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Polygonal Chains & Rigidity Theory
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0:00
[MUSIC PLAYING] Thank you, so much, for the introduction. This is going to be a joint talk, but I'm the one who will set things in motion. So as mathematicians, we like to express things as simply as possible, and I'm going to jump directly into the problem. So this is a polygonal chain. Trivial right? So a bunch of points connected by straight lines-- they can live in 2-D, in 3-D, in arbitrary dimension, if you are adventurous. But for the time being, all the illustrations that I will make
0:41
will be in 2-D. OK, so what is special about it? We view it as a bar-and-joint framework or linkage. We want the length of the distances between the points that have been marked there with an H to retain their lengths. We mark the lengths, and we don't want to change it. However, we've allowed the joints. The joints allow rotation, very much like our arms can rotate about our joints. So this is a bar-and-joint polygonal chain. So in case you think it's too abstract,
1:21
here's a little demo that I picked up from the web. This is a robot arm from the Canadarm from the Space Station. It's a robot arm, rigid links with rotatable joints, and it's performing very important tasks up there. So that's why we refer to this polygonal chain sometimes as robot arms. Now, here's a robot again, very small clip, just to whet your appetite. This doesn't look like a serial linkage, just a polygon. It looks like a branch structure, but the arms, the legs themselves
2:01
are these polygonal linkages. So we are also interested in closed polygonal chains. And at this point, you will think, why? Well, here's an example. This is a robot that carries a tray, and the arms, together with the torso and the tray, form a closed polygonal linkage.
2:26
So what do we want to do with this? Let's say you get your tray, you pick it up, and you want to move it somewhere. You have to reconfigure your polygon from one initial position to a final one. So this is the polygonal chain reconfiguration problem. We have to design some motion, so let me demonstrate one possible reconfiguration. Yes, how it would look like. Remember, we allow rotations, but we want to maintain the edge length. So maybe it looks like this. You start moving a little bit.
2:59
Now, these are lines. OK, maybe that's what I wanted. So another one, maybe. Yeah, but you notice that several move at the same time. And finally, I'm where I plan to be. What I'm trying to argue throughout our talk, what we are trying to argue throughout our talk, is that there is a hidden structure. And in fact, we want to discover this structure. We may not know apriori. In fact, it's not visible. So I will demonstrate it to you, and then I'll tell you how we model it mathematically.
3:34
Although it's not visible, it is what can guide motion, what can help us design trajectories and further on to applications. So on the left, we have the polygon in motion. On the right, I'm going to add extra structure. Remember, the structure is invisible to you, but I'm adding it. So here is what I'm doing first. I'm using extra bars. They are those added red. And as I add them, I eliminate the freedom to move, yes? So it is as if my arms-- all of a sudden, I'm hitting something here.
4:16
So I turn a join that was very flexible into a part that is rigid. OK, so I'm arguing, and then I'll explain in more detail later, that the structure on the right is rigid-- completely rigid. It doesn't move at all. So what I'm going to do instead, I'm going to remove one edge, and it's going to be selected in a particular way-- namely, on the exterior. In mathematical language, on the convex hull of the planar linkage. I'm going to remove it. I still left it there with a dot to suggest
4:56
maybe something like a rubber band, because I will allow that to flex. I'll allow that to move. But the rest, the red bars, are going to keep the structure partially rigid, if you want to describe it that way. So now, with this, I have the freedom to move, and the two things move. And although on the left you don't see it, on the right, you see that it's guided. It's moving in such a way that the red bars retain their lengths. So now I got again get to something that is fully--
5:29
it's close to my target, so I'm going to freeze. I'm going to lock that joint. I'm not going to move it from now on. And now, if again you do a little calculation, you will realize that this one is completely rigid again. So we have to drop another bar from the convex hull, get another structure that moves and moves along a very well-determined trajectory, until it reaches the goal. And again, I locked that joint. I drop another one, and then I conduct the final motion, and I'm done.
6:12
That's my final structure. That's the goal. That's where I wanted to get. OK? So now, this hidden structure can be explained in terms of an area of mathematics called rigidity theory. So let me give you a brief introduction to rigidity theory. So this object living in the plane is rigid. It means that it can do things like this. You can rotate, translate it in the plane, but it doesn't change the shape. The shape remains the same. And in fact, any collection of triangles
6:47
would have this property.
6:51
Well, when you built a structure from bars and joints as big as this one, you definitely want it to be rigid, not flexible. OK, so on the other hand, from bars and joint, we can also build purposefully structures that are flexible mechanisms. So this is the simplest one. It has one degree of freedom in the sense that by rotating that joint about the other one, there is only one way to rotate, yes? So there is only one force or only one motor that will activate the joint in the robot arm.
7:28
So we call it a one degree of freedom structure. But the one degree of freedom structures can be more intricate. So for instance, this is the next, the simplest one, the next one that you can think of, and it is a closed loop a [INAUDIBLE].. And I'm telling you that this one has one degree of freedom [INAUDIBLE]. OK, so it moves roughly like this, and at some point, it may even cross itself, yeah? Can go back. So it has one degree of freedom, and these pieces
7:58
with one degree of freedom are very useful because they move only in one way. Otherwise, if I have too much freedom to move, it's hard to design the motion. Maybe you design it one step at a time, one joint at a time. It's easier when it can go only along a single trajectory. So our goal is to understand those single trajectories, those trajectories that can be described by placing extra bars and getting these one degree of freedom structures. So now, these structures are very famous.
8:31
I'm showing you here a very famous one, the so-called Peaucellier Linkage discovered in 1864 at the peak of the Industrial Revolution. It's the one that converts circular motion to linear motion. It was a famous open problem. So on the right, I'm showing it to you because it's the same linkage, but it's clear that it's planar, at least in the plane. And as it moves, you see that some of those links actually overlap. So it's possible that these linkages are more complex
9:04
than just polygonal chains, than just closed polygonal chains. They may have many links, and in particular, they may even cross. Here's another one. This time, it's in 3-D. And in fact, I have it here. This is the Hoberman Sphere. It's a bar-and-joint linkage. And again, it moves with one degree of freedom. You just pull in any direction, and it expands. It stops at some point, and you move it back. So what I want to do now is to analyze the hidden structure in this reconfiguration motion.
9:43
As I said, the polygon has too many degrees of freedom. But as we add bar, we restrict those degrees of freedom, yeah? So we put people behind bars to restrict their freedom. Sorry for making the pun, but I wanted to retain that. More bars you add, less freedom you have. So if I add enough of them, I can no longer move. I'm completely rigid. This particular one, yeah? So this particular one is rigid, but it's rigid in such a way that if you remove any bar, it becomes flexible.
10:18
It's called minimally rigid. So if I remove one bar from this minimally rigid structure, I get something that is one degree of freedom. But I'm going to remove a special one. This one's from the convex hull, yeah? And I get this one degree of freedom a la Paucellier Linkage. And then, well, it's like I rotated the crank, and then this one keeps moving, yes? So you already have seen this, until an event happens. So the motion now is decide what one degree of freedom structure you want.
10:56
Move it as long as it's possible. I could continue, but I have a hidden reason to stop there. And then, reconfigure. So drop another bar, yeah? So freeze, drop another bar, and continue until the bars align with joint, lock, and continue. So in order to explain, right? So until the whole chain is convex-- in order to explain what is going on, first of all, what do you think with this kind of motions? Are we actually avoiding collisions? Could it be possible that something is so intricate-- not
11:36
in this particular case, but something is so intricate that as I move stuff, maybe I will produce collisions, very much like in the Paucellier Linkage? And so I don't expect you to give me the answer, but remind you that it is possible to get collisions. But in this case, the answer is clear. There are no collisions. And the reason why now has to be explained mathematically, and in order to do the explanation, I need to introduce the concept, yeah? So one was the one degree of freedom motion
12:13
that you have seen. The other is the concept of what an expansive motion means. So let me start with that little definition. So expansive motion-- OK, expansive means when-- it's very much like this one, yes? If I choose two points in this motion, these points move away from each other, but not only these two points move away from each other. Any other pair of points moves away from each other, yeah? So an expansive motion would be one where all the points that I'm keeping an eye on--
12:52
they move away from each other. It's like an expanding universe, yeah? So let's say we may start with three points, and we just move them away from each other. If there are no constraints between the points, I can do that easily. But if there are constraints, like if I have a framework, then let's see what happens, yeah? So let's keep an eye on the evolution of the length. So this one started 6.87. The other is 2.4. And I'm going to increase this big one,
13:24
and you'll notice that the other one also increases simultaneously, yeah? So they increase, increase until this one, the big one, yeah? So reach its maximum-- of course, beyond the maximum, it can no longer expand, so it would start decreasing, although the other one increases. So it's like all of us. We have an expansive period in our life, and then at some points along some dimension, we increase. Along others, we decrease. So mechanisms have one degree of freedom,
14:04
but their trajectories, their full trajectory, is split into intervals. And it's only on one interval that you can fully increase in all possible directions. So that is what happened with this structure, yeah? So if I start in this shape, I'm expanding. I'm young and expanding. And then I reach the maximum. OK. I got the fellowship at Harvard or something, yes? I hope I will continue, but you know, it's maximum on some dimension.
14:43
Good. So these are the definitions, but I give you some very simple examples, but the reality is that we fully understand these structures. So let me tell you quickly about the structure of 2-D expansive motion, and I need to introduce one more concept to the triangle. So we see here a triangle. Everybody knows what a triangle is. We have seen this little object. It's a pseudo-triangle. These also are pseudo-triangles, and here's how you can recognize one.
15:15
You recognize it because it has only three points on the convex hull. So there is sort a hidden triangle behind the scenes. And where it is not on the convex hull, it bends inwards in a concave fashion. So these are pseudo-triangles, yeah? So all of these on the background of their supporting triangle-- and when I put them together, I obtain a pseudo-triangulation. So here's what it means. And this is a game that you can play. And I challenge you. Go home.
15:48
Take your friends, children, whatever, spouses. Draw a bunch of points in the plane randomly, like here. OK, so I draw them. And then, your task is to obtain a tiling of the convex hull of these points into pseudo-triangles, yeah? So you see here, each of these pieces, if I would cut along these bars, I would get pieces that are pseudo-triangles. So let me color them so that you will see them, yeah? So the challenge is how to do that. Maybe it sounds trivial, but let me show you very,
16:24
very quickly how you can play this game. Basically, you play it at random. Just throw it anything, randomly. However, you have to respect some rules. So for instance, I don't allow you to do this. Why that? So the first rule is I want you to keep enough space around each vertex. Around each vertex, you should have an angle larger than 180 degrees. When I added this edge, I violated that rule because now I have angles that are all smaller than 180. So I'm not going to add this one.
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That's the first thing that you have to avoid. The other is don't produce crossings. And if you avoid these two rules, then at some point, there will be nothing you can do. You try this. You try that. Everything will violate. And if you look carefully, all of these faces are pseudo-triangles, OK? So now, this is the structure in an expansive motion that we want to start describing in our project. We discovered this years ago. So here's theory number one. First of all, a pseudo-triangulation
17:39
is always minimally rigid. First of all, it's rigid, and removing any edge, it becomes flexible. And when you remove an edge from the convex hull-- and also, the mystery is clarified now. So when you remove from the convex hull, then it becomes a one degree of freedom mechanism and always, always moves expansively, yeah? So it's not just those little linkages that I have shown you, but all of them. And you can build them tonight. Random points-- your children can build quickly. OK.
18:16
So once again, this is my expansive structure, and I move it. And I want to notice that usually, I have to stop when something aligns because beyond that point, it starts shrinking. So this one is an expansive mechanism, but remember, it's only for a given interval. OK, so that's a structure. Now, the beauty of this is that this is essentially all. You don't have to look further. A one degree of freedom mechanism is essentially any one degree of freedom expansive mechanism is essentially
18:55
a pseudo-triangulation with a convex hull edge removed, yeah? So it's a characterization. And not only that, if now you want to do any motion that it's expansive, in fact, it's a complex combination of such structures. So if you understand the structure, you can describe any possibility in what is called mathematical or convex combination of the original ones. So we have a full characterization. And I would be really happy if I could extend this to dimension three to have a theorem that does
19:37
the same thing in the dimension [INAUDIBLE] higher. Unfortunately, that is not such a thing for various reasons that have to do with famous open problems in mathematics that have been open for over 150 years. But since we are here, putting everything together, the problem with I started with is that every planar polygonal chain can be convexified in a finite number of steps using this expansive pseudo-triangulation mechanism. So that's the first structure in motion that we plan to present.
20:11
So it can be applied not just to simple structures. It can be applied to complex, convex-- sorry, complex, very convoluted polygons like this, to large, very large. And we would like to do it in 3-D, but I let you know what we can do in 3-D. So first of all, remember that this Canadarm on the Space Shuttle is short. It just has a few linkages. We don't need a complicated algorithm to handle it. It's not closed. It's open, but it is in 3-D.
20:42
We'd love to understand how to move this in 3-D. However, for the 21st century, things that seemed completely irrelevant, for instance, for the engineers of the 19th century-- you don't need too many bars in order to build interesting mechanisms. But I remind you of something that Kepler said in a famous writing. "Where there is method, that is geometry." So we are going to look at matter, and these are nature's robots. This is a video that I picked up last night from the internet.
21:27
And what you see here are what are called molecular markers. These are proteins that perform their functions in my cells, in your cells, in millions and trillions of cells as we speak. And they are made of-- they're like little mechanisms at this time with tens and hundreds and thousands of these linkages, and even hundreds of thousands of such links. And we like to understand how they work. So this is an artist's rendering of what happens in the cell with dynein and kinase,
22:04
two of these molecular motors transporting cargo along the microtubules in the cell. Now, these molecules are made of proteins, and the connection with the chains that we will be talking about is that the backbone of a protein, which looks like this big blob, is, in fact, a polygonal chain living in 3-D. So this is one of the motivating problems for the project and for everything that we conduct from there on. So again, let's see, how large could they be? How many could they be?
22:38
So I did a quick search yesterday, and the Protein Databank is the data bank where I-- the software that we have designed in my group. We get the structures, and there were 145,000 structures. Last year about this time, there were about 130,000. So it grows with over 15 structures per year, and accelerating. Just to give you a very, very quick idea about what is going on, this started about 40 years ago with structures that-- each one of them was worth a Nobel Prize, like myoglobin.
23:13
And these by today's standards are small proteins. They were considered huge by those standards. Nowadays, the largest one in the Protein Databank is their atomic-level structure of the HIV-1 virus capsid. This has about a 2.5 million atoms. And this is another virus. Each time I get the flu, I'm thinking about how beautiful they look in these computer-generated images. They are very symmetric, so we have to put some symmetry, some mathematical theory
23:50
to put it to work so that maybe we can analyze them faster, because for the following reason. So what you see here is a generated image. But while the experimental is [INAUDIBLE],, it's more something like this. So this is a cryo-EM image of a virus, and you see it in two states. It's obvious that it flexes. It's obvious that there is motion there. So this is as it enters your body, and this is where it releases the dangerous nucleic acid inside to infect your cell.
24:29
So we'd like to understand how this happens, and here's an engineer's view of this. Parts of the structure seem to be moving rigidly, and others seem to allow freedom to rotate. So very much like our linkages, except that now there seems to be collective motion, pieces, collections of atoms that move simultaneously. So that's what we want to understand next-- collective motion. So actually, we have seen it, but it was hidden. So let me show it again, yeah? So let's get this mechanism that moves.
25:04
Can you track some points that seem to be rigidly attached to each other?
25:11
And not everything is moving. Not all the points are moving away from each other. Those that stay rigidly attached to each other form what are called rigid components, and I highlighted them here. And now I'm doing the same motion with and without the colors, and you can see that the colored pieces move together. That's a collective motion. So here's what we have done. So a couple of years ago, this started as a collaboration with a biophysicist, and we have this software that is called KINARI--
25:46
Kinematics and Rigidity Analysis for Proteins. And what it does, it takes a protein structure from the Protein Databank and does a coloring of the pieces that if the protein will move and it moves, then those will tend to stick together. So it's very precious information about the stability and about the overall behavior of the structure. So let me give you, quickly, two applications. So this can be used to predict, for instance, the effect of binding a ligand. This is a protein-- HIV protease.
26:28
And it's here without the ligand, without the extra molecule that is designed to fit in that little cavity. And here is with that ligand, and we can infer the effect of the ligand on the protein rigidity, and maybe this gives us some functional information about the role of the protein in the organism. In particular, if we do this, we notice that these two flaps here, which the biologists know they have a functional role. They have a functional in the spread, in the division of the--
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in the multiplication of the HIV virus. And when that little drug is injected in the organism and binds there, it's rigid, and those flaps cannot perform their function, which is-- basically, they act like scissors, OK? So we infer functional information from this. This is work done by a group demonstrated in about 2002. Our goal is to be able to provide a tool where this can be used on a larger scale, not just on a few proteins. So we infer. We hope that with this kind of analysis,
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we can infer function from structure on this large data set of proteins. So here is another one. Everybody has heard about mutations. So what you see here is the same protein, but a little mutation has happened so that certain bonds cannot form. And you can see that large, rigid cluster split into two, which tell us that this one actually is more flexible than the original one. So probably, its function might have been affected, OK? So these proteins, the majority of them,
28:12
have been solved with a technique called x-ray crystallography, meaning that they are not just one, single structure that we are looking at. They are many. So the problem becomes large. So how can we algorithmically handle such challenges? So we want to understand the structure in periodic motion. And at this point, Ciprian will take over and tell you about that. [APPLAUSE]
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- Our periodic structures are widespread in nature, in man-made materials. And we see here, by now a famous structure, of course ideally drawn, but this is one sheet of graphene. So it's an illustration in dimension 2 of a periodic structure. Here, I make it stand next to a drawing which suggests three dimensions. So let's look first at a planar one on the left, where you'll notice the color code of the vertices because they will assist to visualize the periodicity.
29:55
So if we translate this structure, imagined extending indefinitely, then we can superimpose it over itself. So periodicity, briefly, is expressing this translational symmetry of the structure. And if we watch, say, a red vertex by this translation, which necessarily takes hexagons to hexagons and triangles to triangles, these symmetries will always take a red vertex to a red vertex, a blue one to a blue one, green one to a green one. And all these collection of vertices,
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which we call orbits and periodicity, express the symmetry of the periodic structure. On the right, we have the diagram of a crystal structure, actually, which is important for artists, because this crystal structure is involved in making pigment for the color called oftentimes Prussian blue, and the famous wave of Hokusai uses that kind of blue.
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Another source of periodic structures comes from sphere packings. So sphere packings have been imagined since the time of Kepler, and the dates here is a summary of the famous conjecture he made that the best, the densest stacking of spheres of the same radius is this kind of stacking. And Gauss showed that as long as you assume periodicity, that's indeed the case. But what if we don't assume periodicity and just allow the stacking to be disordered? And [? Hale's ?] recent contributions
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were to show that even in the absence of assuming periodicity, this is the densest packing. On the right, we see what a mathematician would do with the structure. You'll take into account the contact graph. So each sphere is considered as a node or vertex. If two spheres are in contact, you put the bar between them, and this gives you a contact periodic framework. And you look at its generators of periodicity. So those three red arrows indicate that if you translate with those vectors,
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the structure will go over itself exactly-- of course, imagining it infinite.
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So here's a simple example, which goes back to the quadrilateral used earlier in Illeana's presentation. So we start with a humble quadrilateral. And we are going to adopt the two diagonals, which are the red arrows, as our periods. So I'm going to take this humble quadrilateral and translate it by that vector, then by this vector, and begin to generate its infinite array, which you see progressing over here. I have put these two at the adequate distance
34:43
so that you see that completed to a larger piece. But we want to imagine for mathematical purposes the actually infinite periodic structure. So you see that surreptitiously, there's a doppelganger here. And if we call it that, we'll see the pattern. So this is called the double arrowhead framework because of how I started with a single quadrilateral. The periodicity construction creates that sibling of the image of the quadrilateral, and that's the expression of a theorem
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that in dimension two, if you have just two orbits of vertices, there's always this addition of symmetry. So it's not by accident that the two are essentially congruent. So how is such a structure moving? Well, now that we have seen that a single quadrilateral does have a one degree of freedom, we just say to ourselves, well, I can repeat the same procedure. I take the diagonals as periods and do the thing, and indeed, we will see that it moves because of that.
36:39
And here we go, and it seems to expand both vertically and horizontally. And we look with more intensity, and it does so. It even goes out of the space of the screen. But by now, I'm reaching as in Illeana's illustrations. I'm reaching a point where two edges align, and my quadrilateral has now the shape of a triangle. And beyond this point, I become convex. And you will see now that [INAUDIBLE] vertically, I increase the structure on the horizontal direction. I'm now retreating. You see?
37:41
It comes back. And [INAUDIBLE] I go beyond, and it retreats. So this is leading us to a notion which became very important in modern materials science-- the notion of auxetic behavior. And this diagram will communicate in the most direct terms what this behavior means. So on the left, if we have a conventional material with some elasticity-- and I invite you to imagine a bar of rubber. If you pull it, as the arrows indicate-- so if you want to stretch it, it will stretch.
38:42
And laterally, it will shrink. But this is the most typical behavior of the vast majority of natural materials, but in some instances, this less intuitive type of behavior was observed-- that you may pull and stretch the material-- say, vertically, in this diagram. But at the same time, laterally, instead of shrinking, this type of material expands as well. So this is the short, intuitive definition of auxetic behavior. It involves lateral widening upon stretching.
39:41
And we have now the challenge, if we go back to this view here, now, I indeed increase in both directions. By the way, the word "auxetic" comes from the Greek word which means tendency to increase. So we can take with a less precise definition than before that auxeticity is a form of growth, but it cannot be excessive in one direction and too modest in another. It is a rather balanced form of growth. But growth cannot be indefinite, at least at the bounded rate.
40:46
So its auxetic behavior will occur for some time and then stop. And in the spirit of our search for hidden structure, we asked ourselves, well, what is the necessary geometrical underpinning of this behavior? And we'll show the answer for the structure at hand. So the answer is that ellipse is responsible-- the existence of that ellipse is responsible for the auxetic behavior. I'll keep it for the time a little mysterious how it comes about, because you see, if we
41:47
have just the quadrilateral, the ellipse is not there. So this is a feature of the periodicity. But there are many conics that you can [INAUDIBLE] around, because in geometry, we learn that if we have five points in the plane, there is always a conic through them. This can be an ellipse, a parabola, a hyperbola, or it can be degenerate, two lines, or a double line.
42:28
And that was known at the time of Apollonius, and of course, that was influential for Kepler later on to explain the planetary motion. And here we have an ellipse now. What I want you to observe is that we are at one type of vertex, and we look at all the edges emanating from it. I have these four edges. So with the origin and the extreme of these four edges, I have my five points. And I trace the conic and the ellipse, and that explains the auxeticity. If I move and come to this iteration,
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I watch the same five points. I draw the conic, and it's a hyperbola. So something has changed in-between, and indeed, the auxetic behavior must have stopped before getting to a hyperbola. So this humble conic is the key indicator, and we want to watch now this transition. I am auxetic. I'm expanding both vertically and horizontally until this is now a very elongated ellipse. And now, the ellipse has degenerated into two lines, and this is the transition to the hyperbola.
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So now, although I'm still increasing in one direction laterally, I'm receding. So this is not auxetic anymore. So we see that the finite situation, the pseudo-triangles for finite linkages, reappear here because both the initial quadrilateral and its doppelganger are in the auxetic configuration pseudo-triangles. But now, if we have a tiling like that with a periodicity and pseudo-triangles, then we actually always have one degree of freedom. And as long as the pseudo-triangles remain,
45:16
the pseudo-triangles will expand. When some alignment occurs and we lose that pseudo-triangle property, things may change. Now, to approach our conclusion, I would say that the theory is not restricted to dimension two, which was chosen for the simplicity of the illustrations. It's actually valid in arbitrary dimension. And the fact that we could find this hidden structure related to conics, or in [INAUDIBLE] dimension, ellipsoids-- is that we approach the matter with a purely geometric
46:12
definition of auxeticity. So this is the definition. I admit, it has a more technical flavor, but in essence, it says the following. I have a periodic structure. Well, what is most important for a periodic structure? Its periodicity. How can I express it in finite terms? I take just the generating translations. So that's a bundle of vectors, exactly as many as the dimension. Now I codify them in a matrix, which is called a gram matrix, from a Danish mathematician
47:00
of the 19th century by the name of Gram. So that makes the idea that the Gram matrix is just telling me what happens to the periods. And as a single entity, it traces a curve in the space of symmetric matrices. But this space is not like any space of matrices because of symmetry. We have there a notion of positive definiteness. So in short, the space of symmetric matrices is not bland. It has more structure. It has this cone of positive semi-definite matrices.
47:53
And this is immediately inviting the analogy with a Minkowski space in special relativity, where we have a cone of light. And this matching is literal if we are in dimension two because two-by-two matrices make a Minkowski space. But in higher dimension, this cone, which is depicted here like a Minkowski cone, is actually more complicated because higher dimensional, semi-definite cones have a more complex structure. But it is very well studied mathematically,
48:42
and it's implicated in many branches of mathematics. So it's not an exotic object at all. So this says we can engage auxeticity in arbitrary dimension. We are not tied to the dimensions of experience two and three. For materials, we can fancy auxetic behavior in arbitrary dimension because the geometry works just the same.
49:30
So here, I want to summarize with another drawing what the result is. So I need the initial illustration. If you'll recall, I had an illustration with red, green, and blue vertices. That is the pattern here. It's, by the way, usually called a [INAUDIBLE] periodic structure because of a Japanese pattern of weaving or basket-making.
50:12
And here, I retained only the essentials. So you see some of the red vertices, some of the green, some of the blue. And the key to understanding the structure-- you choose one vertex in one color or width, and you look at another or, say, the green. So I have this point connected to two green points. They are just three points. Now, through three points, I have a two-dimensional family of conics-- in particular, ellipses. So that this is not telling me what ellipse should I draw there.
51:05
But I know that if auxetic, I need an ellipse. If I move now to a green, I will need an ellipse through the green, and it's--
51:23
no. So if I move to a green vertex, I need an ellipse through this, and it's connected blues. And if I move to a blue, I need an ellipse so its connected reds. So I will need three ellipses.
51:46
They must be now-- ready for a mathematical term? Homothetic. They must be homothetic ellipses. This frightful word means something very simple. I can only take an ellipse, and I'm allowed to translate it and rescale it.
52:15
That's homothety-- translation and dilation. That's it. Now it's not frightening at all. In particular, you will have the same orientation of the axis. Well, my three ellipses are conditioned by being in this brotherhood or sisterhood of being in a homothetic family. And moreover, this detail of the diagram says that if I go from my vertex of origin to the center of the ellipse in the three homothetic ellipses, I satisfy a cycle condition. Those vectors add up to zero.
53:15
So these conditions, in fact, are precisely those imposing the unique solution. So with this explanation, you see that ellipses there are uniquely determined, in fact, but they would be totally invisible, totally hidden, if you don't have this disclosing theory behind. And you'll have a slide with formula. Here it is. But generally, if you distill the matter, it can be expressed in these two lines. So what do we know? We know that the bars must preserve their length.
54:21
There's the question expressing it, but in lattice coordinates. So since periodicity is so important, it gives me a system of reference, and I use coordinates in that system of reference, not necessarily the fixed Cartesian background. But I can differentiate in any system of coordinates, and that's the infinitesimal equation. Now, our geometric definition of auxeticity says that this tangent, this omega is that Gram matrix I tried to make friendly a little earlier.
55:10
So this says the tangent to the trajectory of the Gram matrix is positive definite. And then, to a geometer, that equation immediately says, oh, we have a bundle of homothetic ellipsoids. So that's a geometrical explanation of auxetic motions, and I will conclude with a little animation of motion in this very common crystal, quartz. It's silicon dioxide, and in this diagram, what we see is just the tetrahedra formed by oxygen atoms. So the [INAUDIBLE] is roughly at the center of the tetrahedra.
56:18
But for the purpose of motion, the chief element is that these tetrahedra are, for a good approximation, remaining rigid. So we have a bar-and-joint structure, and there is a flotilla of ellipsoids, which will explain the motion we see. But it's quite numerous, so it would be difficult to portray all these zeppelins in the diagram, so the motion is now contracting now, genuine auxetic. And this is something that happens in the so-called alpha to beta phase transition in quartz.
57:23
If you heat quartz, it goes from a less symmetric structure to a higher symmetry at a certain high temperature. And this is the back and forth of the story of quartz and that transition. Thank you. [MUSIC PLAYING] [APPLAUSE]
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