Flexagons are a staple of recreational mathematics, but it's always the
hexaflexagon that you see. One based on a hexagon. It's got six edges.
And so I'd - you know
someone called Vihart has made a whole series of videos about these things. I'm gonna go for the forgotten
flexagon. The tetraflexagon, one with four edges. And to make it we're gonna use a square of paper.
So the first step is to fold it in half
whichever way you want,
they're both the same. And one of the reasons why I love the
tetraflexagon is you don't need any glue. If you're making any of the hexaflexagons
you have to get some glue and stick bits together,
whereas this one is just folding and cutting. So the first half and half has split it into
four sections long ways. I'm now gonna fold it this way and
if I make the creases really, like good creases, because they're gonna kind of be used as
hinges later, I should end up with;
there we go - it's a piece of paper now split into 16 smaller squares. I'm gonna remove the center four squares.
I'm gonna keep this ring of 12 squares intact. I'm going to take these middle ones out.
You could do that by stabbing the scissors in and just cutting around but as
we learnt from Katie Steckles in her fold and cut theorem video,
you can do anything you want in a single cut if you're a bit clever about how you fold it.
So one cut down there, come on,
and we get - that's not bad.
Okay, there's the center four squares gone, we can get rid of those, and
now we've got the twelve round the outside. So this is now going to be three easy steps and one difficult step.
So we start on the left and we go around clockwise. First step is easy,
take the four on the outside, fold them up and over. Everything should line up.
Second step is we come around, take the three at the top, fold them up and over and they come down like that.
Next step as we keep coming around is take these three, fold them up and over, and that is the end of the easy steps.
The only difficult step is when you take up the bottom ones, you want to lift these up as if we did it
before that first fold. Because we want this to be symmetric so there's no beginning or end.
So what you've got to do is move the original fold out of the way without ruining it
then lift this one up and make sure the bit next to it goes underneath.
So we kind of take all of this bit next to it and just feed it through
until it's underneath, and the rest should follow, and just kind of snap into place and there you are. There is your four edged
flexagon - the tetraflexagon. So what's amazing about the tetraflexagon is it's got a lot of faces.
So we've got face one on the front here, which I'm going to number with ones and then if we fold it over. Okay
so ones on the front. We've got twos on the back. So face one and face two.
So you got twos on that face,
okay, you got ones on that face, and then on the other side of the ones we've got - well it's blank so we'll label that with
a three. So that's face three. So this is a shape with at least three faces. We got face three there. We got face
one there, and I got face two there.
And what happens is every time you fold it closed one way - so I'm forwarding the ones closed with the twos on the outside -
I can open it up on the opposite side.
So it closes this way
but then I open it the other way around and I get a new face. And so I'm gonna label that one -
where have I gone so far? I've got one there. I got two there. Let's label this one four
and then let's see, if we fold this this way, no that way,
then go this way - there's another one, that's five.
You're looking for six faces because if we think of the original shape we made - there's my ring of twelve.
So there's 12 on the front. There's 12 on the back. So I've got 24 of these little squares.
I'm using up four for each one
so if I divide that by 4 I get
6. So there are enough squares to make 6, whether or not it does is a slightly different question.
4 is there - oh, there it is, there's 6. Okay. So, I'm gonna label this one here 6.
And if we actually want to check that we've got all of them; face two,
fold it in - face one, fold it in - face three.
Okay, we could pull the whole thing apart. So if we pull it back out again,
disassemble it, there we go. Okay, so there's all the fac-
we've numbered every single square at some point, to make all of them.
There are two different ways to put this back together.
And if you do it the wrong way, you will get the mirror image kind of what we started with.
The numbers won't line up again. But if I carefully put it back together the same way we folded it originally,
we're back to where we started. There we go.
And once you make one of these - it's kind of fun. You play with it for a while. If
you've not made one before it takes a bit to work out where all the faces are.
Obviously, you don't have to - I've just put numbers. I'm not very creative.
If you were one of these doodling mathematicians you would draw pictures or you know be artistic.
I've just numbered it which is a bit sad. I'm sorry. - (Brady: It's Numberphile!) - It is Numberphile,
it's not doodlephile is it so...
I've numbered it. So you could be as creative as you want and you can spend a while trying to find them,
but if you are into numbers, after a while you're like, hang on. I know there are only six faces, but
what combinations of faces can I get at the same time?
Obviously I can get one opposite two, first one I started with. I can get three opposite one,
can I get three opposite four?
Can I get five opposite - like, what combinations of faces can you see at the same time?
And that was the first thing I started wondering
when I played with one of these. And for a while, I thought I - you can get
combinations of faces, but there's always a one and a two.
You never lose those original two faces. But no,
here's a five opposite of four. So you can - you can get reasonably lost right? The original two faces are long gone.
We have two new faces on the outside and I, when I heard it I was like I want to work out the logic behind
this. What is it, which means I can get different pairs? And which ones are possible? And how do I go between them?
And so I drew a diagram.
Okay, diagram. We're gonna use colours, because that's how we roll here. This is as close to artistic as we get so
what face is definitely possible? So I first did this - actually I didn't do colours. What I did
I was kind of lazy. I was like, okay, I'm gonna label that as being one,
Two. So it's not a fraction 1/2. It's a one opposite of two,
I was using my own notation. And then I went okay, I can go from that to one, three.
So I put 1/3 on my diagram. And then I realised, I was like wait a minute.
One face always stays there. So whenever you flex, whichever one goes on the outside
is still there when you open it up. You only change one at a time.
So when you go from 1/3 to 1/2, the 1 doesn't change. I was like, well I can kind of imagine
those ones being linked. And so I can go from 1/3 to 1/2.
If I keep the 3 I can get to 3/6, so over here somewhere.
I've got 6 and 3 and I've kind of linked those up like that,
right, and I started drawing the network to try and work it out. I'm gonna show you the network
I ended up with because I'm particularly pleased with it.
First of all, I'm gonna put this back to its starting state. One's gonna be red. So this - there's one and one can be opposite...
let's do light blue. It can be light two. There we go, there's two. So one is opposite two.
Okay, so, let's see. What else can I get opposite one? Well, I can go from one - that can be opposite three.
Okay, so I can have one three so I'm gonna put one
three down here - that can be purple, or
one can be opposite - It's actually. it's horizontal and vertical.
So if I fold it
vertical fold you get three, if I fold a horizontal fold you get the other one. So it's five in there.
So five, you're gonna be an orangey colour, and we'll put five, uh,
here.
And so what I've done is I have run all the
ones in a line down here. So this is all the ones you get, which have a one in them.
And so I've still kind of got them on either side so you can still see what's going on.
Okay, we'll do two next. So two, if we fold it
horizontally, it's opposite six. So that's two opposite six that way
so that's going to be six with two. And
then if I hadn't folded it horizontal, if I fold it vertically two is opposite four.
So four is gonna be dark blue and so that's gonna be down here.
So now I have lined up all the faces with two and actually we've got another line going this way.
So there are five combinations we found first, and how you can go between them, but there are other ones. So if I go
forward two - let's go back over here. There's one, two in the middle again. Let's go out to one of these.
Let's go out to one opposite five. So we're over here. If I keep five there I can get five
opposite four! Okay four's over here. So what I can actually do is
extend the five line
down here
and then I can extend the full line up here,
and then I've got five, four. And that predicts I can pop down here to four, two. Because I've, like,
preemptively locked that in. But if I keep four on the outside close it - sure enough. There
it is. Four, two. So that's dropped me down there. And I predict, on the opposite side over here if we extend
six down there, and we extend three up there, that predicts we can have a six, three over there. Sure enough
let's get over that side,
as
where should we go? So we're at four, two. Let's go back to two, one.
Oh we're back at two, one. Done that. Let's go,
let's go down. We'll go to one, three. Which way's that? We don't know.
There's one opposite three. And then we should be able to pop up there to be three opposite 6, sure enough, there it is.
There's three and six, right? And then that's it.
These other ones,
they don't fill in. I was able to create - it's kind of a graph, but I've put some structure into it.
I've had straight lines.
So I've got 6 different coloured lines, and whenever two of those lines crossed that combination of faces is possible.
So originally I just kind of sketched them all over the place,
and then I realised there was some logic underneath them, and I was able to then arrange it to show the structure behind
what combinations of faces are possible. And that pleased me immensely.
So I've been throwing some names of flexagons around. The way it works is you've always got the number of
faces, followed by the number of edges, followed by flexagon. So this has got four edges.
So this is a tetraflexagon. And because it's got six faces it's a hexa-tetra-flexagon. Six faces, four edges,
flexagon. So the ones you see online all the time are hexa-flexagons;
because they're made from hexagons,
they've got six edges. And then however many faces. The easy one, the one you see all the time is the
tri-hexa-flexagon, it's only got three faces. This one gives you six, out of the box. With a lot of work you can make a
hexa-hexa-flexagon. So six edges, six faces. My goodness, it's a real pain.
So, the tri-hexa takes a bit of work, making a chain of
nine equilateral triangles? Technically ten 'cause you'll want to glue.
Once you've got those ten
you've got to fold them, and glue them, and you only get three faces. If you make a chain of 19
equilateral triangles, you've got to fold it. You've got to fold it again. You've got to glue it, right.
You've got to realise you did that wrong, pull it apart. In my experience and a lot of effort later you get six faces.
Whereas this - no glue!
No messing around. Cut out a square. Cut out a smaller square. Bit of folding.
Six faces are all yours. And the middle bit - guess what you can do with this!
I think we all know what we're gonna do with this.
You're gonna make yourself a *smaller* tetraflexagon. And the bit you cut out of that - you're gonna make a smaller one again, right?
You're gonna make fractal - you're gonna make gradually smaller and smaller. An infinite
series of
diminishing - they do approach a limit of total amount of paper. Hexatetraflexagons, like, ah! This, the forgotten
flexagon is best flexagon.
I've sat at a conference, albeit a recreational math conference, at the conference dinner. And the person next to me has gone
Do you wanna see some new flexagons? And I've gone yes!