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I'd like to welcome you all to the Mathematical Institute for another Oxford Mathematics Public Lecture.
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My name is I don't go really. And I'm in charge of the lecture for the institute.
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It's great to see so many of you. Just a few point of order.
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I just want to point to the exits. These are what we call full British exits.
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You'll have to be careful if you take this one.
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I think I've seen Boris Johnson's trying to hide from voting in the Commons today.
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So you'll have to step over him. If you go that way today, we'll talk about origami.
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It's a topic that I've always been fascinated by.
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I'm completely rubbish at it, but I always love it and I've always wanted to have a public lecture on it.
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And I'm extremely grateful for Professor James today to give a lecture on this topic.
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If you're interested in this topic,
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and I'm sure you'll be after it tells you all that he has to tell you today there is another event that is organised by both the university,
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I think the engineering department and the British Origami Society.
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There is a British origami society with 700 members, so there'll be an event in September, early September.
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You can find that on the website of the bourse, as it's called, British Origami Society,
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where there'll be talk both about mathematics but also about more the playful aspect of origami.
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So if you're interested in things you should definitely go with.
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I just wanted to advertise and be short.
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I just want to point out that our Oxford Mathematics Public Lecture, funded in part by X markets and we are very grateful for their help.
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It's a financial company with offices in New York, London and Singapore.
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Today, I will not give you a full introduction. I've admired the words of Professor James for many years.
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I've heard it can give many talks, and I've always been fascinated not only by his research, but also what I would call this clarity of thought.
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There are few people, when you hear them, you realise both that they understand the topic extremely deeply,
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but also they can phrase it in a way that makes you feel you also understand deeply.
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And so I always wanted him to give a public lecture here, and particularly.
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But since he has been a very long time friend and collaborator to Professor John Ball, I thought it would be best for him to introduce him.
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So please help me introduce John Bode. We will do a proper introduction.
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So it's my great honour and pleasure to introduce my scientific colleague and good friend,
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Professor Richard James, who is a distinguished McKnight University professor at the University of Minnesota.
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So I think James is a truly remarkable scientist, an undoubted world leader in both theoretical and experimental mechanics.
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And that combination itself makes him unusual.
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And he has a whole string of extraordinarily original contributions over a range of problems and mechanics,
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mostly centred on on on the behaviour of alloys that undergo solid phase transformations, which is a very important practical issue,
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but stretching into other areas such as the structure of viruses and parts of statistical
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physics now I think belongs to is does not belong to a mathematics department,
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but the Department of Aerospace, Engineering and Mechanics.
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And though he will deny it, he's also a very fine mathematician.
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And I think that one thing that I admire of him is that perhaps more than most mathematicians who work and in mathematics departments,
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he believes in the power of mathematics for describing nature.
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So silly special cases that mathematicians would ignore and his hands turn into discoveries of new materials.
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So I think that's a remarkable ability to to to do that.
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And so it's a great pleasure to invite him to give his lecture. They're just number two.
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Okay. Thank you very much for this very kind introduction.
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And everyone can hear me. Is that clear? And it's a great pleasure to deliver this lecture and a great honour.
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I'm going to speak about in particular. It's a it's a it's a great pleasure to speak in this room, this beautiful room with this beautiful ceiling,
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and also to speak in this wonderful math institute and with its many references to mathematics.
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In fact, this glass ceiling that you see there has a nice reference to to the eigenvalues of the plus in their eigenvectors,
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but it also has a connection, a nice connection to origami.
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So, in fact, I'm going to start with a homework problem.
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Is this this you can think of these glass panels as rigid, but I'm going to allow the joints to be flexible.
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And so my question is, can this be folded? Okay.
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So we'll we'll I'll give you enough information that you'll be able to decide whether this can be folded or not.
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And at the end, I'll tell you I'll tell you all about it. Okay.
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So first a little bit about myself. I come from a kind of place maybe I've not heard of.
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It's it's Minnesota. It's in the United States. It's this little red state up there in the very far north.
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And it's known for these things.
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You know, people like to take vacations in Minnesota, and they they they like to go canoeing because it's the land of 10,000 lakes.
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In fact, there are 15,000 lakes. And Minnesota not not not including small ponds, you know, so we really have a lot of lakes.
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So almost everybody has a canoe and they go around in these lakes.
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So just to convince you that origami and folding has penetrated all areas of technology, I want to show you a little video here.
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That was the the Ori Canoe Company was extremely pleased that I was going to show this video.
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But you can even fold the canoe up.
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So if you have a studio apartment, you can own a canoe and then you can go canoeing.
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So. Secondly, I'd like to make an acknowledgement.
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I have actually a project on origami structures, the design of origami structures, and it's one of the participants.
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It's about five or six people. One of the participants is Robert J.
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Lang. You may know him from his many books on origami, and he's an interesting collaborator,
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a very interesting person, and he does something that we are so far away from doing mathematically.
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He does this, he makes a rhinoceros. So you tell him to make a rhinoceros and he will make you a beautiful rhinoceros.
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So he will decide what the fold lines are on a piece of paper, such that when you fold it up, you get a rhinoceros.
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That's what in mathematics we would call the inverse problem. We have no way to solve the inverse problem.
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And in particular, you know,
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he solves the inverse problem in such a way that with a great dose of asceticism and beauty and simplicity that leads to these origami structures.
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So in fact, this collaboration with him is I feel like it's working with a real with a true genius.
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One can say that Robert de Lange is a true, true genius. Okay.
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So this this talk is about origami, but it's inspired by Adam Mystics,
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which may seem like an unusual place to get some inspiration for folding things.
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But that's that's exactly what I'm going to do.
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So I'm going to start with the optimistic and I start there with with one of the most important atomistic structures, a carbon nanotube.
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It's it's one of the three forms of carbon that did not get the Nobel Prize.
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That's of lower dimension. In fact, I think it's the most interesting one.
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Or nanotubes in general are are very interesting. And what I would like to do is point out a feature of its structure.
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So I want you to imagine. So I put a little couple dots there.
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I want you to imagine that you're they actually sitting on one of those atoms and you're looking out at the structure.
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And of course. So what do you say? You see the nearest neighbours and then you see the atoms bit further away and a bit further away and so forth.
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Now you go to another point on the structure. So you're going to get you started here.
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And we did this. We go to this other point and we sit on that point and we reorient ourselves in exactly the right way.
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You see exactly the same picture as you do on the first one.
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And that's true of every atom of the structure.
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If you if you orient yourself in the right way, you take a picture, you see exactly the same environment out to infinity.
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Okay. That's interesting. I mean, the reason I chose the two red dots is because it looks like this can't possibly be true,
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because how can this be the same environment as this? This is.
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So, in fact, what you have to do is on the top one, you have to look down on the bottom and you have to look up.
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And if you do, then you'll see that that's the orientation at which you'll see exactly the same thing.
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And if you turn sideways, you'll see exactly the same thing. It's a property of that particular structure.
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But it's also a property of many, many structures, and particularly these kind of nanostructures that people are discovering these days.
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So here's. So first, I'm going to put it in mathematical terms that I'll show you a bunch of pictures of.
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I call these objective structures. And so let's let's try to put that in mathematical terms.
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So here it is. Here's a so I'm going to think of positions in space.
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You can think of three coordinates of the position or in two dimensions like this.
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It's the idea is in either case and suppose X one is a point of the structure and there's other points,
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many other points, and I'm not drawing them all. I'll just draw some of them. And now I draw with vectors to to every other point.
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So this and so. Okay, fine. Now, now I go to another point of the structure.
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Any other point say XY and I draw vectors to all the points of the structure.
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Okay. So I get this spray of vectors coming from X one and of course you know this there would be a vector from here to here.
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I just have a draw on it, so forth. An objective structure has the property that I can take that spray of vectors coming from x one,
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I can rigidly rotate it and I can rotate it into into the spray of vectors coming
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from X to that's an objective structure and you can write that down mathematically.
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The notation, I mean, I think some of you will understand the notation.
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I'm not sure everybody will,
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but it's you can you can read this on two different levels of the picture level and the linear algebra level, but it's like that.
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It goes like this. So this is the structure.
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That's the full set of points. And they have the property that for any one of the points, for example EXI you can take all the points,
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you can take the, the, the arrows going from X1 to all the points.
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You can reorient them depending on the choice of AI and you can add them back to EXI and you get the same structure back again.
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Okay, so that's exactly what I said in words is that's the mathematical definition.
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Of course you have a mathematical definition.
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You can study it and you can do quantum mechanics with this definition, and you can do many things, many interesting things.
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But what I will show you is a picture. So all these structures are objective structures, and I'll tell you about some of them.
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Of course, these are Nobel Prize winning structures here, graphene and buckyball.
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I've already discussed carbon nanotubes. These are the sort of classic carbon nanotubes.
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This is carbon nanotube with chirality. That's they're all objective structures and here are some helical structures and some viruses and so forth.
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The only thing I should mention is, is like these viruses, that's the bird flu virus, by the way.
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This is this is phosphor green, which is also a very, very interesting structure.
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And again, you might not think that every atom sees the same environment, whether it's on the lower level or the upper level.
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But that's true. Okay.
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And this this amyloid protein is also extremely important protein.
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But this is one of the bird flu viruses.
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And you can see this is made of molecules. So that the definition I gave you was was for an atomic structure and a molecular structure.
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The definition goes like this. You have identical molecules.
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You number the atoms in each molecule, say 1 to 100.
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And then you you might assume the numbering is done in a very good way so that I go to the 27th atom of of this molecule.
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Now I go to the 27th atom of a different molecule, any other molecule in the structure, and I and I look in an appropriate direction.
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Then the 27th atom of every molecule sees the same environment.
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Okay, so that's the definite 26th atom sees a different environment from the 27th, but all the 26th atom sees the same.
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And then there is a huge number of, of molecular structures which which satisfy that definition.
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Most of all do the atomic case because it's a bit easier to to explain things.
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But those are those are objective structures.
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So now we're going to play a game with the periodic table.
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So there's a periodic table. And my little game would fail miserably if I included these radioactive elements down here.
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That's why. So I raised them. Okay. And I also asked the team because ascertain there's only one gram in the entire Earth's
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crust that any crust at any one moment and no one knows the crystal structure.
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Okay. So this is all about the structure of the elements. Okay.
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How do most people think of the periodic table? You can open a book on atomic structure.
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They think this way. They think in terms of Broadway lattices.
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They they build up the structures from from previous lattices.
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So mathematically,
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you take three factors that do not lie on a plane and you take integers coefficients on those factors and you add them up or and more pictorial terms.
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You start with this atom. Those are the three factors in this particular case.
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And, you know, and so this this atom can be shifted to this position by by adding E three.
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So this that means this middle atom is also part of the structure. It and this one is because it's 23 and and so you take integers all possible
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combinations of integers on these three vectors and you get the structure,
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let's call the phase centred cubic structure FCC.
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You might not think if you, if you, if you're not really quick, like,
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like I'm not you might not think you could get this atom, but this atom is obtained by e1e2 minus c three.
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So it's integers on those three vectors. And this is the phase centred cubic structure.
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So as I say, most people think of the periodic table in terms of rough lattices.
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So I want to, I want to take the periodic table again and I'm going to blacking out all the elements that are not of lattices.
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Okay. So there you go. So of course lots of elements are not don't crystallise.
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And I did this kind of rationally somehow I,
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I took the most common structure at room temperature and if the material is not solid at room temperature,
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I took the structure accepted structure at zero temperature. So I tried not to fiddle anything.
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And there's, there's my blackened out. So many elements do not crystallise as a lattices.
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Now, many of those are actually the same crystal structure.
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About a third of the periodic table. So roughly speaking, half of those black, black and out elements prefer the hexagonal close pack structure.
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Okay, so let me explain what hexagonal close packed is.
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We take we take a bunch of balls and we put them down in a close, packed fashion,
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fit together with the closest the smallest volume, smallest mass, and a large volume say.
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In in this array. And now that's that's layer one.
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And now I'm going to put layer to layer two.
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I'm going to put balls in the holes in the depressions of the first layer, and they'll fit together perfectly, as you see there.
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And the same. Same on the right. In fact, I could it doesn't matter what I wear.
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I put the first one. If I put the first one here, I would still get exactly the same picture.
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Okay. So it does doesn't that mean the fact that they look the same is no loss of generality?
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Okay. Now I'll put layer three on, but layer three I have a decision.
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And layer three, I could do that. In other words, I could.
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I could put a ball right in the depression of the previous layer, but not over one of the first layers.
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Or. I could do that. I could put the the atoms directly over the first layer.
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And the left hand structure is is actually phase centre cubic.
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So it's a little hard to see. But if you go back to the picture before.
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Um. Where is it? Right there.
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If you look down the body diagonal of that of that structure, you'll see exactly the picture that I show on this, this other slide.
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So that's the this picture on the left and on the right, of course, it's something different.
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And on the right is is hexagonal, close, packed. And what I mean by that is, if you continue now, you've got the first layer.
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So you only alternate between two different structures.
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You might say a baobab and you might call this stacking ABC, ABC, ABC.
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The structure on the right is not above a lattice. So that's why that's that explains a number of the positions on the periodic table.
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So why is that? What's the proof? It's not a Bravo lattice. It's very easy.
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If you have a very lattice and you have a vector that goes from one atom to another, you have an arrow and you double that arrow.
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You always get to an atom that follows directly from the definition.
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So I'll have a little arrow and that's going from the centre of the bottom atom to the centre of the layer to atom, the green one.
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So it's going from the centre of the black one underneath there to the centre of the, the, the, the red one.
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If I double that vector, you see there's no atom there.
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So hexagonal, close packed is not a break lattice.
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So but hexagonal close packed is an objective structure.
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So every atom and hexagonal close back sees exactly the same environment.
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Okay. All right, good. So there's the definition I already gave of objective structure.
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And I want to know, obviously, what I'm going to do is I'm going to go back to the periodic table and I'm going
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to blacken out all the elements that do not crystallise as objective structures.
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So that's there's an objective structure, a ring of atoms. This is one example, simple example.
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So now I go back to the periodic table, I block it out. All the elements which are not objective structures.
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Okay. There's one. Manganese. Manganese is not an objective structure.
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It's accepted. Crystal structure at room temperature is not an objective structure.
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So that's quite interesting. But there's also many, many cases here.
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And and in addition to these standard crystal structures, boron likes these icosahedron carbon is the diamond lattice is an objective structure.
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The buckyballs are objective structures. Of course I already discussed this carbon nanotubes first.
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Fostering is an object of structure.
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Sulphur likes this kind of double ring, which is and the halogen compounds like these layered structures which which also have this property.
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So you would think that some really smart mathematician would have figured out why this is.
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But this is one of the this is one of the outstanding open problems in mathematics.
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It's one of those problems in mathematics that every couple of years people simplify the problem
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more so they can try to get more information and they get just painfully little information,
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then they simplify it more. And so now it's the problem of showing that that with Leonard with some some you know bad variant of of Leonard
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Jones potential that that that that you get the FCC structure as the ground state the lowest energy structure.
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And so and that's this problem is called the crystallisation problem.
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So it's it's really a fundamental open problem.
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Why should all these elements which have this, you know, they have very directional bonds to to other to other atoms.
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Why should they? Why should they prefer the same environment?
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It's completely unknown. Okay. But it's a it's a very good natured love, subjective structure,
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but it's a very good way to begin to to to be domestically inspired to make our economy.
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So we'll do that. So I have to talk about I symmetries.
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I symmetries are fundamental to the construction of origami.
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I symmetry is people would use this notation and again, you can think of this on several different levels.
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If you want to think about it on the picture level, I'll show you some pictures. And you can also think of R as a rotation matrix and C is a vector.
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And I see matrix consists of a rotation and a translation.
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So in two dimensions, the rotation might look like that, you know.
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So just something and the translation may look like that.
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So that's that's a nice symmetry and it's so simplistic.
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One of the simple there are more general notions of asymmetries, but the simplest example.
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If you have a nice symmetry, you can act on positions in three or two dimensions depending.
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And that's the that's the, the orbit of the point.
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Our that's the position of the point X acted on by that asymmetry.
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So now let's suppose we have two cemeteries and we're going to have our 1c1 on our two C2.
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I want to multiply them. So one of the great things about mathematics is you can you can invent your own multiplication rules.
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Okay, so here's the multiplication rule. I'm going to choose for G one and G2.
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Looks. Looks a bit weird, right? Well, this this kind of makes sense.
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I should multiply the rotations, rotate ones, maybe rotate again.
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But what's this all about? I mean, there's oh, there should be a left.
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A C2 should be a C2 right there. So. Sorry about that.
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Anyway, makes it more bizarre anyway. So what is that multiplication rule?
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It's a it's actually the most natural multiplication rule. It's it's the the fact is, the multiplication of G1.
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G2 is first you do g of. In fact, I switched them the first.
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I should first do G2 and then G1. I'm sorry. So I'm actually showing G2.
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G1. But anyway, you first rotate with with R one and then you translate C one.
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And then you take the result where you are and you rotate again by R two and you translate by see two.
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Okay. So, so, so the that multiplication rule means exactly this.
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You rotate and translate and then you take the result and you rotate and translate.
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Okay. So it's in mathematical terms, it's called composition of mappings.
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So it's exactly this this rule. So it's a very natural rule for for multiplying asymmetries.
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Now there's a there's a very fundamental notion in mathematics.
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The notion of a group is a fantastic idea.
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And always it's it's usually comes up in considerations of symmetry.
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And usually people would show the platonic solids when they want to describe groups.
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I happen to be a big aficionado of groups that describe things that look completely symmetric.
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But anyway, we have two asymmetries we just described.
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In fact, I will not quite described how to multiply them.
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And you can have a list of asymmetries.
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And the definition of a group tells you if you multiply any two by this rule, then you'll get back in the group.
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Okay, so that's just a closure kind of relation. And and groups have to have the I submit the elements of the group have to have inverses.
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It turns out if if this is the isometric rotation our translations see that the inverse of the symmetry is rotation are inverse.
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That just means instead of doing this you go back and, and it turns out minus R,
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minus one C and that's with that product is the inverse and that's the identity.
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And there's a repository for all the groups, all the discrete groups.
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I won't tell you what discrete is. It's a bit hard to explain, but all the sort of reasonable groups that that can be formed with asymmetries,
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it's the international tables of crystallography.
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And so in the library it's this big, it's online and you can look at those groups organised in a bizarre way.
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But anyway, that's where they are, so they're nice to use.
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Okay, so I mentioned two things.
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I mentioned objective structures, that's each atom sees the same environment and I mentioned something completely different.
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I saw symmetries and I saw symmetries can form groups. So the point is that there's a there's a very strong connection.
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Every object of structure you could.
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In other words, you could try to make them by looking at doing these different environments, but that would be very hard.
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But this is a very easy way to construct all objective structures.
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You take a single point, you take a nice symmetry group with the product I just mentioned,
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and you take the you take the each group element and you apply it to X with this rule of g on x axis,
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our x plus cft is our and C, so x is there g one of x might be their g, two of x might be their two.
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Three of x might be it's called the orbit of the point x, which is a very natural one.
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You see this picture. In fact, I made a kind of helical structure, which is an objective structure, and you can get all objective structures that way.
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So that's the very simple way is first you figure out the isometric groups and then
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you you apply them to points and you get many different objective structures.
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Okay. Now objective structures are very important for for I mean I symmetry groups are very important for origami and it's and it's for this reason.
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And there's also an eye I symmetry that I haven't told you about, a very important eye symmetry, which in which it's not exactly a rotation.
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I'll tell you about it now. So so this is the this is the rotation.
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And I let's make the translation equal to zero. So that's a there's another thing you can do with a star.
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You can you can divide it in half by a by a line, this dashed line, and you can flip it over.
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You can you can reflect across that line and you get the structure.
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Now that cannot be obtained by rotation because of the fact you can see this, this, this, this diamond, this here.
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And if you rotated to this position, the diamond would not be pointing down.
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So there's no way you can do this by rotation. But it's incredibly important for origami.
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Why? Because this this kind of asymmetry where this kind of, you know,
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this is what you we could call it a reflection is exactly what happens when you take a piece of paper and you flip it over.
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Okay. That's that's if I if I drew triangles and so forth on this piece of piece of paper.
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That's exactly. Of course, if I can flip it over and also move it, I can allow a translation as well.
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So and those are those are all the things you get to do in at least piecewise rigid origami.
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Okay. So in fact, that's represented by this matrix. The rotation is represented by this matrix.
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So you can make a little diagram. I find this kind of diagram very useful.
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I take and imagine the square is all two by two matrices.
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And again, you can think of this also in intuitive terms, but and this this circle, I mean, there's just a one parameter.
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It's the angle of rotation.
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So as you change the angle of rotation, you can imagine there's there's a curve in the four dimensional space of two by two matrices that that,
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that closes on itself goes from the identity. Each point on that curve is a rotation matrix like that.
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It's just a representation,
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but I find a useful representation and then you can do this flipping and that's this particular flipping is represented by this matrix.
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But once you flip, of course, just as in origami, once you flip, you can also rotate as well.
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Okay. And if you can, you can say, well, maybe it's more complicated.
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Maybe if instead of flipping this way, I flip this way and rotate it, I could get something new.
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No, that's. You don't get anything. So you just get this second. Second circle.
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One parameter family. Okay, of.
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One parameter family of of flips. Okay.
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So those are the three, three matrices. Okay. So that's that's fine.
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So now, of course, I didn't say anything about the C,
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I'm just talking about the R and I'm going to allow the R when I do origami to be either through this one or this one.
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And so at the beginning, I'm just going to talk about folding from two dimensions back to two dimensions.
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In other words, I take the piece of paper, I fold it up, and I put it back into two dimensions.
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Okay. Now, origami requires more than I saw.
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Symmetries the same. I saw symmetry somehow have to fit together.
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So we have to think about what that means. So let's suppose we have the nice symmetry.
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Let's have the trivial asymmetry, which is the identity, which is identity matrix and zero.
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That means that means it doesn't rotate and it doesn't translate.
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So that's the most trivial asymmetry.
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And now let's that's that that's this one. And now let's let's choose this one.
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And I want to choose it to fit. So I draw a line on this piece of paper, and I flip this over the line that's folding and.
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Now that's governed by by an I cemetery and you can write that I cemetery down but it's not any old I cemetery.
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The fact that those two asymmetries agree on this line is a very strong restriction.
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That's called compatibility. Now I'll write that.
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I'm going to write it as a normal. If people are know matrices, it means that first of all, the translations don't matter at all.
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You can always adjust them that are compatible.
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The real condition is that the two matrices, the two rotation and reflection matrices have to differ by a rank one matrix.
329
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So this would be if you wrote this in its components by J minus AJ is high and dry or if you want to think
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of it and completely intuitively then then exactly it's this it's the meaning of that is this picture.
331
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But the point is we can put it into into analytical form.
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Okay. Here's the theorem. Which is which is which we should prove.
333
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It says that every pair. So you take these two orbits.
334
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This was the rotations, this was the reflections plus rotations.
335
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And I choose the matrix over here. I choose the matrix over here.
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And as I say, you can always adjust the the little and the little b the translations to make an origami.
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But but it turns out that B is always compatible with a.
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Okay. Proof. We can do this. Okay, so I.
339
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I take it again, I can without loss. And generally I can take be to be the identity, or I could take it to be a rotation and a translation.
340
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And now I flip this over and before I push down, I can rotate it any way I want and then I can push down.
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And that that construction, which is simple construction is shows that shows that given any matrix over there,
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given any reflection plus possible rotation and any rotation, any pure rotation, they're always compatible.
343
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Can I invent a little notation for that? I'm going to I'm going to draw a blue line between these two matrices, if they're compatible,
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if they satisfy this condition, or more intuitively, if this if this, if the two symmetries can agree on a line.
345
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Okay. So now we do the simplest economy. And I think what I'm trying to do is okay, so first of all, let me let me let me do it on analytically.
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So there's our one. I pick a rotation that's doing this and and I pick a cue two, I pick a reflection.
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So and as I say, I can always adjust the translations to work.
348
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So I pick, pick a rotation reflection. They're always compatible because I said any two are compatible so I can draw a blue line.
349
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Now I pick a rotation over here and it's compatible with.
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Q to pick a I pick a reflection over there, it's compatible with our three and sure enough, Q four is compatible with our one.
351
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Okay, I satisfied all compatibility.
352
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Just I can choose anything I want, you know, and that means analytically that those four conditions are satisfied.
353
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And even if you really don't quite understand those four conditions, if I add them up, the two cancels, the one cancels.
354
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There, the four cancels and three cancels.
355
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And that means that the sum of the right hand side is zero and there's some kind of restriction on the normals to the interfaces.
356
00:36:25,850 --> 00:36:30,830
Okay, so, so now I'm going to make I saw Matrix. Now tell you what the translations are, I'm going to make them all zero.
357
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And I can do that because they meet at this point. Okay.
358
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Now I'm going to do a very, very famous construction in origami.
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I'm going to take a piece of paper and and I'm going to just do like this.
360
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I'm going to push it because I'm describing origami mapping from R2 to R2.
361
00:36:50,720 --> 00:36:54,200
Right. I'm in two dimensions, so I just do this. Okay.
362
00:36:54,320 --> 00:36:59,300
So you can do this at home. And what happens is this is a parchment paper.
363
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Works pretty good for this. And you can see what are our fault lines.
364
00:37:05,750 --> 00:37:08,239
So you have a lot of interesting fold lines.
365
00:37:08,240 --> 00:37:16,310
We have places that are kind of destroyed, but we also have some quite simple places where four folds come together,
366
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like down here, and there's one right there and this one right there.
367
00:37:22,040 --> 00:37:24,530
And there's there's one right there as well.
368
00:37:25,040 --> 00:37:32,600
And if you if you study them for a while, you'd realise that they there's some particular geometric relation for those guys.
369
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And this is a famous theorem in origami. It says that.
370
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You can fold this fully folded back into to to start in two dimensions and fully fold it back into two dimensions.
371
00:37:45,290 --> 00:37:51,919
If, if, and only if. Alpha plus beta equals pi. So the sum of opposite angles is pi.
372
00:37:51,920 --> 00:37:55,790
And of course, the sum of those R is pi two necessarily.
373
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And there's two solutions and those are that. Those are four for two.
374
00:38:00,820 --> 00:38:04,840
Typical case, those are the two solutions. So that's a something you see all the time in origami.
375
00:38:04,840 --> 00:38:10,180
It's a very, very typical construction if you want to fold from two dimensions back into two dimensions.
376
00:38:11,180 --> 00:38:19,430
Okay. And the interesting thing is there's more than that is that if you satisfy alpha plus beta equals pi.
377
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Then not only is there some kind of mapping from two dimensions to two dimensions,
378
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but there's a mapping from two dimensions up into three dimensions and I symmetry all the way and back to back into two dimensions.
379
00:38:32,620 --> 00:38:38,200
In fact, there are two exactly two such contiguous mappings.
380
00:38:39,130 --> 00:38:42,820
There's also two trivial solutions where you fold it and you fold it.
381
00:38:42,820 --> 00:38:48,760
But okay, that's the you can write. Exactly. And you could write formulas for all the information that that you have.
382
00:38:49,670 --> 00:38:52,610
Okay. So that's a well-known thing in our gallery, but it's nice.
383
00:38:52,850 --> 00:38:58,160
I hope it's nice to see you in a now maybe more mathematical way than you would see in origami books.
384
00:38:58,970 --> 00:39:03,320
So here's something that you won't see in origami books. So this is.
385
00:39:03,740 --> 00:39:08,450
So. So now we say some of op at opposite angles is PI.
386
00:39:08,510 --> 00:39:13,940
Right. And so I'm going to make a lot of four fold things and I'm going to arrange that sum of opposite angles as pi.
387
00:39:14,270 --> 00:39:18,830
Can I fold it? So if I cut out one of those little regions, I can fold it.
388
00:39:19,370 --> 00:39:22,579
But turns out I can't fold that and I can fold that.
389
00:39:22,580 --> 00:39:25,640
But in general, I won't be able to do it on a piece of paper.
390
00:39:25,670 --> 00:39:30,160
You won't be able to fold it. So what is the what is the restriction?
391
00:39:30,310 --> 00:39:33,400
It's it's very nice, actually. It's this.
392
00:39:34,470 --> 00:39:39,090
So I look on the edge and I look at these guys, those four fold guys.
393
00:39:39,100 --> 00:39:46,200
They're. And I arrange, arrange, arrange them to have any, any angles.
394
00:39:46,560 --> 00:39:51,450
But samba of opposite angles is pie for this point, for this point, for all the points on that edge.
395
00:39:52,840 --> 00:40:01,780
And now I arrange. Now I pick this edge and I arrange the sum of opposite angles as pie for for all the points along this edge.
396
00:40:02,950 --> 00:40:11,140
And the theorem says. And okay, one more thing you get to choose for each point on this edge, you get to choose a plus or minus one as well.
397
00:40:12,580 --> 00:40:14,020
And with that amount of freedom,
398
00:40:14,410 --> 00:40:24,610
then the entire fold crease pattern is determined that these are all ways to fold it and you can write formulas for everything.
399
00:40:24,610 --> 00:40:34,659
This is a particularly simple one, but you may not be able to decide how it gets folded, but that's the situation.
400
00:40:34,660 --> 00:40:40,299
So it's very restricted. You can't just go ahead and put these together in some arbitrary way.
401
00:40:40,300 --> 00:40:45,820
With Alpha plus equals pi, you can only choose them here and here, and then everything's determined.
402
00:40:46,780 --> 00:40:54,129
It's a little bit like differential equations. You kind of assign boundary conditions and then you you see that, you see the possible,
403
00:40:54,130 --> 00:40:58,990
you get one solution, unique solution, if you would, those plus or minus, as I've mentioned.
404
00:41:00,830 --> 00:41:06,320
Okay. So that's that's what that folds now that that's that structure falls in this way.
405
00:41:08,880 --> 00:41:13,500
In fact, if it falls flat, I just it just doesn't the movie doesn't go all the way.
406
00:41:14,970 --> 00:41:20,520
And you can do more complicated things like this. So you can turn this case along this edge.
407
00:41:20,520 --> 00:41:27,390
So see those guys right there? They were assigned such an alpha plus spider equals pi and some kind of quasi random way.
408
00:41:27,660 --> 00:41:33,150
And these were assigned along this edge. And then all these all these fold lines were determined.
409
00:41:34,070 --> 00:41:37,070
And then it can be folded. That's.
410
00:41:38,090 --> 00:41:41,570
Okay. So that's gives you an idea about that.
411
00:41:41,610 --> 00:41:45,690
You can see it again, that fold that folds flat, by the way. Okay.
412
00:41:46,050 --> 00:41:50,550
So nature has another has there's all kinds of folding in nature.
413
00:41:50,590 --> 00:42:02,990
And this was mentioned by John Paul, but it turns out that that phase transformations and crystals are very closely related at some level to origami.
414
00:42:03,000 --> 00:42:07,920
So what is a phase transformation in crystals and how does it differ from origami and how is it the same?
415
00:42:08,880 --> 00:42:17,760
So many transformations in crystals are transformations where where the crystal structure changes at some particular temperature.
416
00:42:18,290 --> 00:42:24,780
Okay. So maybe very often this cubic structure like this body centred cubic structure, which, by the way, this very lattice.
417
00:42:25,760 --> 00:42:28,610
Is the stable structure at high temperature.
418
00:42:29,390 --> 00:42:37,280
And what happens when you reach a you cool this material, you reach a certain temperature and the and the material distorts.
419
00:42:37,280 --> 00:42:45,860
And that just shows one example in this particular material, which is here it starts by shortening on this vertical axis.
420
00:42:46,130 --> 00:42:49,730
And the top face here, which is a square here, becomes a rhombus.
421
00:42:51,380 --> 00:43:00,200
And now you can think by symmetry that there should be actually six ways that can distort in this with this kind of face transformation.
422
00:43:00,200 --> 00:43:03,590
And those are the six matrices that describe those distortion.
423
00:43:03,800 --> 00:43:09,020
So what are the six ways? So instead of shortening along this axis, I could shorten along this axis sucker.
424
00:43:09,020 --> 00:43:13,250
I shorten along this axis. And for each one of those, I can make two rhombus.
425
00:43:13,550 --> 00:43:17,990
I can pull it out like this or I can pull it out like this. And those are the six distortions.
426
00:43:19,290 --> 00:43:25,860
So now these are described not exactly like like like I saw cemeteries, but quite similar in some way.
427
00:43:26,250 --> 00:43:31,970
You know, you first have to you first have to distort and that's the end.
428
00:43:31,990 --> 00:43:39,330
You do a linear transformation described by one of those matrices, and then you're allowed to rotate because, of course, it's a crystal.
429
00:43:39,630 --> 00:43:46,470
You can you can rotate it however you want and you're not going to it's going to be equally possible before and after rotation.
430
00:43:47,840 --> 00:43:53,140
So. But then you can also have translations, of course. So it's a little bit more complicated than origami.
431
00:43:53,150 --> 00:43:58,760
We have not only a rotation here or a part or a reflection in a a translation.
432
00:43:58,760 --> 00:44:02,110
We have also a distortion, but the distortions are discrete.
433
00:44:02,120 --> 00:44:09,710
There's only six of them. Okay. And exactly the same mathematics work.
434
00:44:09,740 --> 00:44:15,620
So instead of two of those orbits, we have three by three matrices, we have six orbits.
435
00:44:15,620 --> 00:44:25,130
We have this free rotation. This is kind of just a representation of of all the distortion plus rotations the crystal can undergo.
436
00:44:25,820 --> 00:44:35,719
And there's a particular microstructure of this crystal. And if you if you go in here and you you notice this, in fact, it's a very unusual situation.
437
00:44:35,720 --> 00:44:39,680
You find out that this this matrix is compatible with this matrix.
438
00:44:39,680 --> 00:44:45,990
And this is compatible with this one. This is compatible with this one. The normal cases and this would not be compatible back to air.
439
00:44:46,040 --> 00:44:49,069
So in that sense, it's very different from origami.
440
00:44:49,070 --> 00:44:55,890
But for this material, it's this fact be it goes right back to a and that means all those four interfaces are compatible.
441
00:44:56,330 --> 00:44:58,610
And once you realise that you can make the whole structure.
442
00:44:58,610 --> 00:45:05,899
So and as John was saying, I mean, these, these, these phase transformations are incredibly interesting.
443
00:45:05,900 --> 00:45:13,370
And I'm going to show you I'm going to show you one in which you can imagine that, you know, just like origami,
444
00:45:13,370 --> 00:45:21,200
if you get to play with those six matrices by changing these so-called lattice parameters, alpha, beta, gamma, you get to change the distortions.
445
00:45:21,770 --> 00:45:27,559
Then maybe you could have very special distortions in which the two phases fit together in many, many ways.
446
00:45:27,560 --> 00:45:31,670
You got many, many origami. So, in fact, that is the truth.
447
00:45:32,120 --> 00:45:40,669
That is exactly the truth, is if you tune Alphabet and cannot have some special values, then you'll get many.
448
00:45:40,670 --> 00:45:43,870
Origami is possible with these crystal structures. Okay.
449
00:45:44,890 --> 00:45:48,580
So how do you tune your tune? By changing the composition of the material.
450
00:45:48,910 --> 00:45:53,310
So if the material is is some alloy, then you change the composition.
451
00:45:53,320 --> 00:46:01,180
You very carefully measure these things, this, and you go towards your mathematically determined lattice parameters.
452
00:46:02,140 --> 00:46:05,860
So. Very good. So. That's that's how you do it.
453
00:46:05,860 --> 00:46:10,200
And I want to show you material that's been tuned. This is one of the most or.
454
00:46:11,290 --> 00:46:13,449
There are two sort of spectacular materials.
455
00:46:13,450 --> 00:46:20,979
This is this is one of them that had been obtained by this kind of tuning of lattice parameters so that there were many,
456
00:46:20,980 --> 00:46:24,460
many ways to fit together completely non generic conditions.
457
00:46:24,790 --> 00:46:27,850
And it's fun to watch a movie of it transforming back and forth.
458
00:46:27,880 --> 00:46:31,960
By the way, I had six distortions in the case I mentioned before.
459
00:46:32,080 --> 00:46:38,260
This material is one of the distortions, but it's very similar, very similar otherwise to to what I was saying before.
460
00:46:39,210 --> 00:46:46,000
So this is a movie where you were I'm just heating and cooling the specimens.
461
00:46:46,000 --> 00:46:50,400
So there's a there's a there's a little thin film heater underneath this this piece of material.
462
00:46:50,730 --> 00:46:54,390
It's heating it uniformly as possible, and it's cooling.
463
00:46:54,870 --> 00:46:59,140
And one of the wonderful things about this movie is that the every picture is the different.
464
00:46:59,340 --> 00:47:05,000
Different. So in normal phase transforming materials, every picture is the same.
465
00:47:05,240 --> 00:47:11,330
But this particular material somehow has so many different origami to make that it can do a different each time.
466
00:47:11,330 --> 00:47:17,809
And this this particular material is one of the most reversible materials that that's known.
467
00:47:17,810 --> 00:47:24,830
As I say, it's one of two materials that okay, to explain it a bit more in that phase.
468
00:47:24,830 --> 00:47:27,650
That's the cubic phase, the boring grey one.
469
00:47:29,040 --> 00:47:34,920
You can raise the temperature to some higher temperature and you're always in the grey phase, that cubic phase.
470
00:47:35,280 --> 00:47:43,230
But you can cause it to transform by putting stress. And that's the most demanding test of a phase transformation is do it under stress.
471
00:47:43,740 --> 00:47:49,200
This material will, even though it's zinc, gold, copper, which is never expected to do this,
472
00:47:49,830 --> 00:47:57,030
this material can go 100,000 cycles with a stress which would make the steel in this building fail.
473
00:47:57,450 --> 00:48:02,730
Okay. It's way about twice the failure stress of the steel in this building.
474
00:48:04,560 --> 00:48:13,890
So, okay, so okay. Now let's go back to two to origami and let's try to be again, Adam mystically inspired.
475
00:48:14,820 --> 00:48:20,400
And so I arrange I arrange that, that the sum of those two angles is PI.
476
00:48:21,060 --> 00:48:22,920
And I'm also going to do it on a parallelogram.
477
00:48:22,920 --> 00:48:29,130
So I'm going to arrange that, that, that this length equals this length and this length equals this length.
478
00:48:31,380 --> 00:48:37,410
And you can follow that because it's a it's got four regions and some have opposite angles as pie.
479
00:48:37,440 --> 00:48:41,520
There's the two solutions I was describing earlier. This is partly folded.
480
00:48:42,480 --> 00:48:46,740
You can you can fully fold it. And as I say, those are the two ways.
481
00:48:47,340 --> 00:48:51,149
So if you have that, you could do so you can think of doing something interesting.
482
00:48:51,150 --> 00:48:57,480
So let's let's partly fold it. So we take this guy, we take maybe this guy here, maybe that's what I did.
483
00:48:57,750 --> 00:48:59,700
Now I guess it's the top one. But anyway, it doesn't matter.
484
00:49:00,180 --> 00:49:09,050
And remember, after it's been folded, this length equals equals this length here, and this length here equals this length.
485
00:49:09,550 --> 00:49:14,750
And in fact, it's down here. The way I hope it's clear, the way the folding goes, the perspective.
486
00:49:14,760 --> 00:49:23,620
But anyway, this this length equals this one, and this one equals this one. You know, I saw cemeteries preserve length.
487
00:49:24,580 --> 00:49:32,230
So you can imagine you may be able to find a nice symmetry that maps this line into this line and this line into this line.
488
00:49:32,830 --> 00:49:37,820
Turns out you can do that. That's not that hard. I'll tell you what they are in a minute.
489
00:49:38,620 --> 00:49:42,680
And not only that, you can find these two asymmetries and they commute.
490
00:49:42,700 --> 00:49:46,580
Actually. So G1. G2 is G2. G1 we're using the same product for.
491
00:49:48,090 --> 00:49:55,530
If two asymmetries compute and you can just take powers of them and that automatically forms a group.
492
00:49:55,530 --> 00:50:03,340
So this sapele in case the the community elements is just wonderful because if you take two things of the form G one to the page,
493
00:50:03,360 --> 00:50:08,790
two to the Q and you take another one from here like ag1 to the one to the as you write it out,
494
00:50:09,360 --> 00:50:13,590
since they come here, you can switch these two guys and then of course you realise you can switch,
495
00:50:13,860 --> 00:50:17,970
you can push all the g ones to the left and put push all the details to the right.
496
00:50:18,300 --> 00:50:25,710
You get something like this, it's back in this set. Okay. So a very easy way to form a group if you have if you have the elements compute.
497
00:50:26,370 --> 00:50:32,460
So in fact, I can make asymmetries which both which both do that oops, there it is.
498
00:50:33,000 --> 00:50:36,330
Which both map the side and to the side and which form a group.
499
00:50:37,380 --> 00:50:44,100
And they look something like that. But this is the wonderful thing about groups is is.
500
00:50:45,200 --> 00:50:49,279
This this isometric G1, for example, maps this side into this side,
501
00:50:49,280 --> 00:50:54,830
but I can apply it to the whole partly folded origami and it fits on there perfectly.
502
00:50:54,840 --> 00:51:01,310
It satisfies compatibility. And then I can take G2 and G2 fits on there perfectly.
503
00:51:01,700 --> 00:51:05,930
And the wonder of a group is that G1 times G2.
504
00:51:06,990 --> 00:51:13,260
Applied to this original or partly folded origami perfectly fits with all its neighbours.
505
00:51:13,590 --> 00:51:19,260
And I can keep going and going and going and make big origami so we can do that.
506
00:51:19,440 --> 00:51:26,830
And let me tell you about the asymmetries that that so we have to be sure that they compute and they they.
507
00:51:26,850 --> 00:51:31,080
So the symmetries that do that are our members of the helical groups.
508
00:51:31,590 --> 00:51:34,229
And there's a very interesting theorem about the helical groups.
509
00:51:34,230 --> 00:51:41,340
I won't go into detail about it, but those are the four helical groups and they're the details of the asymmetries are there.
510
00:51:41,340 --> 00:51:47,250
But you don't want to read that. I mean, it's just it's just it's just explaining to you that you can explicitly write them down.
511
00:51:49,370 --> 00:51:57,049
This group, for example. So I explain it by I take this blue ball and I take it to orbit under this group and the age is there.
512
00:51:57,050 --> 00:51:59,750
So age to the piece. So I get I get that structure.
513
00:52:00,350 --> 00:52:08,389
And in this case, I took I took the group B, I took the orbit of this under this group and and I coloured,
514
00:52:08,390 --> 00:52:12,680
I actually coloured the atoms according to the power. M Which could be one or two.
515
00:52:12,680 --> 00:52:20,479
That's why you got the colours. And C Is, is, is the orbit of, of one of these points and it's coloured according to the power.
516
00:52:20,480 --> 00:52:27,440
Q And so forth and so on. It turns out that we're going to the groups that satisfy the conditions on the previous slide.
517
00:52:27,440 --> 00:52:32,510
They, they, they compute and they, they can map edges into edges.
518
00:52:33,820 --> 00:52:40,560
Our members of this group see. So it shows that I can write them down explicitly.
519
00:52:41,280 --> 00:52:45,510
So then immediately you can just form lots and lots of our gummies and that's them.
520
00:52:47,100 --> 00:52:53,219
So and this is this is one, for example, where the where the thing is folded and then it folded a little bit more.
521
00:52:53,220 --> 00:53:00,170
And you get the structure here. But you can imagine, as I said, you apply these group elements.
522
00:53:00,170 --> 00:53:03,560
You're building up the structure when you get around the other side. Of course, they may not meet.
523
00:53:04,470 --> 00:53:08,670
And so there's additional conditions and actually they're quite restrictive.
524
00:53:09,000 --> 00:53:15,630
So I won't go into detail. Now, this is a this is a bi stable case like I was just describing.
525
00:53:15,640 --> 00:53:20,820
You have folding and you get this structure, you fold it a little bit more and you can get this structure.
526
00:53:20,910 --> 00:53:25,770
So it's kind of bi stable and in fact, you can write the solutions.
527
00:53:25,770 --> 00:53:31,020
You can write solutions in terms of two parameters. And the solutions are represented by these dots.
528
00:53:31,740 --> 00:53:40,350
These dots are isolated. And that means, in fact, there's a little picture there so you can get so that's jumping between different solutions.
529
00:53:40,380 --> 00:53:49,980
You can do it again here and you can see you can't continuously, you know, morph this helical structure by, by, by any method.
530
00:53:51,160 --> 00:53:57,130
So of course. So that's the situation I just was.
531
00:53:57,850 --> 00:54:01,420
So. But it would be fun to morph it. So. So let's try.
532
00:54:01,660 --> 00:54:08,530
Let's try another method of morphing. So and again, we'll be we'll be out of mystically inspired.
533
00:54:08,680 --> 00:54:12,640
So this is a this is a virus actually studied this virus at one point.
534
00:54:13,150 --> 00:54:17,490
This is bacteriophage T4. It has a capsid.
535
00:54:17,880 --> 00:54:26,100
It has its very complicated virus and attacks. Bacteria like E Coli is a favourite target and it has.
536
00:54:26,380 --> 00:54:29,670
There is the real thing. It attacks them in the following way.
537
00:54:30,120 --> 00:54:33,390
It lands on the surface of the bacterium.
538
00:54:33,750 --> 00:54:41,250
And these these these tail fibres are sticky to the surface of the of the bacterium.
539
00:54:41,460 --> 00:54:44,790
And these long tail fibres are are also sticky.
540
00:54:44,790 --> 00:54:53,280
So they come down. And what they do is they and as soon as those come down, a nucleus, a phase transformation in the tail of this virus.
541
00:54:53,730 --> 00:55:01,530
And so the phase transformation and there's a picture of theoretic from a from a theory, but quite, quite,
542
00:55:01,680 --> 00:55:07,110
quite accurate representation of the phase transformation which occurs in the tail sheath of bacteriophage.
543
00:55:07,110 --> 00:55:12,720
T4 Inside this tail sheath is a very stiff tail tube.
544
00:55:13,050 --> 00:55:17,220
So this is joined very rigidly onto the surface of the bacterium.
545
00:55:17,250 --> 00:55:23,160
So when it shortens, it drives the tail tube through the cell wall.
546
00:55:23,550 --> 00:55:28,590
In fact, it not only drives it through, but it twists it. It twists the knife as it's going through.
547
00:55:28,980 --> 00:55:35,820
And then the DNA of the of the bacteriophage T4 enters the cell and then it does its usual thing of replication.
548
00:55:37,210 --> 00:55:42,310
So we could be we could be optimistically inspired and we could say, okay,
549
00:55:42,310 --> 00:55:47,350
there could be phase transformations and helical structures, kind of like the phase transformations I showed before.
550
00:55:47,740 --> 00:55:57,549
And it's not hard to work out all possible compatible structures which undergo phase transformations and you're seeing representatives of Bose,
551
00:55:57,550 --> 00:56:02,920
they're very similar to the idea of compatibility is exactly the same.
552
00:56:02,920 --> 00:56:07,480
In fact, idea of compatibility we're using for origami, but it's for two helical structures.
553
00:56:09,010 --> 00:56:17,600
Okay. So that provides a nice way of of morphing and a face morphing origami structures.
554
00:56:17,620 --> 00:56:25,299
We can have we can arrange a compatible interface and then we can by moving that interface, we can do morphing of the structure.
555
00:56:25,300 --> 00:56:32,980
I'll show you that again. So this is twisting and also increasing its length.
556
00:56:34,080 --> 00:56:40,620
So that's that's all I had to say, except I have to come back to the homework problem.
557
00:56:41,040 --> 00:56:46,020
Before I do, I want to acknowledge. Fan Fang is a graduate student published pinsky there.
558
00:56:46,140 --> 00:56:51,150
They did a lot of important work in this area very recently. But I particularly want to acknowledge Frank Yu.
559
00:56:51,150 --> 00:56:57,870
So Frank, who is a high school student from Wayzata High School in Minnesota, it's near Minneapolis.
560
00:56:58,540 --> 00:57:01,950
It's and Frank Yu is 16 years old.
561
00:57:02,340 --> 00:57:11,070
And Frank, who knows all the theory I presented, including the mathematical part, and I can show you some nice 3-D printed structures that he made.
562
00:57:11,730 --> 00:57:16,380
Okay. Back to this structure in the Math Institute that have that I've ratio out here.
563
00:57:16,590 --> 00:57:20,990
So it turns out the following is true. There's a four dimensional space.
564
00:57:21,740 --> 00:57:25,090
There are four. And there's a there's a region in that space.
565
00:57:25,100 --> 00:57:28,880
You can define that region absolutely precisely. It's not even very difficult.
566
00:57:29,360 --> 00:57:35,210
And you can take any curve in that four dimensional region, and that will define an origami structure.
567
00:57:35,480 --> 00:57:42,290
So that can be folded a huge number of ways. So, for example, this is one of the ways it can be folded.
568
00:57:45,900 --> 00:57:49,410
You can see it again, I guess. Oop. Okay.
569
00:57:49,410 --> 00:57:55,500
I guess you can't see it, but there's a great many ways that can be folded here.
570
00:57:58,820 --> 00:58:02,820
There it is right there. So.
571
00:58:03,770 --> 00:58:07,240
So I hope this doesn't make you nervous when you leave the room.
572
00:58:07,320 --> 00:58:13,080
You know, maybe you don't want to walk under this glass ceiling, but thank you very much for your your interest.
573
00:58:13,080 --> 00:58:13,350
And.