Hexaflexagons

Now that the PSLE is over, I can finally do more interesting things! Like making hexaflexagons.

Hexaflexagons look like this.

Hexaflexagon

J. Nathan Matias

The difference between this and a normal paper hexagon is that it can flex. If you make one of these and push down every other crease(there should be three of these creases) as far down as you can, the inside will open up and reveal a new face. You can do this again and again infinity times, although flexing the simplest type of hexaflexagon three times will cause it to go back to the first face. Note that this only works one way for the simplest type of hexaflexagon(the trihexaflexagon, named because of its three faces); if the three creases you choose do not cause the inside to open up, the other three creases will.

To make the trihexaflexagon, you need to make a strip of nine equilateral triangles and some tape, or ten if you want to use glue. It’s a bit difficult to explain in words, so here are the instructions. You can draw designs on them too.

There are instructions online on how to make a double trihexaflexagon as well, also known as a hexahexaflexagon because it has six faces. You need twice as many triangles though. I also made a quadruple trihexaflexagon(a dodecahexaflexagon?) but when I was flexing it, it somehow turned into two square pyramids without a bottom, which also happens with the hexahexaflexagon. I tried to get it back to a hexagon but I couldn’t so I kept flexing it into weird polyhedra made of equilateral triangles, such as square pyramids stuck to a tetrahedron, until it finally turned back into a hexagon! Success!

Or so I thought until I realised that two of the triangles had somehow flipped over and now showed a different design! Aargh! It also happened to many other ‘sides’ of the quadruple trihexaflexagon but with different numbers of triangles switched. Some of them had 3 triangles switched, some had only 1, and lucky sides escaped this horrible mutilation of hexagons with their designs still all on one face. I had to find the point where I glued the quadruple trihexaflexagon together and carefully peel it apart, then re-fold the whole thing. So be careful with these things!

If you’ve read the instructions for the hexahexaflexagon you might know that it’s pretty easy to make this family of hexaflexagons by just doubling the number of triangles for each new hexaflexagon. I have not, however, tried making a triple trihexaflexagon. Maybe I should, but I can see some problems with it. Oh well.

For more about the history of hexaflexagons, see Vi Hart’s video on hexaflexagons and the sequel, and also the safety guide.

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