A "flexagon" is a way of folding and rejoining paper to make a flat object which appears to have more than two sides. It can only show two of those sides at once, so you "flex" it - folding it and unfolding it in a particular way - to switch out which two sides appear at a time.
The word "flexagon" was coined by one of four people, all of whom were graduate students in physics or mathematics at Princeton in 1939. They were Arthur Stone, who apparently discovered the hexagonal format you're going to see in this demo; Bryant Tuckerman, Richard Feynman, and John Tukey. All four of them achieved some fame in their respective fields and one of them achieved a fair amount of fame, period. These four did all the initial "research" into the flexagon (read: they farted around with them a lot when they should have been studying.) You can find the whole story, and a lot more detail than this demo goes into, in Martin Gardner's The Scientific American Book of Mathematical Puzzles and Diversions (Simon and Schuster, 1959) - if you can manage to find a copy.
We're going to make two hexaflexagons (the hexa- just refers to the shape of the finished object; there are other kinds). We'll start with the simpler three-sided "trihexaflexagon," because it is easy and shows the basic principles; then we'll try for the six-sided "hexahexaflexagon."
You will need:
- a roll of adding machine paper (not thermal paper)
- a means of measuring a sixty-degree angle
- something to cut the paper with
- something to glue two surfaces of paper together with that will only glue those surfaces (e.g. won't seep through and glue anything else in a stack of layers); I used double-sided tape because 1) I had some and 2) you don't have to wait for that to dry.
Begin by cutting off the end of the tape at a sixty-degree angle. Discard the scrap end or use it as a bookmark.
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Using the sixty-degree angle you cut, you can now start folding the tape into equilateral triangles in a zigzag, using each triangle as a guide for where to fold the next one. The first two are shown in the picture. Although this is not precision science, do try to keep the corners en point and the triangles as exact as possible (which will get harder as the stack gets taller). Sloppy triangles make the structure harder to flex when it's done.
Of course you could take your protractor and mark out triangles on the tape and then fold them, for extra precision, and if you prefer that, be my guest. I'm lazy and this way is much faster.
You'll want to make at least ten triangles this way.
Cut off a section containing ten triangles; no more, no less.
I have pre-colored the triangles on my strip to make it easier for you to see where the folds will be. Each of these two steps shows one side of the colored strip. The side shown in this step has two triangles that aren't colored (there are pencil hatchmarks on them); those will get glued together later.
The flash obscured the colors a bit so I'll read them off to you, left to right: Glue triangle, red, yellow, yellow, blue, blue, red, red, yellow, glue triangle.
Here's the other side. I have flipped the strip over vertically, bottom to top.
The colors are, left to right: yellow, blue, blue, red, red, yellow, yellow, blue, blue, red.
The first fold we're going to make is between the two blue and two yellow panels at the right end in this photo. We're going to fold it up (mountain-fold, if you know origami) and bring the left end of the paper under, so it looks like the picture in the next step.
What we're doing here is folding the strip into thirds, with one end overlapping the other, so that the colors on a single face correspond. This is actually easier done than said. Here I have made the first fold, as described in the previous step, to bring four blue triangles to the same side.
Notice the glue panel at the far left, so you can tell which side you're looking at there.
The next fold will be between the blue panels and the yellow ones in this photo.
Here is the fold described in the previous step taking place. Notice I am bringing the strip under. As you will see when you do this yourself, in order to get all six blue triangles on the same side, that end I'm bringing under is going to have to come up over the red triangle seen here.
I've flipped the model over from the previous step so you can see that the stray triangle poking out (the yellow triangle in the previous image) happens to be a "glue panel" on its other side. And, hey, another glue panel is right next to it. How about that. Get your tape or your glue and fold that stray triangle over, bringing the two glue surfaces together forever.
Once your glue dries, you have made a trihexaflexagon.
Your brand-new flexagon is currently showing blue on one side and yellow on the other. I flipped it over, so you're seeing the blue side in my flex demonstration, but it would work just as well with the yellow side.
Your hexagon has six creases between triangles, right? Bring every alternating one together (down) and the others up, into the "cootie catcher" formation.
Now, obviously, there's two ways to do that, and one of them may not flex. You'll know if you've got the way that flexes because you will be able to "open" the thing at the top. Don't force the one that won't open; especially a flexagon you've just made, which may not open smoothly. It sometimes takes a little coaxing, but you'll definitely be able to tell the difference between "opens stiffly" and "I'm about to tear the paper."
You now have an object that is blue on one side and red on the other side. Continue flexing to cycle between the colors over and over.
Now, there's no special voodoo here. If you count triangles on each face, you get 3 faces x 6 triangles = 18 triangles. Add our two glue triangles, which cancel each other out because they're stuck together, and that's 20 triangles ... which is exactly what we had in the two-sided strip of ten triangles we began with.
Doing the math, if we want to make a six-sided hexaflexagon, we need 6 faces x 6 triangles + 2 glue = 38 triangles ... which, divided by two, is a strip of nineteen triangles.
Here is one side of the strip of nineteen you presumably made between these steps.
As before, I have pre-colored it to make life easier. The colors here, left to right, go red-blue-yellow six times, with a glue triangle at the right end.
(Because this strip has an odd number of triangles, the two glue panels will not be on the same side of the strip.)
Flipping the strip over top to bottom, the colors on the other side are: first a glue triangle; then orange, orange, green, green, purple, purple three times.
Still looking at this side, working left to right, bring the two orange panels together, then the two green panels together, then the two purple panels together, then the next two orange panels together, and so on. This step is "rolling it up."
This is what you get after you roll it up. Those two purple panels at the end shouldn't be showing; that flap should be down.
Here's why I did the trihexa first: What you have here, after "rolling it up," is an exact analogue of the trihexa strip, and it gets folded exactly like making a trihexa; you just have to be careful because of all the folds you've already made in it as you rolled it up. At no point in this process should any of the orange,green, or purple panels become exposed from where you've hidden them. That part's a done deal.
In these photos I'm going to try to put all the yellows on one side. Start by bringing the two blues in the middle together.
We're two folds along now. The end of the strip nearest you in this photo - with that glue panel that just won't stay folded down - was at the right side of the previous photo. Those two blue triangles at the back should be meeting each other now too.
Last bit of assembly, showing how the glue panels should end up positioned to be glued together.
Once your glue dries, you have made a hexahexa. Congratulations.
You will find that flexing this isn't as simple as the trihexa was. For one thing, the orange, green, and purple faces will not want to come out as often as the other three. In fact the Princeton team did some analysis of this (of course they did) and they found that in the most orderly algorithm for traversing the faces, the 1/2/3 faces appeared three times as often as the 4/5/6 faces did. You will have to go find the book I cited if you want more details on that.
I think of the 4/5/6 faces as the "inner faces" - once you get to them there is only one way to flex that side, whereas the outer faces can be flexed in multiple ways.
There are apparently ways to make hexaflexagons with other numbers of sides, but they don't involve straight strips and I've never been able to figure out how to fold them. Straight strips can only be used to make ones with a number of sides which is a multiple of three.
If you're daring, you could try to make a twelve-sided one (I guess that would be a dodecahexaflexagon). Calculate and make the correct length of strip (hint: 37) , and roll it up - you now have a strip analogous to the starting strip of a hexahexa. Roll it up again, as described here, and proceed, and good luck.
Now, an interesting quirk ...
What I've done here, because I was too lazy to make a second hexahexa, is I've divided each yellow triangle into thirds and put a distinctive mark in each of those. Notice all the circles are at the center.
Now I'm going to flex for a while, traversing through sides in no particular order ...
... and back to yellow HEY WAIT A MINUTE.
No, I didn't take it apart and redo it.
It turns out the triangles on some of the faces rotate (with respect to one another) as the thing is flexed.
A visual artist with time on his/her hands could decorate each side with more interesting designs than I've chosen here and get some very nice kaleidoscope-like effects when the thing is played with.
And that's about all I have to say about hexaflexagons. I'll make a tetraflexagon for you one of these days, but that's a different demo.