it’s a pleasure today have been key to Randy’s boner randy has worked on Australia’s in his hand experiment breaking horses in theory of topics about which college in geometry from packing membranes crystals because degree from Cal Tech as PhD performer is visiting studies pen and he’s going to talk today about Smith’s explanations well thank you thank you for inviting me and have a pleasant day so far I’m always allowed I understand that you’re allowed to change your title so i added to make it more interesting purified smectic my original abstract was considered to jargony so you saw the new abstract sounded fun and light and happy so let’s see what we do is their way to turn all the lights off I don’t have to press this button that says okay okay good so I’m going to tell you about connecting with the crystals and I’m going to tell us they were before I hope you will now or after the words but I want you to see this is what a spectacle Luke rissalah looks like a real experiment so you follow some recipe you you make some molecule and you share a slide and you’ll get in the microscope and you see this now look what you see you see ellipsis okay and those are ellipses and sometimes you see the ellipses and sometimes there are pieces of hyperbola sometimes or even pieces of parabola and these are perfect he’s a really nice ellipses in these points where the lights coming out that’s one of the faux side that’s a focus of the ellipse okay and the question is why you see that right and what does this have to do with liquid crystals with high museum displays well they’re all related and this is the typical texture you see it’s not organized so let me give you a little historical account a liquid crystals there you know 120 hundred fifty years old they were discovered by right uh sir oops oops come on the families okay they were discovered by right answer and characterized by lesbains which is why they have all these crazy names there’s liquid crystals are phases of matter their other phases are not fluids or not solids and there’s a language thing I know the pollo agree with me about I’m going to call the phase with a crystal and I called molecules liquid crystal also just like Colin’s okay and so here is a little bit crystal molecule it’s you think of it as a rock I think um everyone you can think of it as a rock to all right and what is that at first you get this phase where the rogers falls down any which way so now they are the Centers of massive random so it’s a fluid it’s not a crystal is no periodic order the directions are also random but blue crystal is neat because you can concentrate it or you can lower the temperature and it’ll start to line up the molecule centers are still random and still fluid and said now it’s a fluid that has a direction in it and because there’s a direction in it can change them in a tour if I have this whole room filled with this liquid crystal are all mine in this nematic phase I can twist this side and that side with twists with it and I’m interested slowly it’s not like how we know an automatic transmission works i could twist it very slowly and we twist over there and that’s not a case where water is the river flowed with water if i did this nothing wouldn’t happen over there fighting it slowly okay and you can have the other kind of pneumatic what you call these you call these plain evidence oh you can enter mattox made quite ever have the peanut M&MS where instead of having the mirage their discs those were what supports okay all right what are these like he’s like a demand so what happened is you cool the system more you concentrate it you get something that starts to be like a crystal and what you get is you get these phases called Smith dicks smectic comes from the greek word for soap snake

most and the greek word for soap it’s interesting Ivory soap means the smectic if you dissolve in water it actually does this so a smectic is something where the molecules are all raised in some fewer direction and they start making players and each layers are fluid I look down from the top the center’s are just random so their fluids from the point of view of two-dimensional layers but the layers are stacked neatly it’s like a very messily stacked deck of card rather messily staff pad and paper alright so here’s a weird thing if you push the paper on the top the paper on the body doesn’t move for the same reason they can’t transmit cheer but they can’t transmit sheer cross again if this rumor filled with smectic in the layers of pointing so that the normals were up and down in this room if I try to live this layer move all the way over here they can come up over there also for if you imagine these walls made a spectacle over like Samson push this wall on that all new with it or not okay and there were various barriers of aspectx they are not named in the order they were discovered so there’s gaps like they’re smectic there’s no slightly be there’s smectic acts exactly half an eye and L but not all the letters are there but I’m going to talk about these so what a little crystals look like when you look at them sorry show you the picture of the smectic this is what animatic looks like all right and these fine lines are where the word neva the matter came from they mean threads and this is under cross polarisers and I’ll explain it a little bit this is called ash weird texture which just means a like street right even max excuse the mess okay here is a picture of some DNA that makes a little crystal it makes a chiral of your crystal and it makes layers and you can see that it makes layers this is opalescent right so it reflects light in different directions right depending on you know the orientation of the stacks here is hysterical ooh crystal which is very similar to not exactly the same as a smectic it is something where we have some periodic order and they’re crossed polarizers and so you see stripes and I’ll get back to that here is another speck that this much more Mesa organized it again you get this periodicity by the way these pictures are painted with a microscope okay and this microscope is like the kind of you buy toys r us okay and these microscopes that means the pictures you know 100 microns baby right it’s the same wafer and that means with these stripes or maybe a micron part so you ask yourself what can molecules the molecules that make these look at crystals up are usually a few federal rings or something they make structures their molecules that are a few nanometers thick and they main structure that a micron periodicity where does that mean scale come from that’s a missionary I may be a series we have series at explain why it should be big but no boo theory of why they should be as because they are or how to predict them there’s another phase which I won’t talk about very much it’s called the blue phase the blue phase this thing you’re looking at I know you think it’s a crystal believe is a crystal now John your block okay so so this has facets like the crystal has facets it grows the wolf construction all that stuff but this is a liquid the thing that’s crystal it is not where the molecule sit but which way they point and then organized in three dimensions into a structure how they point in route so in that block the molecules know how to point and it’s a 3d modulation of the orientation not Lindsay right and can anyone tell me at what wavelength or what weight scale that modulation blue but you know is nothing so you gotta choose little bit right yeah bluish exactly but there are a few phases we move is right there’s less colors this is a smectic these are called focal comic fan like textures and you start to see things if you look carefully and use your imagination that look like those ellipses that I was talking about and we’ll explain that later so let’s go back to the sharing section the first thing that people saw this is animatic into two dimensions your eyes go immediately to the fact that there are these black lines and none of the black lines with their black

lines that come together at for full junctions and sometimes the twofold junctions and you ask yourself what do you so this is the setup it’s a very simple setup you have at least a leap or incandescent lights be equal to me only big souls so here’s this incandescent there’s this like and you have a polarizer you have another polarizer what you call the analyzer because it’s perpendicular and then you have these look at crystal molecules in between the molecules you do some chemistry you get them to lie in the plane of the polarizing the analyzer so you have two planes now what happens if light goes through there were no allu crystal there what do you see you see nothing right nobody gets to because they’re crossed polarizers but these molecules are not only long rods physically there long rods from the point of view of life they are birefringence may a different dielectric constants along the long axis and perpendicular to the long axis and so what happens light comes in if the molecules is pointy in some generic direction it breaks the light up into the ordinary way of the extraordinary wave which move at different phase velocities and the gate to the upside and a half cancel and so some way kids through they don’t cancel they don’t come out in the same polarization that they started in so when the molecules are lying in the plate light gets through yo I’m confused about the two dimensions and the 50 here what were these before you take some years eight there are these two planes defined by the place yes now what is the plane defined by the pneumatics in two dimensions it’s the same plane so they all lined the molecules to the chemistry of the surface of molecules all like tangent to the loose lengths exactly I’m a two points are parallel now if it should be that the molecule happens to point along the polarizer direction then there’s only an ordinary way that comes in the way it comes in you’re only getting one dielectric constant because one of those one of the light one of the motivations or cygwin and it goes through the other side of the polarization unchanged okay and when that happens you get black everything so the black lives are precisely the places where the molecules are either pointing up and down or left and right alright so the black lives in mathematics where to say this is the pre image of these two directions this tells you any place where the molecule is either along the polarizer the analyze you get black one so what I’m going to tell you is those places where four things come together come from these things that are topological defects vortices here’s a vortex this is the time you might imagine where the molecules go around at the center your honor which rating points just like you know the old coordinates okay which way does the thing point the center but there’s other kinds like this which looks more like an elongation Algeria nor everything sticking out and here’s something else these are all situations where if you look the molecule rotates around by 2 pi as you go around so what would you see under cross polarisers well I would see everywhere where the molecules are off the floor on direction I’d see a black line so here I would see a black line but now I also get a black line wherever they’re parallel to the analyzer so get a black line I don’t know coming this ready and so what happens you get for white lines each time why for because it’s a molecule rotates by 2 pi it has to because when you get back sabo his party well I haven’t gotten there yet but they can be done okay right if they have to rotate so that they’re back to where they were they rotate by 2 pi they’re certainly back to what they were so you get four brushes because it goes to the polarizer Direction twice and the animalyzer affection choice and of course take a guest you also see these ones where there’s two when there’s two is more exciting because that means that they only rotate it around my pie so

without knowing anything about the molecules with no electron diffraction or you know high-resolution scattering or whatever technology one I already know the molecules and the multiple pointing the other way are the same because here i can go what’s in the polarized direction and once in the analyzer direction i can only rotate around one pi naught 2 pi and when I rotate around 1 PI system still knows what to do the multiple are still pointing the same direction so without knowing anything about the molecule I now know that the molecules for this phase having something symmetry but the molecule pointing this way this way is the same all right to me that’s beautiful I look at a micron size picture between polarizers that were invented in the 1830s okay bye be on the same bi-rite biot-savart to people right feelings of our two different people right bo is the one who knew invented who invented this okay all the same different temperatures yes their mom mom both both but there are many molecules that go through all the phases they’re multiples you know I pick the ones so this is cool i look at the topology of these defects and i can tell you things about the geometry of the molecules you can have higher charges too we have lots of brushes here you have oops here you have a situation where you looks like you have six brushes it’s a lot because this brush goes back on this up but here’s where we really do have eight brushes and here’s one you really do have ten brushes and if you have eight brushes it means it goes around 4-5 times and if you have 10 brushes it means it goes around on 55 oh I’m sorry 12 brushes means go around six x times so you can have whitey and multiple whiny and the cool thing is all you need to do is count how many brushes come out of the defect you count the number of brushes that tells you the charge of the defect right Tech brushings come out and must have rotated around by a 55 all right 22 rushes come out of 11 lot so that’s neat because it means i don’t even have to resolve that everywhere except a good I just kind of all the brushes on the inside the charges add they have just like electric charges and in fact if you get really excited about it you can look in the book that we like to refer to as com events Keith book and you can then calculate the energetic interaction between these charges and you write out the logarithmic just like just like electric charges the plate isn’t that great right we’d not yet another problem to electrostatics right once you understand electrostatics you will profoundly understand physics it’s like wait okay so what about the spending so the spec that gives this layer system i should have said that so membranes makes metrics to to get these ice cream sandwiches of ice cream your oily ice cream and charge cookie right and then charge of the key is foggy in water and they make these nice stacks and for my purposes i’m going to describe the smectic by saying a smectic is a little set or if you like the electrostatic it word for level set equipotential right of sub function x when x is 0 that’s the first layer would fly as aids the next label 28 the following layer you notice that if I have a function Phi of X Y Z and I saw that the layers will never cross into each other this is a good way of representing multiple surfaces the other thing is is I can then say there’s a density wave which is proportional to cosine of that five so the layers sit at the peaks i can write down the normal to the surfaces right remember grad Phi is perpendicular feel the perpendicular potentials this is the unit normal and i can write down some energy this is the last time a member of a great line of energy all right because i’m interested in the ground states and all i care about is these two terms this term tells me that the layer are equally spaced the grandpa wants to be man to do one the other is the bed the energy it’s telling you how much energy costs the bed no letters the

ground state is a state where the layers are equally spaced and don’t bend deck cars but now of course what I’m going to do to you is I’m going to put them in a more complicated geometry where they can’t make a deck of cards and I say what happens how does this the deal with that nope these Smith dicks you notice there’s these layers but they also have this normal the normal points in some direction north is just like the dematic the pneumatic work we talked about that sub factor you notice that the normal doesn’t know which way it points these layers are up down the symmetric so you can ask yourself conspectus also have defects like the batteries remember the defects that you show you they can but there’s a theorem so they can make defects so here’s a situation where you have the layers going around in a circle you see at the center the normal to the layers doesn’t know which way to point so perhaps around my goodbyes looky here this guy here right here’s the situation with a normal you say ho as you go around from go give you on top and you go round and now you’re pointing down oops so the normal rotated by PI as I go around that point here’s one where nothing happens here’s one where rotates by minus PI naught plus live relative to the orientation are going right I have to pick an orientation to go around the circle but there’s a clear I’m regular mathematician the room and they probably know it within her means right the mirror basically says you can have a bless one defect in a half and then nothing and a minus a half and minus one and a minus three-halves you can keep going you get a minus two and minus 50 left but you can’t ever go class plus one you can’t have a question and this has to do with the thing that has to always have layers and you know this really bothered us we didn’t I mean we could read the theorem we can read each step in the theorem but isn’t really understood the theorem and then on top of it it really bothered us in a much more profound way it maybe you could end effects together I just showed you the nematic I could by Counting brushes they added together what charges we can look in the vents me at all and see then you can tell it’s how charges add charge this cat but we started playing with it we made it much simpler system well not just a 1 a 2 dimensional system so we have a two-dimensional smectic that lives in this XY plane ok I had that five field which I said equipotentials up fine I’m going to draw as a graph as a function over the excellence of surface this is not the smectic surface this service is just a made-up service living in three dimensions above the excellent ok equipotentials or level sets here just like all those times you intersect this surface with the set x equals zero and you get this line I can do it with x equals one thing at the next line so the way that I get the layers of the smectic is I just take level sets of that green surface and so what is it there’s not a word for this this is a tough map and topographical maps are spectacle fact it’s 101 very second is a topographical map every topographical map of the smectic topographical maps have a nice property that the lines that cross just like snap it’s the layers don’t have to be will be space I’m not worried about the ground stand only learning about the topology the layers don’t cross there’s something else about maps or topographical maps it’s cool you know that you’re at the top or the bottom by every which way you look is down or every which way will his hop and that means that there’s a contour line that circles around you then your topeak or a basin a contour lines circles around you those r plus 1 defects that’s a place for the director field rotates around 2 pi as you go around the same direction notice the peaks basins are the same charge now there’s a theorem in Morse theory earring it’s a theorem in calculus okay it’s called it’s called is a fear that says that between every two mountain peaks there’s a mountain pass

it’s called the mountain pasture and the mountain pass look at that that’s a place where the director rotates but it rotates the wrong way around it rotates x minus 2 pi if you go around clowder clockwise able to see the director rotates clockwise that’s the minus one defect so the minus one there’s always a mic this won’t be treaty plus ones so now imagine that I try and take two puzzling defects bring them together to make that plus two which I can do it in an attic well and they’re bringing together this run past is always with that there’s my unison deep I can’t never avoid the mountain pass in other words if you put a mountain on top of them out of them still just about right if you Los Angeles what they do if you want to be more space as they cut the tops off of two miles and make one week one toe but that’s just what not right you have to remove the past energy mountains okay where I’m from the Midwest where there’s no mountains they make sounds attractions true okay then pile it up and then they put dirt on top of it he was seen okay and and they can do they contain two little mountain so they can filled it with more stuff and then also remove the pass but then you’ve also we have one now so you can never put them out without amount you can do something weird some of you I don’t have gone to ask them some of you who like to Aspen the best part Aspen as a place called la paz what your pass is a place where you go hiking where there are three valleys going out from one single pass okay you go three with so there are three mountains usually if you have three mountains you have three passes between each air but there’s nothing to prevent you from having one single pass that goes into the three valleys that’s why you can have any negative charge money a three of passive the three dollars will be minus three halves but you can’t have any positive truck so you can’t put moms on Mother’s so that’s the difference alright and so is the problem let me tell you why it’s a problem usually when we talk about topological defects we say what do we do we have a sample it has defects we have some ground state manifold this is all the possible ground stakes when we go around some loop of the sample we go around some loop in the manifold Bradley’s parenthesis is you know complicated has handles if it has a handle it’s possible to get trapped on the handle and there’s no way to smoothly go from a situation where I’m caught on me on the handle of this Davis called a climb bagel and and the place where I’m knock knock I can’t contract’ this little each track on the handle and there’s this whole beautiful Theory called home Adobe theory that allows you to discuss how maps behave from closed circles here to closed paths here and you can classify topological defects by talking about maps from this group to that group is the group of loops well groups have this interesting property what’s the definition of the group John ok i’ll give you one thing has never identity that’s for you do that ok everything i said in verse you that’s it but what’s the most important thing about it yes you could take two elements and multiply them together are at that or whatever want to call it and get a new element of the group well if the topological defects are classified by max from one group to another I’ve got a problem because I can’t add two plus one defects together if I try to make two books once I don’t get something new right and that means that group theory fails the characterizes beef in the smack day so home a trophy theory which is the standard practice doesn’t work it’s okay this works thinking about tube surfaces works alright it’s just it’s not appropriate anymore use the standard tools you have to use this thing which is you know some kind of sloppy version of war spirit I guess but cymatics can have another kind of defect which I haven’t told you about yet they’re kind of defect that you learn about regular crystals they’re called dislocations right this location is at a

very fond of dislocations I’m sure there are buildings on this campus they’re like this so there’s like a pre-war building I mean one or two and a post-war building and they’re attached but I know they used to be more space for the pipes in the pre-war buildings and even taller back then right let me look look good fine and so if you ever have the empower people and the game for space and so you attach this pretty warm building to this post-war building and you get this extra floor and you have this problem you know you go you’re walking around on the first floor and take the elevator up nozomi few stories here just goes the ball right to save energy so you walk up two floors and you walk over here to the new building which is all neat and glassy looking and you go down two stories and you end up here you started here and you ended their your location messed up you’ve been dislocated all right you don’t know where you are right and throw the real things ok you can end this war is always called the mezzanine right they say one this is two three give an extra name right there is a whole building at Penn which is devoted just to connecting all their buildings together I guess all stairwells and overhears right and when you look at it if you guys wearing your back looks like one of those impossible things so dislocations happen and these are the kind of defects that you see in crystals they’re a different kind of defect they’re going to defect where I don’t know which way the director points the normal point their effect I don’t know what the value of fires so this is a place that violates the rules it’s a place where my rule is violated the layers do cross but there’s a way around this ok the way around it is instead of saying that that surface that limit of the XY plane that resurfaced lived on the real line if you live on a circle so I can go on a thousand feet and come back for us started all the stuff in between is different but a thousand feet is the same as your feet so i canno periodically its Fourier series so i’m going to show you what would happen i’m going to start with a nice helicoid this doesn’t work for my Mountain this one my mountain range we’d have a problem because if i took level sets way out here they would all crashing I don’t have layers in the other hand this would be a nice crowd saved this would be about what is the ground state in my sector just a plane at an angle and then when you take level sets and you get equally spaced layers so i’m going to make student ok so here’s the helicoid and i’m going to stretch it I stretched it in a funny way it’s still periodic in the Z direction so when you go from here to here that’s really the same place right goes on forever so I’m saying this and this and this and this and this mr. old sequence and you say why you telling you this here’s the helicoid the tilted helicoid watch as i take sections on little sets through software that does this ok you ready I keep going look do you see the dislocation two extra layers when you cut helicoid an angle you get dislocations right you can keep going to get the other geometry ok so helicoid fixes we can actually have dislocations and discipline asians all the one story by having mountain ranges on circles ok and there’s like a mathematical thing I think it’s called more Sonoma coffee or something which is probably we probably useful for us alarm okay to stay this morning but we have over one there are some extra things that once you start talking about the surface you realize something cool about the surface which is the normal to the surface the only thing that controls the spacing is the direction of the normal and if the normals at 45 degrees then when I take sections of it the layers are going to be equally spaced so I really care about is that the normal is always at 45 degrees to the z axis so any surface it has a normal with which is 45 everywhere 45 degrees to the to the z axis will happen equally spaced layers okay but now doing low dimensional geometry some

of you may know sorry it could be any number you I ok all right I’m going to work in do this we’re 45 degrees is one ok so here I have this I know this is like the tropic of the tropic of Boston 45 degrees right and here’s the cool thing there’s a theorem that says if you look at a surface and you watch it’s normal and you watch how it’s normal moves around on this unit sphere the area that normal sweeps out is the gaussian curvature of the surface but you notice that a surface that has a normal that only lives on this circle doesn’t sleep out in here stays on the line doesn’t fill in that gap this is on the are not married no Cassie curvature it turns out two dimensions is enough stuff to show you that if it’s no gaussian curvature it must be I symmetric to the point you can have this or you can have this for again this you can have a plane cones which also have no gaussian curvature except it’s a little point there or you take half a code and attach it to planes on the back yes I’m sorry what are you doing with this here to give I’m going to get the area that the normal sweeps out of the sphere and it doesn’t sweep a generic suits that a lot and you thought about the normal to a seer some other stream out that sphere he asked what does that model do on that sphere it touches every point on that red sphere as I move around the youth with the this other sphere in my head right it’s normal ends up pointing in every direction a master but here if the normals constrained and pointing along 23 degrees or 45 degrees then it doesn’t sleep on variable and then this is the definition of the Gaussian curve so these are the only services that can actually have that in and you take sections of them to get the no defect a half defect in the plus one defect if you try to make me minus one-half e vector minus five-halves defect and you wanted two layers to be equally spaced you could not do it’s impossible so these have very little compression it turns out that i just hope you the line there’s another surface which we call mr. surface because we read about in her paper right this is a piece of paper that you can twist around and you kind of make a cone out of it but there’s a hole down the middle you know if you eat that stuff that stuff with a small candy floss kidding right I can do come on the speaker cone thing with their long as a whole you can never twist it up really tight so these are like you know wrapping these maybe you do this at home you do a thought of paper tonight okay go on wrap and you get the structure and if you take sections of it you get a spoiler but the spiral is equally spaced these distances are equal space because this end of gaussian curvature he has a problem has a hole in the middle but if you know might have a hole in the middle these are also equally spaced surfaces and their related to something which I bet many of you own and you don’t know they’re related to things called envelopes and moves are very natural right and one of those inside these new is going very loose okay so here’s a circle I’m gonna do first is I’m going to draw 5c engines to the circle okay now what I’m going to do is I’m going to draw curves that are everywhere perpendicular to those tangents now this is really easy to do how easy is this to do that are you can you know really easy now he’s a dog because what you do is you take a dog and you put it on leash and you wrap it around the tree okay and then it done wraps itself always and the strata always a high extension right and when it does that that means it that this the line connecting the tree to where the dog is will be perpendicular to its path this is always a nice tension and as this unwind this will be the path the dog days and by construction you notice these are all equally spaced so here’s your another set of equally spaced layers that come from these in balloons now it turns out maybe you own in balloons all right many of you may be wearing some right now okay because it turns out gears and it

moves all right this is the most genius thing when I learnt read about this about these guys in 1810 you everything right you want to have key on two gears that don’t rub against each other you want them to touch exactly normal all right not only does this shape which is the involute shape of the teeth make sure that they touch wrongly and don’t shred on each other it also means the force that’s acting across this thing is always in the same direction so you can reinforce these two ears and you know exactly weekly the thrust infinity in fact circular saw blades are the shape also so that they always pushes in the same direction and here’s the best part even if it start this way they would end this way alright David Wright himself down until they stopped raining but they do back in the eighteen hundreds to make trains this way and if you go to Europe and movie called trains you’re going to put the cogwheels right and the cogwheels are all good movies so we’re almost done I’m going to go back to those conic sections so here’s a picture this is a picture taken with a camera phone okay and i’m not talking about eight neither of those motorola razr foot phones all right from the from the last century okay and so here are these public domains here are these ellipses a chain of ellipses and if you look carefully you see lines going through them which turn out to be hyperbolic going through the more in this case is degenerate it can be a change of circles with wives going straight up and down why do you see ellipses remember i showed you down the first slide here’s a more standard picture that you look in their book here’s a better picture that has these otherwise going through them these are my pergola these are ellipses why do you see that these have been seen forever 1910 ok ok that’s me ok so no but I actually looked at the paper ok maybe had pictures ok there’s like dog misses so why do you see a conic section at all and it turns out that the surface thing which I used to explain an apology is actually useful device for understanding the geometry also so I remember this is called pssc physics who learned from dssc physics nobody nobody oh ok right in the back of the back of the book had a picture this is the most terrifying thing a high school physics this is when we thought physics lab ok and you have this table with a glass tray there’s water there is two motors water making waves you know light shining down on paper the trees the pictures wait right here 16 year old water which is the hall under my direction and this is what you get you get these wait so this is my smectic concentric circles the mountains that make them are these cones now if it were late I know exactly what happened when it’s like you get interference it’s all very exciting but neither sickness these are real materials so instead they just crash stop okay so look what happens you have concentric circles they come together they hit on a line what was happening in my space up above in the space of above I have to come and when the two cones intersect they intersect of a hyperbola this is home to the same angles and what happens if one man is taller than the other so these started before leaves I still opiate centric circles they exactly crash into each other on a real-life program on a non-agenda hyperbola that’s coming from this I throw away the top naps I just need the bottom lines and here’s a the same client this yes these are the two but the top of it I throw up look it’s nothing about the mountain ranges so we realize something any other seems kind of dumb and this is why I set 45 degrees to be one it turns out that’s the same thing as in the speed of life to me because we realize that this picture of services was wrong say that these services lived in Euclidean space was wrong and we should do instead is we should write the equation for a cone in

this stupid way minus x squared plus x squared plus y squared equals 0 fine and when I read it this way I realize of course that this is the way like satisfies in hospi space so if instead of thinking of this as Euclidean I think of this is mycoskie space this is a null service a language service and in eighteen something or other this Dutch guys are rinse derive a full set of transformations they were just they were they were derived specifically to preserve white cones in fact to preserve no services and so that means that if you give me one mole surface i can give you another null surface which is also an equally spaced becca right because every l surface could mean equally spaced smectic it has the speed of might as well under sun a 45-degree angle of course what happens when you lose the cone when you get becomes boring but it’s not going to be usually comes so here are two columns okay how would you describe this to a student taking special tip music ah I have these two points here and these two points are space like separated and their time interval the delta T is doing there at the same time space like separated and now you do a boost and there’s still space etc but now because you’re a moving frame there are different times and so that means that one’s taller than the other so it means that this and this are the same structure now you’re going to say to me when does this ever come up when are you ever studying smectic so while we go to near light-speed I’m not I’m never doing that but I’m interested in calculating energetics so is if you need for me that I can use this simple coordinate system to study something this is just the winner transformation so I can make a linear transformation now study among hop-up unit system better yet I can study for instance the volcano how we describe this to a special relativity student by the way i was on the seasons defense as an aside all other seats of defense i discovered there are two kinds of observers it’s what’s relativity there there are the observers who understands anyhow here are two columns they are the two the two events are time like separated their time light they’re exactly the same spatial point I do a boost there’s still time I’ve separated and now there’s a loose that’s what he looks comes from it comes from the intersection of combs that I have to tell you that the guys Radel who first saw that was a genius he knew when he saw loops as a hyperbola that it must be because with intersections of equally spaced surfaces so he could do suspected from that geometry you see under a rainbow microscope alright not from seeing lawyers it turns out so these are related by a boost those our way by this and since my fake is not really special relativity I give you things that mess up on cassette causality and there’s another transformation called the special conformal translation turns out all of these are the same such a mess cause oddity Isis distance has been look who’s that what do you mean it precisely lat phone yes what name is Fisher doesn’t it change t minus t ken things become would not be connected ok but this bike okay all right well I stand corrected it does something but we don’t do it right pointy special relativity why don’t we do it because of a system at home with the distance okay thank you their salts all right here’s the transformation all these structures companies so I can start with this very simple geometry and then evaluate the energies or anything I want a lot of these things if I want to study fluctuations around this structure it can be linear information backwards now what happens if you want to go to three dimensions because that’s the real dimension ring well one thing you can do is you can take those things that you can rotate them around the line of symmetry so that you have now instead of two circles intersecting and I purple to sewers intersect the honor of a boy or you could have two sets of concentric spheres intercepting on an ellipsoid but there’s something better that happens in three dimensions because this is terrible because now I have a whole wall a whole surface where the energy is bad when there’s a place where the molecules

don’t know which way the point by the way that’s why you can see them those ellipses you saw you saw them without the aid of crossbow Rogers because it’s a place where the violent your constant jumps because the molecule direction jumps suddenly and that’s what scatters light that’s the focal and that’s the color the colorful comic dub games but you can do something else you can actually take a structure and you can show using special relativity that if you take an ellipse and hyperbola and the ocean hyperbole ellipse goes through the focus of thy her dog I her moves through the focus of the ellipse and they’re perpendicular plate and that’s all represented here in this nice you know special relativity way you can show that these layers that that I’m drawing these are not really these are the actual layers in three-dimensional space are level sets of a four-dimensional I prefer not to post which i’m not going to drop as i can but it turns out of course this is just the Lorenz transformation of this this would completely understand consent your tour I I can do this with a donut right you take a donut you keep dipping in chocolate eventually the chocolate gets stuck in the middle and you get this other cusp which is that line and we do the Lorentz transformation the line of the hyperbola on the circle become the lips so that’s what you see and this is called a focal conic domain so here’s a picture of one from a real experiment right 20 microns it’s big and here’s Mathematica version of it okay and what’s it good for it turns out that you can take this focal comic domain and you can put in a coma and watford l understood a long time ago was that you can take concentric spheres and you can take a wedge a conical wedge other considers and shove this thing in and when you shove it in the layers here max lawyers here then candid they’re parallel to each other so there’s no discontinuity in the direction of the molecules so it’s smooth you don’t see anything all you see is that there’s a region where now something’s different and you can then build spheres around it and you can take as many skiers as you want pluck out cold coat and shove back in these focal chronic domains and all match very slowly and nicely here’s a picture book right so here’s a sphere and the sphere it turns out the services like to have negative gaussian curvature and send positive now see curvature but you get this fear that studded with all these and those are those are the things coming home you can actually take a bunch of them together and build a grain boundary on in a second you don’t have to just have a wall but get a crystal you can actually glue them together these are ones that are boosted so that you have these two asymptotic directions which is I hyperbole and you can actually build layers that way symmetric layers so you can have a brain matter with layers like this here and look like that there and they are joined by these local time now you might say to me well what goes in between these holes smaller fumble and beat you know that even smaller ones you get an apple team but we did something else here’s the life we took a single sphere and we put it in a spectacle beta meniscus and now you have a situation where the top surface wants to heaven will your parents because my bottom surface ones that have perpendicular called hybrid anger and so the layers have to go from being with this on the bottom two deadening up on the top but they have to bed different amounts as you go along the surface of the services and what you end up getting is you end up getting a completely self assembled set of focal conic domains that are bigger in the middle because they have to have a morir centricity and they get smaller and smaller as you go out and ventually relation so you can’t rebuild this right and you can look at it across polarizers you can see where those points are so we’re going to those British kind of lines above lower level figure that’s just sitting there you got like it as a hot liquid up against the spheres is probably some microscope this is this is no sorry this is a high-profile probably buy some kind of probably so my student dan going came up

with this very brilliant way of constructing arbitrarily complex surfaces so you see here the concentric spheres you cut out a wedge you replace it with these pieces of focal common domain so they said oh but I can then take that and make another wedge and fill it back in with spears and I can keep doing that I could make again again i can actually pattern any arbitrary surface shape and doing that he actually came up with a very nice model of what the structure was so that we could control it and so here we did something we use them to make lenses so this is worth it Francesca Sarah wed see the letter P it’s for penny ok letter P we get the letter P this where Jamie’s we’re changing where we’re focusing so when we focus down low the ones that are closer to being small they’re closer to being the flat surface we get to be here it’s the same under feet there’s only 100 b then as we focus out and we go to higher heights we get the peas focused around here and then finally they come out here so we actually have a multifocal lens that self-assembles all right if you mess it up it reveals itself and if you heat it up and cool it back down it heals itself and you don’t have to do it on a flat surface you get an IT service you want we can transfer numberless capillary capillary c3 pocket even better because these focal konechno beings are elliptical they actually are sensitive to pulverize light so we put polarized smiley faces in and you can see that on the side willing and polarization this way so we have a picture to smiley face and then on the top up here we only get it with a polar illusions like that so we made a compound lens multifocal compound lens at the text polarization is it how about doesn’t know but it’s something you do okay we’re not love so these are the main people who work on the theory with me like I said Francesca Sarah led the last experiment and thank you so the whole discussion worth emphasizing the was ignoring the gradient entrance that’s right maybe some structures well so so there’s bad answer which is the experiments of Cena but the video answer is yes that’s true but it’s only going to happen in short distance scale and what will happen is you get closer right the relatives could we resolve that and psychics are our fortunate materials you can you freeze fracture so people should go out may nice and long expect things with local comic domains and we ought to do it we just put a bead and you get all these local Conor Coady go ahead do that and then we should look at the core to see how the pores are different right where the curvature gets very hot if you recall that’s a long ago at the grande institute of man you look hot spoke of smacking systems and also didn’t seem to follow all the rules and that was presumably also because the director field we’ve got a serious form community today for whatever to you or resolve a comic that means it’s an excellent question let’s see when i can show you when we took this picture this was a sample that it’s at for 20 years these are big these are quite big and see them in your eye alright if they could have formed in second no I did you have to wait a long time they course it very slowly but these pictures that I showed you the very first picture that I showed you this one that’s that’s the more standard that you just see right and if when we look at when we look at the movies of our eyes we can film them tempio raising lowering the temperature at a forum a few seconds three seconds in the movie would you vote for eventual well what do you mean it’s a short time

it depends takes long yeah that’s the cool has pulled out from the nematic phase into the symmetric phase next order there’s a week for that and then symbols again back repeatedly into that one structure but I don’t know very much about the day that’s correct it’s not as exotic as a pitch drop experiment but it’s good because this picture that I showed you from Boulder that was 20 years old I took it but that’s not 10 years picture has a chain because the sample earth rotating other people who investigate the dynamics for example by if you drove a very fine wire for this picture and I’m thinking about like the food dynamic stuff in this it’s done here in something said can you get can you get dynamical formation of things that such as our sins so they do that a pneumatically crystals which have this orientation right and a lot of people who study active flowing things that he acted pneumatics I don’t know you yet study actress windex right I guess it would be very difficult to study because what would happen is the crystal or keep breaking up it was just melted into the matter and pull the wire but you could certainly ask what happens if you try to flow as method in the directions that it doesn’t support could you have a nice align thing to have it flow past them to begin cook okay you buddy Hagen is gonna flow past it and come back together and not and not be disturbed on the other side you design the I service’ since this is something that happens directly so so we have a we have is a very very closed off excellent hands Mohammed Garvey who you know I’ve no no I’m not sure he did it on purpose even you just want to know what would happened he put this drop of grain of sand right on the thing and he saw the shrimp he’s very careful very clever girl we do design a smiley face the smile that was that was overhearing Hopkins no no no it wasn’t me I want to make a smiley face so we should retire upstairs and have some