Well, this object is made from a single sheet of paper
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.