Twist and Wonder

Aim / Overview

  1. This activity asks a child to think carefully about the question of whether spinning something right around 360 degrees always leaves that something the same as it was before, and shows how there are examples of things which do not behave in the everyday way.
  2. The children build and test a very simple object that takes two full turns to get it back to how it was before. The simplicity, the sheer surprise at the behaviour and its likeness to a magic trick (which takes very little skill to reproduce) make it enthralling to many children of roughly six years old and over.
  3. The children then turn their own bodies into an object that has this same surprising behaviour.
  4. The activity leads into a simple discussion of electrons and other fundamental particles in modern physics which also have the weird behaviour explored in the exercise.

Materials

You will need:

  1. About a ½ to 1 metre length of ribbon about 4 centimetres wide. Crepe paper works excellently and bright colours are lots of fun. Fairly stiff fabric like some organzas used as gift wrapping ribbons or synthetic sheer curtain material work quite well too;
  2. A small doll, maybe 20 centimetres tall. Actually any object that has a “front” and a “back” (e.g. like a playing card) will work, but a doll or a “character” (toy animal) is best. We are social animals, hardwired to ken a face, and this activity calls for the children to keep track of twists of an object. A doll thus makes the geometry unmistakable. So if you use a playing card, a court card is best;
  3. A see-through plastic cup of about 250ml capacity (one from a drink machine will do) with a large dot or other obvious feature on one side. You can paint a big dot on it to make its rotation very clear;
  4. Stickytape;
  5. A4 white paper sheet;
  6. Scissors;
  7. Two pencils of different, contasting colour;
  8. A little water (about 100ml).

Age Group

6 to 206

Prelude

Ask a child to stand up and turn around on the spot one half turn. Ask them: are they the same as they were before turning around? What’s different? The point being that the child now is facing a different way – her or his orientiation – their relationship with the things around them – has changed. Now ask them to turn around a full turn. Ask them again – are they the same as they were before turning around. Is anything different? Now a thoughtful child may give many subtle answers which are wholly right – “I’m a teeny-tiny bit older now”, or “I’m now Ava / Sam after having spun around, whereas I was Ava / Sam before having spun around” (time relationships) or even “I’m a little bit giddy now, whereas I wasn’t before” – my own daughter answered me that she had burnt up a tiny bit of sugar from her breakfast to give herself the energy to turn! So encourage them to have fun with these ideas until bored with them – when you sense they are ready to shift on, try to direct the questions towards the idea of “orientation” – maybe even explain this word if you think they are old enough. Maybe ask them about turning other things like a ball – actually this one might bring up a subtlety, for if the ball is perfectly round and smooth, any turn through any angle about any axis will leave it the same – if so, ask about a ball with a dot painted on it – or something irregular like a shoe or a lumpy cupcake or a chocolate crackle! The main idea here is that, after a full turn about any axis, everyday things come back to being pretty much the selfsame as they were before the turn.

The Big Question

Now ask the big question: are there any things in the World which do not follow the rule that a full (360 degree) turn brings them back to being the same as they were before the turn? This question is likely to seem truly bizarre to many if not most children, as anything that doesn’t follow this rule is utterly outside their – and our – everyday experience. It may even upset some children – the question doesn’t make sense to them and, depending on how their peers react, they may feel “dumb” for not grasping it, for some others may boast that such a question is wholly obvious and not really worth thinking about, whilst others will say they “get” the question not to boast but because they genuinely believe it is an easy question and have overlooked the subtlety – if so, be ready to stress Small Things Amuse Great Minds! The great discoveries and findings in science have, almost always, been made by someone looking at something in a slightly oblique way, something that everyone else thought was thoroughly understood and mundane. So be ready to reassure that the question is indeed a very hard one, that people didn’t even know about what such a question might even mean until only about one hundred years ago. The question is only likely to make sense by actually witnessing something that behaves in such a weird way – and (i) building such an object and (ii) talking about how we wouldn’t even be alive if it weren’t for objects that behave in this way is exactly what we are about to do!

Enter Amber, the Quaternion Doll

Amber

Figure 1: Amber

This is Amber the Quaternion Doll. I choose the name Amber because our word “electron” comes from the Greek word for amber – the ancient Greeks witnessed that a piece of amber would draw small bits of dust and chaff to itself after it had been rubbed with fur, so this indeed was how electricity was discovered. Amber looks as though she comes from Russia – maybe she went there on a long holiday once – but I like to think of her as coming from Norway, which is where the mathematician Sophus Lie (pronounced “Lee”) was born in 1842 to go on to found, in roughly 1870, the branch of mathematics called Lie Theory, which gives us a precise description of the remarkable behaviour we about to explore. The particular part of Lie theory that gives us this description was discovered by Otto Schreier in 1925, so Amber maybe could be said to have a German parent too.

Bringing Amber to Life

Now we stickytape one end of our metre long ribbon to Amber, and tether the other end to a tabletop as in my drawing below. That’s it! Amber is ready to show us something pretty remarkable. Quaternion Doll

Figure 2: Bringing Amber to Life

The first thing to do is ask Amber to spin around on the spot one full turn. We do this of course by winding Amber’s ribbon once around as drawn below.

Once Twisted Amber

Figure 3: Once twisted Amber

Ask her whether she is the same as before we began turning. Is she? Of course not! Her ribbon is all twisted up!

Twice Twisted Amber

Figure 4: Twice twisted Amber

So now let’s wind her once around again, so that she has made two full turns on the spot. What now? Surely she’s not the same as when we began. Her ribbon is even more twisted now! But in a hidden way we are about to show, Amber has indeed come back to how she was at the beginning of the experiment. For we can undo all these twists without twisting Amber at all!

Path for Undoing Double Twist

Figure 5: Path for undoing a double twist

Slacken the ribbon with the twists still in it and, holding Amber in the selfsame, constant orientation, pass her under the ribbon by taking her along a path like the broken line path in my drawing above. As shown in the sequence of drawings below, this translation undoes the double twist.

Undoing Double Twist 1 Undoing Double Twist 2
Undoing Double Twist 3 Undoing Double Twist 4
Undoing Double Twist 5 Undoing Double Twist 6

Figure 6: Undoing the double twist

In motion, this sequence looks like the figure below.

Dirac Belt +1 Homotopy

Figure 7: Undoing the double twist in motion (Click here to download high resolution version of this movie (approx 10MB))

Hold on a second! What kind of hocus pocus is this? Surely if we began with one twist in Amber’s ribbon, we could undo it in the same way? This is a good point – we don’t yet know what would happen in that case. As scientists, we need to test this. So let’s go back to Amber with one twist in her ribbon as in Figure 3: “Once twisted Amber”. From here, if Amber follows the path shown in Figure 5: “Figure 5: Path for undoing a double twist”, the “untwisting” move looks like the sequence below.

Lone Twist 1 Lone Twist 2
Lone Twist 3 Lone Twist 4
Lone Twist 5 Lone Twist 6

Figure 8: Trying to undo the lone twist

Dirac Belt -1 Homotopy

Figure 9: Trying to undo the lone twist in motion (Click here to download high resolution version of this movie (approx 10MB))

If you check carefully, Amber’s ribbon still has one twist in it but now the twist is in the opposite sense! Check this: in Figure 3: “Once twisted Amber” we wound Amber anticlockwise; in Figure 8: “Trying to undo the lone twist” and Figure 9 , we see that she has been wound clockwise and it takes one turn anticlockwise to undo her. So the two turns is rather a special number. Look again at Figure 5: “Path for undoing a double twist” and look carefully at the gathered twists along the two roughly vertical sections of the place where the ribbon “droops” and think of these gathered twists as “swivvels”; you take Amber around the path in Figure 5, each swivvel adds one full twist to the ribbon, so that each trip around the path in Figure 5 adds two, and not one, twist to the ribbon. So we see that, without twisting Amber, we can undo two, four, six or any even number of twists in her ribbon. We can do this for twists in either sense: two clockwise turns of Amber in Figure 4: “Twice twisted Amber” can be undone by taking Amber around the path in Figure 5: “Path for undoing a double twist” but in the opposite direction to that shown there. So we see that if Amber makes two full turns on the spot, she truly does come back to where she began, whereas if she makes one full turn, her ribbon has a twist that cannot be undone without making Amber twist back the opposite way. We have indeed found something in the World that (i) is not the same after one full turn and (ii) needs two full turns to come back to where it began!

Amber in the Middle of an Endlessly Long Ribbon

Now it may seem that we need the end of the ribbon where Amber is to be a “free” end so that we can pass Amber under her ribbon and undo the twist. This is not so! Amber’s behaviour is exactly the same even if we put her in the middle of an endlessly long ribbon. For example, you can attach a ribbon to two tables at either end, leaving a bit of slack, so that the ribbon becomes a bridge between two tables. It must have no twists in it when you stick it to the tables. Then sticky tape Amber to the middle. Actually, a good way to do this version of the demonstration is to use a book instead of Amber: “grasp” the ribbon in the book halfway between the tables by using the ribbon as a bookmark. Now put the twists in the ribbon and hold the book still, looping the ends around it. This is quite tricky and needs practice, but look carefully at Figure 10: “Untwist Sequence for Amber at the Middle of a Ribbon” to understand that it can be done. In practice, instead of looping both ends at once (a trick more readily done if you’re an octopus) as in Figure 10, 11 and 12 you unloop one end (seemingly tangling the other end) and then unloop the “tangled” end next. As I said, this needs a bit of practice, but it makes the demonstration even more compelling.

 Infinite Ribbon 1 Infinite Ribbon 2
 Infinite Ribbon 3  Infinite Ribbon 4
 Infinite Ribbon 5  Infinite Ribbon 6

Figure 10: Untwist Sequence for Amber at the Middle of a Ribbon

In motion, the two-twist Amber on an endlessly long ribbon looks like Figure 11.

Twin Dirac Belt +1 Homotopy

Figure 11: Untwist Sequence for Two-Twisted Amber at the Middle of a Ribbon (Click here to download high resolution version of this movie (approx 10MB))

In motion, the two-twist Amber on an endlessly long ribbon looks like Figure 12.

Twin Dirac Belt -1 Homotopy

Figure 12: Untwist Sequence for Once-Twisted Amber at the Middle of a Ribbon (Click here to download high resolution version of this movie (approx 10MB))

You can run a simulation of all of the above demonstration in the program Wolfram Mathematica using a free downloaded Mathematica notebook player. This download happens very easily and is altogether safe. Go to the Appendix below to run this simulation.

I am Amber!

We can actually turn our own bodies into a ribbon and doll! Can you rotate your hand continuously as many turns as you like about an axis in your forearm in the same direction whilst keeping your feet flat on the ground, and without twisting your arm off? Indeed you can, as I show in the short video.  Some dance twirls, wherein a pair of dancers join hands but do not let their grip go whilst one partner twirls, work in this same way. Likewise for some baton, firestick and sword twirling displays.

Weird Surfaces

Some fancy words and jargon. Mathematicians call things that are transformed by rotations in the odd way we have just seen “spinors” or “spinor objects” (pronounced “spin – or”, not “spine-or”). A second showing of such behaviour that children will enjoy is the Möbius strip (after its finder August Ferdinand Möbius 1790 – 1768). To demonstrate this, we need some paper strips (a quarter of an A4 sheet cut lengthwise works well). Begin with the questions How many sides does a sheet of paper have? You’ll undoubtedly get the answer two – a front and a back. Now for the big question: Do all sheets of paper have two sides? What would a sheet of paper with ONE side look like? Let the children grapple with this one themselves for a little while. Most likely they will decide that a sheet of paper with one side makes no sense at all. As with Amber and her spinorial twists, this is an extremely simple but fiendishly subtle idea, so some children again may find the idea upsetting and may need re-assurance that they are in no way dumb if a one sided sheet of paper does seem an impossible concept to them. So now ask them to count the sides on one of a paper strip. The answer you get will of course be two. Now join the ends of the strip together to make a ring like a link for a paper chain. Check how many sides this ring has. Also check how many edges this object has. The answer is two in both cases. Next, make another ring but this time make full twist in the strip and then join the ends together. The object you will get looks like the one in my drawing below.

Ring Full Twist

Figure 13: Ring Link with Full Twist

Check this object carefully and ask how many sides it has. To help this check, the children may like to colour each side of the strip two different colours before making the ring with the full twist. I have also emphasised the edges in my drawing with blue and orange, and if you look carefully you can see that these two edges stay separate nothwithstanding the full twist. The two coloured sides also stay separate – this object still has two sides and two edges, just like the untwisted ring. Next, make another ring but this time make only a half twist in the strip and then join the ends together. The object you will get looks like the one in my drawing below – check the number of edges and sides.

Ring Half Twist

Figure 14: Ring Link with Half Twist

Astonishingly, this thing – the Möbius strip – has only one side and one edge. This is how a piece of paper with only a front and no back side looks! Again, the use of two different colours can help clarify things. Also look carefully where I have drawn the arrows in my drawing above – you see that the two edge colours meet so that the formerly two edges become one. Imagine being a little bug walking on one edge. For the ring with a whole number of twists, the little bug has to walk around the ring once to get back to the same point – they will weave in and out a bit, but their journey is still 360 degrees about the ring’s centre. For a ring with whole plus a half number of twists, the little bug’s journey has doubled in length and they must make two full circuits – 720 degrees – around the centre of the ring object to get back to the start. Now make one more half twist and another full twist ring, this time drawing three evenly spaced lengthways lines along both sides of each strip before twisting and joining. For both objects, pierce them on the line at one point and try to cut them in two by cutting along the centreline as in my picture below.

Cutting Moebius

Figure 15: Cutting a Möbius Strip in Two

If you do this with the Möbius strip – the half twist ring – we get the object below. We haven’t cut it into two at all! The object stays one connected whole! But now look carefully. Count the sides and edges of the cut band. Now the former Möbius strip has two sides and two edges. It is indeed the same as a band with two full twists in it. If we make a new one-twist band and do the same trick with it as before we get what you see in Figure 17: “One Twist Srip Cut into Two”. The one-twist band does split into two objects and indeed each of them is a copy: they are both bands with two full twists in them. They are also linked. So, what do you think will happen if we do the same trick again as before to the cut-in-two Möbius strip? Figure 18: “Möbius Strip Cut into Four” shows this. We again have two bands, but now each has two full twists in it and they are knotted together in a different way from the two one-full strip bands linked together in Figure 17: “One Twist Strip Cut into Two”.

Moebius Strip Cut

Figure 16: Möbius Strip Cut into Two

One Twist Ring Cut

Figure 17: One Twist Strip Cut into Two

 Moebius Strip Cut In Four

Figure 18: Möbius Strip Cut into Four

  In general, if you do this cutting trick with a band with an odd number of half twists in it, it becomes a band twice as long as the first but with twice the number of *full* twists as the beginning band had half twists. A band with an even number of half twists (this is the same as a band with a whole number of twists in it) splits into two copies: the two linked copies have the same number of full twists as the beginning object had.

Amber Particles

What about fancier arrangements of ribbons? Could we make Amber so that she needed to make three full turns, four full turns or some number other than two full turns before she reached the same state as she was at the beginning? It turns out that the answer, as can be shown from Lie theory, is that in three and higher dimensions, the answer is an emphatic no! There are objects in the World that rotate one full turn to come back to their beginning state, there are objects like Amber that need two full turns to do so and there is nothing else! This is as long as our object must stay a one-piece connected whole. Of course, we can make fancy arrangements with gears and cogwheels that take any number of turns to bring them back to their beginning state, but these objects are made of separate wheels and the machine does not stay a connected whole. If you tried to do the same with your arms they would twist off. The spinor behaviour we have explored is highly important in modern science, particularly when we deal with subatomic particles. The electrons that take up most of the space in the atoms of your body, the protons and neutrons that make up these atoms’ nucleusses and the quarks that make up these protons and neutrons are all spinor objects. They must all be spun through two full turns to be unchanged by the rotation and such a particle rotated through only one full turn has fundamentally different behaviours from those it had before the rotation. It turns out, although unfortunately I don’t know a nontechnical explanation of this, that this spinor behaviour is precisely what makes these particles take up space. Particles of light, for example, behave under rotations just like everyday objects do, and so beams of light can pass effortlessly through one another. If it weren’t for the spinor behaviour, every electron in every atom would slump into its nucleus, there would be no different chemical behaviours for different elements and indeed it is unlikely that the idea of an element would mean anything at all. Life could not even be were it not for the spinor behaviour of the electrons in your body and the bodies of all living things. We can end this activity by talking about a last question. Electrons and Amber have the same behaviour. Does this mean that electrons, quarks and all the other like particles have little ribbons on them, somehow linking them to the “background space” around them? On one hand, the answer to this is easy and an emphatic no. I don’t think there would be one scientist on the Earth who would seriously believe that, if somehow we could put an electron under a fantastically powerful microscope, that we would see little ribbons there! The problem is that even the idea of “seeing” anything at this level probably makes no sense at all. “Space” is a notion that begins to break down at this level, and the particles certainly are nothing like tiny billiard balls. Rather, modern science thinks of empty space itself as being defined by and filled by fundamental objects like “electron fields”, “quark fields”, “light fields” and so forth. In a very real sense, there is only one electron: the one electron field that fills the whole World. The particles themselves are not so much like billiard balls but rather more like “messages” that are swapped between the more fundamental fields as they interact to give rise to the behaviours that define our World. So there’s not really anything to “see” at this level. So, on another deeper level, the answer to this basic question is in a sense “yes”. The mathematics that describes an electron’s behaviour and that which describes Amber and her ribbon is exactly the selfsame. Furthermore, mathematics can even tell us the no other description is even possible! (without giving rise to contradictions). So the mathematics of Amber’s twirls and those of the electron is not simply a bit alike – it is exactly the selfsame mathematics – there aren’t even separate descriptions with the same words! So if there’s nothing to see at this level, then the descriptions are all that there is. Imagine you and a friend were great scientists and had found out for the first time about this weird spinor behaviour of something you cannot see, and that you can only sense how this thing behaves in relation to other things. You would spend a great deal of time thinking about this and describing what your experiments told you. You would try to foretell what bevaviour would arise from situations and experiments that you haven’t yet done and then do them and find that you would be wrong most of the time. You would be utterly bewildered by such other worldly behaviour, it would be like groping in the dark for a light switch. At first you wouldn’t have the words to describe what you seem to have found. After many years of thought your descriptions would slowly become more and more precise. The other worldly “thing” would become more and more wonted to you. You would slowly be understanding its behaviour. By this time you would have invented a name for this new thing and the weird science it brings. You and your friend would begin forming new thoughts in the language of mathematics. You may not think of mathematics as a language but it most certainly is, just like English or Chinese or German. It is the language for precisely describing the relationships between the things and processes in the World. Number, for example, is only one part of mathematics, it gives us the nouns and adjectives to describe the relationships of size and order between things. And sooner or later someone would point out to you that your mathematics describing your discovery is the selfsame mathematics that describes Amber. Most likely you wouldn’t work this out yourself because you would be too busy and overwhelmed thinking about your discovery. Or maybe you and your friend would never reach this point, maybe you would both grow old and die and leave your children scientists to begin where you left off, and it would be they who would grasp this gift. With this thought, this new dazzling insight, in mind, you would now begin to foretell the results of new experiments you’d never even dreamt of because your new powerful description would let you see so clearly. And when you did so, you would at last be right! It is as though after so many years of listening carefully to Nature, She at last trusts you and has given you a key to effortlessly open up and see into the most beautiful jewel box one could behold, whereas before you spoke the right mathematics you were clumsy and staggering and endlessly frustrated. Descriptions of the new science become now so smooth and effortless and correct that is seems madness to think that the new science could be separate from the right mathematics. They would seem altogether unseparable. It would be as though they were the very selfsame thing. So in this sense, electrons truly do have their own tiny ribbons!

Appendix: The Dirac Belt Simulator

Below is a simulation of Amber and her ribbon. If for some reason this is not working on your browser, visit the copy I have uploaded to the Wolfram Demonstrations Project Site.

Click here to go to the Wolfram Demonstrations version of the simulator

Use the control “homotopy” to set whether Amber begins her untwist sequence with two full twists in the ribbon (as in Figure 6: set “homotopy” to 1) or with one full twist in the ribbon (as in Figure 7: set “homotopy” to -1). Click the “+” symbol next to the main “time” slider to expand the controls: you can use these to slow the simulation down (double down arrow) or speed it up (double up arrow), begin, halt or one-step (“+” and “-” buttons) the simulation. You can also use the “belts” control to put one or two belts on Amber. The two belt simulation shows that the untwisting sequence works whether there are one or two belts. The two belts simulation shows that Amber’s behaviour is the same even if she is at the middle of an endlessly long ribbon, this is even though we might otherwise think that a “free” end to Amber is needed to allow her to pass under the ribbon. We can see with the two belt trick that a free end is not needed.

You can also grab the Amber-and-Ribbon object, strech and shrink it and re-orient it in space to get a better view – it has full 3D controls. The simulation’s size can also be zoomed by grabbing the edge of its window.