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Nonahexaflexagons

A regular piece of paper has two faces.

And you are familiar with the Mobius strip - that is a strip of paper twisted a half turn and pasted into a loop, so that it has only one face.

But have you ever seen a piece of paper that has been folded and glued so that it has nine faces? Meet the nonahexaflexagon...

Martin Gardner

After thinking about Martin Gardner (here's a nice interview) thanks to the Dragon illusion cutout, I went back to read an old compilation of his columns.

Gardner's very first "Mathematical Games" column for Scientific American was on the topic of hexaflexagons, a curious paper folding trick that had been discovered by Arthur Stone while a student at Princeton. Before Gardner's column, the joys of hexaflexagons were not widely known; Gardner had heard about hexaflexagons through the magic community.

Six-Faced-Hexagon

Although I had read Gardner's column on hexaflexagons long ago, I had never tried folding one. A hexahexaflexagon is a foldable paper polygon with six faces.

You may be saying, "what's the big deal about a six-faced hexagon - all hexagons have six sides!" Ah hah, yes, six sides, but an ordinary hexagon cut out of paper has only two faces, front and back. A hexahexaflexagon is different: within its folded innards it keeps four additional hidden faces that you can reveal by flexing the shape. To color the faces of a hexahexaflexagon completely, in other words, you will need six crayons, one for each hexagonal-shaped face.

So when hunting for an art activity for the family this morning I eyed our huge box of crayons and decided it was time, together with my kids, to try to make a hexahexa.

Printable HexaHexa PDF

Wikipedia contains a nice printable PDF template for a hexahexaflexagon that we followed successfully. My recommendation, when following the instructions, is to color the front face (with the 1-2-3-1-2-3-1-2-3 colors), then fold and glue everything together and then color the remaining three faces after it's been folded. You can't mess up the front, but if you are off-by-one on the back side, half the faces won't be solid colored. Additionally, it is worthwhile to fold over and glue the paper strip once to double it before beginning your flexagon origami, so that your flexagon is double-ply and a bit more tear-resistant.

My daughter Piper took a real liking to our first hexahexaflexagon, for which she chose the colors. "My hexa-flower," she calls it, appropriately. To flex a hexaflexagon, you need to bunch it into a three-pointed shape and open it like a flower.

More Hexaflexagonation

The first hexaflexagon discovered by Stone was a three-face trihexaflexagon, and he soon discovered the six-faced hexahexaflexagon that is described in the PDF above.

But there is more to flexagons. According to Gardner, the Princeton Flexagon Committee (a group of Stone's mathematical friends at Princeton at the time including Tuckerman, Tukey, and Feynman) discovered that every number of faces should be possible to achieve by making hexaflexagons out of longer, somtimes more serpentine, strips of triangles. It is possible to fully enumerate the set of possible hexaflexagons, and Gardner describes patterns for four-, five-, six-, and seven- faced hexaflexagons.

Googling for More

However, it is hard to find on the internet any instructions for building hexaflexagons with more than seven faces. I had my heart set on building a nine-faced nona-hexaflexagon, yet all Google was able to come up with on a search for the term [nona-hexaflexagon] was a cryptic flexagon newsgroup message from a person who expressed his own surprise that he was able to make one, but without hints as to how. Of course, as the web grows, Google gets smarter. Probably in a few days this article will also come up as a hit.

And yet I found the lack of an explanation of hexaflexagons intriguing. The Princeton flexagon club never published their Complete Theory of Flexagonation, and so the general knowledge of flexagons seemed to stop where Martin Gardner stopped.

The Nonahexaflexagon

Eventually I did find my nonahexaflexagon. Antonio Carlos M. de Queiroz posted an exhaustive catalog of hexaflexagons up to order 10, and several interesting nonahexaflexagon patterns are found here. The most intriguing one is the one de Queiroz labels "14-9", and that is the one that I decided to cut out.

Here is a video of some flexing of the nonaflexahexagon:

If you want to try making some of these flavors of hexaflexagons, you may find a triangular graph paper useful.

I have posted a PDF file for 8.5x11 triangle graph paper, with 1-inch triangles, here. The PDF was generated using this graph-paper generation website, but I hand-modified the PDF file directly to work around a bug in the generator.

An Undocumented Theory

But we are still left with the puzzle: how did de Queiroz generate his exhaustive list of hexaflexagons? What did Stone, Tuckerman, Tukey, and Feynman discover about the secrets of flexagons? There is little in the way of explicit explanation on the internet. What if I want to make an 11-flexagon?

An enticing tidbit from Google came from Australian teachers Rob Smith and Gordon Cowling, who had also scrutinized Gardner's complete list of patterns for 3-through 7-faced hexaflexagons.

Why did they look like this, and why were there no others? There seems to be no pattern at all.... I also scoured the internet, but invariably found that the author either knew no more than me, or seemed to assume that it was so obvious that it did not warrant explanation. Finally, I saw the pattern. It is so simple that I wonder it took me so long to see it.

But then what? In their hexaflexagon tutorial, Smith and Cowling maintain the tradition of omitting any clarification of the complete theory of flexagons. The details are left to us, dear readers, to figure out for ourselves.

Update: in response to an email query from Mark, I have posted a few photos of the steps I took to fold my nonahexaflexagon. If you look carefully, you will see that "folding in the same direction" means folding it in the same direction twistwise...

Posted by David at August 20, 2006 07:00 PM