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So say you just moved from England to the US
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and you've got your old school supplies from England
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and your new school supplies from the US
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and it's your first day of school and you get to class
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and find that your new American paper doesn't fit in your
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old English binder.
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The paper is too wide, and hangs out.
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So you cut off the extra and end up with all these strips of paper.
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And to keep yourself amused during your math class
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you start playing with them.
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And by you, I mean
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Arthur H. Stone in 1939.
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Anyway, there's lots of cool things
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you do with a strip of paper. You can fold it into Shapes
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and more shapes.
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Maybe spiral it around snugly like this.
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Maybe make it into a square.
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Maybe wrap it into a hexagon with
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a nice symmetric sort of cycle to the flappy parts.
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In fact, there's enough space here to keep wrapping the strip,
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and the your hexagon is pretty stable.
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and you're like. "I don't know, hexagons aren't too exciting,
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but I guess it has symmetry or something."
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Maybe you could kinda fold it
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so the flappy parts are down and the unflappy parts are up.
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That's symmetric, and it collapses down into these three triangles,
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which collapse down into one triangle, and collapsible hexagons are,
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you suppose, cool enough to at least amuse you a little but during your class.
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And then, since hexagons have six-way symmetry,
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you decide to try this three-way fold the other way,
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with flappy parts up, and are collapsing it down
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when suddenly the inside of your hexagon decides to open right up
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What, you close it back up and undo it.
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Everything seems the same as before,
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the center is not open-uppable.
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But when you fold it that way again,
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it, like, flips inside-out. Weird.
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This time, instead of going backwards,
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you try doing it again and again and again and again.
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And you want to make one that's a little less messy,
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so you try with another strip and tape it nicely
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into a twisty-foldy loop. You decide
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that it would be cool to colour the sides,
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so you get out a highlighter and make one yellow.
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Now you can flip from yellow side to white side.
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Yellow side, white side, yellow side, white side
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Hmm. White side? What? Where did the yellow side go?
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So you go back and this time you colour the white side green,
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and find that your piece of paper has three sides.
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Yellow, white and green.
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Now this thing is definitely cool.
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Therefore, you need to name it.
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And since it's shaped like a hexagon and you flex it
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and flex rhymes with hex, hexaflexagon it is.
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That night, you can't sleep because you keep thinking
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about hexaflexagons.
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And the next day, as soon as you get to your math class
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you pull out your paper strips.
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You had made this sort of spirally folded paper
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that folds into again, the shape of a piece of paper,
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and you decide to take that
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and use it like a strip of paper to make a hexaflexagon.
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Which would totally work, but it feels sturdier
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with the extra paper.
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And you color the three sides and are like,
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orange, yellow, pink.
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And you're sort of trying to pay attention to class.
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Math, yeah. Orange, yellow, pink.
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Orange, yellow, white? Wait a second.
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Okay, so you colour that one green.
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And now it;s orange, yellow, green, Orange, yellow, green.
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Who knows where the pink side went?
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Oh, there it is. Now it's back to orange, yellow, pink.
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Orange, yellow, pink. Hmm. Blue.
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Yellow, pink, blue. Yellow, pink, blue. Yellow, pink, huh.
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With the old flexagon, you could only flex it one way,
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flappy way up.
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But now there's more flaps. So maybe you can fold it both ways.
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Yes, one goes from pink to blue,
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but the other, from pink to orange.
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And now, one way goes from orange to yellow,
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but the other way goes from orange to neon yellow.
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During lunch you want to show this off
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to one of your new friends, Bryant Tuckerman.
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You start with the original, simple, three-faced hexaflexagon,
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which you call the trihexaflexagon.
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and he's like, whoa!
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and wants to learn how to make one.
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and you are like, it's easy! Just start with a paper strip,
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fold it into equilateral traingles,
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and you'll need nine of them, and you fold them around
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into this cycle and make sure it's all symmetric.
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The flat parts are diamonds, and if they're not,
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then you're doing it wrong.
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And then you just tape the first triangle to the last
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along the edge, and you're good.
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But Tuckerman doesn't have tape.
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After all, it was invented only 10 years ago.
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So he cuts out ten triangles instead of nine,
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and then glues the first to the last.
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Then you show him how to flex it by pinching around a
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flappy part and pushing in on the opposite side to make it
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flat and traingly, and then opening from the centre.
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You decide to start a flexagon committee together
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to explore the mysteries of flexagotion,
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But that will have to wait until next time.