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>> Thank you for coming out tonight our speaker tonight professors at Brooklyn. Joined Georgia Tech 2017 so he's a relatively new faculty member here before that he was a undergraduate at Cal Tech he did his Ph D. work under a previous Dean Paul Gilbert at the University of Illinois about a champagne is a research fellow at the University of Michigan and a post doc at Cornell So he's going to be talking to us about the science of work I mean I think it's a really interesting topic that you will enjoy at the end of the talk we will have some time for questions and so myself and James will be walking around with these microphones so if you do have a question just raise your hand and wait for one of us to come to you with a microphone and then you can speak so that everyone can hear the question and Professor Auckland's answer without further ado Thank you all right thank you all right thank you very much Dr Garko sound levels are right around here OK great so I've been to a lot of these myself as an audience member and I just want to do quickly tell you a little about the series it's been around since at least 2012 it's organized by Dr Greco Peter saw and stuff you need her both of Georgia Tech's physics department and the action item here is that if you go to detect physics You Tube channel you can see many of the past lectures so it's a craft it's a hobby and I'm betting that most of you in the audience got a chance to play around with that when you were young maybe some of you still play around with it if you're really good you might call them art it's certainly an art not particularly qualified to comment on art but there will be lots of pretty origami in this talk.

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It's part of history it's been around for almost 2000 years and it's part of culture as well of course it's most closely associated with the Japanese culture but it probably started in China and many cultures have employed it for various purposes since then so this the left hand side is kind of the human side and I'm going to be focusing more on the other side of origami so it creates a lot of interesting math puzzles that can reveal to us the structure of curved surfaces it's useful for engineering it's a great way of creating lightweight but strong structures that can collapse and fold up when they're not in use and can deploy to one or more useful shapes there's also science or something in between math and engineering or we try to understand the fundamental properties that allow us to make origami and what the absolute limits of what we can do with it are so I want to start off with something that most of us take for granted which is solid objects so what does it mean for matter to be solid Well what we mean is that it has a particular shape to it some shape that it prefers of this break as this prismatic shape In contrast most of the matter in the universe is not solid it's something like a fluid so water it will take the shape of its container you can pour it you can share it you can pull it from side to side and it'll move along with you it'll only resist you if you pull really fast you know if you slap the surface of the water it hurts but if you go in slowly you will feel very little resistance so there's lots of different kinds of solids from the foams this axis is the youngest models a measure of the elastic stiffness right axis is the density phones are very lightweight but Week 2 medals in technical ceramics which are about the strongest that you can have.

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And right here we have composites a little narrower region of the spot here and that's closer lighted what we want to talk about today because in fact the mechanical response of the structure is not based purely on its composition it's not based only on its chemical structure or its microscopic structure it's based on its shape it's based on things that we can see and we can feel with our bare hands so if we take a solid object and we pull on it since it has a preferred shape we're going to feel a force resisting our strain that we're applying to it so when we pull this out and we stretch this out there's a force that's coming back and acting on us to return it to its original shape this is called hoax law and after Robert Hooke a contemporary of Isaac Newton's in the Royal Society and hoax law states that the force increases the more you Paula if you stretch it twice as much you get twice as much force as if you stretch it only a little bit from its original shape and there's another factor here which again is the thickness of the object if I add 3 more layers here so that it's twice as thick as can be twice as much force from falling apart from the same distance that's not too surprising but if we try to do something even a little bit more complicated it gets a little hard to visualize in particular I want to talk about bending so I suppose and so just simple stretching we bend our object so if we bend or object then now different things are happening in different layers so if we look at this plot we can see once we've bent it and suddenly this layer is shorter than the flare as I go out from this point each layer stretches more and more.

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So in particular we can have a neutral layer in the middle and we can have something that's compressed on one side of it and something that's extended on the other side so not only is it the case that the more event that the more we have something that's compressed and stretched but now things get a little more complicated if we add more layers because if we add more layers Well this next layer if we add this will be compressed even more than the layer before it and compressed even more so suddenly the resistance to bending the force we get back as we start to bend it is not just increasing linearly with the thickness is actually increasing with the cube of the thickness and what that means is if we take our initial structure are thick elastic slabs of matter we make it 10 times thinner it's not going to get 10 times easier to bend it's going to get a 1000 times easier to so what we can do is we can take our structure and we can make it thinner and thinner and as we make it thinner and thinner it goes from something like a thick lastic block which is roughly as easy to stretch as it does to bend to a thin sheet that becomes very very easy to bend even easier than it is just stretch so by controlling the shape of the thing the macroscopic shape we can drastically modify what kind of mechanical properties of how what kind of mechanical response so the key material here of course is paper so prior to paper we didn't have anything that's going to fold in the way we need it to fold to have origami paper strong it's stiff and another useful feature of course is that when we bend it when I bend it very tightly we form a creates a permanent create so we can permanently alter its structure we can program its structure just by bending it and folding it increasing in the paper was invented almost 2000 years ago in China and the history as far as I can tell is not terribly well known certainly I can't make much of that and in fact.

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After paper was invented we know that people started to fold it very early on still in China and shortly after that in Japan as far as I know we don't know for sure whether the Chinese took paper folding to Japanese or whether it was discovered independently there but before long it was occurring in Korea and eventually in Europe as well what's really surprising here is that if you look at the 1st origami book that was in Japan in the year 1800 current era so paper 100 current era the 1st book on paper folding 800 so that's a big gap and that makes me a little uncertain on the history but this is still quite an old book so this is a book from over 200 years ago and it shows something like the traditional Japanese crane here on the right we have something more like traditional Chinese paper folding which is a little more curved a little less angular here and again all of this relies on the property of paper that is foldable so we can ask how foldable is it so for a long time people thought that if you had a sheet of regular paper not some sort of space age material but just regular cellulose regular would matter if you could only fold it about 7 or 8 times it just keeps getting thicker and thicker each time you fold it it gets twice as thick so if you fold it 10 times be over a 1000 times that her and it was relatively recently that Brittany Galvin what the time was a high school student sat down worked out the math carefully and worked out what you need to do to break that record and she didn't just break it she shattered so going from 8 fold to 12 folds that's for additional folds let's 32 times 16 times that or if you go from 832 times that her if you go from 7 to long clear with the previous record was.

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And what she did was she ordered some fancy but not too fancy it was commercially available toilet paper about 4000 feet those about a mile of toilet paper and working with a team of her fellow high school students they were able to fold it 12 times easily making it into the Guinness Book of World Records and this would have been impossible as paper had acted like the classics lab here but the trick is with paper there's nothing holding one layer of the other so the layers can slide against each other so there wasn't an enormous force that was generated when they were trying to fold this thing so paper is already quite the wonder material of course has come up with other materials that are useful for folding in different ways.

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But it can't do everything and in particular some of the things that we're interested in for engineering applications are what we would call functional materials so solar panels of course very useful for capturing energy for getting electrical energy and they're particularly useful in outer space so in outer space you can't just plug in to wall power and moreover it doesn't work too well if you want to bring up chemical fuel because chemical fuel weighs a lot you have to keep going up more and more as a solar panel can continue to capture energy from the sun so solar panels are quite useful for space based missions in a way ground based solar panels of course are awful for sending up into space because they're these big broad flat things that's how they work they want to be as broad as possible to capture as much of the sun's energy as possible but that is not good for sending things up in a rocket for sending things up in Iraq of course you want to fold them up as much as possible and the nice thing with origami is that you're only going to have creases of the few places so you can bend of the few places you can design a new solar panel that will bend at the hinges here and you can collapse it now and that's just what people do when they have solar panels for space based missions this creates a problem so origami is not trivial.

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So on the International Space Station in 2006 they ran into a problem they were trying to extend the solar panels automatically so outside the living quarters of the astronauts and the panels got stuck and that's a problem with the panels don't employ they don't get power the mission can fail so it wasn't big deal because this was a manned mission so they could always send people out to manually unfold the panels but they want to avoid that so one thing they tried was to get an astronaut to exercise vigorously so they have exercise equipment there that the astronauts use to keep up their muscle mass and 0 gravity and the idea was just to exercise their grossly try to shake the space station get them loose and they were reduced to this sort of silly maneuver which ended up not working so they needed to do the spacewalk after all.

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Because they're trying to avoid the space walk which is hard but not impossible for them but it would have been impossible in unmanned missions so an unmanned mission if you're origami doesn't fold right and your origami solar panels the mission can fail sorghum is a serious business you can take us to places farther than we've been before at least more convenient than past ways of getting power up into outer space and people have worked it out so people have come up with a number of origami designs so I think this one in particular was developed at Brigham Young University and the idea that we want is that again it should collapse into a small space as possible and then expand out into as broad flat and low mass a surface as possible and then should be able to do this in only one way so if you've ever played around with origami my eyes know this is easy to Miss Falbe things you miss a crease somewhere it gets a little off things aren't quite where you want it to be.

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And you have to go back and fix it which is fine with your human hands but if you're in outer space if you're lying on something that's essentially a very very simple robot it's not so easy to deal with misfolded So you want this to be as robust and simple mechanism as possible while meeting your criteria as being able to collapse and expand.

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So there are actually a lot of foil structures most of them are quite ordinary things that we take for granted so a cardboard box you can fold it up nice and flat for delivery and then when the on it puts up a new expand it occupies a much larger volume and is pretty strong it's mostly made of paper but it has these nice corrugations in it that make the panels stiff Hoberman go just a novelty but it's generally collapses in expands Hence again collapsing expands and draws very useful collapse down fit in your bag expand out just a simple membrane to protect you from the rain and I used to mention for a long time that sort of the holy grail of things was to get to foldable electronics to get to something like a screen you could fold but know that nobody had figured out how to do that yet so just very recently a few companies have announced that they have worked out how to do foldable phones I don't know how well they work I guess we'll see but the idea is that you know if you want to pay a couple $1000.00 for a phone soon you will apparently be able to get a phone that will unfold so it folds up twice so you get I guess Troy twice the screen size so maybe in 10 years we'll all be able to unfold a laptop from our pocket which should be really cool if it works.

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There's lots of other things that you can do with origami so especially if you're not limiting yourself to what appears we call origami which of course should be paper so if you're thinking instead more about foldable surfaces where you're allowed to put in the hinges by hand and you're allowed to use different materials so the same group that came up with one of those solar panel designs also developed a Kevlar system so this caviar system collapses down it's lightweight and compact enough that a single person can easily carry carry it and kevlar is a chance eyeball proof nothing's really bulletproof it's bullet resistant so it can stop a bullet and this is intended to be used as a shield to protect police officers when they're going into situations where they.

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Expect to be shot at and it's broader and lighter weight than existing shield so one person can carry it out and can protect 3 people at least 3 people crouching down there other uses in Dubai again this is not what a purist would really call origami there's lots of open space between the panels and it's not made of paper but it does have these foldable panels and what they're doing here this is a building in full hard to see the scale here but each layer of this is a level of the building a floor of the building and in Dubai of course it gets very hot the sunlight is very bright and they open and close these panels in order to control the light and heat that's being applied into the building so it's fully computerized it can expand and collapse opening up for more light or less light as the day goes by and as the year goes by then.

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This structure as far as I know is not really in use now but it's been demonstrated that it could be effective what this is is a metal heart stent So a heart stent. Is meant to apply force and hold things open in the human heart and of course when you're inserting it since you have to move the person's body and organs out of the way you want to be as small as possible so again that's a job for origami with origami you can collapse into this little thing that is mostly solid metal and then expanded out into something that has lots of empty space in between with a thin metal tube around it so there's lots of different applications you can have where the general idea again is that you want to be able to get small and then get big get small get big foot back and forth between different states just unfold things around the edges here let's think about the best way to understand this mathematically So when people started out doing origami they were not so far as we know concerned with its mathematical properties they're concerned with it's as Benteke qualities but eventually they developed a language that evolved into diagrams look like this and these diagrams are very useful for practical purposes they tell you how to make the origami they say fold here now fold here fold this unfold this do this turnaround like this and the end of the day you're going to have an origami bird here.

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Mathematicians don't like that so much so little complicated for them what they want to do is they want to boil origami down into the simplest possible description that they can find so instead of step one step 2 step 16 what they say is at the end of the day just tell me where are the crease patterns are and then tell me one more thing so we have 2 different kinds of creases here we have this is shown as red creases and blue creases and these are called Mountain folds and valuable so Mountain fold if you look down on it from above looks like a mountain it's bending away from you valley fold it looks like the edge of the valley sloping up towards you.

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So these 2 bits of information where the creases are these one dimensional lines over a 2 dimensional surface and this is completely abstract and sort of thinking of this is completely 2 dimensional completely flat before we start folding completely one dimensional line infinitely thin and all we're telling you about them is which ones fold up and which ones fall down this particular design is not the same as this one this one is called Darwin's or kid by Robert Lang if you've been reading the names on the bottom here you have already seen his name we're going to see a lot he's a great organist I'm a big fan of his work as many people are and the question for us now is what can we do with these creatures patterns what can we say about them mathematically how can we.

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Make this quantitative and precise and we want to narrow things down we want to talk about a single vertex here so vertex is just what we call the point where the creases come together so here we have a vertex with 6 creases attached here we have one with 12345 vertices attached if we count the outer edges They're here with 4 vertices attached so at that point we can ask ourselves I'm not going to worry about the whole sheet and not going to worry about Paul how all these viruses work together I'm not going to worry about all the folds I can put in I'm just going to start asking about a single vertex and ask the question that mathematics mathematicians like to ask the turn that turns out to be quite revealing is is it flat full and by flat foldable I mean it starts off flat so it starts off flat but can I put folds in here and I fold each of these creases around the vertex and make it flat again so.

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If I take something like this if I just fold it up this is flat fold so I've given it some fold to have a single vertex here and it still flat it still is very thin thing as opposed to some configuration like this where I've put folds I have I've changed the angles here of change the angles of the faces against each other but it's not flat at this point so the question that we're going to ask ourselves is is the vertex flat foldable Partly this is for the reason that we already talked about that being able fold flat is very useful for deployment B. can claps it into a tiny little shape and put it away in a tiny little space until we need it but partly it's just leads to a lot of subtle and rich math that reveal a lot about the structure of this crease pattern so the 1st theorem so this is a mathematical theorem and it's named after my colleague it was discovered more or less independently by my colleague Justin and Radha and not in that order and what they found out is that if you take a vertex if it can be folded flat then the difference between the number of Mountain Valley Falls that come together that vertex must be to the number of Mt folds number minus the number value folds must be 2 or negative 2 and the reason if you're good at 3 D. visualization is not too hard to see so if we take a vertex like this and we go around and we go over a mountain fold and we look at when it's in the flat state when it's flat folded we look at how much return the mountain fold is going to take us on say half turn and if we go through another mountain called we're going to have another half turn but if we had a bellyful it's going to be a half turn the other way.

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So if we add up all those half turns and we come back to where we started on the origami so we make a full circle come back to where we started as we've gone around we must make a full turn in one direction or a full turn in the other direction so we must have netted a total of 2 have turns one way or 2 have turned the other way and since man falls or have turned one way Valley Falls or have turned the other way that means that the number of Mt falls in Valley fold as to differ by 2 So let me try to show you this so I should say sometimes people do this is a demo so if you happen to have a sheet of paper you can try it in the audience they can get a little tricky though it doesn't always work out quite the way we expected so I wanted to just do it once when I knew it was going to work so start off with a flat sheet so something like this nothing special and just start folding it and just keep folding it till you get to a point where it's still folded flat so I just take this I fold it once I fold a trice I fold it 3 times that seems good enough and now I am told so I'm fold it and look at my creases So if at all my creases night I'm going to label them and say which one of these is a mountain fold which ones of I fall so this is clearly a mountain fold increase is closer to us than the sides here this one is a valley fold the valley is further away from us in the sides and either side of it so if I go through and I label the mountain folds and valley folds it's a little hard to see here but on 5 of these I have solid lines the solid lines indicate mountain folds dash lines indicate Valley Falls 5 miles 3 miles too so there we have it no matter how many times you do it no matter what tricks to try to play if you play by the rules are going to have that the number of mountain folds and the valley folds are going to differ by 2 but is this enough if I just choose the numbers of mountain folds in Valley fold it turns out that that doesn't guarantee that it's flat foldable so this is an if statement but not an if and only a statement if it's flat foldable then this is true but if this is true it's not necessarily flat foldable.

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So we have to go a bit further than my Kahlo and we have to go to college office there so callous Aki starts off by labeling mountain and valley falls and we see that we have 3 mountain folds in one valley fold here you know what he labels the 2 different kinds differently here but he also looks at the angles between the creases so I can look at the angle that I rotate through as I go from one crease to the other so I get these 4 angles in this case because for creases and cause the sock use their own says that a vertex is flat foldable if only a so this is a more powerful statement this is a guarantee now we can alternately add and subtract angles between creases to get 0 so in this case we say altho on minus Alpha 2 plus out of 3 minus out before that has to add up to 0 is this going to be flat foldable it doesn't always add up to 0 there's no guarantee that's going to add up to 0 but if it's going to be flat for the ball as opposed to just fold the ball it has to add up to 0 and we can see this from this diagram again if we're good at visualizing how things go so as we fold this some of these angles are sort of face up and some of them are face down and if we look at how we rock back and forth as we go through the angles as we go through the 1st one I might rotate by Alpha one to the right and then the other one is flipped so I'm going to rotate by alpha to the left and Alpha through to the right and Alpha 4 to the left and when I do that I have to come back to where I started because again I've gone around in a circle around my vertex come back to where I started and it was flat folded I have to come back to where I started so I have to have Alpha one plus Alpha 3 is equal Alpha 2 plus Alpha 4 and we can check that too.

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So I can keep going with my same bit of origami here and again it's a little hard to see but what I've done is I've just labeled alternating angles so I have this angle in this next angle than this next single and I'm going to break a cardinal rule of origami and break out the scissors so I break out with scissors and I cut it up and I just see what happens when I take all of the alternating angles and balance them against all the other alternating that's what I get is that indeed I have half the angles together add up to exactly $180.00 degrees that's not always going to be true of course with always has to be true if it's flat foldable and the other half also has a $280.00 degrees so this is now a guarantee if this is true I know there must be some way to flat fold so that's the vertex the vertex we can do the vertex at least for flat fold ability is not too bad if we look at general folding if we say what are all the holes we can do that are not initially flat folds you can get rather complicated but is still just a single vertex But if we have something more complicated if we have this big crease pattern here we could ask is this flat fall ball so now it's not just a question of Can I get one vertex flat it's can I get all of them flat at the same time this is a harder problem and it leads to another favorite of mathematicians which is a coloring problem so we can do this we can take the faces and we can color so we can take 2 colors so this is a 2 coloring and we can say I'm going to color every face by one of 2 colors and my rule is going to be that if 2 faces share a crease then they can't be the same color.

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So what I can do is I can start off with one face and say OK all the faces of charities with that I'm going to color the other color and then all the faces that share creases with them I'm going to call the color of the 1st color and if I go through this it turns out that a flat foldable she saw not just a single vertex has to have this property that no 2 faces the chair crease are the same color and that's because if I do fold it flat then in a sense each of my face is either face up or face down so I can label the face up ones with one color and the face up down ones with the other color and if 2 faces share crease then one of them must be face up and one of them must be face down so I can do those 2 colors so now I have 2 things I can say if I want to make my whole sheet flat fold a ball if I want to completely controlled in this way I need the individual vertices to be flat foldable which I can do from college sockets there and if I also need to have this additional structure that says the faces have to be 2 color of all the question is Is this enough and the answer is no it turns out that we can have all the virtues working and we can have those 2 color ability and it's still no guarantee that the whole thing will ever fold flat in fact generically it won't and it turns out the problem of working out when sheets fold flat and when they don't is quite difficult it's computationally difficult it's what's called N.P. hard it's something that we can check in a good amount of time so if I tell you here's how you assign the mountain and valley folds then you can check to see if it works pretty easily you can just go through and say OK do I have 2 more Mountain Falls in Valley fold to the adverts X. no problem but if I want to find something that actually does that it takes a very long time on the computer to actually do that so there are limits to what we can do with computers so I want to shift now back to the as that excited.

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And this is another one of Robert Lang's pieces at the time he made it it was certainly his most well known piece so this is a Black Forest cuckoo clock and it doesn't show up on this image but even if you zoom in very closely and you look at the fine details you see details on details you see these nice curved leaves you see this nice pendulum and it's even a little functional in that if you fall on the pendulum because in the cuckoo clock comes out.

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It tells the time of course the sun actually turned but still right twice a day so you can have all of this complicated structure that again this was just a single sheet single flat sheet nothing special no glued no cutting. That Lange was able to make and he's able to make a lot of stuff if you go to his website you can find yourself wasting a lot of time looking at the different pieces is realized so this is the Emperor's scarf and some of you might know that story but it's not important to my or the piece So again a lot of fine detail.

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And you might say well actually a scorpion is not so hard a cuckoo clock is not so hard I mean the Scorpion has a sort of hard care Appice can we do something that feels like it has a little more personality and perhaps to show that Lange did human face so I think this is all one piece except maybe for the glasses the glasses I think might have been a different sheet to start with this is C P S Now some of you might know the name so I think he's most known these days for he was a scientist who also wrote books and he complained that people who love the arts and humanities didn't talk to the people who love science and math and vice versa so perhaps origami is an attempt these days to bring those 2 different cultures and called them the 2 cultures together so that's probably I'm guessing why Robert Lang chose his face to sculpt to make out of origami I should say and how do you do this in general.

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So there's an old joke. That's in this bill that cartoon says How do you fold it how do you sculpt an all of Delta says it's easy to all you do is you start with a chunk of marble and move everything that doesn't look like an elephant then actually work out from just as chunk of marbles and have an elephant in it so of course in practice sculpture is not easy but in theory at least it's hard if I say here's the shape I want you to sculpt than the algorithm is just remove everything on one side of that shape and keep everything on the other side obviously that's not how you do origami so if you want to make something like this origami elephant that Bernie Peyton made so again a single sheet here the tusks are just from the other side the other side is white the main side is great and then there's a baby elephant here all things kind of hard you know the story of the blind man in the elephant the elephant has lots of different parts to it as the pillar like legs the snake like tail the broad flanks the shark tusks the flappy ears and then the trunk so the trunk has these cool wrinkles in it so there's a professor here David who loves talking about how these wrinkles enable the trunk to be so sensitive and so dextrous but that doesn't tell us how to make an origami elephant and.

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There's a problem with this the problem is that if we look at the elephant and think of it as a 2 dimensional surface and we look at our sheet of paper we have a problem the elephant is very curved pretty much anything you want to make is very curved so some origami like the origami crane it's pretty angular it's not very correct at all but a lot of the modern shapes that people do like to make are curved so how do you make a curve from a flat surface it's a hard problem it's a problem that has the devil people for a long time.

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So could we fold a sphere can we fold up our paper and make a sphere. And this is a problem that map makers have encountered so the problem they have of course is that we know that the earth is round but we like our maps deployable we like to be able to fold them up we like to put them on sheets of paper so we need to make our maps flap so maps take the round earth and map it that's what mathematicians say they got it from the map makers they map it to a flat sheet and the repeater projection is one of many you can again you can look all these up they'll have their advantages they all have their disadvantages there's no killer projection because there's no projection it can do all the things that we'd like it to do so in particular the Mercator projection preserves the angles between things so like this map if we start off with the square grid and map it like here if we look at it the 90 degree angle say 90 degree angles but some of the squares stretch and some of them contract and that's what's happening here so if we take the Mercator projection things are really getting stretch of places specially near the poles so it's not too bad if you're at the same latitude but at the poles like Greenland things are getting off the place so if you look at this Mercator map you compare Greenland which is purple to Africa which is red here they look like they're about the same size maybe Greenland even a little bigger actually of course Greenland is tiny compared to Africa so this is stretching things all the place which creates its own issues for accurately representing the world which is part of the reason that Google maps recently like recently in the last few years switched from having flat maps to any zoomed out enough making it look like the surface of a sphere.

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But we have this problem for you can't do that we have this problem that says how do we get from our flat space to our sphere without trying to do this awful stretching thing so with origami the problem of course is that we can stretch paper right paper just does not have a lot of stretch in it so if you really pull the heck out of it it might stretch by about 5 percent but it won't be good for much afterwards and I will tear soon after that.

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So there are 2 different kinds of bending it turns out some of you may know this so it turns out of this is take a sheet of paper invented up into a cylinder right now probably all. But it's not easy at all to get to a sphere in fact it's impossible to do it purely by bending the paper you need to stretch it and what's the difference between the cylinder and the sphere Well it turns out there's this thing called Galaxy and curvature and the way we see it is we pick a point on our 2 dimensional surface and then we draw 2 perpendicular lines and we're drawing them along the surface a little start to curve and we choose an angle so that one curves as much as possible so we get to the point what's called the principle axes and we just measure how curvy these are and in particular what we do is we measure how big a circle it would take to make this curve and we take the inverse of its radius of the bigger the circle the less curvy it is and to get the gas in curvature we just multiply these 2 things together so lots of different ways we can do this and what it reveals actually is that by this measure when I've taken this flat sheet and fold up into a solid or I haven't done anything at all so the reason being that even though I've got one cut type of curvature Another one is perfectly flat so when I multiply them together I still get 0 In contrast the sphere has positive coverage or something that's shaped like a saddle has negative coverage because one bends one way and one bends the other and if we make our sphere smaller it's actually more curve because the smaller this circle here the higher we define that curvature this is a useful exercise because it gets us to what's called Gauss's Grecia.

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And gals who was arguably the greatest mathematician in history said that if you take a curved surface and develop it upon any other service whatever which is to say if you bend it the measure of curvature so he means the measure of dowsing curvature but he can't call it that because it's not an egomaniac.

[00:36:50]
In each point remains unchanged so what that saying is no matter how much we bend our paper how much we fold it we can never really honestly go from a flat sheet to something like an elephant which has gotten curvature so we have to cheat origami is all about this kind of cheating and how to do this so to do this we're going to turn to Tony R Toci who's a true artist and also a scientist OK we don't need the sound.

[00:37:21]
OK whatever OK So we have the sound playing softly here this is the crease pattern so this is the crease pattern and the way the guys like to hear it touch and Robert Lang get their keys patterns is they don't really do that with our puny human minds these days it is computer assisted So we have a computer assisted way of generating prefabs like this which is a large part of the reason that we're able to generate more sophisticated origami than we could in previous generations really if you look at his technique is just using his hands and these little clips so even the purists are fine with having this little clip he's going to take them out by the end and he's just going through and he's generating this crease pattern on the sheet of paper and the clips of course to hold in place because there are many more creases than he has fingers and it's taking him some time it takes quite a bit of time even once you have this pattern even once you've done a few test runs it takes a long time to get all these because the computer doesn't tell you what order to fold these and the computer doesn't worry about the fact that you can't just take paper and pass paper through paper that will bump into itself so it's a problem and a half just to figure out how you actually get things slipped through here and thought she has chosen to try to make a bunny so the bunny is a little like C.P. Snow's face it's harder than a scorpion in that it has all these nice soft smooth curves it's also an allusion to something that some computer scientists around here are probably familiar with which is the Stanford Bunny which is just an image that was used as sort of a benchmark for how good you are at creating and storing computer image so this is a harder problem of course in the sense that we're trying to create a physical object and a surface but it's going at it it takes them again quite some time so if you watch carefully you have to be quite careful at one point he changes a straight out for a similar shirt because it takes 2 days to do this so he's doing this and if you look at this you actually start to see a curved surface.

[00:39:17]
And the reason this works even though gals with their arms says it shouldn't is that we aren't removing the material so if we could remove material we could change the gas in curvature we are moving it but what we're doing is we're creating a flap and tucking it away so the bunny has a lot of extra material that is tucked away inside the surface which is why the bunny ends up being so much smaller than the sheet that we started with if you just look at the surface area of the bunny that showing it's much less than the surface area of the sheet because the bunny has a lot of paper just tucked in inside it so it's getting close here I think the ears are the hardest part I think the years here are the ones here with the greatest density of fold here and we're almost to.

[00:40:02]
The end here and I'm just going to skip to the OK We just got a few seconds left so there we go so so funny so we get this nice curved surface coming from this nasty nasty full pattern and really there's no way to do this except with the human hand there's no technology we have they can equal what nature has given us.

[00:40:34]
Now I want to switch to something that is kind of at the opposite extreme of that which is not this complicated intricate pattern where every bit is different from every other bit but instead something that we might even think of as a material the nice repeating pattern so we have this repeating structure that kind of looks like they're fitting structure of a crystal of course on a much larger scale this is due to Curio Meera is a Japanese astrophysicist and he designed it as an astrophysicist aid in space missions and indeed in 1906 it was deployed the solar panels unfolded successfully so successful use of origami here and the nice thing about this is so symmetric Each panel is a parallelogram and it's really just the same parallelogram translated and flipped around over and over so in a sense what we have is just one vertex between 4 parallelograms that repeat it over and over so you can fold a single vertex with 4 creases coming out of it so you can fold the whole thing it's really easy to make even if you have no particular account for working with your hands you just have this nice pattern that you put in and it collapses so this is definitely what we call flat foldable I can get it down into just a single panel all sitting on top of each other Is it a do is it to make big or is it making different contexts so for you know how I mentioned before has folded up in the tube so you have these nice tubes where you just sort of squeeze them and they extend you can make things like bridges and walls out of them and then they again just collapse right down make them at the human scale are a bit larger.

[00:42:07]
You can do more things with the you're already so the mirror Henri I like to say is a little like the fruit fly is for geneticists Where is just the most useful simple basic pattern that we can start adding stuff to and saying different things so you can do things like Here we have different layers of the mirror Henri and then loops this is what you'd call a defect except the people who made it put it in on purpose we have suddenly 3 layers we come to layers here so we have this defect that radically changes how folds and unfolds we can also have interfaces between 2 different kinds of your Henri the interface is active hinges so we started a modified mechanic or a sponsor started to take this one thing that it had and add in more and more things by changing the structure So each little bit is still the mirror Henri the same pattern but now writing patterns on top of patterns and you can keep going you can say how many shapes can I program and the answer is pretty close to all of them you can get pretty arbitrary shapes so a number of people this was published a few years ago and Elmo Hi David was here recently and he spoke about this borrowed some of this graphics for the promo figure you might have seen advertising this talk and what he and his group do is they start with the mirror Ori Let's see OK so I'm missing a graphic here but what they do is they start with something that looks like the mirror already here except And so this regular painting pattern is slowly modified so that the mirror already here isn't quite the same as the mirror already over here and when they do that.

[00:43:41]
They're able to tuck things away so they're able to tuck extra material away just like Toci did with his money and they're able to tuck more material away in different places so what they can do is they can get this nice curved surface they can get essential to any curved surface they want so if it's symmetrical like this if it's the same as we rotate around it's axis that certainly makes these are in practice but in theory it doesn't need to be so what they do is they start with a flat sheet and again in seriously the doing practice it's a little complicated they will do is pop it up turn into a vase and then they cheat a little origami wise they have to actually tape it together at the end here or otherwise and be able to stand here and they're able to get pretty much any shape they want in this way the way they're doing it is with a computer algorithm that allows them to approximate any curved surface and to do this it's useful to be able to go small so the more intricate we can make this what you might call metal material in that it has the structure on top of its microscopic structure that's changing how it's mechanical response works if you can make it finer you can get closer the surface you want so this is 10 strips per cell and you can see it's a curved surface but you can really see that it's angular it's block it doesn't look exactly like that 100 strips at least in the graphic with this quality it's almost perfectly like a smooth surface when thousands of miles per strip it's pretty much perfect so the more intricate You can make it the more control you can have over different kinds of shapes you can get with your origami surface and I want to switch a little bit so we've been talking about origami as a human endeavor and if you think about it then what human beings want to do with origami.

[00:45:28]
Is are things that nature's want to do with origami So nature has wanted broad flat surfaces for things like bird wings so this is origami but it's also meant to suggest an actual bird with actual wings so for bird wings and before them insect wings. And hey what about plants so you have things like petals you have things like leaves what is a leaf but a natural solar cell and like the artificial solar cell it wants to be broad and flat and thin and strong and it wants to get its functional material wants a good score fell out there exposed to the sunlight so origami is useful for Nature and Nature seems to figure out how to do it again if we're using the term origami a little loosely here so if we look at a fold in a wreath we can see indeed this looks very much like something we can make with origami this is nice so what Richard Fineman the great physicist a great explainer of science said it is what I cannot create I do not understand he was doing quantum electrodynamics So that was a problem for him it's hard to create an electron but for a folded leaf.

[00:46:34]
The quote is not about quantum electrodynamics of course. With Luis it's not so bad we can figure out how to make the macroscopic structure of a leaf or this is a sketch of an insect wing and it turns out so some people just keep coming up we're back to Dave and again and Dave and showed that you could get origami without anyone doing the folding nature could figure out how to fold it without some sort of central plan or during the fall or so panel a here is just regular Oregon this is something that an actual human has to do with his or her hands the natural process is also reminiscent of the leaf again of this quality of graphics it could almost be an origami object and down here and see this is something quite different this is something that's on a smaller scale if you know how small Exactly and it's also something where you had an experiment where no one was doing the folding all you had was interactions that were happening on the surface of this object and it pulled it into this sort of wrinkled pattern that if you squint a little it's not perfect and squint a little it does look very much like the mirror already full.

[00:47:41]
And even simpler David showed that if you take a simple equation so simply equation a little bit like you know things roll down hill that you want to minimize energy and you put in simple interactions and you ask these simple interactions to create a surface that minimizes the energy what you get is this wavy pattern here which again looks rather like the mirror Henri So even very simple action principles very simple equations are the types of equations that we know that nature can obey without any great complication can generate things like origami and again as we've seen with wings and leaves that's often a useful thing to do and in a sense this is definitely not strictly origami in a strict in a sense though folding at least is something that is fundamental to nature so if you take functional biopolymers proteins they form the structures of the human body and of course with other organisms bodies and even if you got the structure right even if you put all the molecules along the line here where you wanted them to go it's not going slow going to function properly so there are quite a few diseases that are associated with protein Miss fallings So you also need to fold it properly and if you can fold it properly you can get of course a wide variety of biological functions and if you can't you can't so one thing that animates people is to try to at least approach we're very far from it this level of mastery that nature has of being able to take these molecules and not only make these molecules down at this tiny scale but fold them to put them into the shape that we want in fact is it's quite difficult it's difficult even to simulate a protein folding takes a supercomputer quite a bit of effort to simulate for a few seconds.

[00:49:26]
So I'm going to step away a bit from origami and get to something related so this is Jurgen me so origami is paper folding from the Japanese of course Kerrigan me is cutting paper so we're breaking the cardinal rule of origami we have to leave the scissors in the drawer and we get more freedom in the origami and also less structure so I mentioned that origami and Kerrigan can be a cultural event hopefully many of you recognize the cultural phenomenon this is this is the Millennium Falcon and I guess Darth Vader and other want to know the fighting in the corner there.

[00:50:04]
So if we have the freedom to cut we can do some interesting things so what people at Cornell did was they cut pieces out of the paper and they pulled on it and suddenly it acted like a spring the paper starts to rotate it can really stretch quite a bit that's quite different from origami right no matter how I fold this I'm never going to get it to go longer than this this is as long as I can go before I start cutting those Kerry got me we can go further and what people did the Cornell group wasn't just interested in paper they're interested and finish paper possible graphing I'm sure many of you heard of graphene So it's a single atom thick carbon so carbon is also what paper is made of but the structure is quite different here so it's made every time you take a pencil to paper but it was a nice little 2004 by Gaiman novus all of earning that in physics Nobel Prize for major advances also occurred here at Georgia Tech It has incredible electronic properties but for discussion today we have the question there are more interested can we bend it and that turned out to be a hard problem so the Cornell group.

[00:51:13]
The 1st author was the only. Senior author Paul McHugh on figured out this is very hard to see in the scan here but if you look carefully you'll see something like this here the same pattern so they took the exact same pattern and they cut it in here and part of the reason they did that was it turns out the graphing is not actually as easily foldable as you might think it's very very thin and we saw that very thin things should fold easily but it turns out the grafting always has little ripples in it so instead it acts sort of corrugated so that's much thicker than actually is so it's not so easy the bend so they had to cut holes in it and when they did they're able to stretch it out and form these tiny ultra thin Springs so the idea here is that by cutting out graphing we can create these new patterns and these new structures they give us some of the smallest mechanical parts anyone's ever seen and then later some of the same people including Mark Meskin showed me this in the lab Cornell is quite awesome to see did figure out how to fold so the way they figured out how to fall goes back to our when our very 1st slides here they took grafting and then they stuck silicon dioxide on glass and then they also stuck some rigid panels on it and why did they do that well they're going to heat it up and when they heated up glass expands graphene also expands and theory but not as much and we know that when one layer is expanding and one layer is not expanding so much that's bending that's bending right there when one is expanding and the other is not in they're stuck together so just by getting it out by shining a laser on it they're able to fold up objects so they can take this initially flat thing heated up and they'll fold up into a Q So again similar to macroscopic graphing we can fold things out start from a flat object make a 3 dimensional object.

[00:52:55]
This thing is really cool so this thing they don't put in the rigid panels they just have a graphing strip it's hard to see here but they have a graphing strip with the silicon dioxide and they just fire a laser at it they fire a laser at it goes wild coils up and this can act a little grabber if you have an object there you want to grab just shine the laser on the strip crawl peddle coil around it grab and hold on to it a bit from the laser off it will release some about our time here but there's other stuff I wish I could tell you about so there's the notion of memory so it turns out that paper when you crumple it remembers that you crumpled it in fact it even remembers how you crumpled it and it can remember it for a surprising long time so one of the things that I did with my research that I really love is that we took a system that was not at all origami we took a system of magnetic spins microscopic magnetic spins or interacting and we mapped that system and origami it was an origami But we said hey the way that these spins can move is equivalent to the folding of paper it's exactly equivalent and moreover we know a lot about how we can fold paper so by crafting this origami analog or magnet system we're actually made able to make new predictions about the new ways that the degeneracy of the spins the new ways the spin system could form itself that was equivalent to folding the origami system so it's just about time here so thank you very much.

[00:54:32]
We have time for a few questions. Well OK So the question about the folding part of the origami so that I noticed it took 2 days from the full that that body is there any sort of algorithm or mathematical assistance required to figure out how you can fold papers folding through itself I would imagine that would take that a lot of time in the process yeah so that's a good question I don't actually know the answer so the algorithm is that it can tell you here's the curved surface I want here's how you need to put the creases in and I know that very much about something of what someone like Taji does is really just eyeball it by looking at what's happening locally and try to modify things locally and the idea is if you're doing something off in one corner it's not going to affect things off all the way over here so you can kind of eyeball the notion of there being a geometrical distance that is going to assure that you don't have to worry so much about what's happening there in practice I don't know if there's also a computer algorithm that is going to give him some clue as to say start on this area now start in this area and I'm over here.

[00:56:00]
So you mention that when he was folding a large piece of paper that the only way to really do it was with human hand do you have any examples like I have somebody tried to automate the process yeah so that's that's a big topic right now so partly because people are very interested in microscopic origami so for macroscopic origami doing it by hand works all right you know it's labor intensive it's not something that you would use for mass market for the origami applications macroscopically it works microscopically it's quite hard to actually better at making really small sheets than we are at bending really small sheet.

[00:56:41]
So something like I showed in the last couple slides where what they did was they pasted panels onto it and they pasted silicon dioxide onto the graphene and then they heated it up that's one way of doing cell folding origami there's other ways to control it other than temperature there are things like adding oyster more stricken also causes differential swelling that causes one substance to bend in a way that others don't there's chemistry there salt content there's all sorts of different knobs that you can try turning their feel their There's origami that couples to magnetic fields so there are different ways they have their advantage and their disadvantage is that there's origami robots where the rope origami is actually actuated and has little pieces that are hooked up to batteries that can make the origami go from a flat state to picking up and walking on by itself so there's lots of different ways to it it's really interesting area I have a question for you Professor up with How did you 1st get interested in origami in a scientific or mathematical sense yeah so it kind of grew out of my more general interest which are in like civil structures in the mechanical response of things that have complicated structures not at the chemical level not at them like a level but at larger scale so some of the things I look at our.

[00:58:03]
Systems that you could think of as a bunch of point particles connected by rocks and for some of these. They look very much like origami and some of these we realize these are actually mathematically equivalent to origami and one of the things that really surprised us something that I'm working on with my student James there is that adding that extra structure just creates this incredible new layer of math where you get these new results that you don't have you have things that work before that no longer work and then you have new knobs that turn on new things you can control one of the things I study is topology so there are things called Top logical invariance new numbers that you can have with origami that you can't have with the other mechanical structures we're looking at so it just ended up being something that in addition to I think things that we can all appreciate on the political level and being a very fruitful area for us to look at Great OK let's think Professor Rocklin one which I.

[00:59:01]
Thank you thanks.