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• 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 objectbius 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 gluedbius strips.

• So there are four possibilities: straight against straight

• Straight againstbius, andbius againstbius.

• And now you're thinking that Tadashi can't count because there are only three cases but actually there are four because when you dobius againstbius

• Those twobius 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 againstbius

• 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 thebius

• 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 againstbius: flat square again.

• Let's now dobius againstbius. But now we'll do twobiuses 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.

• Twobius 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.

We'll start very slowly, but don't worry we'll speed up in a moment.

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B1 INT möbius centre cut straight glue cutting

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