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• So say you just moved from England to the US

• and you've got your old school supplies from England

• and your new school supplies from the US

• and it's your first day of school and you get to class

• and find that your new American paper doesn't fit in your

• old English binder.

• The paper is too wide, and hangs out.

• So you cut off the extra and end up with all these strips of paper.

• And to keep yourself amused during your math class

• you start playing with them.

• And by you, I mean

• Arthur H. Stone in 1939.

• Anyway, there's lots of cool things

• you can do with a strip of paper. You can fold it into shapes.

• And more shapes.

• Maybe spiral it around snugly like this.

• Maybe make it into a square.

• Maybe wrap it into a hexagon with

• a nice symmetric sort of cycle to the flappy parts.

• In fact, there's enough space here to keep wrapping the strip,

• and then your hexagon is pretty stable.

• And you're like, "I don't know, hexagons aren't too exciting,

• but I guess it has symmetry or something."

• Maybe you could kinda fold it

• so the flappy parts are down and the unflappy parts are up.

• That's symmetric, and it collapses down into these three triangles,

• which collapse down into one triangle, and collapsible hexagons are,

• you suppose, cool enough to at least amuse you a little bit during your class.

• And then, since hexagons have six-way symmetry,

• you decide to try this three-way fold the other way,

• with flappy parts up, and are collapsing it down

• when suddenly the inside of your hexagon decides to open right up.

• What? You close it back up and undo it.

• Everything seems the same as before,

• the center is not open-uppable.

• But when you fold it that way again,

• it, like, flips inside-out. Weird.

• This time, instead of going backwards,

• you try doing it again. And again. And again. And again.

• And you want to make one that's a little less messy,

• so you try again with another strip and tape it nicely

• into a twisty-foldy loop. You decide

• that it would be cool to color the sides,

• so you get out a highlighter and make one yellow.

• Now you can flip from yellow side to white side.

• Yellow side, white side, yellow side, white side

• Hmm. White side? What? Where did the yellow side go?

• So you go back, and this time you color the white side green,

• and find that your paper has three sides.

• Yellow, white, and green.

• Now this thing is definitely cool.

• Therefore, you need to name it.

• And since it's shaped like a hexagon and you flex it

• and flex rhymes with hex, hexaflexagon it is.

• That night, you can't sleep because you keep thinking

• And the next day, as soon as you get to your math class

• you pull out your paper strips.

• that folds into again, the shape of a piece of paper,

• and you decide to take that

• And use it like a strip of paper to make a hexaflexagon.

• Which would totally work, but it feels sturdier

• with the extra paper.

• And you color the three sides and are like,

• Orange, yellow, pink.

• And you're sort of trying to pay attention to class.

• Math, yeah. Orange, yellow, pink.

• Orange, yellow, white? Wait a second.

• Okay, so you color that one green.

• And now it's orange, yellow, green. Orange, yellow, green.

• Who knows where the pink side went?

• Oh, there it is. Now it's back to orange, yellow, pink.

• Orange, yellow, pink. Hmm. Blue.

• Yellow, pink, blue. Yellow, pink, blue. Yellow, pink, huh.

• With the old flexagon, you could only flex it one way,

• flappy way up.

• But now there's more flaps. So maybe you can fold it both ways.

• Yes, one goes from pink to blue,

• but the other, from pink to orange.

• And now, one way goes from orange to yellow,

• but the other way goes from orange to...neon yellow.

• During lunch you want to show this off

• to one of your new friends, Bryant Tuckerman.

• which you call the trihexaflexagon.

• And he's like, whoa!

• and wants to learn how to make one.

• And you're like, it's easy! Just start with a paper strip,

• fold it into equilateral triangles,

• and you'll need nine of them, and you fold them around

• into this cycle and make sure it's all symmetric.

• The flat parts are diamonds, and if they're not,

• then you're doing it wrong.

• And then you just tape the first triangle to the last

• along the edge, and you're good.

• But Tuckerman doesn't have tape.

• After all, it was invented only 10 years ago.

• So he cuts out ten triangles instead of nine,

• and then glues the first to the last.

• Then you show him how to flex it by pinching around a

• flappy part and pushing in on the opposite side to make it

• flat and trianglly, and then opening from the center.

• You decide to start a flexagon commitee together

• to explore the mysteries of flexagation.

• But that will have to wait until next time.

So say you just moved from England to the US

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B1 yellow pink orange flappy fold paper

# Hexaflexagons

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林宜悉 posted on 2020/03/30
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