Hexaflexagons 2 This video is based on, and in honor of, Martin Gardner's column from 1956, "Hexaflexagons," which can be found here: http://maa.org/pubs/focus/Gardner_Hexaflexagons12_1956.pdf. For more information, see the top comment below.
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- So say you're Arthur Stone and
- you're showing your hexaflexagon to your friend Tuckerman
- and you already blow his mind by showing him it has 3 sides.
- Orange, yellow, pink. orange, yellow, pink.
- but now you're about to extra-super-blow his mind
- by showing him there is even more colors
- and he's like, 'wow! Where did the blue side come from?'
- but you're having trouble of finding all six. Like,
- you know that there is a green side somewhere, but where is it?
- you're like, "OK Tuckerman, I found the green side"
- It's right in....here. Hmm.
- Anyway Tuckerman immediately decided that he needs to discover the fastest way to get to all the colors
- which he called "The Tuckerman Traverse"
- so you and Tuckerman are working on that
- And there's hexaflexagons all over the lunch table and then
- 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 a hexahexaflexagon by first
- folding a strip of 18 triangles with a 19th for gluing.
- You and Stone have just figured out how to number the faces
- by dissecting a perfect specimen.
- You number them 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 1s and 2s and 3s on the outside, like:
- And then fold THAT around into a hexaflexagon so that all the 2s are on the front.
- And then flip it and glue the two "GLUE" parts together, so that all 3s 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 backwards, flexing in reverse.
- Now he's all intrigued about all the flexing possibilities and you're like
- "Let me show you the Tuckerman Traverse!"
- But Feynman, being Feyynman, is like "We must create a diagram!"
- and Tuckerman is like "Really, it's not that hard-"
- "NO! Diagram!"
- So you're Feynman and you've already seen you can cycle from 1 to 2 to 3,
- So you write that down with arrows and stuff.
- Or you can go backwards. But from 1, 2, 3 you can also flex the other way, in which
- 1 goes to 6, or 2 to 5, or 3 to 4. And if you did 1 to 6, once you're at 6 you can only flex 1 way because the other doesn't work
- You HAVE to go to 3. Or backwards, back to 1.
- But then you notice that if you go to 3 you can only flex one way and the other
- is un-open-up-able but before when you're on 3 you can go either to 1 or 4
- but now you can only go to 1.
- And you can go backwards to 6 but not backwards to 2.
- Which means that this 3 isn't the same 3 as the first 3. Somehow, it's just the same color,
- but in a different "state".
- You show that to your friend, John Tukey, and he's like:
- "Oh, yeah, that makes sense."
- And he draws a star in the middle of your 3 and sits back as if that explained everything.
- So you're like "whatever."
- and flip it back around to get to the other 3 and check it - the star turns into a...not star.
- And from this alternate 3 there's this 1-6-3 loop that connects to the main loop at 1,
- which is the same 1 as 1 has always been.
- But there's a different 1 off of the main 2 in the 2-5-1 loop.
- And, of course, everything looks different when you flip it over,
- And these 3s are also different because they have different numbers on their other side.
- But you complete a diagram of possibilities, which allows you to find the optimal Tuckerman's Traverse.
- You also diagram the original trihexaflexagon, which is pretty simple.
- The Flexagon Committee approves of your diagram 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.
- Okay, now fast-forward 15 years and be Martin Gardner. You're an amateur magician,
- and you're hanging out at your friend's place, talking about magician stuff.
- Anyway, your friend shows you something you've never seen before - a big flexagon he's made out of cloth.
- And you're thinking "Hey, this is awesome! maybe other people would like to know about this "flexagon" thing."
- So you write an article for Scientific American and since you've lended yourself a gig
- writing a regular column about recreational mathematics called "Mathematical Games",
- And it's a HUGE success, and it 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 works of M. C. Escher,
- and other things like that.
- Now fast forward 50 years and say you're me and the generation of people inspired by Martin Gardner
- are now 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.
- Okay, now say you are....you.
- If you think Hexaflexagons are cool that is just column No. 1,
- And I invite you to joining with the hundreds of people who celebrate Martin Gardner's birthday every October 21st.
- This year there will be hexaflexagon parties in homes and schools all over the world and if you want to attend
- our 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, falling out of your pockets,
- Stuck in your head 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.
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