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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

Unexpected Shapes (Part 1) - Numberphile

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    林宜悉 posted on 2020/03/30
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