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• So say you're Arthur Stone, and you're

• and you've already blown his mind by showing him

• it has three sides-- orange, yellow, pink, orange, yellow,

• pink-- but now you're about to extra super blow

• his mind by showing him that there's even more colors.

• And he's like, whoa, where did the blue side come from?

• But you're having trouble finding all six.

• Like, you know there's a green side somewhere in here,

• but where is it?

• You're all like, OK, Tuckerman, I think I found the green side.

• It's right in here.

• Anyway, Tuckerman immediately decides

• he needs to discover the fastest way

• to get to all the colors, which he calls the Tuckerman

• traverse.

• So you and Tuckerman are working on that,

• and there's hexaflexagons all over the lunch table,

• and another student is curious about what you're doing

• and wants to join your committee.

• His name is Richard Feynman.

• So stop being Arthur Stone, and start being Brian Tuckerman.

• So you're Tuckerman, and you teach Feynman

• how to make the hexa-hexaflexagon

• by first folding a strip of 18 triangles with the 19th

• for gluing.

• You and Stone have just figured out how to number

• the faces before you fold them by dissecting

• a perfect specimen.

• You number them 1-2-3, 1-2-3, 1-2-3, 1-2-3, 1-2-3, 1-2-3.

• Glue on one side.

• Flip it, and glue 4, 4, 5, 5, 6, 6, 4, 4, 5, 5, 6, 6, 4, 4, 5,

• 5, 6, 6 on the other.

• You coil it around so that you get ones and twos

• and threes on the outside like 1, 2, 2, 3, 3, 1, 1 2, 2, 3, 3,

• and then fold that around into a hexagon,

• so that all the twos are on the front.

• And then flip it, and glue the two blue parts together,

• so that all threes are on the back.

• Feynman has some trouble flexing it,

• but you show him how to pinch two triangles together and then

• push in the opposite side.

• He somehow still does it wrong and ends up doing it backwards,

• flexing in reverse.

• Now he's all intrigued by all the flexing possibilities,

• and you're like, let me show you the Tuckerman traverse.

• But Feynman, being Feynman, is like, we must create a diagram.

• And Tuckerman's like, really, it's not that hard.

• No, diagram.

• So you're Feynman, and you've already

• seen you can cycle from one to two to three, one, two, three.

• So you write that down with arrows and stuff.

• Or you can go backwards, but from one, two, and three, you

• can also flex the other way, in which case one goes to six,

• or two to five, or three to four.

• And if you did one to six, once you're at six,

• you can only flex one way, because the other doesn't work.

• You have to go to three or backwards back to one.

• But then you notice that if you go to three,

• you can only flex one way, and the other is un-open-up-able.

• But before when you were on three,

• you could go either to one or four,

• but now you can only go to one.

• And you can go backwards to six, but not

• backwards to two, which means that this three isn't

• the same three as the first three.

• Somehow it's the same color, but in a different state.

• You show this to your friend John Tukey,

• and he's like, oh yeah, that makes sense.

• And he draws a star in the middle of your three

• and sits back as if that explained everything.

• So you're like, whatever, and flip it back around

• to get to the other three and check it.

• The star turns into a not star.

• And from this alternate three, there's

• this 1-6-3 loop that connects to the main loop at one, which

• is the same one as one has always been.

• But there's a different one off of the main two

• in the 2-5-1 loop.

• And of course, everything looks different if you flip it over.

• And these threes are also different,

• because they have different numbers on the other side.

• And you complete a diagram of possibilities,

• which allows you to find the optimal Tuckerman traverse.

• You also diagram the original trihexaflexagon,

• which is pretty simple.

• The flexagon committee approves your diagrams

• and decides to call them Feynman diagrams.

• Everything is going great until 1941,

• because suddenly there's important war stuff to do,

• and flexagons are largely forgotten.

• OK.

• Now fast forward 15 years, and be Martin Gardner.

• You're an amateur magician, and you're

• hanging out at your friend's place

• Anyway your friend shows you something

• you've never seen before-- a big flexagon he's

• And you're thinking, hey, this is awesome.

• So you write an article for Scientific American,

• and soon you've landed yourself a gig writing a regular column

• about recreational mathematics called "Mathematical Games,"

• and it's a huge success and gets hundreds of comments.

• I mean, letters, and there's nothing else like your column.

• And all the cool people are inspired by you,

• and you're pretty much the reason

• why people know about things like tangrams,

• and Conway's Game of Life, and the work of MC Escher,

• and other things like that.

• Now fast forward 50 years, and say you're

• me in the generation of people inspired by Martin Gardner

• are now the people inspiring you.

• So he's your math inspiration grandfather.

• And now you yourself are in the business of mathematically

• inspiring people, and you want them

• to be aware of their math inspiration heritage.

• OK, now say you are you.

• If you think hexaflexagons are cool

• that was just column number one.

• And I invite you to join in with the hundreds of people

• to celebrate Martin Gardner's birthday every October 21.

• This year, there will be hexaflexagon

• parties in homes and schools all over the world.

• And if you want to attend or host one,

• check the description.

• I'm celebrating by making these videos,

• and also I just like the image of flexagons everywhere--

• floating around lunch tables, spilling out of your pockets,

• lost in your couch cushions.

• I like to keep some ready to deploy out

• of my wallet or tiny yellow purse,

• in case of a flexagon emergency.

• And then there's more recent innovations

• in flexagon technology, and all the cool ways to color

• them, and other stuff.

• But that will have to wait until next time.

So say you're Arthur Stone, and you're

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林宜悉 posted on 2020/03/30
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