We'll start very slowly, but don't worry we'll speed up in a moment.
A strip of paper and I glue the ends into a loop. A straight loop.
And I cut it along the centre line. Well, what's going to happen? Everyone can guess what's going to happen.
In fact everyone knows with total conviction what's going to happen.
It splits into two halves.
Next, we're going to make the famous object Möbius strip.
Now that's one twist - 180 degrees - and glue the ends.
And this is the well-known object which only has one face and so forth.
What's going to happen if we cut this along the centre line?
And unlike the previous case it does not fall into two pieces.
Now, so far so good. And people might say I've already done that and so forth, I've seen it in books.
But even professionals mathematicians often don't realize how many twists there are in this resulting object.
In this configuration we see all the twists are in the upper half of the strip and the lower half has no twists.
Let's count the number of twists this object has.
In order to do so I'm going to apply the violence and I'm not going to move.
Cutting the lower half and start untwisting the top one.
Still twisted obviously.
Two, three, it's side but still twisted.
Four, and finally it's straight.
That's not what I wanted to show you.
What I wanted to show you is something different.
What we want to do is to glue together those strips.
For example that's straight against straight.
glued at right angles to each other.
and what we're going to do is cut this kind of object
all the way around the centre line, all the way around the centre line
Not only straight ones but also with glued Möbius strips.
So there are four possibilities: straight against straight
Straight against Möbius, and Möbius against Möbius.
And now you're thinking that Tadashi can't count because there are only three cases but actually there are four because when you do Möbius against Möbius
Those two Möbius strips could be of the same chirality, that is they might be twisted in the same direction
or they can have opposite chirality, that is they are twisted in opposite cases.
Clearly, maybe they will produce different results so we have to be careful.
Straight against straight. Here is a cross and I glue the ends, that's one straight strip for us.
And the other strip
like so
By the way a little bit of engineering advice if you want to show this to friends and family it's very tempting
to make the strips loop and then glue them at right angles, but then it become really difficult actually.
You have to go in there and glue and it's nasty. So it's much better to make a cross and then glue the ends.
By cutting all the way around one centre line and all the way around the other centre line
somewhere in the middle you have to make a cross cut
But in order to make the suspense last, I'm going to leave the cross cut until the end.
So here I started cutting one of the loops all the way along the centre line
And the other loop along the centre line.
Also all the way around but leaving the cross cut until the end.
This is the picture just as we are about to finish cutting. You can see this was one untwisted loop
cut along the centre line and this was another untwisted loop cut along the centre line
Okay, now, earlier when we had a single untwisted ordinary loop and when we cut it along the centre line
it just split in two halves, the results of which are visible here.
What's going to happen this time?
Easily, I'm going to cut this, finish cutting it and chopping it, chop, chop.
And what I manage is a flat square!
Hmm, that's funny. Flat square. What happened?
Okay. We'll come back to that in a moment. But let's try the next case.
Let's try straight against Möbius
That's a straight strip. And then the other one I'm going to twist once into a Möbius strip.
And by which I mean twist one and then glue the ends, okay.
This object is generally different from the previous one so when I cut this object along the centre line like this.
It should produce something, hopefully, different.
Let's try it again. As usual in order to make the suspense last I'm going to leave the cross cut until the end.
And then now the other loop
Let's cut along the centre line
Cut, cut, cut. And that's the picture that we have.
As we are about to finish cutting so we can see that this was a straight loop
which has been cut along the centre line. And that was the Möbius
twisted, and has been cut along the centre line.
And this, as we say, is different from the previous one
straight against straight because of this twist.
Now what's going to happen when I cut finish cutting everything?
Are you ready? Am finally finish cutting and what we've managed is
again a flat square!
Well that was disappointing or interestingly disappointing.
Well maybe we are getting a flat square every single time.
Straight against straight: flat square. Straight against Möbius: flat square again.
Let's now do Möbius against Möbius. But now we'll do two Möbiuses of opposite chiralities,
that is twisted in opposite ways.
Now if you want to show this off to friends and family you have to remember that this is opposite.
And in order to do so you have to exercise your short term memory.
Here, first let's glue this by twisting this piece
clockwise, in a right handed screw fashion.
and make a Möbius.
And you have to remember this.
So that's one.
In a moment, when I'm about to close the other one.
Wait what have I just done? Is it clockwise or anti-clockwise?
What was it?
Well it was clockwise, so this time you have to spin it counter clockwise
or anti-clockwise if you're looking at it from the other side of the Atlantic.
In a left handed screw fashion and then close the ends.
Glue the ends together and that's the other piece.
Two Möbius strips that have opposite chiralities
that are glued at right angles to each other
It's a rather beautiful thing.
And although this is strickly not necessary from the mathematical point of view,
I'm going to reinforce these pieces from the back because
that is going to be better for the resulting sculpture later on
And this is a present for everyone.
I cut all the way around one.
And all the way around the other loop.
Leaving the cross cut until the end as usual.
Are you ready? So nothing up here, nothing up here. And what the image is is a present for all of you.
And I finish cutting. I finish cutting and I told you this is applicable, an applied mathematics,
what the emerges is amazingly,
a pair of linked hearts!
So you can see, it is an applicable mathematics as you can see and you can apply it in all sorts of contexts.
And I hope you make good use of it.
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, a good way to think about this is to cut one of the pieces completely around.
four and finally it's straight.
So the object had four twists not two.
Where do the extra two twists come from? Do you know? As I say, we seemed to have proved, a moment ago,
that the object should have two twists but it had four twists it comes from the following interesting effect.