Hexaflexagons 2

So say you're Arthur Stone, and you're showing your hexaflexagon to your friend, Tuckerman, 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 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 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.