And you're going to complain, that uh, wait a minute, we haven't done two Möbiuses of the same chirality.

But we'll visit that because that one is, shall I say, disappointingly interesting.

Or is it, interestingly disappointing?

It's not going to be something as romantic as this but, it is nonetheless, quite baffling.

But in the first instance, let's get to grips with why you get flat squares.

There is a good way to think abut this.

You see as usual in science, in general and in mathematics in particular, sometimes if you

take the, a good point of view, everything becomes clean and amusing. But if you take the wrong point of view

everything becomes ugly and really painful.

So, here is a straight against straight and we saw that if you cut this along the center line,

so you get the flat square, but now, let's understand why we get the flat square.

Until now, in a silly attempt at fair tricks, I was leaving the cross cut until the end.

A good point of view, however, is to cut one of the pieces completely around without any suspense.

And then, of course it will remain for us to cut the remaining piece along the centre line

and we'll attempt that in a moment.

When we cut one of the pieces all the way around the centre line, what we get is this.

We get a pair of handcuff-like objects connected by a straight belt

between. Untwisted belt.

So what we have to do is cut this belt but along the centre line

and at this stage I think you can already visualize what's happening.

What are we going to get when we finish cutting this?

Well, because when we are in this configuration and when we open it we get a square.

So you see where it comes from? It came from

this. You can see that was straight against straight, cut open,

and cut open, that's a square.

So we now understand, we can visualize in our heads how this square comes about.

How about straight against Möbius?

That's actually an interesting case. We shouldn't

make the suspense thus. We should cut one of the pieces all the way around.

Which piece, it turns out that it's helpful to cut this straight part all the way around.

And we know that we get the flat square and our job is to understand why we get the flat square.

Okay, after we cut the straight part, what you get is this.

You can visualize right?

It's a pair of handcuffs again, but this time connected by a belt which is twisted.

Or if you like

you can have an untwisted belt but now handcuffs on opposite sides of the belt.

So now-- So the previous case, straight against straight had handcuffs on both sides, both on the same sides

but here on this end and on this end, okay.

Now what's going to happen if we cut this along the centre line? That's the finishing job but

before we do this, let me work a piece of magic.

I take this and then I turn this around

So I get exactly the same thing as the result

of two straight against straight. Two straight strips.

-And you're allowed to do that?

-That's not breaking any rules?

Well, who knows but it's the same piece of paper.

And I can start and finish where ever I like so if I cut this, I am going to get a square

This operation that I have just, this shenanigan that I have just performed in front of you

corresponds to the following thing.

If I cut this into, along the centre line

what you would get is not in the first instance

a flat square but you'd get something like this

Kind of a twisted version of flat square.

Like that. And

all you have to do is to

untwist it and then it becomes flat.

and that corresponds to this operation of

taking this and then

turn around and here we are.

You get another flat square.

Okay, that's very nice!

Well it remains for us to try two Möbiuses

with the same chirality or the same twist.

Let's remember which way we are making the Möbius.

Let's twist it clockwise and the other one also

let's twist it clockwise.

By the way some of you might be worried, well, this is clockwise seen from this side but if you see it from this side it turns out to be the same thing.

So two of them clockwise, so we have

two Möbius strips

exactly of the same chirality

glued against each other.

I'd like to emphasize that this object is very very symmetric.

Two copies of the same Möbius strips, same including orientation

twisted the same way, glued at right angles

So if I position this way and turn it around this way you obtain exactly the same thing.

There's a symmetry there. A rotational symmetry.

On the other hand cutting this object along the centre lines is of course a symmetric operation.

I mean, it doesn't distinguish left or right.

So we are applying a symmetric operation to a symmetric object.

And there's a widespread believe, indeed a conviction in science that if you have a symmetric situation and apply a symmetric operation what you get out is symmetric.

Okay, let's see if that's going to happen.

I'm going to cut this along the centre lines, respective centre lines.

A totally symmetric object and I'm doing a symmetric operation.

And you see why I call it interestingly disappointing, or disappointly interesting, in a moment.

And I cut it all the way around .

So here I go.

I finish cutting

everything.

And what emerges is first of all disappointing

because they separate. Oooh! How un-romantic, you know!

As opposed to the linked hearts before.

But there's something really wrong about this

unromantic object. You see I got two pieces and one of them looks like a boat

and this is what mathematicians like to call a 2-gon, you see. You know n-gon because it only has two vertices it's a 2-gon.

And it looks like a boat.

The other one is also a 2-gon but it's twisted in space

so you can not actually untwist it

however hard you try

into this shape.

So, from a symmetric object cut along in a symmetric fashion we got two

objects that are not the same as symmetric object. How did this happen? It's really strange.