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### Course: Math for fun and glory > Unit 1

Lesson 3: Hexaflexagons# 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. Created by Vi Hart.

## Want to join the conversation?

- She is amazing WOW! (And its funny)(6 votes)
- Wait, why are we celebrating Martin Gardener's birthday instead of Arthur Stone's birthday? After all, all Martin Gardener did was popularize hexaflexagons, while Arthur Stone
**invented them**.(5 votes) - why did she color them?(3 votes)
- She colored them so she knew what side she was on.(1 vote)

- kind of confusing and you show show how to make them(3 votes)
- How do you make a hexaflexagon?(2 votes)
- http://www.puzzles.com/hexaflexagon/activities.html

This website has a very good explanation on the trihexaflexagon.

If you don't trust my link, I understand, I've been hacked too. To get another link like this one just google "how to make a hexaflexagon". That's what I did. Vi also gives some recommendations for websites as well.(2 votes)

- my friend Cooper just made a hexa-hexaflexagon with only 18 triangles how is this possible.

#hexaflexaperplexia(3 votes) - Do you think hexaflexagons snow flakes are a good idea?(3 votes)
- can i get an instructional guide to make these i cant do itt

or maybe someone just tell me i dont know(3 votes) - I have a question , no mater how hard I try I can't make a hexaflexagon with 6 sides(3 votes)
- What is the most unusual material anyone here has made a flexagon out of?(2 votes)

## Video transcript

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.