I'd like to welcome you all to the Mathematical Institute for another Oxford Mathematics Public Lecture.
My name is I don't go really. And I'm in charge of the lecture for the institute.
It's great to see so many of you. Just a few point of order.
I just want to point to the exits. These are what we call full British exits.
You'll have to be careful if you take this one.
I think I've seen Boris Johnson's trying to hide from voting in the Commons today.
So you'll have to step over him. If you go that way today, we'll talk about origami.
It's a topic that I've always been fascinated by.
I'm completely rubbish at it, but I always love it and I've always wanted to have a public lecture on it.
And I'm extremely grateful for Professor James today to give a lecture on this topic.
If you're interested in this topic,
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,
I think the engineering department and the British Origami Society.
There is a British origami society with 700 members, so there'll be an event in September, early September.
You can find that on the website of the bourse, as it's called, British Origami Society,
where there'll be talk both about mathematics but also about more the playful aspect of origami.
So if you're interested in things you should definitely go with.
I just wanted to advertise and be short.
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.
It's a financial company with offices in New York, London and Singapore.
Today, I will not give you a full introduction. I've admired the words of Professor James for many years.
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.
There are few people, when you hear them, you realise both that they understand the topic extremely deeply,
but also they can phrase it in a way that makes you feel you also understand deeply.
And so I always wanted him to give a public lecture here, and particularly.
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.
So please help me introduce John Bode. We will do a proper introduction.
So it's my great honour and pleasure to introduce my scientific colleague and good friend,
Professor Richard James, who is a distinguished McKnight University professor at the University of Minnesota.
So I think James is a truly remarkable scientist, an undoubted world leader in both theoretical and experimental mechanics.
And that combination itself makes him unusual.
And he has a whole string of extraordinarily original contributions over a range of problems and mechanics,
mostly centred on on on the behaviour of alloys that undergo solid phase transformations, which is a very important practical issue,
but stretching into other areas such as the structure of viruses and parts of statistical
physics now I think belongs to is does not belong to a mathematics department,
but the Department of Aerospace, Engineering and Mechanics.
And though he will deny it, he's also a very fine mathematician.
And I think that one thing that I admire of him is that perhaps more than most mathematicians who work and in mathematics departments,
he believes in the power of mathematics for describing nature.
So silly special cases that mathematicians would ignore and his hands turn into discoveries of new materials.
So I think that's a remarkable ability to to to do that.
And so it's a great pleasure to invite him to give his lecture. They're just number two.
Okay. Thank you very much for this very kind introduction.
And everyone can hear me. Is that clear? And it's a great pleasure to deliver this lecture and a great honour.
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,
and also to speak in this wonderful math institute and with its many references to mathematics.
In fact, this glass ceiling that you see there has a nice reference to to the eigenvalues of the plus in their eigenvectors,
but it also has a connection, a nice connection to origami.
So, in fact, I'm going to start with a homework problem.
Is this this you can think of these glass panels as rigid, but I'm going to allow the joints to be flexible.
And so my question is, can this be folded? Okay.
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.
And at the end, I'll tell you I'll tell you all about it. Okay.
So first a little bit about myself. I come from a kind of place maybe I've not heard of.
It's it's Minnesota. It's in the United States. It's this little red state up there in the very far north.
And it's known for these things.
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.
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.
So almost everybody has a canoe and they go around in these lakes.
So just to convince you that origami and folding has penetrated all areas of technology, I want to show you a little video here.
That was the the Ori Canoe Company was extremely pleased that I was going to show this video.
But you can even fold the canoe up.
So if you have a studio apartment, you can own a canoe and then you can go canoeing.
So. Secondly, I'd like to make an acknowledgement.
I have actually a project on origami structures, the design of origami structures, and it's one of the participants.
It's about five or six people. One of the participants is Robert J.
Lang. You may know him from his many books on origami, and he's an interesting collaborator,
a very interesting person, and he does something that we are so far away from doing mathematically.
He does this, he makes a rhinoceros. So you tell him to make a rhinoceros and he will make you a beautiful rhinoceros.
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.
That's what in mathematics we would call the inverse problem. We have no way to solve the inverse problem.
And in particular, you know,
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.
So in fact, this collaboration with him is I feel like it's working with a real with a true genius.
One can say that Robert de Lange is a true, true genius. Okay.
So this this talk is about origami, but it's inspired by Adam Mystics,
which may seem like an unusual place to get some inspiration for folding things.
But that's that's exactly what I'm going to do.
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.
It's it's one of the three forms of carbon that did not get the Nobel Prize.
That's of lower dimension. In fact, I think it's the most interesting one.
Or nanotubes in general are are very interesting. And what I would like to do is point out a feature of its structure.
So I want you to imagine. So I put a little couple dots there.
I want you to imagine that you're they actually sitting on one of those atoms and you're looking out at the structure.
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.
Now you go to another point on the structure. So you're going to get you started here.
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.
You see exactly the same picture as you do on the first one.
And that's true of every atom of the structure.
If you if you orient yourself in the right way, you take a picture, you see exactly the same environment out to infinity.
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,
because how can this be the same environment as this? This is.
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.
And if you do, then you'll see that that's the orientation at which you'll see exactly the same thing.
And if you turn sideways, you'll see exactly the same thing. It's a property of that particular structure.
But it's also a property of many, many structures, and particularly these kind of nanostructures that people are discovering these days.
So here's. So first, I'm going to put it in mathematical terms that I'll show you a bunch of pictures of.
I call these objective structures. And so let's let's try to put that in mathematical terms.
So here it is. Here's a so I'm going to think of positions in space.
You can think of three coordinates of the position or in two dimensions like this.
It's the idea is in either case and suppose X one is a point of the structure and there's other points,
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.
So this and so. Okay, fine. Now, now I go to another point of the structure.
Any other point say XY and I draw vectors to all the points of the structure.
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.
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,
I can rigidly rotate it and I can rotate it into into the spray of vectors coming
from X to that's an objective structure and you can write that down mathematically.
The notation, I mean, I think some of you will understand the notation.
I'm not sure everybody will,
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.
It goes like this. So this is the structure.
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,
you can take the, the, the arrows going from X1 to all the points.
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.
Okay, so that's exactly what I said in words is that's the mathematical definition.
Of course you have a mathematical definition.
You can study it and you can do quantum mechanics with this definition, and you can do many things, many interesting things.
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.
Of course, these are Nobel Prize winning structures here, graphene and buckyball.
I've already discussed carbon nanotubes. These are the sort of classic carbon nanotubes.
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.
The only thing I should mention is, is like these viruses, that's the bird flu virus, by the way.
This is this is phosphor green, which is also a very, very interesting structure.
And again, you might not think that every atom sees the same environment, whether it's on the lower level or the upper level.
But that's true. Okay.
And this this amyloid protein is also extremely important protein.
But this is one of the bird flu viruses.
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.
The definition goes like this. You have identical molecules.
You number the atoms in each molecule, say 1 to 100.
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.
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.
Then the 27th atom of every molecule sees the same environment.
Okay, so that's the definite 26th atom sees a different environment from the 27th, but all the 26th atom sees the same.
And then there is a huge number of, of molecular structures which which satisfy that definition.
Most of all do the atomic case because it's a bit easier to to explain things.
But those are those are objective structures.
So now we're going to play a game with the periodic table.
So there's a periodic table. And my little game would fail miserably if I included these radioactive elements down here.
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
crust that any crust at any one moment and no one knows the crystal structure.
Okay. So this is all about the structure of the elements. Okay.
How do most people think of the periodic table? You can open a book on atomic structure.
They think this way. They think in terms of Broadway lattices.
They they build up the structures from from previous lattices.
So mathematically,
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.
You start with this atom. Those are the three factors in this particular case.
And, you know, and so this this atom can be shifted to this position by by adding E three.
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
combinations of integers on these three vectors and you get the structure,
let's call the phase centred cubic structure FCC.
You might not think if you, if you, if you're not really quick, like,
like I'm not you might not think you could get this atom, but this atom is obtained by e1e2 minus c three.
So it's integers on those three vectors. And this is the phase centred cubic structure.
So as I say, most people think of the periodic table in terms of rough lattices.
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.
Okay. So there you go. So of course lots of elements are not don't crystallise.
And I did this kind of rationally somehow I,
I took the most common structure at room temperature and if the material is not solid at room temperature,
I took the structure accepted structure at zero temperature. So I tried not to fiddle anything.
And there's, there's my blackened out. So many elements do not crystallise as a lattices.
Now, many of those are actually the same crystal structure.
About a third of the periodic table. So roughly speaking, half of those black, black and out elements prefer the hexagonal close pack structure.
Okay, so let me explain what hexagonal close packed is.
We take we take a bunch of balls and we put them down in a close, packed fashion,
fit together with the closest the smallest volume, smallest mass, and a large volume say.
In in this array. And now that's that's layer one.
And now I'm going to put layer to layer two.
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.
And the same. Same on the right. In fact, I could it doesn't matter what I wear.
I put the first one. If I put the first one here, I would still get exactly the same picture.
Okay. So it does doesn't that mean the fact that they look the same is no loss of generality?
Okay. Now I'll put layer three on, but layer three I have a decision.
And layer three, I could do that. In other words, I could.
I could put a ball right in the depression of the previous layer, but not over one of the first layers.
Or. I could do that. I could put the the atoms directly over the first layer.
And the left hand structure is is actually phase centre cubic.
So it's a little hard to see. But if you go back to the picture before.
Um. Where is it? Right there.
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.
So that's the this picture on the left and on the right, of course, it's something different.
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.
So you only alternate between two different structures.
You might say a baobab and you might call this stacking ABC, ABC, ABC.
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.
So why is that? What's the proof? It's not a Bravo lattice. It's very easy.
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.
You always get to an atom that follows directly from the definition.
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.
So it's going from the centre of the black one underneath there to the centre of the, the, the, the red one.
If I double that vector, you see there's no atom there.
So hexagonal, close packed is not a break lattice.
So but hexagonal close packed is an objective structure.
So every atom and hexagonal close back sees exactly the same environment.
Okay. All right, good. So there's the definition I already gave of objective structure.
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
to blacken out all the elements that do not crystallise as objective structures.
So that's there's an objective structure, a ring of atoms. This is one example, simple example.
So now I go back to the periodic table, I block it out. All the elements which are not objective structures.
Okay. There's one. Manganese. Manganese is not an objective structure.
It's accepted. Crystal structure at room temperature is not an objective structure.
So that's quite interesting. But there's also many, many cases here.
And and in addition to these standard crystal structures, boron likes these icosahedron carbon is the diamond lattice is an objective structure.
The buckyballs are objective structures. Of course I already discussed this carbon nanotubes first.
Fostering is an object of structure.
Sulphur likes this kind of double ring, which is and the halogen compounds like these layered structures which which also have this property.
So you would think that some really smart mathematician would have figured out why this is.
But this is one of the this is one of the outstanding open problems in mathematics.
It's one of those problems in mathematics that every couple of years people simplify the problem
more so they can try to get more information and they get just painfully little information,
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
Jones potential that that that that you get the FCC structure as the ground state the lowest energy structure.
And so and that's this problem is called the crystallisation problem.
So it's it's really a fundamental open problem.
Why should all these elements which have this, you know, they have very directional bonds to to other to other atoms.
Why should they? Why should they prefer the same environment?
It's completely unknown. Okay. But it's a it's a very good natured love, subjective structure,
but it's a very good way to begin to to to be domestically inspired to make our economy.
So we'll do that. So I have to talk about I symmetries.
I symmetries are fundamental to the construction of origami.
I symmetry is people would use this notation and again, you can think of this on several different levels.
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.
And I see matrix consists of a rotation and a translation.
So in two dimensions, the rotation might look like that, you know.
So just something and the translation may look like that.
So that's that's a nice symmetry and it's so simplistic.
One of the simple there are more general notions of asymmetries, but the simplest example.
If you have a nice symmetry, you can act on positions in three or two dimensions depending.
And that's the that's the, the orbit of the point.
Our that's the position of the point X acted on by that asymmetry.
So now let's suppose we have two cemeteries and we're going to have our 1c1 on our two C2.
I want to multiply them. So one of the great things about mathematics is you can you can invent your own multiplication rules.
Okay, so here's the multiplication rule. I'm going to choose for G one and G2.
Looks. Looks a bit weird, right? Well, this this kind of makes sense.
I should multiply the rotations, rotate ones, maybe rotate again.
But what's this all about? I mean, there's oh, there should be a left.
A C2 should be a C2 right there. So. Sorry about that.
Anyway, makes it more bizarre anyway. So what is that multiplication rule?
It's a it's actually the most natural multiplication rule. It's it's the the fact is, the multiplication of G1.
G2 is first you do g of. In fact, I switched them the first.
I should first do G2 and then G1. I'm sorry. So I'm actually showing G2.
G1. But anyway, you first rotate with with R one and then you translate C one.
And then you take the result where you are and you rotate again by R two and you translate by see two.
Okay. So, so, so the that multiplication rule means exactly this.
You rotate and translate and then you take the result and you rotate and translate.
Okay. So it's in mathematical terms, it's called composition of mappings.
So it's exactly this this rule. So it's a very natural rule for for multiplying asymmetries.
Now there's a there's a very fundamental notion in mathematics.
The notion of a group is a fantastic idea.
And always it's it's usually comes up in considerations of symmetry.
And usually people would show the platonic solids when they want to describe groups.
I happen to be a big aficionado of groups that describe things that look completely symmetric.
But anyway, we have two asymmetries we just described.
In fact, I will not quite described how to multiply them.
And you can have a list of asymmetries.
And the definition of a group tells you if you multiply any two by this rule, then you'll get back in the group.
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.
It turns out if if this is the isometric rotation our translations see that the inverse of the symmetry is rotation are inverse.
That just means instead of doing this you go back and, and it turns out minus R,
minus one C and that's with that product is the inverse and that's the identity.
And there's a repository for all the groups, all the discrete groups.
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,
it's the international tables of crystallography.
And so in the library it's this big, it's online and you can look at those groups organised in a bizarre way.
But anyway, that's where they are, so they're nice to use.
Okay, so I mentioned two things.
I mentioned objective structures, that's each atom sees the same environment and I mentioned something completely different.
I saw symmetries and I saw symmetries can form groups. So the point is that there's a there's a very strong connection.
Every object of structure you could.
In other words, you could try to make them by looking at doing these different environments, but that would be very hard.
But this is a very easy way to construct all objective structures.
You take a single point, you take a nice symmetry group with the product I just mentioned,
and you take the you take the each group element and you apply it to X with this rule of g on x axis,
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.
Three of x might be it's called the orbit of the point x, which is a very natural one.
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.
So that's the very simple way is first you figure out the isometric groups and then
you you apply them to points and you get many different objective structures.
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.
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.
I'll tell you about it now. So so this is the this is the rotation.
And I let's make the translation equal to zero. So that's a there's another thing you can do with a star.
You can you can divide it in half by a by a line, this dashed line, and you can flip it over.
You can you can reflect across that line and you get the structure.
Now that cannot be obtained by rotation because of the fact you can see this, this, this, this diamond, this here.
And if you rotated to this position, the diamond would not be pointing down.
So there's no way you can do this by rotation. But it's incredibly important for origami.
Why? Because this this kind of asymmetry where this kind of, you know,
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.
Okay. That's that's if I if I drew triangles and so forth on this piece of piece of paper.
That's exactly. Of course, if I can flip it over and also move it, I can allow a translation as well.
So and those are those are all the things you get to do in at least piecewise rigid origami.
Okay. So in fact, that's represented by this matrix. The rotation is represented by this matrix.
So you can make a little diagram. I find this kind of diagram very useful.
I take and imagine the square is all two by two matrices.
And again, you can think of this also in intuitive terms, but and this this circle, I mean, there's just a one parameter.
It's the angle of rotation.
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,
that closes on itself goes from the identity. Each point on that curve is a rotation matrix like that.
It's just a representation,
but I find a useful representation and then you can do this flipping and that's this particular flipping is represented by this matrix.
But once you flip, of course, just as in origami, once you flip, you can also rotate as well.
Okay. And if you can, you can say, well, maybe it's more complicated.
Maybe if instead of flipping this way, I flip this way and rotate it, I could get something new.
No, that's. You don't get anything. So you just get this second. Second circle.
One parameter family. Okay, of.
One parameter family of of flips. Okay.
So those are the three, three matrices. Okay. So that's that's fine.
So now, of course, I didn't say anything about the C,
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.
And so at the beginning, I'm just going to talk about folding from two dimensions back to two dimensions.
In other words, I take the piece of paper, I fold it up, and I put it back into two dimensions.
Okay. Now, origami requires more than I saw.
Symmetries the same. I saw symmetry somehow have to fit together.
So we have to think about what that means. So let's suppose we have the nice symmetry.
Let's have the trivial asymmetry, which is the identity, which is identity matrix and zero.
That means that means it doesn't rotate and it doesn't translate.
So that's the most trivial asymmetry.
And now let's that's that that's this one. And now let's let's choose this one.
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.
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.
The fact that those two asymmetries agree on this line is a very strong restriction.
That's called compatibility. Now I'll write that.
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.
You can always adjust them that are compatible.
The real condition is that the two matrices, the two rotation and reflection matrices have to differ by a rank one matrix.
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
of it and completely intuitively then then exactly it's this it's the meaning of that is this picture.
But the point is we can put it into into analytical form.
Okay. Here's the theorem. Which is which is which we should prove.
It says that every pair. So you take these two orbits.
This was the rotations, this was the reflections plus rotations.
And I choose the matrix over here. I choose the matrix over here.
And as I say, you can always adjust the the little and the little b the translations to make an origami.
But but it turns out that B is always compatible with a.
Okay. Proof. We can do this. Okay, so I.
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.
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.
And that that construction, which is simple construction is shows that shows that given any matrix over there,
given any reflection plus possible rotation and any rotation, any pure rotation, they're always compatible.
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,
if they satisfy this condition, or more intuitively, if this if this, if the two symmetries can agree on a line.
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.
So there's our one. I pick a rotation that's doing this and and I pick a cue two, I pick a reflection.
So and as I say, I can always adjust the translations to work.
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.
Now I pick a rotation over here and it's compatible with.
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.
Okay, I satisfied all compatibility.
Just I can choose anything I want, you know, and that means analytically that those four conditions are satisfied.
And even if you really don't quite understand those four conditions, if I add them up, the two cancels, the one cancels.
There, the four cancels and three cancels.
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.
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.
And I can do that because they meet at this point. Okay.
Now I'm going to do a very, very famous construction in origami.
I'm going to take a piece of paper and and I'm going to just do like this.
I'm going to push it because I'm describing origami mapping from R2 to R2.
Right. I'm in two dimensions, so I just do this. Okay.
So you can do this at home. And what happens is this is a parchment paper.
Works pretty good for this. And you can see what are our fault lines.
So you have a lot of interesting fold lines.
We have places that are kind of destroyed, but we also have some quite simple places where four folds come together,
like down here, and there's one right there and this one right there.
And there's there's one right there as well.
And if you if you study them for a while, you'd realise that they there's some particular geometric relation for those guys.
And this is a famous theorem in origami. It says that.
You can fold this fully folded back into to to start in two dimensions and fully fold it back into two dimensions.
If, if, and only if. Alpha plus beta equals pi. So the sum of opposite angles is pi.
And of course, the sum of those R is pi two necessarily.
And there's two solutions and those are that. Those are four for two.
Typical case, those are the two solutions. So that's a something you see all the time in origami.
It's a very, very typical construction if you want to fold from two dimensions back into two dimensions.
Okay. And the interesting thing is there's more than that is that if you satisfy alpha plus beta equals pi.
Then not only is there some kind of mapping from two dimensions to two dimensions,
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.
In fact, there are two exactly two such contiguous mappings.
There's also two trivial solutions where you fold it and you fold it.
But okay, that's the you can write. Exactly. And you could write formulas for all the information that that you have.
Okay. So that's a well-known thing in our gallery, but it's nice.
I hope it's nice to see you in a now maybe more mathematical way than you would see in origami books.
So here's something that you won't see in origami books. So this is.
So. So now we say some of op at opposite angles is PI.
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.
Can I fold it? So if I cut out one of those little regions, I can fold it.
But turns out I can't fold that and I can fold that.
But in general, I won't be able to do it on a piece of paper.
You won't be able to fold it. So what is the what is the restriction?
It's it's very nice, actually. It's this.
So I look on the edge and I look at these guys, those four fold guys.
They're. And I arrange, arrange, arrange them to have any, any angles.
But samba of opposite angles is pie for this point, for this point, for all the points on that edge.
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.
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.
And with that amount of freedom,
then the entire fold crease pattern is determined that these are all ways to fold it and you can write formulas for everything.
This is a particularly simple one, but you may not be able to decide how it gets folded, but that's the situation.
So it's very restricted. You can't just go ahead and put these together in some arbitrary way.
With Alpha plus equals pi, you can only choose them here and here, and then everything's determined.
It's a little bit like differential equations. You kind of assign boundary conditions and then you you see that, you see the possible,
you get one solution, unique solution, if you would, those plus or minus, as I've mentioned.
Okay. So that's that's what that folds now that that's that structure falls in this way.
In fact, if it falls flat, I just it just doesn't the movie doesn't go all the way.
And you can do more complicated things like this. So you can turn this case along this edge.
So see those guys right there? They were assigned such an alpha plus spider equals pi and some kind of quasi random way.
And these were assigned along this edge. And then all these all these fold lines were determined.
And then it can be folded. That's.
Okay. So that's gives you an idea about that.
You can see it again, that fold that folds flat, by the way. Okay.
So nature has another has there's all kinds of folding in nature.
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.
So what is a phase transformation in crystals and how does it differ from origami and how is it the same?
So many transformations in crystals are transformations where where the crystal structure changes at some particular temperature.
Okay. So maybe very often this cubic structure like this body centred cubic structure, which, by the way, this very lattice.
Is the stable structure at high temperature.
And what happens when you reach a you cool this material, you reach a certain temperature and the and the material distorts.
And that just shows one example in this particular material, which is here it starts by shortening on this vertical axis.
And the top face here, which is a square here, becomes a rhombus.
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.
And those are the six matrices that describe those distortion.
So what are the six ways? So instead of shortening along this axis, I could shorten along this axis sucker.
I shorten along this axis. And for each one of those, I can make two rhombus.
I can pull it out like this or I can pull it out like this. And those are the six distortions.
So now these are described not exactly like like like I saw cemeteries, but quite similar in some way.
You know, you first have to you first have to distort and that's the end.
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.
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.
So. But then you can also have translations, of course. So it's a little bit more complicated than origami.
We have not only a rotation here or a part or a reflection in a a translation.
We have also a distortion, but the distortions are discrete.
There's only six of them. Okay. And exactly the same mathematics work.
So instead of two of those orbits, we have three by three matrices, we have six orbits.
We have this free rotation. This is kind of just a representation of of all the distortion plus rotations the crystal can undergo.
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.
You find out that this this matrix is compatible with this matrix.
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.
So in that sense, it's very different from origami.
But for this material, it's this fact be it goes right back to a and that means all those four interfaces are compatible.
And once you realise that you can make the whole structure.
So and as John was saying, I mean, these, these, these phase transformations are incredibly interesting.
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,
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.
Then maybe you could have very special distortions in which the two phases fit together in many, many ways.
You got many, many origami. So, in fact, that is the truth.
That is exactly the truth, is if you tune Alphabet and cannot have some special values, then you'll get many.
Origami is possible with these crystal structures. Okay.
So how do you tune your tune? By changing the composition of the material.
So if the material is is some alloy, then you change the composition.
You very carefully measure these things, this, and you go towards your mathematically determined lattice parameters.
So. Very good. So. That's that's how you do it.
And I want to show you material that's been tuned. This is one of the most or.
There are two sort of spectacular materials.
This is this is one of them that had been obtained by this kind of tuning of lattice parameters so that there were many,
many ways to fit together completely non generic conditions.
And it's fun to watch a movie of it transforming back and forth.
By the way, I had six distortions in the case I mentioned before.
This material is one of the distortions, but it's very similar, very similar otherwise to to what I was saying before.
So this is a movie where you were I'm just heating and cooling the specimens.
So there's a there's a there's a little thin film heater underneath this this piece of material.
It's heating it uniformly as possible, and it's cooling.
And one of the wonderful things about this movie is that the every picture is the different.
Different. So in normal phase transforming materials, every picture is the same.
But this particular material somehow has so many different origami to make that it can do a different each time.
And this this particular material is one of the most reversible materials that that's known.
As I say, it's one of two materials that okay, to explain it a bit more in that phase.
That's the cubic phase, the boring grey one.
You can raise the temperature to some higher temperature and you're always in the grey phase, that cubic phase.
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.
This material will, even though it's zinc, gold, copper, which is never expected to do this,
this material can go 100,000 cycles with a stress which would make the steel in this building fail.
Okay. It's way about twice the failure stress of the steel in this building.
So, okay, so okay. Now let's go back to two to origami and let's try to be again, Adam mystically inspired.
And so I arrange I arrange that, that the sum of those two angles is PI.
And I'm also going to do it on a parallelogram.
So I'm going to arrange that, that, that this length equals this length and this length equals this length.
And you can follow that because it's a it's got four regions and some have opposite angles as pie.
There's the two solutions I was describing earlier. This is partly folded.
You can you can fully fold it. And as I say, those are the two ways.
So if you have that, you could do so you can think of doing something interesting.
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.
Now I guess it's the top one. But anyway, it doesn't matter.
And remember, after it's been folded, this length equals equals this length here, and this length here equals this length.
And in fact, it's down here. The way I hope it's clear, the way the folding goes, the perspective.
But anyway, this this length equals this one, and this one equals this one. You know, I saw cemeteries preserve length.
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.
Turns out you can do that. That's not that hard. I'll tell you what they are in a minute.
And not only that, you can find these two asymmetries and they commute.
Actually. So G1. G2 is G2. G1 we're using the same product for.
If two asymmetries compute and you can just take powers of them and that automatically forms a group.
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,
two to the Q and you take another one from here like ag1 to the one to the as you write it out,
since they come here, you can switch these two guys and then of course you realise you can switch,
you can push all the g ones to the left and put push all the details to the right.
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.
So in fact, I can make asymmetries which both which both do that oops, there it is.
Which both map the side and to the side and which form a group.
And they look something like that. But this is the wonderful thing about groups is is.
This this isometric G1, for example, maps this side into this side,
but I can apply it to the whole partly folded origami and it fits on there perfectly.
It satisfies compatibility. And then I can take G2 and G2 fits on there perfectly.
And the wonder of a group is that G1 times G2.
Applied to this original or partly folded origami perfectly fits with all its neighbours.
And I can keep going and going and going and make big origami so we can do that.
And let me tell you about the asymmetries that that so we have to be sure that they compute and they they.
So the symmetries that do that are our members of the helical groups.
And there's a very interesting theorem about the helical groups.
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.
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.
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.
So age to the piece. So I get I get that structure.
And in this case, I took I took the group B, I took the orbit of this under this group and and I coloured,
I actually coloured the atoms according to the power. M Which could be one or two.
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.
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.
They, they, they compute and they, they can map edges into edges.
Our members of this group see. So it shows that I can write them down explicitly.
So then immediately you can just form lots and lots of our gummies and that's them.
So and this is this is one, for example, where the where the thing is folded and then it folded a little bit more.
And you get the structure here. But you can imagine, as I said, you apply these group elements.
You're building up the structure when you get around the other side. Of course, they may not meet.
And so there's additional conditions and actually they're quite restrictive.
So I won't go into detail. Now, this is a this is a bi stable case like I was just describing.
You have folding and you get this structure, you fold it a little bit more and you can get this structure.
So it's kind of bi stable and in fact, you can write the solutions.
You can write solutions in terms of two parameters. And the solutions are represented by these dots.
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.
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.
So of course. So that's the situation I just was.
So. But it would be fun to morph it. So. So let's try.
Let's try another method of morphing. So and again, we'll be we'll be out of mystically inspired.
So this is a this is a virus actually studied this virus at one point.
This is bacteriophage T4. It has a capsid.
It has its very complicated virus and attacks. Bacteria like E Coli is a favourite target and it has.
There is the real thing. It attacks them in the following way.
It lands on the surface of the bacterium.
And these these these tail fibres are sticky to the surface of the of the bacterium.
And these long tail fibres are are also sticky.
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.
And so the phase transformation and there's a picture of theoretic from a from a theory, but quite, quite,
quite accurate representation of the phase transformation which occurs in the tail sheath of bacteriophage.
T4 Inside this tail sheath is a very stiff tail tube.
So this is joined very rigidly onto the surface of the bacterium.
So when it shortens, it drives the tail tube through the cell wall.
In fact, it not only drives it through, but it twists it. It twists the knife as it's going through.
And then the DNA of the of the bacteriophage T4 enters the cell and then it does its usual thing of replication.
So we could be we could be optimistically inspired and we could say, okay,
there could be phase transformations and helical structures, kind of like the phase transformations I showed before.
And it's not hard to work out all possible compatible structures which undergo phase transformations and you're seeing representatives of Bose,
they're very similar to the idea of compatibility is exactly the same.
In fact, idea of compatibility we're using for origami, but it's for two helical structures.
Okay. So that provides a nice way of of morphing and a face morphing origami structures.
We can have we can arrange a compatible interface and then we can by moving that interface, we can do morphing of the structure.
I'll show you that again. So this is twisting and also increasing its length.
So that's that's all I had to say, except I have to come back to the homework problem.
Before I do, I want to acknowledge. Fan Fang is a graduate student published pinsky there.
They did a lot of important work in this area very recently. But I particularly want to acknowledge Frank Yu.
So Frank, who is a high school student from Wayzata High School in Minnesota, it's near Minneapolis.
It's and Frank Yu is 16 years old.
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.
Okay. Back to this structure in the Math Institute that have that I've ratio out here.
So it turns out the following is true. There's a four dimensional space.
There are four. And there's a there's a region in that space.
You can define that region absolutely precisely. It's not even very difficult.
And you can take any curve in that four dimensional region, and that will define an origami structure.
So that can be folded a huge number of ways. So, for example, this is one of the ways it can be folded.
You can see it again, I guess. Oop. Okay.
I guess you can't see it, but there's a great many ways that can be folded here.
There it is right there. So.
So I hope this doesn't make you nervous when you leave the room.
You know, maybe you don't want to walk under this glass ceiling, but thank you very much for your your interest.
And.