and just cut loose like this.

And Brady, can you hold this flap, please?

Ok, we shall call this the "first state",

so, this flap is, um, above what he's holding.

Now please release it.

He can hold it like this and this is going to be the "second state", ok?

This flap is below what he's holding.

Now, you cannot go from the first state to the second state

because he's in the way, yeah?

And you cannot, you know, go from the first to the second.

Well, if these are made of rubber, something flexible,

then I can stretch it and pass it over his body and come back underneath

But it's not rubber, I mean, it's real paper, so you can't do this.

So, you cannot go from first to second.

Now, please close your eyes for a moment, and...

Okay, keep it closed...

Now, I'm going to do something

And open your eyes

What!? It went from the first to the second

And he was holding this all the time and he didn't let it go

so how did it happen?

That's really strange.

From this state

to this state

while this is always being held

So, let's do it with eyes open this time

Can you hold it again?

Now, you can keep the eyes open

First I'm going to roll it like this

and that's like the pair of handcuffs

Or, if you want, you can think of

circle here and then imagine another imaginary circle which makes it a torus

And I, mathematician like to say I "reverse" the orientation of both generators of the torus

loops of the torus

And they do this, and then that keeps the orientation and I'm on the other side.