night we are very very honored to introduce dr mohamad of on is the developing professor of applied mathematics he's also professor of organismic and evolutionary biology and also professor of physics all at harvard university after graduating from the indian institute of technology he moved to the united states where he received two master's degrees and  then took his ph d at stanford university in one nine hundred ninety five there after he taught at oxford and cambridge universities and then at mit where he became a tenured professor in mechanical engineering and then he was lured to harvard with all his titles there dr my heart of a list of awards and distinctions are very extensive i'll just mention a few you can see more on his website there's a link to that on our website for tonight's event page he received a two thousand and nine macarthur genius a grant he's received the guggenheim morial fellowship and harvard's george ledle prize he has presented many times prick prestigious invited lectures and he has served on numerous editorial and advisory boards i should say has he has continues to do this and is probably one of the busiest people on the planet distinguishing feature of dr mahadevan and genius is his ability to discover and explain complex relationships between form and function in nature especially the underlying geometry of structures and processes at the very smallest the very largest scales most of what we see we see superficially d underlying secrets of how things fold how they flow how they drape how they grow and many other every day processes that if they'd our senses tonight he explains h a type of folding generally associated with art is present in many natural structures and it has numerous applications you will i think be surprised at how beautiful nature really is when he finishes talking tonight how the understanding of the relationship between form and function makes us appreciate a much deeper level of nature and please welcome d yes and generous introduction so for now it's downhill ok so what i'd like to do over the next hour or so is to try and give you a flavor of the world of origami as i see it i'm going to talk nationally about some the work that i've been involved in but i'd also like to essentially give you a sense of what other people around there are doing the title that i gave  website so if you think about it from the point of view of art it's quite remarkable you can take a piece of paper which is x starts out being featured less and end up over there if you think about it from a point of view of science i'm going to spend some time on it you can actually see it in very different places you see it over here in the context of a leaf you see it over here in the context of just a drying piece of jell o and this of course takes a lot more effort to make then this this is a piece of paper which has been folded into a very simple kind of origami which is named for the inventor choreo mira who invented this a long time ago and this is essentially a mathematical model which has been rendered on the computer so you have it in art i'm going to show you that you actually see origami appearing very naturally in a number of scientific settings and then i'm going to tell you a little bit about the work that i haven't directly been involved in but it has become one of the primary how being thought of today which is in the context of technology and so this is a video actually due to an artist but inspired by science so you can what you see over here is happening in real time so there's a little piece of metal or plastic for oil which has hinges and you put it on water and these hinges essentially swell in the presence of water and this thing starts to open up and you can see three or four generations of it opening up so there is clearly an artistic sense that appeals to it as soon as you see it but there is also a scientific question associated with how one might engineer how one might control and how one might make functional some of these patterns so sort of as i said i'm going to make you well take you for a little walk through some of these examples over the next forty five minutes or so but i want to start by telling you a little bit about the history of this beautiful art because in fact the artists kept it alive for a very long time and a theme that i hope you will see right through is that science and technology are usually the last in the room artists been thinking about these problems for a very long time and it's only very recently in the last forty fifty years that people south thinking about it scientifically in fact and in the context of technology to not even the last ten years i would say ok so origami as  she it came to japan a few centuries later people think about for six hundred years later and in both china and japan origami was associated with a number of different valley falls and you need to essentially put these falls in place at the right place at the right time so you need this particular sequence otherwise you'll get frustrated and in fact if you're able to do this and i think children do this far better than adults then you can get shapes like the crane or the butterfly and this is an example of a very large no photograph a very large number of cranes inspired by this very sad and moving story of  japanese child who was essentially a survivor of the atomic bomb one of the atomic bomb explosion one hundred forty five cycle society and and she essentially wished for a thousand cranes to be built and she survived about ten years  beyond the war and there's a wonderful book which i encourage you to read because that's a little bit of the cultural history of origami compressed into three or four minutes and as i said this whole art form had persisted had been sustained by artists but really took off in japan over the last hundred years or so thanks to this person of the here are here are your she's who passed away just a few years ago and he  used his abilities to fold to create a very large number of patterns which he then put down in a large number of books and also introduced some new techniques that origami that most of you are probably familiar with associated with what i easier to fold and the folds also are not so strong so they're much much more rounded ok so for example you see this is bull and you can see there are a lot more rounded surfaces rather than in a crane which might very important because essentially it was sustained by artists in artisans this this form if i think about it from the point of view of science i think we're from the point of view of mathematics and i ask what is it that makes origami possible what are the elements  origami in fact the first in the most important thing is that you have to start with an object which is like a sheet of paper has lateral dimensions which are large and one dimension which is extremely small that's critical so even though it's typically done with paper as an art form in fact it can be done with anything it can be done with a thin sheet of aluminum it can be done with a thin sheet of steel if you have enough ability to apply forces so it's approximately two dimensional i see approximately two dimensional because even though mathematically you might be able to idealize it as a surface in fact its thickness is present it's very small but it's present and that's important because that essentially is responsible for the second aspect associated with the elements origami which i can break up into a few different parts because it makes it easy to follow so a sheet of paper or a piece of fabric such as the one that you wear is relatively easy to ben out of the plane but essentially impossible to stretch you can apply a large force on it but it doesn't essentially change its length and it's very hard to essentially try and change angles in it so you can't also shear it so that's an important part of approximately two dimensional you can take a piece of metal you can take a piece of paper indicates a plastic and they all have this fundamental characteristic that it's very hard to actually change the elements in length or angles in the piece but it's very easy to bend it out of the plane as i said it very easy to bend if i bend it smoothly and i let it go at this comes right back so it has no memory of a bend it very strongly than i can essentially get creases and we need these creases because without these creases we're not going to get shapes which persist and a critical part of origami is of course that once a essentially bend it substantially i form the sharp creases and those creases have a memory ok that crease individual having a memory is essentially the beginning of the origins of shape because i not going to be able to make shape with a thin sheet of paper unless after i did something to it it remembered what i did it doesn't have to have the it doesn't have a perfect memory but has to have some so this is essentially the fundamental reason why we need a thin sheet in the context of having the elements of origami and there is one bit of terminology which if you open in any origami book you will find very early on which is what are the fundamentals  and the fundamental units of origami are basically mountains and valleys and in fact you can really think of them as the same because every mountain becomes a valley and every valley becomes a mountain when you flip it around and if you look at the structure i'm sorry that this picture is faded a little bit if you look at the structure of a pattern for folding what you will see primarily first thing is the lines along which you fold a thread could correspond for example to mountains and blue corresponds to valleys and then in addition somebody has to also tell you the order that's of course not there but those are the two things that you see when you open up a book and with those two in principle you should be able to build complex origami is and i again want to tell you a little bit about the etymology because it's kind of fun the word complex in the english language comes from the latin root com which means together and click area which means false it's interesting because we use the word complex all the time but it has a very together so it is important that it's been done sequin shelley and therefore the order in space in the order in time matters otherwise you get frustrated physically and intellectually ok you do this like a person  alright i see some people nodding actually so i ask the question and i'm going to first show you an answer in fact the code for the very common you just haven't perhaps paid attention to them ok the first and very common cu essentially forming a kind of a wing that's example number one very very common for example number two these are very pedestrian examples this one i like particularly if you've ever been to essentially  mcdonalds and bought a box of fries you see that it starts out the box starts out flat but then just before the person puts the flap fries inside he or she opens up the box and when they open up the box i'm sorry i don't have a the flattened version of it you can see now that the bottom is curved in one direction and the sides are curved in the opposite direction and that gives it now volume and of course that volume allows you to put fries in it now of course curveballs don't exhibited at the moment new york so this is a very beautiful example of how you can take a piece of paper and fold it with a lot of effort along curves and outcomes something really beautiful but it also has these extraordinarily wonderful everyday origins so sufficiently and it's hard to stretch and hard to bend so i want to make that a little bit mathematical and quantitative i apologize for that but i simply want to say that every piece of origami that you essentially have for get these equations  every piece of origami that you have can be thought of as constructed of little pieces of flat paper linked up to cones and those cones are linked up to cylinders we just basically think of any piece of origami and it will have these three elements so just like if you go to a tailor or fancy could turn here and they'll tell you that you have to basically have these kinds of cuts in those kinds of stitches in order to essentially drape the body every piece of origami has as elements pieces of cones of pieces of cylinders and pieces of flats particular so the ingredients are very simple but the way they come together is quite remarkable all right so i want to now tell you about a wonderful little increase but of course these lines are not all apart from each other because of they were all apart from each other all i'm going to get is a set of mountains and valleys these lines then meet and when they meet now a question if you can ask is what happens at the locations where these meet and in particular you can ask are there any interesting criteria associated with taking a piece of paper which started out flat and then i folded in this case i just folded in a very simple form it's also a flat ok so are there any criteria which distinguish when and if you can do this i'll give you an example over here with this piece of paper i have another example which i'll come back to in a minute which is this  bag it starts out flat i open it initially this way and then i open it this way and then eventually it is capable of storing whatever you need to store in it and these four walls are allow you to essentially go from one set of flat conditions to another set of flat conditions and the question you can ask is is there anything special about that the crane started out flat featureless you folded it and it had the shape of now a bird  also flat so what happens when creases meet is a natural question that you can ask them in an architectural setting if i want to cover a roof i want to know is it possible to essentially open it so that it becomes flat and then is it possible to close or do something else which also perhaps might keep it flat ok so in the crane for example this is the formed crane this is the set of folds which essentially gives you the crane and one can ask is there something specific about flat fo ok so this is now two little results which i'm going to tell you i'm going to show you basically demonstrating the questions are there just so that you know that one can make th the concept is very simple so i have here so this was a flat sheet of paper initially i folded it and when i folded it i got one two three mountain folds and one valley fold and of course it's exactly the same for flip it add up all the angles i must have all the angles add up to the angles associated moving around the circle which means one hundred eighty degrees at three hundred sixty degrees so if i add up all the angles they must add up to three hundred fifty degrees so that's a simple little theorem but there is another much more non trivial theorem and that says the following that says that the part of this fault which is visible over here can be obtained by going forward in along the back and then backwards or it's also it's the difference between this angle the angle of the back and what is visible over here but it's also the difference between that angle and this one because we really see that ok so what is visible is essentially the difference of two angles on one side are also two angles on the opposite side or equivalently i can when i go back and forth so i go along one fold then i come back along the second fall and i go along the third one and then i come back at the end of essentially going around the hole ok and when i write that out what happens is that i get that the alternate angles must add up and the alternate angles must add up to one hundred eighty degrees this is an important condition because these two conditions essentially define for me when i can take a sheet of paper folded up and then end up with a folded structure which might you might appreciate but now i was actually quantify this mathematically or i say i thought if it is we've known about this for a few tens of years ok so then one can ask the following question if i start with a sheet of paper which is flat and i end up with a complicated origami structure sculpture which is also flat can you fold destruction for example the crane from a flat sheet moving through a sequence of moves so that you end up with the crane and never have to stretch the paper anywhere because this condition that i gave you is a condition which is true at every one of the vertices so it's true or that vertex at that vertex so on but it doesn't say how they all talk to each other which is exactly what you need to understand if you want to figure out how to take this flat sheet of paper and convert it to something like an origami fold mathematically that's saying that if i satisfy this condition at that's local flat full ability does it imply a global flood full of means or can i go from here to here without ever stretching the paper ok so this is this was for a long time open and why is this interesting from the point of view of science or even application it's interesting because i want to construct complicated shapes i need to know how to start from a flat sheet and end up with this complex shape and if i can't do that there are some obstruction this is this is important to know in fact for the flat for the paper bag it turns out that you can't go from something which is open to something which is closed without actually stretching it  so i this is not possible the answer is no you can't do that and what that means is that the mathematics is very powerful in telling you what you can do but if you want to go ahead and try to do that you have to worry about the fact that the sheet of paper is actually made of a real material and real materials cannot really never be empty infinitely rigid and therefore they will stretch a little more shear a little so i have to no account for this level of realism in accounting for the fact the one of the most popular well studied origami examples is this mountain valley fold the set of mountain valley falls where i have as i said one two three mountains in one valley and now in your mind's eye i'd like you to imagine take that and just repeat it in this direction and repeated in that direction repeating that direction in for a number of times so i have a sequence mountain valley falls all of which look exactly the same as a way to pack solar sails so that you can essentially take it up into space and then deploy very easily because you had a large surface area in fact you also built a version of this which you can buy in tokyo as a subway map ok i'm going to show you how you can essentially deploy this you can open the entire map in one move and you can close the entire map in one move and i want to point out why this is interesting and also i should emphasize why it ought to surprise angle to each other then they will all open simultaneously and then they will all close family and sleep and it took man a few thousand years to realize after the invention of paper that what you should have done was to follow korea muda because now the whole thing opens and closes in one move so mathematically that means that you have just one degree of freedom rather than having to the power and if ok so so this is a beautiful structure it is even as i said starting to become used in very different in many different engineering applications but in fact it turns out that also can be seen spontaneously i showed you an image of this  it dry first dries on the surface first and so the surface forms a little skin and then the inside starts to essentially dry and as the inside dries it also shrinks so it puts the surface film into compression and when it puts the surface film into compression you can essentially see exactly this pattern forming but it's fully self organize or completely self organize in the sense that it's one of these look ma no hands kind of situations spontaneously you get this pattern and we saw this a few years ago and we got quite excited because this is essentially a simple way to create at least the structure and then i was looking around in the literature and i found that in  fact it's not just that you see it in these simple settings in the laboratory but you can actually also see it in spring when you look at leaves of not all trees but in this case i've shown you the leaf of a horn am tree you can also see this in beech leaves and if you look at the bud over here you can tell what the leaf is like inside but you can see that as the leaves deploys just like solar sail except now on a human scale you look at the leaf any seed has exactly the same structure as that just got a single made rib and then you have these periodic fours and what that allows essentially is that the entire leaf can deploy in one move so to speak because there is nobody to come and essentially unfold the leaf ok so you see this in leaves you also see this in fact an insect wings and i want to share with you a video from a research group that just published this actually a few months ago last year so this is from the rove beautiful and you see so you saw that the that the under which delicate wings are folded when it's ready to fly it opens the shields out deploy the wings and it flies away and soon you will see how it actually falls it ok all right so in fact origami has been around for forty years even though we've thought about it for forty years it's been around for millions of years probably more than fifty million years and it has come at different places it has appeared at different places it has appeared in plants it has appeared in animals i'll show you a third place where it appeared in animals in a minute why and the answer is provided by some mathematics which i'm not going to be able to gold into but i showed you this image it turns out that this structure this doubly periodic folded structure is the natural solution that you get if i take a thin sheet and i squeeze it in two  ections and when i squeeze it in two directions this is attached to substrate that's the that's the fundamental most stable solution that you century get so this is a demonstration of exactly that so the mathematics shows that this is genetic and in fact you see this as a consequence not surprisingly in very different settings in artificial settings and in biological settings as well ok so i told you about a bit of a digression mathematically now i want to sensibly digress to tell you about some biology and in fact in fact tell you about how your gut and mine also have origami like structures so these were experiments that we did collaboration with actually a group at harvard in the medical school a couple of years ago and in cliff tape in this group they basically study how embryos essentially develop various aspects of that in particular  that gut has become rather long and all convoluted ok and you can see progression as a function of time that's not what i really want to tell you about except to point out an important principle which allows us to explain allows us to explain how this calling occurs and it occurred because two different tissues which were connected to each other were growing but they were not growing at the same rate so the gut is a tube the tube is attached to the body by a piece of tissue the gut grows at  one rate the two goes at a different rate of the tube is growing faster in the gut than the gutters and going to be put into some sort of compression and when you put in the compression this is the pattern that you get and to show that that is the case if you take the gut and you cut away the tissue in the gut straightens out ok so that's basically the message that i'd like you to take away from this because it's going to come up in a couple of minutes different at different locations and you can even build physical models of this to show that with a piece of rubber  q you can actually create something which looks exactly like the biological system so you understood something you can essentially make this mathematical i won't go into that but why am i showing you this because i'm showing you this as a part of how the outside looks like and i want to emphasize the outside looks like that because this object grows relative to that so if i'm connecting two objects one growing more than the other than the one that's going faster it's going to be squeezing the one which is growing slower is going to be pull so a consequence of that is some sort of compromise and this is a compromise what happens inside the gut so what happens inside the guts of anaamika cut this is the gut and if i make a cut along any cross section then is initially very smooth after a while you find that muscles start to develop the muscles which essentially allow food to propagate along and these muscles are regionally oriented a circumferential direction so you can see actually squeeze this way and when that happens the inside of the gut which is originally smooth starts to become bridge like ok so i'd like you to imagine that i have the cylinder and it's trying to grow but the outside prevents it from growing and therefore you have all this excess material and it's going to start to essentially form forwards so very similar to what i showed you what happens outside but now inside the guts so you form ridges you little longer and in addition to forming muscles this way you also form muscles that way and when you have these muscles forming that way these ridges we think of now long lines and these ridges are squeezed in the other direction and when they get squeezed where do you see you see this ok you see this because initially so i'm going to have to ask you to use your imagination initially each one of the zig zags was straight and it was all rolled around and i was the god and then you squeezed it in one direction or equivalently the inside of the gut grew relative to the outside and when that happened these straight ridges started to become zigzag like and of course we now see origami inside your god ok so this happens in in the chick it happens in humans so here is a larger view and you see it's exactly the same structure and again why does it appear it appears because the inside the inner layer of the gut is essentially growing faster than the outer layer and it's growing faster in the circumferential direction it's also growing faster in the exactly like this thin sheet on a piece of the surface of the piece of jell o which is being squeezed so exactly the same thing happening both ways ok and then little later what happens is that the zigzags form cylinders and computational model of how this is happening so the whole gut is growing the red layer corresponds to the inner part of the gut and now you see exactly what i said before you start to see these exactly initially form these  ges like this if i make a cut you'll see a lot of ridges and as it grows and now it's starting to be constrained in the direction along and now it's actually see all right ok so i've shown you know examples in very different biological systems where you see or examine occurring and origami occurs because you have thin sheets which are growing and these thin sheets which are growing are constrained they are prevented from growing in different directions and the consequence of being prevent a consequence of that constraint is that you start to fold and that's exactly what the basic principles of origami are except that in classical origami the way we think about it we do the folding and over here it happens spontaneously it happens on its own ok all right so i've shown you a little bit about what happens when you think about origami in these natural systems and as i said over the last few years there's been  a fever pitch activity from the point of technology you may have heard about metamaterials metamaterials materials which have properties which you might not have essentially initially thought w were possible for example very famous class a mathematician else in the news associated with trying to make cloaks which are invisible ok so you can basically now take science fiction ideas from harry potter and elsewhere and create cloaks which are invisible and people have made some progress in making invisibility cloak but they work only in certain wavelengths associated with certain levels of light similarly you can ask whether not you can cloak an object from the point of view of sound so you don't have sound which propagates through or you can ask whether you can make properties or they can make material which are very bizarre property from the point of view of mechanical properties for example if i pull on an object typically five pull on a piece of rubber i don't have one with me but for pulling a piece of rubber a cylinder of rubber it will pull it will stretch in the direction which i pull it and it will get squeezed the other directions because constant volume of a pull in one direction then i have extended this dimensions with my squeeze down in the other direction now i can ask is it possible to construct materials where they pull in one direction and actually structures in the other direction and in fact origami is exactly such a material in fact i already showed it to you but  not you should not think of the paper as a material if you think of the folded piece of paper as the material because i pull on it and rather than squeeze down it's essentially opening in the direction perpendicular  ok so that's an unusual property because most materials don't have that but just because of the geometry of the folds you actually see that in fact you have to do something fancy like origami to be able to demonstrate that you just take a piece of crumpled paper you take a crumpled paper the result of most of my calculations and if i pull on it expanse except that this is very disorderly and not very beautiful and perhaps not even functional ok so one can start these properties and are simply tell you what i've said before that this object the mutare for example is flat almost everywhere except for these creases it has only one degree of freedom if i pull the whole thing it opens and if i squeeze in it the whole thing crawled closes it has this property of shrinking sorry this is wrong and mistake it stretches laterally when stretched actually so it is a normal material would shrink laterally when i stretch it and the object this origami object when i stretch it in one direction it also stretches so that should be stretches rather than shrinks now it's also equally funny when i say  bend it i take a piece of paper and i can take a piece of paper and i can bend it in one direction or i can bend it in the other direction and that's about it i can't bend it in two directions simultaneously you try to close to close to the top it's partly like a sphere the neck has curvature in one direction and at any location the curvature in the other direction has the opposite sign ok so it's like a saddle and with piece of paper you can't that but as soon as i have folded the piece of paper into this origami structure i bend in one direction and it starts to create a saddle so now again you've got these interesting properties where the actual material does not have any of this once you put the folds in now you can essentially do funny things you can build a saddle for example ok so in the context of bending it it actually can do bizarre things again i'm not going into the details you can essentially draper saddle and and you can make this mathematical or x so that means that you can take these folded materials and create folded objects and create metamaterials which have properties which go beyond what the basic piece of paper actually originally had just by virtue of its german tree and what people have been doing is essentially using this as a basis for it and when they actually did the whole thing this flat object essentially folded up and walked away ok so people have started to build self folding origami days to robots you can not only situation where the folds are present and then you have little elements which can essentially shrink or swell as a function of temperature or if you put it in a solvent they absorb the solvent and do different things and you just let it be and when you let it be if you program it appropriately so that certain forms appear first and certain falls appear second this flat sheet of paper will fall itself into a boat ok this is also recent work coming out of harvard and mit from different groups and because i told you that origami is not limited to large scale what really matters is not the absolute scale what matters is relative scale i need something which is thin and i need something which is thin compared to the other two essentially a form of carbon which is one atom thick and it can be very large in the other two dimensions and so it's something which is one atom thick again the thickness is small compared to the other dimensions over here the other dimensions are of the order of tens of m you can essentially folded in once you can fold it and if you can actuated by putting molecules which will swell or shrink you can make graphene do wonderful things ok i have about five minutes yeah so i want to finish by now a few years ago we looked at pine cones inspired partly where we're examining but just out of curiosity how many of you know that pine cones can essentially close everybody knows that they will be open because you see them outside but a few people over here know they can close you just put them in water and they will close right now let me show you here's a pine cone you just throw it in the water and this is a time lapse image which essentially associated with doing it over many hours and it closes and then you take it out and it opens and you can do this as many times as you want ok the reason that this is possible is associated with the fact that at the base of every pine cone brac that's the name for these leaf length structures you have little fibers which swell when they centrally are in contact with water and then they shrink when they dry out and that's exactly what we've been showing you over here so ok so knowing that in fact and knowing that paper is the same is would just treated wood and so would can well and shrinking you know that you know that of your paper back when you leave them in the kitchen or in the bathroom for too long there is this wonderful architect christian grant who basically now created a printer where the printer essentially paints or prints the folds and also puts water and as this sheet of paper comes out it folds itself ok so here is an example where knowing what that paper was well and shrink and doing a little bit of very clever engineering now you can essentially get these things to fold up on their own so this is an example i'm now going backwards using the math instead of one and the science to understand nature turning that around and then coming back and saying we can create things which are useful ok i want to opposite angles must sum up to one hundred eighty degrees and so we have to respect those rules and now we want to ask and i think this object and create a tesselation so some sort of tiling of the plane and use that to program shape ask can i take a flat sheet of paper and make it anything ok in particular the way we wanted to do this is you give me a shape that you would like so you give me my head for example and i'm going to take this and we're t ake a few of these nodes colored in blue and i stick them to my head and then i have a bunch of red ones and the red ones are above my head because it's folded and i allow them to move and i ask how do i move those red ones so that i satisfy these rules and at the same time i also drape my head ok so i want to do what i just told you i want to play some on the surface that i want to approximate i want others to move i want to preserve these constraints and i want to solve a math can create spirals you can create a sphere is i told you that with this cheat you can do it but now you can because i've very where the forces are i can create saddles i can create a pill box i can create a candlestick i can create a flower ways and a student in our group leave my duty folded all of them and so on at the bottom you see i'm sorry you can't even see the falls but these are real paper models of each one of those so we designed these mathematically and then we basically  them ok what does that allow you to do it allows you for example to take a flat sheet of paper fold it let me play that again in one direction and then fold it in the other direction and i can just completely collapse it so i can do what i said we asked the question in general about paper bags and cranes and so on we can do this in general but for some specific cases in this case associate with cylindrical shapes so the recall shapes by definition vary in one direction not in the other you can do it so there are natural implications for architecture on the small and the large scale you can also mathematically than design structures which have memory this object has no memory because you can keep it anywhere in between and it's happy the object i'm going to show you next where i start with a flat sheet of paper and i can fold it up into the saddle and as i folded it costs me some energy because i have to stretch my facets a little bit and i'll play this again you will actually see that the energy goes up and then goes down so it's flat foldable over here which means all the conditions are satisfied there in there but in between after stretches ok lot of time or equivalently very small facets then i can make a very very good approximation of a covering for my head or yours but if i don't give you that and i don't and you don't have the patience and i can get a very coarse view so there is this trade off between accuracy and effort and i'm showing you this over here for what you would get when i had low accuracy but i can do it very fast and as i go up i've shown you what happens when i increase the accuracy with smaller and smaller facets and you can make this quantitative you can make this mathematical but i want to get get into the details of that all right sam i'm done so i want to basically finish i told you about these examples which started out in art in fact i deliberately kept this one for the last i think this is the first example at least that i know of in european traditions of origami appearing you may have to peer a little hard to see it but this is essentially a painting where bronzy me and on this the school of architecture is designing now interesting pieces of origami inspired textiles i told you about the mathematics and the science there are some very very beautiful ideas  are associated with asking questions like the following i can i take a flat sheet of paper and convert it into the shape of a donor you can easily converted to a cylinder no problem and it can do so  so here's my cylinder and i want to try and roll it up but you see i can do so it's impossible essentially starts to form creases so there's a very interesting mathematical question which asks can i give up the ability to do so smoothly but nevertheless manage to do it so i want to give up as little as i possibly can in terms of smoothness i want to have as much smoothness as possible can i do it and there's a very very interesting  of mathematical results associated with that switch which goes back actually to people like john nash from almost fifty years ago and very recently have been realized by mathematicians sean over here and  completely come tracked in compressed and then you put it into an artery for example and if it's able to swell in response to the blood and the plasma and so on it can swell and as it swells it will essentially create a stand ok so you go from that structure to that so there are these very interesting classes of of devices that people are building stop with my favorite example which is the crumpled sheet of paper ok because it is this tension between order and disorder which is the root of everything that i have told you perfect order is a flat sheet there's nothing in it complete disorder is a crumpled piece of paper equally there's very little in it is totally random it's this tension between order and disorder which gave rise and which gives rise in which we continue to do so our how and what we can do with recovery thank you very much this probably ten years ago and so i am very fond of it but it took me i don't know many hours i don't even remember but the question that you ask is a very very good and a very important one because i think there is a sense as to if i want to create something out of it in reproducible way how do i do it i've given you two or three different alternatives i've given you an alternative which is biological which is this it just happens you know i mean so you use not the biology but  use the physical and the mathematical basis for that that's one possibility what people typically do nowadays is play ok how do you create new designs you play how do you essentially emphasize we knew what to do we we basically did the mathematics and then we wrote a program we leave by duty the student work with me essentially did that wrote a program and then out not completely but partially and then leave i sat down and it's actually built so that was designed rather than empedocles and that's the advantage so you take the mathematics exactly thank you  a very good yeah so the reason i showed you these is precisely because they're all the possible simple combinations so this object is curved only in one direction or the curvature negative curvature and then these are mixtures so over here the center is like this the top and bottom like that over here the center is the same the top and bottom are like that and over here the top is like this and the bottom is like that or that ok so i basically given you now all the combinations so now you become a tailor ok you take these objects and you stitch them together and in principle you should be able to do a lot if you i'm not sure what you mean with this anything missing we have the elements now putting them together my suspicion is no there's nothing missing we can do everything a great age yeah basically that's so that's why i'm sure you know i showed you these because these are the basic building blocks and then you put them together to be able to create any shape that's the part if there is so i don't have a theorem but i do have an intuition based on all the calculations that we have so far that we can do almost i think anything few papers but there isn't anything right now which is out there which is actually synthesize them that's exactly what we showed over here i didn't have a chance to go through those passes exactly that question so how how approximate and then you also tell me how much effort you're willing to put in you're willing to put in effort you can approximate well you're not going to put in effort you can't so it's the usual tradeoff accuracy and  which embodies the physical limits of the sheet of paper being impossibly hard to stretch and impossibly hard to share and that leaves to be used to conditions as soon as you move away from that for a real yeah yeah we build them and you know we build a myth yeah we build them in because we say that we created out of facets and the condition at the locations where the facets meet at every vertex we must satisfy these conditions but we and so the so the number of facets is something that we prescribe and then the angle is something that we essentially calculate and the number is prescribe it based on the fact that how same laws of physics as simple materials and so what you're seeing over there is just a consequence of the fact that one part of the material in that case the inner part of the gut is growing relative to another part of the material the outside of the gut in the case of the wing we don't know i don't know enough about the morphogenesis of how that happens but it's almost certainly the case that when the wing is forming during development in the insect it's very likely that subject to the same laws and differential growth plus constraints give you those patterns again and again but there is nothing which is being i want to emphasize there is nothing which is being optimize maximize area to basically maximize your ability to absorb nutrients in the context of the wing you want to pack away the wing when you basically are foraging through a forest but you want to deploy it when you want to fly so it's very different conditions the packing constraint is generic so you want to have a lot of surface where you don't you want to give it a lot of volume so that aspect is very generic in fact all these folding patterns that you see not just in  origami situations but you look at proteins you look at your surface of your brain you look at as i said before we talked about it you look at the i've got all these i have a fundamental question associated with trying to maximize in some sense or the other surface but trying not to have a lot of volume so that aspect sort of comes in again and again but the functional consequence of having large areas relative to volumes is very different for the brain because this was