating geometric puzzles in origami similar to "wooden burr puzzles" and fell upon this specific subset of geometric constructs, which he dubbed polypolyhedra. He first started writing on them in 1999 and after exhaustively identifying the set of possible models (and implementing and diagramming several in the process) documented the full set on his website in 2004. He also discovered that the general concept of entangled polyhedra had been extensively explored by Alan Holden, who described and showed wooden dowel models of many of them in his book, Orderly Tangles (Columbia University Press, 1983), and a notable origami example of the concept already existed in Tom Hull's 1995 creation, Five Intersecting Tetrahedra. Holden, in fact, presented a wooden dowel model of six entangled pentagrams (figure 7.10 in his book), as well as many similar structures, both polypolyhedra and others.

I came to learn that SIP is #7 of these 54 forms and that it, and several others, have been implemented and documented over the years by various people, often simultaneously and without knowledge of the others' efforts. In his own implementations, Lang gave names to several of them (after mountain peaks in the Himalayas). SIP was one of them, which he called “Shishapangma.”

Here is a listing of the current list of which of the polypolyhedra have been executed by whom and when:

And which have been diagrammed and the place of their original publication:

In addition to that, there are several other polypolyhedra which are close cousins to the 54 polypolyhedra Robert Lang found*, including Daniel Kwan's Six Intersecting Pentagonal Prisms and Kawashima Hideaki's Crystal Star (Origami Tanteidan #106, 11/2007) as well as a full set of explorations of the utility of the CrOw 22.5 unit by Mancini including Star Icosidodecahedron, Star Prism, Star Holes and Borromean Stars.

With various aspects of it being created by various individuals independently, as well as using each other for jumping off points it's difficult to create a simple, straight forward timeline. But since there has been SO much fascinating material produced around this model and various aspects of its and its cousins' development, I'll attempt one – just for this particular model.

Attribution: The Final Verdict

So at this point, having spent DOZENS of hours researching the proper attribution of the piece, I can say with some certainty that I am still completely uncertain as to who might "get credit" for the model I diagrammed. Lang clearly was the mathematical "archeologist", who unearthed the model and presented it to the origami masses, though it had been discovered previously in different contexts and forms at an earlier time by several people. Four different folders have all implemented the model in 3 different CrOw angles, two different unit lengths and 3 different rectangle sizes. Evan Zodl had videotaped one version of the Mancini CrOw unit on YouTube while I created a completely different method of folding the unit Mukerji taught me and created the assembly algorithm.

I recall waaaay back in the day when I was writing video games that a given game was written by ONE person and one could make statements such as "I LOVE the work of Silas Warner." But as games (and budgets) expanded in scope this shifted and it became teams of people rather who created the work: artists, programmers, writers, producers, etc. And, perhaps, as things evolve in origami this will become the case as well and models will require credits rather than a simple attribution.

So in that spirit we roll credits:

- JC Nolan, July '12

*Lang found the 54 uniform-edge polypolyhedra, which are those in which each edge and its immediate neighborhood is the same as every other. There are many other polypolyhedra with more than one type of edge, such as Kwan's SIPP.