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all right lecture 17 was about folding  polygons into polyhedra and today we  will do it with real pieces of paper uh  but before we get there i want to talk  about some things we could make  um  there's uh in lecture i demonstrated  this uh perimeter halving technique  where you take any convex polygon  and you pick any point on the perimeter  then you measure out the perimeter  halfway on each side you get the  antipodal point y  and then just glue  locally you you glue everything around x  and around y  uh so i mean you just start we call the  zipping where you don't glue any extra  material in here you just start zipping  these guys together i guess  these circles are when you happen to hit  this vertex but then you just keep going  zip zip zip zip zip zip  zip zip zip zip zip zip  all right so that's one thing we could  make with convex polygons slightly more  general  is called pitaform  this is defined in the textbook  and the the idea is instead of a convex  polygon  you could take any convex body it could  have some corners or it could be smooth  so this is what we call a convex or what  i'm going to call today convex 2d  body  meaning it can be smooth in addition to  polygonal  uh and then you pick any point on the  boundary and you know you measure out  the perimeter halfway  x and y and you  uh  glue  in exactly the same way  okay now  this does it does not follow from  alexandra's theorem that this will make  a convex thing  but it follows from a slightly more  general form of alexandro's theorem  called alexandro pogrilov theorem  so pogo love was uh  or is was a student of alexandrov  uh and he  he edited some things that alexandrov  wrote but then also uh pogrelov proved  some wrote his own papers and proved  some stronger versions of alexandrov's  theorem that hold for smooth bodies not  just polygonal folding so alexandra's  theorem you recall if you have a convex  polyhedral metric homeomorphic to a  sphere  then  it's realized by a unique convex  polyhedron  so the pogrilov's extension is that if  you have  everything except the polyhedral part  so if you have a convex  metric  homeomorphic to a sphere  so topologically it's a sphere  uh but we omit the polyhedral part so  polyhedral was that you have finitely  many vertices now i can have  in some sense i have infinitely many  vertices i mean just just a little bit  of  i mean infinitesimal curvature at every  point here  then it's realized  by  a unique  convex  3d body  so  same concept of a convex  object  uh well  i should say uh  this is really the surface of  so a convex body i should probably  define that convex body is one where you  take any two points and you draw the  straight line and you stay within the  body for the entire straight line  so that's convex you take any two points  like this one and this one draw a line  between them you stay inside  uh  so here we have us i mean the body i  mean the body is the piece of paper  we're going up here uh  we have a 3d body uh but then we just  look at the surface of it the boundary  and that's going to be the surface we  make so  uh convexity i mean technically that  surface is not convex it's the interior  that's convex because you can draw lines  through it and you stay inside  okay so uh this is exactly the same  thing but instead of polyhedron here we  have body because we didn't have  finitely many vertices  and i'm not going to prove this theorem  but uh existence is basically the same  proof you just take a limit so you can  polygonalize  uh with you know lots of little edges  here add little  points  and take the limit as that approximation  gets very very close to the actual curve  at all times you have a polyhedron you  will it's easy to show you can you will  converge to a convex body  the uniqueness is the harder part i  believe i haven't read the proof but  uh  it was more challenging i think that  took  20 years or so to settle uniqueness  alexandra was like 1950 this theorem i  think is  1973 the final version  but uh the version without unique was  approved back in 1950  so uh that's something else we can do  instead of cutting out a polygon we can  cut out a nice smooth thing  or you can add some kinks and be smooth  elsewhere as long as it's convex  then  you're guaranteed that you're we're only  g we're only gluing together two convex  points could be flat or could be  strictly convex  which means we'll have at most 360  degrees of material at any point because  always going to at most 180s  and so we will have a convex metric  meaning always at most 360 of material  everywhere so then this theorem applies  so we'll get a unique thing  another thing we can make  is called a d form  and i have some  examples of d forms here so the idea  with the d form is you take two convex  2d bodies so here they have some  straight parts and some curved parts  of the same perimeter  i pick one point from one body and i  pick another point from the other body i  attach them there  and then i zip  so i'll be zip zip zip zip zip zip zip  zip zip zip  zip zip zip zip zip that's where  wherever my fingers were at the same  time those are points that get glued  together and this is what the convex  body you get again alexandre pogarilov  applies because we're still only gluing  two convex points together at any point  as long as these guys have matching  perimeter we'll be okay and so we will  get a unique convex 3d body  in this case we get this fun  fun kind of shape  so this is what i suggest we make  deforms tend to look a little bit cooler  uh somehow because they have two  polygons and the  one that's particularly easy to build is  two copies of the same shape  but then you don't want to glue  corresponding points like if i glued  this this point to that point i would  just get a flat doubly covered thing and  that's boring but you just pick some  other point and glue it to glue  non-matching points then it it acts like  you have two very different convex  bodies it's kind of cool i have some  images of  d forms here there's actually a whole  book on d forms by john sharp  they're invented by this guy tony wills  who's an artist at the top the quote is  there's no such thing as an ugly deform  so we are guaranteed success here  and also it says they surprise confused  can be addictive they're an intellectual  virus beware  so you've been warned  i think i have these are some this is  tony wills the inventor uh now  for the  from the artistic perspective you don't  have to start from convex polygons we're  going to start from convex convex bodies  because they're guaranteed to work if  you're careful to satisfy alexander  pogarilov  you can have a convex metric even though  you start from non-convex bodies  although in this case he's actually  getting non-convex shapes as well more  typical d-form is something like this  this is made from two ellipses  and he builds them out of metal  this is a simulation  by this guy  kenneth brack  who has this software called surface  evolver  and it's usually used for computing  minimal surfaces soap bubbles and things  like that but here it's computing what d  forms look like approximately  and so this is an example of taking two  ellipses  and  instead of if you glued matching points  they would be flat but then you change  the rotation so here they're rotated by  22 and a half degrees rotated by 45  degrees rotated by 90 degrees so you get  this nice kind of continuum of different  things that you could  zip together  so i propose we make some  and so you have your scissors and tape  what we  if you don't have two sheets of paper  get some from the front  and so everyone have scissors tape  everything  gonna have to share  a bit  um  so what we wanna do is make two  identical convex shapes  so you have some here two rectangles or  valid convex shapes you could cut not at  all  everyone should do something different  i'm just gonna make sure they're nicely  aligned hold on tight because you don't  want them to let go and then just cut  always turn left with your cutting  you can be smoother  just always turn left  and once you have your two shapes  uh you pull them apart make sure you do  not glue corresponding points pick some  other points  and  attach them together  with tape here's where it gets a little  bit challenging  smooth curves it's a little harder to  tape them together  so you probably have to use a bunch of  points of tape  so to speak  not literal points  and then just tick tick tick  before you get all the way you wanna  make sure you can  i can do something  all right here's my deform  mostly convex  kind of fun  so a funny thing you'll notice about  deforms if you if you made yours smooth  so if you don't have any kinks on the  outer boundary then the surface looks  smooth  it's kind of nice  um  the other thing is i mean there's this  noticeable seam  where you taped things  there you're not smooth obviously you've  got a crease along the seam  uh but also it looks like all that the  surface is doing  is kind of taking the convex hull the  envelope  of that seam  so there's two properties we observe one  is that it looks smooth  except at the seam and the other is that  the whole surface is the convex hall of  the 3d  seam  and  i want to prove both of those things  those are both true  facts  but first i need to define some things  but so we're going to prove that  d forms are smooth  and other good things this is a paper  that was originally a class project  in this class i think  in 2007  sounds about right submitted during  class it looks like or just after  semester  with greg price  d forms have no spurious creases  so what's a d form uh you take two  for us is going to be two convex shapes  of equal perimeter you glue two points  two corresponding points p and q  together and then you zip from there  we can actually talk about something  even more general which we'll call a  seam form  the seam form you can have multiple  convex shapes  a little harder to imagine but you can  join multiple points together maybe you  zip or you join all three of those  corners together as long as you have at  most 360 of material everywhere be fine  maybe do some zipping  i'm not going to try to figure out  exactly how this gluing pattern works  but you find a gluing pattern where you  don't violate anything so it looks like  if i do this one i better have these  guys also a little bit sharp  so that  they don't add up to too much angle  there  so these three points would also join  together this would be like three petals  in a  uh like a seed pod or something  you know like those leaves that wrap  around uh little fruits or something  i won't try to draw it but  i guess i will try to draw it  something like this  so that's that's a thing this could even  fold not exactly a sphere but something  like a sphere so seam forms you can have  any number of convex shapes and glue  them together however you want provided  at all points uh you satisfy alexander  of pogo reel off so you only have at  most 360 degrees of material and you are  topologically a sphere  then you have a seam form so most of the  theorems i'm going to talk about apply  to general seam forms but we're  interested in d forms and petaforms in  particular  so  let me tell you  some theorems about seam forms  theorem one  is that  seam form  equals  a convex hull  of  its seams  so the seams in general are these parts  where you did gluing  wherever you tape stuff the boundaries  of the convex polygons those are the  seams they map to some curves in 3d if  you take the convex hall claim as you  get the entire seam form  [Applause]  that's actually pretty easy to prove  then the second claim is there aren't  many creases  other than  the seams of course the seams are  usually going to be creased  and these creases are going to be  line segments  and not just any line segments but their  endpoints are pretty special  so the claim is that the  the creases have to be line segments  connecting  basically connecting strict vertices  what do i mean by strict vertex  something like  these three points  which come together provided the total  angle here is strictly less than 360. i  call that a strict vertex whereas  these points  connecting from here to here  uh those don't count as strict vertices  they are they do have curvature but only  infinitesimal curvature because i mean  this is essentially 180 degrees of  material just slightly less because it's  curved  so only where i have kinks  can i potentially have strictly less  than 180 and if i join them together to  be  strictly less than 360 that's a strict  vertex so potentially i have a seam  coming from here  uh also at the other end point so in  this picture  i read  i might have a seam line i might have a  crease like this i might have a crease  like this i might have a crease like  this  potentially  but in something like a deform  there are no strict vertices if if these  were smooth  so i'll assume here  these are smooth  you could call it a smooth d form if you  like then there's no  vertices no strict vertices  only these sort of barely  barely vertices  uh so there can be no creases  okay there's one other situation which  is creases could be tangent to seams  uh this should seem impossible because  the way i've defined things it is  impossible if your tangent to a seam  that would be like you're like this how  am i supposed to have a crease inside  that's tangent to the seam the answer is  you can't if  if  these regions are convex you can't have  them  this statement is actually about a more  general form of seam forms where you can  have a non-convex  piece so in general seam form you have a  bunch of flat pieces  and you somehow suture them together so  that you satisfy alexander progresov  and now you could have a tangent right i  can i could for example have a crease  that emanates from there or crease it  emanates from there but right now it has  nowhere to stop the only way for these  creases to actually exist is if there's  a a kink here  then  this could go from there to there so  this is starting at a vertex  and ending tangent you could of course  also start tangent and n tangent if  there's another bend  okay but claim is all creases look like  that which means  if you carefully design your thing or  vaguely carefully such as a smooth d  form then you're guaranteed there are no  creases which is what we observe so this  is like  justifying our intuition from these  examples  one other case  is the pitiform  so p deform does actually make vertices  at this point x is going to be a strict  vertex because it has less than 180 of  material so definitely strictly less  than 360. same with y  so p deform is  potentially going to have a seam from x  to y but that's it has it most sorry a  crease from x to y has at most one  crease  i think most examples we've made do have  such a crease  from x to y  okay  uh  let's prove these serums  or at least sketch the proofs  don't want to get too technical here  so  to prove the first part that the scene  form is convicts all of it seems uh it's  helpful to have a tool here which  is a theorem by minkowski we'll call it  minkowski's theorem although he has a  bunch  it relates convex things to convex hulls  it says any convex body is the convex  hall of its extreme points what are  extreme points  extreme points are points on the surface  where you can touch just that point with  a tangent plane  so if you think of polyhedra these are  vertices of the polyhedra but i want to  handle things that are smooth so they  may not might not really have any  vertices in general i have some convex  body if i can draw  imagine this is in two dimensions if i  can draw a tangent plane  it just touches at a single point then i  call that point extreme  let's look at an example here  here i believe  every seam point is going to be an  extreme point because i can put a  tangent plane it just touches at that  point  okay what am i distinguishing from well  for example this point this surface is  developable we know it's ruled so  there's a straight line here you can see  it in the shadow in the silhouette  if i try if i said is this point extreme  i try to put a tangent plane on the only  way to get a tension plane  is to include this entire line so it's  impossible to just hit this point  or any point along that line in fact  every interior point i claim you cannot  just hit that point you've got to hit an  entire line  whereas at the seam i can do it i can  angle between this  and this  there's a tangent plane because this is  curving that only hits at one point if i  had a straight segment then the  endpoints of the straight segment of the  seam would be extreme but the points in  the middle of the segment would not be  extreme  okay so that's the meaning of extreme  and we're going to use this  because i claim  that  these interior points can never be  extreme  so it could only be the  uh  it can only be the seam points that are  extreme and therefore we are the convex  hall of the seam points  because we are the convex hall of the  extreme points  okay how do we  argue that  so i  claim  an extreme point  can't  be locally flat  and at the same time be convex  because  convex in 3d  that's what we need if you had a point  and it just meets a tangent plane  at that single point that means locally  you're kind of going down  from that plane if you think of that  plane as vertical  so  here here so this is a situation okay i  want to prove this uh to prove it i need  to introduce another tool which is a  generally good thing for you guys to  know about so kind of an excuse to tell  you about the gauss sphere  which i don't think we actually cover in  lectures but  might come up again  uh gauss sphere is a simple idea for  every tangent plane let's just look at  think of this single point okay i want  to look at the it's called the tangent  space look at all the tangent planes  that touch this point because this  tangent plane this particular tension  plane i drew only touches that one point  it has some wiggle room right i can  i can pick any direction just wiggle  and and rotate the plane around this  point and it won't immediately hit the  surface i've got a little bit of time  before it hits the surface so there's uh  there's a two-dimensional kind of space  of tangent planes  that touch just this point  uh because there's at least one i've  gotta have some lego room so i wanna  draw that space and a clean way to draw  the space is for every such plane to  draw a normal vector  perpendicular to the plane  take the direction of that normal vector  and draw it on a sphere  say unit sphere so this is a sphere  this one looks pretty vertical so that  would correspond to the north pole of  the sphere  so that direction becomes a point on the  sphere think of it as this vector from  the center of the sphere to the north  pole  okay but we'll just draw it as a point  and because i've got a two-dimensional  space of maneuverability or rotatability  of this plane i'm going to get some  some region which maybe i should draw  like this  of the sphere those are all the possible  tangent  directions to  or say normal vectors of tangent planes  at that point  so this is called the gauss map where  you map all these normals to the gauss  sphere which got sphere just unit sphere  fun fact  the area  of  that thing  which is the map of all the tangent  normal directions  equals the curvature at that point  you may recall at some point curvature  is actually gaussian curvature so that's  why gauss is all over the place here uh  now because this i claim this is a  two-dimensional space this thing will  have positive area therefore this point  that vertex has positive curvature on  the surface  and yet it was supposed to be  in the middle  of one of these flat shapes that's  supposed to have zero curvature  contradiction  okay or stated more positively if i have  a point that is an extreme point it's  only touched by one of these planes i  have this flexibility therefore i get  area therefore it is not  uh  locally flat  so it must be a seam point  could be one like this where i'm barely  non-flat but i am non-flat there's kind  of infinitesimal curvature here or it  could be one of these points where i'm  very non-flat or i guess p and q x and y  here are other examples of  uh here i'm very not flat i've only got  180 degrees of curvature roughly  so it's got to be one of those if you're  going to be the extreme point therefore  all the extreme points are seam points  therefore  convex hall  of the extreme points is the convex hall  of the seams and that is your seam form  okay so that's part one  it's pretty easy  let me tell you about  part two part two builds on part one  that's one reason  why we care about it  so part two is that there are no  spurious creases  uh unless you have strict vertices then  you could connect those up  all right  so let's look at  a locally flat  crease point  actually at this point i should probably  mention  this paper i think of as a forerunner to  the  uh  how paper folds between creases paper  which is the  that hyperbolic paraboloids don't exist  you may recall there are theorems about  what creases look like what world  surfaces look like and so on between  creases  in that setting we prove things like  straight creases stay straight and and  all these good things here it's a little  bit different because we have kind of  two levels of creases there's the seam  which is special  and then we're imagining hypothetical  creases between the seam and you know  could you have a curved crease claim is  no  uh claim is all the creases in between  the seams have to be straight  uh and in fact they can't look like this  they have to be at corners somehow  so that's what we're going to prove but  a lot of the same techniques most of the  same definitions this paper was kind of  a warm-up i feel like for the  non-existence of hyperbolic paraboloids  although  they're  they use a lot of same tools they have  two of the same authors but uh they're  not  there's no theorem that really is shared  between the two they're not identical  okay so we're looking we're looking  inside a flat region we're imagining  let's look at a crease point it lies on  some crease  locally  and it's a folded crease so  uh  i'm gonna  wave my hands a little bit but imagine a  folded crease maybe it's a curve we  don't know  it's folded by some angle some non-zero  angle otherwise it's not a crease  so locally  it looks like two kinds of kind of  planes at least to the first order  they're going to be planes maybe  something like this  i'm going to draw the creases straight  because to the first order it is  straight  but it might be slightly curved  so we'll look we're looking at a point  here and my point is it's it's bent by  some angle  so i'm going to draw the two tangent  planes there's a tangent plane on the  right  whatever the surface is doing on the  right there's a tangent plane over there  whatever the surface is doing on the  left of the crease there's a tangent  plane there they are not the same plane  because we are creased whatever the  crease angle is that's the angle between  these two planes  okay  i want to look at this tangent space  again a little bit you can think of it  the tangent space at the least it has  this tangent plane for this point you  have at least this tangent plane and at  least this other tangent plane and so in  particular you have to have  the sweep between them  so that you get at least a  one-dimensional set there now this is  this can happen so we're not going to  get a contradiction we're not going to  suddenly discover there's a  two-dimensional area and get that it  wasn't flat but at least we've got a  one-dimensional arc  on the on the gaussian sphere here we've  got something like  this so far this is one extreme  tangent plane on the left this is  extreme  tension plane on the right  these two points correspond to those two  and then we've got the arc of  you can sweep the tangent plane around  and still stay outside the surface  okay  well that's interesting it's interesting  because all of those tangent planes  share  a line  right which is going to be  it's this line that i drew here i mean  you look at these two planes  there's a line that they share  and if you rotate the plane through that  line  you'll still pass through this point and  you won't hit the surface so in fact all  those tangent planes share this line i  think we actually just need these two so  if you don't see that don't worry there  are these two planes  there's a line there  they're tangent which means  uh that line is on or outside the  surface  these tangent planes don't hit the  surface i mean they touch the surface  barely but they don't go interior  okay  so what  well  i claim that in fact the surface  must  touch this line  claim the surface  at least locally  it's got to follow along  that uh  intersection line  of the left and right tangent planes  why  it comes down to this guessing sphere  thing  we know that the surface lies inside  these two planes there's this wedge here  and the surface must be below we also  know the surface comes and touches it at  this point  could it from here kind of dip down away  from this line  i claim no if it dipped down  then there'd be a third tangent plane  which would be you know the tangent  plane could follow and dip down as well  and then we've got not just a  one-dimensional arc but we're actually  going to get a whole triangle  in the gauss sphere  once you have a whole triangle that  means you weren't flat you have positive  curvature  so we're looking at a locally flat  crease point somewhere in the middle of  one of these polygons  so in fact you you can't afford to  dip down because then you'd have at  least a little bit of curvature there  because you have zero curvature you've  got to have you've got to stay straight  so that means that the surface locally  has to exist along the segment  in both directions  until something happens now we assumed  that our point was locally flat and a  crease point  uh so  this the surface must continue in this  direction until we reach a point that is  either not locally flat or not a crease  point  i claim  it's got to remain a crease point  because look you're following along this  intersection line of these two tangent  planes these remain tangent planes these  are not only tangent planes of this  point they're also tension planes at  this point and potentially any point  along this segment as long as the  surface is here they are tangent planes  and therefore you are creased here i  mean you can't be smooth if you're come  if you're butting up against these two  tangent planes  okay so crease must be preserved so the  only issue is locally flat  it could be at some point you become not  locally flat  and that indeed happens that's when you  hit  the seam you hit the seam then you're no  longer locally flat  so this proves the creases must be line  segments between two points  on the seam  there's a little bit more to prove which  is  uh  either hits a strict vertex or it's  tangent to seams  and this basically follows the same  argument that if you hit something other  than a strict vertex or a seam  you would have  positive curvature where you shouldn't  basically so i'll just i'll leave it at  that  i think that's enough of the proof get  the idea why  creases can't exist  any questions about that  all right um i have  a few more questions  um one question is  what exactly  uh can we go see more examples of  rolling belts  and  in some sense  this example  is an example of a rolling belt  so deforms you take two say ellipses  and the seam that connects them is a  rolling belt  and this is an illustration of the  rolling action  as you  change  which point glues to which point let's  say i fixed this point i could glue to  this one or to this one or to this one  or to this one and then i zip around  that's the rolling of a belt  okay more generally i can draw a picture  so more generally  a rolling belt from the viewpoint of a  gluing tree  is that you have basically it's like  it's like a  conveyor belt like on a  uh  at the airport or on a tank or something  you've got treads here  and this can roll so if i mark a point  let's say i mark this point and i roll  it around this way a little bit  this point maybe stays stationary  and i end up with something  like this  and what that means is whatever this guy  glued to over here is now over here so  we're changing the gluing  this guy is now getting glued to some  point like this  because it kind of rolls around so in  general you're rolling this but then  you're always just gluing across this  path for this to work  every angle out here must be less than  or equal to 180. like the material  that's out on the outside remember the  gluing tree has the outside of the  polygon in here the inside of the  polygons out here so all the materials  on the outside here  all of these things must be less than  equal to 180. if you can find a path in  your gluing tree where you've got less  than 180 material all around you then  you can do this rolling and no matter  how you roll and then just glue across  you will still have a valid gluing a  valid alexandrite gluing so that's what  rolling belts look like in general we  see them all over the place with  these deforms  the simple example i showed in classes  when you take a rectangle you glue  into a cylinder  and now up here i've got a rolling belt  because if you look at the sides here  i've got at most i've got actually  exactly 180 degrees of material all  around the belt  and so i could collapse it like this or  i could collapse it  let me tape it  a little hard to hold everything  [Applause]  so i could uh if you look at this belt i  could collapse it like this or i could  roll it a little bit and collapse it  like that or i could roll it a little  bit collapse it like that roll it a  little more collapse it like that all of  these are valid gluings because they're  always gluing 180 to 180  and that's what rolling belts  are  okay  here's  yeah  not worry about that  uh does the broken applet work now i  don't know why it wasn't working  probably some  conflict with suspension or just the job  installation was bad but here it is it's  it's freely available online you draw  your graph  it's a it's a little hard to just use  you can't just it's hard to draw an  exciting example here i'll draw a  tetrahedron because that's simple uh the  in this case it's abstract so all of  these edges are unit length it's kind of  like tree maker you separately specify  the lengths then you say compute and it  will find  uh in this case it's a tetrahedron  an equilateral regular tetrahedron it's  a little hard to see because it has got  a really either weird field of view or a  lot of perspective here  so but this is a regular tetrahedron you  can verify that by say making these  lengths  some of these lengths longer i'll make  three of them  five  so then  this is like a pyramid  of course we're just essentially  applying koshi's rigidity theorem here  and say oh this  uniquely assembles into this  spiky tetrahedron but if you have some  you could in principle draw in a cross  here  with the appropriate gluing it's a  little tricky to draw in because you  have to also triangulate  this be hard to do from scratch to  compute gluings but in theory especially  if you use software to compute the  gluings and compute shortest paths in a  glued polygon  which we'll be getting more to next  class uh next lecture is about  algorithms for finding gluings  then computing shortest paths then uh  you could use that as input to this  software which will then compute what  the 3d polyhedron is  so that's the  alexander implementation this is by i  think a student of bubenko and esmestiev  and  all right one quick question about sort  of out of time  about the non-convex case so we're doing  all this work about making convex  surfaces i said non-non-convex case is  trivial  in a sense the decision problem is  trivial it's not an easy theorem to  prove  and uh i already tipped it up  uh it follows from what's called the  bergo zalgaler theorem  so  and it says that if you have any  polyhedral metric  so that means you take a polygon you  glue it however you want it can't have  oh you could even have boundary take any  polyhedral metrics you take any polygon  glue it however you want could have  handles you know it could be a donut it  could be whatever  then it has an isometric polyhedral  realization in 3d  so there is a way so we're doing it  abstractly you know we're gluing stuff  together it's kind of topologically we  don't know what we're doing  but then there is a way to actually  embed it as a polyhedral surface in 3d  without stretching any of the lengths so  it's a folding of the piece of paper  could have creases in it will have  actually tons of creases in it  and it has  exactly the right it connects all the  things you wanted to connect together  which is kind of crazy  furthermore  so  it could be self-crossing  but if your metric your surface is  either orientable so it doesn't have any  cross caps or  mobius strips in it  or it has boundary then there's even a  way to do it without crossings  so that's basically all the cases we  care about uh you could glue this  together into a sphere or into a into a  disc or into a taurus or to to handle  taurus or whatever no matter what you do  there is a way to embed it in 3d as a  presumably non-convex  very creased  surface  this is a hard theorem to prove it uses  if you're familiar with nash's embedding  theorem about embedding riemann surfaces  it uses a bunch of techniques from there  called spiling perturbations which i  don't know and the one description i  read calls the resulting surfaces  strongly corrugated which i think means  tons of creases they are it is finitely  many creases that's the theorem there's  no bound on them as far as i know  there's no algorithm to compute this  thing i've never seen a picture of them  otherwise i'd show it to you  so lots of open problems in making this  real  but at least the decision problem is  kind of boring you do any gluing it will  make a non-convex thing  to find that thing is still pretty  interesting question though  that's it  you

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all right lecture 17 was about folding  polygons into polyhedra and today we  will do it with real pieces of paper uh  but before we get there i want to talk  about some things we could make  um  there's uh in lecture i demonstrated  this uh perimeter halving technique  where you take any convex polygon  and you pick any point on the perimeter  then you measure out the perimeter  halfway on each side you get the  antipodal point y  and then just glue  locally you you glue everything around x  and around y  uh so i mean you just start we call the  zipping where you don't glue any extra  material in here you just start zipping  these guys together i guess  these circles are when you happen to hit  this vertex but then you just keep going  zip zip zip zip zip zip  zip zip zip zip zip zip  all right so that's one thing we could  make with convex polygons slightly more  general  is called pitaform  this is defined in the textbook  and the the idea is instead of a convex  polygon  you could take any convex body it could  have some corners or it could be smooth  so this is what we call a convex or what  i'm going to call today convex 2d  body  meaning it can be smooth in addition to  polygonal  uh and then you pick any point on the  boundary and you know you measure out  the perimeter halfway  x and y and you  uh  glue  in exactly the same way  okay now  this does it does not follow from  alexandra's theorem that this will make  a convex thing  but it follows from a slightly more  general form of alexandro's theorem  called alexandro pogrilov theorem  so pogo love was uh  or is was a student of alexandrov  uh and he  he edited some things that alexandrov  wrote but then also uh pogrelov proved  some wrote his own papers and proved  some stronger versions of alexandrov's  theorem that hold for smooth bodies not  just polygonal folding so alexandra's  theorem you recall if you have a convex  polyhedral metric homeomorphic to a  sphere  then  it's realized by a unique convex  polyhedron  so the pogrilov's extension is that if  you have  everything except the polyhedral part  so if you have a convex  metric  homeomorphic to a sphere  so topologically it's a sphere  uh but we omit the polyhedral part so  polyhedral was that you have finitely  many vertices now i can have  in some sense i have infinitely many  vertices i mean just just a little bit  of  i mean infinitesimal curvature at every  point here  then it's realized  by  a unique  convex  3d body  so  same concept of a convex  object  uh well  i should say uh  this is really the surface of  so a convex body i should probably  define that convex body is one where you  take any two points and you draw the  straight line and you stay within the  body for the entire straight line  so that's convex you take any two points  like this one and this one draw a line  between them you stay inside  uh  so here we have us i mean the body i  mean the body is the piece of paper  we're going up here uh  we have a 3d body uh but then we just  look at the surface of it the boundary  and that's going to be the surface we  make so  uh convexity i mean technically that  surface is not convex it's the interior  that's convex because you can draw lines  through it and you stay inside  okay so uh this is exactly the same  thing but instead of polyhedron here we  have body because we didn't have  finitely many vertices  and i'm not going to prove this theorem  but uh existence is basically the same  proof you just take a limit so you can  polygonalize  uh with you know lots of little edges  here add little  points  and take the limit as that approximation  gets very very close to the actual curve  at all times you have a polyhedron you  will it's easy to show you can you will  converge to a convex body  the uniqueness is the harder part i  believe i haven't read the proof but  uh  it was more challenging i think that  took  20 years or so to settle uniqueness  alexandra was like 1950 this theorem i  think is  1973 the final version  but uh the version without unique was  approved back in 1950  so uh that's something else we can do  instead of cutting out a polygon we can  cut out a nice smooth thing  or you can add some kinks and be smooth  elsewhere as long as it's convex  then  you're guaranteed that you're we're only  g we're only gluing together two convex  points could be flat or could be  strictly convex  which means we'll have at most 360  degrees of material at any point because  always going to at most 180s  and so we will have a convex metric  meaning always at most 360 of material  everywhere so then this theorem applies  so we'll get a unique thing  another thing we can make  is called a d form  and i have some  examples of d forms here so the idea  with the d form is you take two convex  2d bodies so here they have some  straight parts and some curved parts  of the same perimeter  i pick one point from one body and i  pick another point from the other body i  attach them there  and then i zip  so i'll be zip zip zip zip zip zip zip  zip zip zip  zip zip zip zip zip that's where  wherever my fingers were at the same  time those are points that get glued  together and this is what the convex  body you get again alexandre pogarilov  applies because we're still only gluing  two convex points together at any point  as long as these guys have matching  perimeter we'll be okay and so we will  get a unique convex 3d body  in this case we get this fun  fun kind of shape  so this is what i suggest we make  deforms tend to look a little bit cooler  uh somehow because they have two  polygons and the  one that's particularly easy to build is  two copies of the same shape  but then you don't want to glue  corresponding points like if i glued  this this point to that point i would  just get a flat doubly covered thing and  that's boring but you just pick some  other point and glue it to glue  non-matching points then it it acts like  you have two very different convex  bodies it's kind of cool i have some  images of  d forms here there's actually a whole  book on d forms by john sharp  they're invented by this guy tony wills  who's an artist at the top the quote is  there's no such thing as an ugly deform  so we are guaranteed success here  and also it says they surprise confused  can be addictive they're an intellectual  virus beware  so you've been warned  i think i have these are some this is  tony wills the inventor uh now  for the  from the artistic perspective you don't  have to start from convex polygons we're  going to start from convex convex bodies  because they're guaranteed to work if  you're careful to satisfy alexander  pogarilov  you can have a convex metric even though  you start from non-convex bodies  although in this case he's actually  getting non-convex shapes as well more  typical d-form is something like this  this is made from two ellipses  and he builds them out of metal  this is a simulation  by this guy  kenneth brack  who has this software called surface  evolver  and it's usually used for computing  minimal surfaces soap bubbles and things  like that but here it's computing what d  forms look like approximately  and so this is an example of taking two  ellipses  and  instead of if you glued matching points  they would be flat but then you change  the rotation so here they're rotated by  22 and a half degrees rotated by 45  degrees rotated by 90 degrees so you get  this nice kind of continuum of different  things that you could  zip together  so i propose we make some  and so you have your scissors and tape  what we  if you don't have two sheets of paper  get some from the front  and so everyone have scissors tape  everything  gonna have to share  a bit  um  so what we wanna do is make two  identical convex shapes  so you have some here two rectangles or  valid convex shapes you could cut not at  all  everyone should do something different  i'm just gonna make sure they're nicely  aligned hold on tight because you don't  want them to let go and then just cut  always turn left with your cutting  you can be smoother  just always turn left  and once you have your two shapes  uh you pull them apart make sure you do  not glue corresponding points pick some  other points  and  attach them together  with tape here's where it gets a little  bit challenging  smooth curves it's a little harder to  tape them together  so you probably have to use a bunch of  points of tape  so to speak  not literal points  and then just tick tick tick  before you get all the way you wanna  make sure you can  i can do something  all right here's my deform  mostly convex  kind of fun  so a funny thing you'll notice about  deforms if you if you made yours smooth  so if you don't have any kinks on the  outer boundary then the surface looks  smooth  it's kind of nice  um  the other thing is i mean there's this  noticeable seam  where you taped things  there you're not smooth obviously you've  got a crease along the seam  uh but also it looks like all that the  surface is doing  is kind of taking the convex hull the  envelope  of that seam  so there's two properties we observe one  is that it looks smooth  except at the seam and the other is that  the whole surface is the convex hall of  the 3d  seam  and  i want to prove both of those things  those are both true  facts  but first i need to define some things  but so we're going to prove that  d forms are smooth  and other good things this is a paper  that was originally a class project  in this class i think  in 2007  sounds about right submitted during  class it looks like or just after  semester  with greg price  d forms have no spurious creases  so what's a d form uh you take two  for us is going to be two convex shapes  of equal perimeter you glue two points  two corresponding points p and q  together and then you zip from there  we can actually talk about something  even more general which we'll call a  seam form  the seam form you can have multiple  convex shapes  a little harder to imagine but you can  join multiple points together maybe you  zip or you join all three of those  corners together as long as you have at  most 360 of material everywhere be fine  maybe do some zipping  i'm not going to try to figure out  exactly how this gluing pattern works  but you find a gluing pattern where you  don't violate anything so it looks like  if i do this one i better have these  guys also a little bit sharp  so that  they don't add up to too much angle  there  so these three points would also join  together this would be like three petals  in a  uh like a seed pod or something  you know like those leaves that wrap  around uh little fruits or something  i won't try to draw it but  i guess i will try to draw it  something like this  so that's that's a thing this could even  fold not exactly a sphere but something  like a sphere so seam forms you can have  any number of convex shapes and glue  them together however you want provided  at all points uh you satisfy alexander  of pogo reel off so you only have at  most 360 degrees of material and you are  topologically a sphere  then you have a seam form so most of the  theorems i'm going to talk about apply  to general seam forms but we're  interested in d forms and petaforms in  particular  so  let me tell you  some theorems about seam forms  theorem one  is that  seam form  equals  a convex hull  of  its seams  so the seams in general are these parts  where you did gluing  wherever you tape stuff the boundaries  of the convex polygons those are the  seams they map to some curves in 3d if  you take the convex hall claim as you  get the entire seam form  [Applause]  that's actually pretty easy to prove  then the second claim is there aren't  many creases  other than  the seams of course the seams are  usually going to be creased  and these creases are going to be  line segments  and not just any line segments but their  endpoints are pretty special  so the claim is that the  the creases have to be line segments  connecting  basically connecting strict vertices  what do i mean by strict vertex  something like  these three points  which come together provided the total  angle here is strictly less than 360. i  call that a strict vertex whereas  these points  connecting from here to here  uh those don't count as strict vertices  they are they do have curvature but only  infinitesimal curvature because i mean  this is essentially 180 degrees of  material just slightly less because it's  curved  so only where i have kinks  can i potentially have strictly less  than 180 and if i join them together to  be  strictly less than 360 that's a strict  vertex so potentially i have a seam  coming from here  uh also at the other end point so in  this picture  i read  i might have a seam line i might have a  crease like this i might have a crease  like this i might have a crease like  this  potentially  but in something like a deform  there are no strict vertices if if these  were smooth  so i'll assume here  these are smooth  you could call it a smooth d form if you  like then there's no  vertices no strict vertices  only these sort of barely  barely vertices  uh so there can be no creases  okay there's one other situation which  is creases could be tangent to seams  uh this should seem impossible because  the way i've defined things it is  impossible if your tangent to a seam  that would be like you're like this how  am i supposed to have a crease inside  that's tangent to the seam the answer is  you can't if  if  these regions are convex you can't have  them  this statement is actually about a more  general form of seam forms where you can  have a non-convex  piece so in general seam form you have a  bunch of flat pieces  and you somehow suture them together so  that you satisfy alexander progresov  and now you could have a tangent right i  can i could for example have a crease  that emanates from there or crease it  emanates from there but right now it has  nowhere to stop the only way for these  creases to actually exist is if there's  a a kink here  then  this could go from there to there so  this is starting at a vertex  and ending tangent you could of course  also start tangent and n tangent if  there's another bend  okay but claim is all creases look like  that which means  if you carefully design your thing or  vaguely carefully such as a smooth d  form then you're guaranteed there are no  creases which is what we observe so this  is like  justifying our intuition from these  examples  one other case  is the pitiform  so p deform does actually make vertices  at this point x is going to be a strict  vertex because it has less than 180 of  material so definitely strictly less  than 360. same with y  so p deform is  potentially going to have a seam from x  to y but that's it has it most sorry a  crease from x to y has at most one  crease  i think most examples we've made do have  such a crease  from x to y  okay  uh  let's prove these serums  or at least sketch the proofs  don't want to get too technical here  so  to prove the first part that the scene  form is convicts all of it seems uh it's  helpful to have a tool here which  is a theorem by minkowski we'll call it  minkowski's theorem although he has a  bunch  it relates convex things to convex hulls  it says any convex body is the convex  hall of its extreme points what are  extreme points  extreme points are points on the surface  where you can touch just that point with  a tangent plane  so if you think of polyhedra these are  vertices of the polyhedra but i want to  handle things that are smooth so they  may not might not really have any  vertices in general i have some convex  body if i can draw  imagine this is in two dimensions if i  can draw a tangent plane  it just touches at a single point then i  call that point extreme  let's look at an example here  here i believe  every seam point is going to be an  extreme point because i can put a  tangent plane it just touches at that  point  okay what am i distinguishing from well  for example this point this surface is  developable we know it's ruled so  there's a straight line here you can see  it in the shadow in the silhouette  if i try if i said is this point extreme  i try to put a tangent plane on the only  way to get a tension plane  is to include this entire line so it's  impossible to just hit this point  or any point along that line in fact  every interior point i claim you cannot  just hit that point you've got to hit an  entire line  whereas at the seam i can do it i can  angle between this  and this  there's a tangent plane because this is  curving that only hits at one point if i  had a straight segment then the  endpoints of the straight segment of the  seam would be extreme but the points in  the middle of the segment would not be  extreme  okay so that's the meaning of extreme  and we're going to use this  because i claim  that  these interior points can never be  extreme  so it could only be the  uh  it can only be the seam points that are  extreme and therefore we are the convex  hall of the seam points  because we are the convex hall of the  extreme points  okay how do we  argue that  so i  claim  an extreme point  can't  be locally flat  and at the same time be convex  because  convex in 3d  that's what we need if you had a point  and it just meets a tangent plane  at that single point that means locally  you're kind of going down  from that plane if you think of that  plane as vertical  so  here here so this is a situation okay i  want to prove this uh to prove it i need  to introduce another tool which is a  generally good thing for you guys to  know about so kind of an excuse to tell  you about the gauss sphere  which i don't think we actually cover in  lectures but  might come up again  uh gauss sphere is a simple idea for  every tangent plane let's just look at  think of this single point okay i want  to look at the it's called the tangent  space look at all the tangent planes  that touch this point because this  tangent plane this particular tension  plane i drew only touches that one point  it has some wiggle room right i can  i can pick any direction just wiggle  and and rotate the plane around this  point and it won't immediately hit the  surface i've got a little bit of time  before it hits the surface so there's uh  there's a two-dimensional kind of space  of tangent planes  that touch just this point  uh because there's at least one i've  gotta have some lego room so i wanna  draw that space and a clean way to draw  the space is for every such plane to  draw a normal vector  perpendicular to the plane  take the direction of that normal vector  and draw it on a sphere  say unit sphere so this is a sphere  this one looks pretty vertical so that  would correspond to the north pole of  the sphere  so that direction becomes a point on the  sphere think of it as this vector from  the center of the sphere to the north  pole  okay but we'll just draw it as a point  and because i've got a two-dimensional  space of maneuverability or rotatability  of this plane i'm going to get some  some region which maybe i should draw  like this  of the sphere those are all the possible  tangent  directions to  or say normal vectors of tangent planes  at that point  so this is called the gauss map where  you map all these normals to the gauss  sphere which got sphere just unit sphere  fun fact  the area  of  that thing  which is the map of all the tangent  normal directions  equals the curvature at that point  you may recall at some point curvature  is actually gaussian curvature so that's  why gauss is all over the place here uh  now because this i claim this is a  two-dimensional space this thing will  have positive area therefore this point  that vertex has positive curvature on  the surface  and yet it was supposed to be  in the middle  of one of these flat shapes that's  supposed to have zero curvature  contradiction  okay or stated more positively if i have  a point that is an extreme point it's  only touched by one of these planes i  have this flexibility therefore i get  area therefore it is not  uh  locally flat  so it must be a seam point  could be one like this where i'm barely  non-flat but i am non-flat there's kind  of infinitesimal curvature here or it  could be one of these points where i'm  very non-flat or i guess p and q x and y  here are other examples of  uh here i'm very not flat i've only got  180 degrees of curvature roughly  so it's got to be one of those if you're  going to be the extreme point therefore  all the extreme points are seam points  therefore  convex hall  of the extreme points is the convex hall  of the seams and that is your seam form  okay so that's part one  it's pretty easy  let me tell you about  part two part two builds on part one  that's one reason  why we care about it  so part two is that there are no  spurious creases  uh unless you have strict vertices then  you could connect those up  all right  so let's look at  a locally flat  crease point  actually at this point i should probably  mention  this paper i think of as a forerunner to  the  uh  how paper folds between creases paper  which is the  that hyperbolic paraboloids don't exist  you may recall there are theorems about  what creases look like what world  surfaces look like and so on between  creases  in that setting we prove things like  straight creases stay straight and and  all these good things here it's a little  bit different because we have kind of  two levels of creases there's the seam  which is special  and then we're imagining hypothetical  creases between the seam and you know  could you have a curved crease claim is  no  uh claim is all the creases in between  the seams have to be straight  uh and in fact they can't look like this  they have to be at corners somehow  so that's what we're going to prove but  a lot of the same techniques most of the  same definitions this paper was kind of  a warm-up i feel like for the  non-existence of hyperbolic paraboloids  although  they're  they use a lot of same tools they have  two of the same authors but uh they're  not  there's no theorem that really is shared  between the two they're not identical  okay so we're looking we're looking  inside a flat region we're imagining  let's look at a crease point it lies on  some crease  locally  and it's a folded crease so  uh  i'm gonna  wave my hands a little bit but imagine a  folded crease maybe it's a curve we  don't know  it's folded by some angle some non-zero  angle otherwise it's not a crease  so locally  it looks like two kinds of kind of  planes at least to the first order  they're going to be planes maybe  something like this  i'm going to draw the creases straight  because to the first order it is  straight  but it might be slightly curved  so we'll look we're looking at a point  here and my point is it's it's bent by  some angle  so i'm going to draw the two tangent  planes there's a tangent plane on the  right  whatever the surface is doing on the  right there's a tangent plane over there  whatever the surface is doing on the  left of the crease there's a tangent  plane there they are not the same plane  because we are creased whatever the  crease angle is that's the angle between  these two planes  okay  i want to look at this tangent space  again a little bit you can think of it  the tangent space at the least it has  this tangent plane for this point you  have at least this tangent plane and at  least this other tangent plane and so in  particular you have to have  the sweep between them  so that you get at least a  one-dimensional set there now this is  this can happen so we're not going to  get a contradiction we're not going to  suddenly discover there's a  two-dimensional area and get that it  wasn't flat but at least we've got a  one-dimensional arc  on the on the gaussian sphere here we've  got something like  this so far this is one extreme  tangent plane on the left this is  extreme  tension plane on the right  these two points correspond to those two  and then we've got the arc of  you can sweep the tangent plane around  and still stay outside the surface  okay  well that's interesting it's interesting  because all of those tangent planes  share  a line  right which is going to be  it's this line that i drew here i mean  you look at these two planes  there's a line that they share  and if you rotate the plane through that  line  you'll still pass through this point and  you won't hit the surface so in fact all  those tangent planes share this line i  think we actually just need these two so  if you don't see that don't worry there  are these two planes  there's a line there  they're tangent which means  uh that line is on or outside the  surface  these tangent planes don't hit the  surface i mean they touch the surface  barely but they don't go interior  okay  so what  well  i claim that in fact the surface  must  touch this line  claim the surface  at least locally  it's got to follow along  that uh  intersection line  of the left and right tangent planes  why  it comes down to this guessing sphere  thing  we know that the surface lies inside  these two planes there's this wedge here  and the surface must be below we also  know the surface comes and touches it at  this point  could it from here kind of dip down away  from this line  i claim no if it dipped down  then there'd be a third tangent plane  which would be you know the tangent  plane could follow and dip down as well  and then we've got not just a  one-dimensional arc but we're actually  going to get a whole triangle  in the gauss sphere  once you have a whole triangle that  means you weren't flat you have positive  curvature  so we're looking at a locally flat  crease point somewhere in the middle of  one of these polygons  so in fact you you can't afford to  dip down because then you'd have at  least a little bit of curvature there  because you have zero curvature you've  got to have you've got to stay straight  so that means that the surface locally  has to exist along the segment  in both directions  until something happens now we assumed  that our point was locally flat and a  crease point  uh so  this the surface must continue in this  direction until we reach a point that is  either not locally flat or not a crease  point  i claim  it's got to remain a crease point  because look you're following along this  intersection line of these two tangent  planes these remain tangent planes these  are not only tangent planes of this  point they're also tension planes at  this point and potentially any point  along this segment as long as the  surface is here they are tangent planes  and therefore you are creased here i  mean you can't be smooth if you're come  if you're butting up against these two  tangent planes  okay so crease must be preserved so the  only issue is locally flat  it could be at some point you become not  locally flat  and that indeed happens that's when you  hit  the seam you hit the seam then you're no  longer locally flat  so this proves the creases must be line  segments between two points  on the seam  there's a little bit more to prove which  is  uh  either hits a strict vertex or it's  tangent to seams  and this basically follows the same  argument that if you hit something other  than a strict vertex or a seam  you would have  positive curvature where you shouldn't  basically so i'll just i'll leave it at  that  i think that's enough of the proof get  the idea why  creases can't exist  any questions about that  all right um i have  a few more questions  um one question is  what exactly  uh can we go see more examples of  rolling belts  and  in some sense  this example  is an example of a rolling belt  so deforms you take two say ellipses  and the seam that connects them is a  rolling belt  and this is an illustration of the  rolling action  as you  change  which point glues to which point let's  say i fixed this point i could glue to  this one or to this one or to this one  or to this one and then i zip around  that's the rolling of a belt  okay more generally i can draw a picture  so more generally  a rolling belt from the viewpoint of a  gluing tree  is that you have basically it's like  it's like a  conveyor belt like on a  uh  at the airport or on a tank or something  you've got treads here  and this can roll so if i mark a point  let's say i mark this point and i roll  it around this way a little bit  this point maybe stays stationary  and i end up with something  like this  and what that means is whatever this guy  glued to over here is now over here so  we're changing the gluing  this guy is now getting glued to some  point like this  because it kind of rolls around so in  general you're rolling this but then  you're always just gluing across this  path for this to work  every angle out here must be less than  or equal to 180. like the material  that's out on the outside remember the  gluing tree has the outside of the  polygon in here the inside of the  polygons out here so all the materials  on the outside here  all of these things must be less than  equal to 180. if you can find a path in  your gluing tree where you've got less  than 180 material all around you then  you can do this rolling and no matter  how you roll and then just glue across  you will still have a valid gluing a  valid alexandrite gluing so that's what  rolling belts look like in general we  see them all over the place with  these deforms  the simple example i showed in classes  when you take a rectangle you glue  into a cylinder  and now up here i've got a rolling belt  because if you look at the sides here  i've got at most i've got actually  exactly 180 degrees of material all  around the belt  and so i could collapse it like this or  i could collapse it  let me tape it  a little hard to hold everything  [Applause]  so i could uh if you look at this belt i  could collapse it like this or i could  roll it a little bit and collapse it  like that or i could roll it a little  bit collapse it like that roll it a  little more collapse it like that all of  these are valid gluings because they're  always gluing 180 to 180  and that's what rolling belts  are  okay  here's  yeah  not worry about that  uh does the broken applet work now i  don't know why it wasn't working  probably some  conflict with suspension or just the job  installation was bad but here it is it's  it's freely available online you draw  your graph  it's a it's a little hard to just use  you can't just it's hard to draw an  exciting example here i'll draw a  tetrahedron because that's simple uh the  in this case it's abstract so all of  these edges are unit length it's kind of  like tree maker you separately specify  the lengths then you say compute and it  will find  uh in this case it's a tetrahedron  an equilateral regular tetrahedron it's  a little hard to see because it has got  a really either weird field of view or a  lot of perspective here  so but this is a regular tetrahedron you  can verify that by say making these  lengths  some of these lengths longer i'll make  three of them  five  so then  this is like a pyramid  of course we're just essentially  applying koshi's rigidity theorem here  and say oh this  uniquely assembles into this  spiky tetrahedron but if you have some  you could in principle draw in a cross  here  with the appropriate gluing it's a  little tricky to draw in because you  have to also triangulate  this be hard to do from scratch to  compute gluings but in theory especially  if you use software to compute the  gluings and compute shortest paths in a  glued polygon  which we'll be getting more to next  class uh next lecture is about  algorithms for finding gluings  then computing shortest paths then uh  you could use that as input to this  software which will then compute what  the 3d polyhedron is  so that's the  alexander implementation this is by i  think a student of bubenko and esmestiev  and  all right one quick question about sort  of out of time  about the non-convex case so we're doing  all this work about making convex  surfaces i said non-non-convex case is  trivial  in a sense the decision problem is  trivial it's not an easy theorem to  prove  and uh i already tipped it up  uh it follows from what's called the  bergo zalgaler theorem  so  and it says that if you have any  polyhedral metric  so that means you take a polygon you  glue it however you want it can't have  oh you could even have boundary take any  polyhedral metrics you take any polygon  glue it however you want could have  handles you know it could be a donut it  could be whatever  then it has an isometric polyhedral  realization in 3d  so there is a way so we're doing it  abstractly you know we're gluing stuff  together it's kind of topologically we  don't know what we're doing  but then there is a way to actually  embed it as a polyhedral surface in 3d  without stretching any of the lengths so  it's a folding of the piece of paper  could have creases in it will have  actually tons of creases in it  and it has  exactly the right it connects all the  things you wanted to connect together  which is kind of crazy  furthermore  so  it could be self-crossing  but if your metric your surface is  either orientable so it doesn't have any  cross caps or  mobius strips in it  or it has boundary then there's even a  way to do it without crossings  so that's basically all the cases we  care about uh you could glue this  together into a sphere or into a into a  disc or into a taurus or to to handle  taurus or whatever no matter what you do  there is a way to embed it in 3d as a  presumably non-convex  very creased  surface  this is a hard theorem to prove it uses  if you're familiar with nash's embedding  theorem about embedding riemann surfaces  it uses a bunch of techniques from there  called spiling perturbations which i  don't know and the one description i  read calls the resulting surfaces  strongly corrugated which i think means  tons of creases they are it is finitely  many creases that's the theorem there's  no bound on them as far as i know  there's no algorithm to compute this  thing i've never seen a picture of them  otherwise i'd show it to you  so lots of open problems in making this  real  but at least the decision problem is  kind of boring you do any gluing it will  make a non-convex thing  to find that thing is still pretty  interesting question though  that's it  you
 

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313
00:11:25,829 --> 00:11:26,630

 

314
00:11:26,630 --> 00:11:36,470

 

315
00:11:36,470 --> 00:11:49,190

 

316
00:11:49,190 --> 00:11:52,069

 

317
00:11:52,069 --> 00:11:54,710

 

318
00:11:54,710 --> 00:12:04,550

 

319
00:12:04,550 --> 00:12:06,310

 

320
00:12:06,310 --> 00:12:09,350

 

321
00:12:09,350 --> 00:12:10,949

 

322
00:12:10,949 --> 00:12:13,590

 

323
00:12:13,590 --> 00:12:13,600

 

324
00:12:13,600 --> 00:12:14,710


325
00:12:14,710 --> 00:12:16,389

 

326
00:12:16,389 --> 00:12:16,399

 

327
00:12:16,399 --> 00:12:17,509


328
00:12:17,509 --> 00:12:18,870

 

329
00:12:18,870 --> 00:12:20,550

 

330
00:12:20,550 --> 00:12:22,389

 

331
00:12:22,389 --> 00:12:23,910

 

332
00:12:23,910 --> 00:12:26,710

 

333
00:12:26,710 --> 00:12:29,750

 

334
00:12:29,750 --> 00:12:31,509

 

335
00:12:31,509 --> 00:12:33,829

 

336
00:12:33,829 --> 00:12:33,839

 

337
00:12:33,839 --> 00:12:35,350


338
00:12:35,350 --> 00:12:37,350

 

339
00:12:37,350 --> 00:12:38,949

 

340
00:12:38,949 --> 00:12:40,470

 

341
00:12:40,470 --> 00:12:42,550

 

342
00:12:42,550 --> 00:12:45,430

 

343
00:12:45,430 --> 00:12:46,629

 

344
00:12:46,629 --> 00:12:46,639

 

345
00:12:46,639 --> 00:12:47,990


346
00:12:47,990 --> 00:12:48,000

 

347
00:12:48,000 --> 00:12:49,430


348
00:12:49,430 --> 00:12:51,670

 

349
00:12:51,670 --> 00:12:53,110

 

350
00:12:53,110 --> 00:12:53,120

 

351
00:12:53,120 --> 00:12:55,030


352
00:12:55,030 --> 00:12:57,829

 

353
00:12:57,829 --> 00:13:00,470

 

354
00:13:00,470 --> 00:13:03,670

 

355
00:13:03,670 --> 00:13:05,670

 

356
00:13:05,670 --> 00:13:08,470

 

357
00:13:08,470 --> 00:13:10,550

 

358
00:13:10,550 --> 00:13:12,629

 

359
00:13:12,629 --> 00:13:14,389

 

360
00:13:14,389 --> 00:13:15,910

 

361
00:13:15,910 --> 00:13:15,920

 

362
00:13:15,920 --> 00:13:17,030


363
00:13:17,030 --> 00:13:18,870

 

364
00:13:18,870 --> 00:13:21,430

 

365
00:13:21,430 --> 00:13:24,470

 

366
00:13:24,470 --> 00:13:27,110

 

367
00:13:27,110 --> 00:13:29,990

 

368
00:13:29,990 --> 00:13:31,750

 

369
00:13:31,750 --> 00:13:36,389

 

370
00:13:36,389 --> 00:13:37,670

 

371
00:13:37,670 --> 00:13:39,189

 

372
00:13:39,189 --> 00:13:44,550

 

373
00:13:44,550 --> 00:13:48,790

 

374
00:13:48,790 --> 00:13:54,550

 

375
00:13:54,550 --> 00:13:57,030

 

376
00:13:57,030 --> 00:13:59,670

 

377
00:13:59,670 --> 00:14:01,750

 

378
00:14:01,750 --> 00:14:03,750

 

379
00:14:03,750 --> 00:14:06,550

 

380
00:14:06,550 --> 00:14:09,430

 

381
00:14:09,430 --> 00:14:10,710

 

382
00:14:10,710 --> 00:14:12,389

 

383
00:14:12,389 --> 00:14:14,069

 

384
00:14:14,069 --> 00:14:16,629

 

385
00:14:16,629 --> 00:14:18,150

 

386
00:14:18,150 --> 00:14:20,790

 

387
00:14:20,790 --> 00:14:22,629

 

388
00:14:22,629 --> 00:14:24,629

 

389
00:14:24,629 --> 00:14:24,639

 

390
00:14:24,639 --> 00:14:25,670


391
00:14:25,670 --> 00:14:27,030

 

392
00:14:27,030 --> 00:14:29,350

 

393
00:14:29,350 --> 00:14:30,870

 

394
00:14:30,870 --> 00:14:33,430

 

395
00:14:33,430 --> 00:14:34,870

 

396
00:14:34,870 --> 00:14:38,230

 

397
00:14:38,230 --> 00:14:40,550

 

398
00:14:40,550 --> 00:14:45,990

 

399
00:14:45,990 --> 00:14:50,470

 

400
00:14:50,470 --> 00:14:52,790

 

401
00:14:52,790 --> 00:14:54,470

 

402
00:14:54,470 --> 00:14:56,710

 

403
00:14:56,710 --> 00:14:58,949

 

404
00:14:58,949 --> 00:15:00,790

 

405
00:15:00,790 --> 00:15:02,870

 

406
00:15:02,870 --> 00:15:04,470

 

407
00:15:04,470 --> 00:15:06,949

 

408
00:15:06,949 --> 00:15:09,110

 

409
00:15:09,110 --> 00:15:11,350

 

410
00:15:11,350 --> 00:15:12,470

 

411
00:15:12,470 --> 00:15:13,990

 

412
00:15:13,990 --> 00:15:16,550

 

413
00:15:16,550 --> 00:15:16,560

 

414
00:15:16,560 --> 00:15:19,189


415
00:15:19,189 --> 00:15:19,199

 

416
00:15:19,199 --> 00:15:20,069


417
00:15:20,069 --> 00:15:21,509

 

418
00:15:21,509 --> 00:15:25,189

 

419
00:15:25,189 --> 00:15:27,110

 

420
00:15:27,110 --> 00:15:30,949

 

421
00:15:30,949 --> 00:15:33,430

 

422
00:15:33,430 --> 00:15:33,440

 

423
00:15:33,440 --> 00:15:35,269


424
00:15:35,269 --> 00:15:40,069

 

425
00:15:40,069 --> 00:15:40,079

 

426
00:15:40,079 --> 00:15:42,949


427
00:15:42,949 --> 00:15:45,189

 

428
00:15:45,189 --> 00:15:47,189

 

429
00:15:47,189 --> 00:15:49,030

 

430
00:15:49,030 --> 00:15:51,110

 

431
00:15:51,110 --> 00:15:52,550

 

432
00:15:52,550 --> 00:15:55,910

 

433
00:15:55,910 --> 00:15:57,269

 

434
00:15:57,269 --> 00:15:58,440

 

435
00:15:58,440 --> 00:15:58,450

 

436
00:15:58,450 --> 00:16:00,150


437
00:16:00,150 --> 00:16:08,150

 

438
00:16:08,150 --> 00:16:09,430

 

439
00:16:09,430 --> 00:16:10,710

 

440
00:16:10,710 --> 00:16:11,910

 

441
00:16:11,910 --> 00:16:13,670

 

442
00:16:13,670 --> 00:16:18,550

 

443
00:16:18,550 --> 00:16:21,430

 

444
00:16:21,430 --> 00:16:25,829

 

445
00:16:25,829 --> 00:16:27,509

 

446
00:16:27,509 --> 00:16:59,030

 

447
00:16:59,030 --> 00:17:01,189

 

448
00:17:01,189 --> 00:17:03,269

 

449
00:17:03,269 --> 00:17:03,279

 

450
00:17:03,279 --> 00:17:05,110


451
00:17:05,110 --> 00:17:07,270

 

452
00:17:07,270 --> 00:17:09,829

 

453
00:17:09,829 --> 00:17:11,110

 

454
00:17:11,110 --> 00:17:12,870

 

455
00:17:12,870 --> 00:17:14,630

 

456
00:17:14,630 --> 00:17:17,429

 

457
00:17:17,429 --> 00:17:19,829

 

458
00:17:19,829 --> 00:17:21,750

 

459
00:17:21,750 --> 00:17:23,990

 

460
00:17:23,990 --> 00:17:26,150

 

461
00:17:26,150 --> 00:17:27,990

 

462
00:17:27,990 --> 00:17:30,310

 

463
00:17:30,310 --> 00:17:32,150

 

464
00:17:32,150 --> 00:17:33,750

 

465
00:17:33,750 --> 00:17:33,760

 

466
00:17:33,760 --> 00:17:34,950


467
00:17:34,950 --> 00:17:37,190

 

468
00:17:37,190 --> 00:17:39,669

 

469
00:17:39,669 --> 00:17:42,470

 

470
00:17:42,470 --> 00:17:42,480

 

471
00:17:42,480 --> 00:17:43,909


472
00:17:43,909 --> 00:17:45,990

 

473
00:17:45,990 --> 00:17:48,310

 

474
00:17:48,310 --> 00:17:49,669

 

475
00:17:49,669 --> 00:17:51,430

 

476
00:17:51,430 --> 00:17:52,870

 

477
00:17:52,870 --> 00:17:53,990

 

478
00:17:53,990 --> 00:17:55,590

 

479
00:17:55,590 --> 00:17:56,950

 

480
00:17:56,950 --> 00:17:58,230

 

481
00:17:58,230 --> 00:17:58,240

 

482
00:17:58,240 --> 00:17:59,190


483
00:17:59,190 --> 00:17:59,200

 

484
00:17:59,200 --> 00:18:01,830


485
00:18:01,830 --> 00:18:04,070

 

486
00:18:04,070 --> 00:18:06,150

 

487
00:18:06,150 --> 00:18:07,510

 

488
00:18:07,510 --> 00:18:09,510

 

489
00:18:09,510 --> 00:18:11,990

 

490
00:18:11,990 --> 00:18:13,590

 

491
00:18:13,590 --> 00:18:15,830

 

492
00:18:15,830 --> 00:18:18,549

 

493
00:18:18,549 --> 00:18:20,549

 

494
00:18:20,549 --> 00:18:22,070

 

495
00:18:22,070 --> 00:18:25,270

 

496
00:18:25,270 --> 00:18:27,190

 

497
00:18:27,190 --> 00:18:31,750

 

498
00:18:31,750 --> 00:18:33,510

 

499
00:18:33,510 --> 00:18:34,630

 

500
00:18:34,630 --> 00:18:36,549

 

501
00:18:36,549 --> 00:18:38,390

 

502
00:18:38,390 --> 00:18:40,390

 

503
00:18:40,390 --> 00:18:42,789

 

504
00:18:42,789 --> 00:18:44,549

 

505
00:18:44,549 --> 00:18:44,559

 

506
00:18:44,559 --> 00:18:45,430


507
00:18:45,430 --> 00:18:47,830

 

508
00:18:47,830 --> 00:18:47,840

 

509
00:18:47,840 --> 00:18:48,870


510
00:18:48,870 --> 00:18:50,549

 

511
00:18:50,549 --> 00:18:52,789

 

512
00:18:52,789 --> 00:18:54,830

 

513
00:18:54,830 --> 00:18:57,669

 

514
00:18:57,669 --> 00:18:59,909

 

515
00:18:59,909 --> 00:19:01,990

 

516
00:19:01,990 --> 00:19:06,470

 

517
00:19:06,470 --> 00:19:08,710

 

518
00:19:08,710 --> 00:19:11,750

 

519
00:19:11,750 --> 00:19:13,750

 

520
00:19:13,750 --> 00:19:15,590

 

521
00:19:15,590 --> 00:19:17,350

 

522
00:19:17,350 --> 00:19:19,430

 

523
00:19:19,430 --> 00:19:22,549

 

524
00:19:22,549 --> 00:19:22,559

 

525
00:19:22,559 --> 00:19:23,430


526
00:19:23,430 --> 00:19:25,350

 

527
00:19:25,350 --> 00:19:27,510

 

528
00:19:27,510 --> 00:19:29,750

 

529
00:19:29,750 --> 00:19:31,270

 

530
00:19:31,270 --> 00:19:32,870

 

531
00:19:32,870 --> 00:19:35,830

 

532
00:19:35,830 --> 00:19:37,430

 

533
00:19:37,430 --> 00:19:39,110

 

534
00:19:39,110 --> 00:19:41,510

 

535
00:19:41,510 --> 00:19:43,270

 

536
00:19:43,270 --> 00:19:45,270

 

537
00:19:45,270 --> 00:19:46,310

 

538
00:19:46,310 --> 00:19:50,830

 

539
00:19:50,830 --> 00:19:50,840

 

540
00:19:50,840 --> 00:19:52,390


541
00:19:52,390 --> 00:19:53,909

 

542
00:19:53,909 --> 00:19:56,150

 

543
00:19:56,150 --> 00:19:59,190

 

544
00:19:59,190 --> 00:20:00,870

 

545
00:20:00,870 --> 00:20:03,190

 

546
00:20:03,190 --> 00:20:04,710

 

547
00:20:04,710 --> 00:20:07,270

 

548
00:20:07,270 --> 00:20:10,310

 

549
00:20:10,310 --> 00:20:12,230

 

550
00:20:12,230 --> 00:20:14,630

 

551
00:20:14,630 --> 00:20:17,110

 

552
00:20:17,110 --> 00:20:17,120

 

553
00:20:17,120 --> 00:20:18,149


554
00:20:18,149 --> 00:20:20,470

 

555
00:20:20,470 --> 00:20:21,990

 

556
00:20:21,990 --> 00:20:24,950

 

557
00:20:24,950 --> 00:20:24,960

 

558
00:20:24,960 --> 00:20:27,350


559
00:20:27,350 --> 00:20:27,360

 

560
00:20:27,360 --> 00:20:28,230


561
00:20:28,230 --> 00:20:34,390

 

562
00:20:34,390 --> 00:20:36,549

 

563
00:20:36,549 --> 00:20:45,029

 

564
00:20:45,029 --> 00:20:45,039

 

565
00:20:45,039 --> 00:20:47,430


566
00:20:47,430 --> 00:20:49,510

 

567
00:20:49,510 --> 00:20:52,149

 

568
00:20:52,149 --> 00:20:55,990

 

569
00:20:55,990 --> 00:20:57,990

 

570
00:20:57,990 --> 00:20:59,510

 

571
00:20:59,510 --> 00:20:59,520

 

572
00:20:59,520 --> 00:21:14,230


573
00:21:14,230 --> 00:21:34,789

 

574
00:21:34,789 --> 00:21:36,630

 

575
00:21:36,630 --> 00:21:38,630

 

576
00:21:38,630 --> 00:21:40,390

 

577
00:21:40,390 --> 00:21:43,270

 

578
00:21:43,270 --> 00:21:45,669

 

579
00:21:45,669 --> 00:21:47,029

 

580
00:21:47,029 --> 00:21:48,470

 

581
00:21:48,470 --> 00:21:50,630

 

582
00:21:50,630 --> 00:21:52,149

 

583
00:21:52,149 --> 00:21:53,669

 

584
00:21:53,669 --> 00:21:56,549

 

585
00:21:56,549 --> 00:21:59,350

 

586
00:21:59,350 --> 00:22:00,870

 

587
00:22:00,870 --> 00:22:02,870

 

588
00:22:02,870 --> 00:22:04,870

 

589
00:22:04,870 --> 00:22:10,390

 

590
00:22:10,390 --> 00:22:13,270

 

591
00:22:13,270 --> 00:22:14,630

 

592
00:22:14,630 --> 00:22:17,350

 

593
00:22:17,350 --> 00:22:18,470

 

594
00:22:18,470 --> 00:22:20,310

 

595
00:22:20,310 --> 00:22:20,320

 

596
00:22:20,320 --> 00:22:21,510


597
00:22:21,510 --> 00:22:23,430

 

598
00:22:23,430 --> 00:22:26,630

 

599
00:22:26,630 --> 00:22:28,549

 

600
00:22:28,549 --> 00:22:30,070

 

601
00:22:30,070 --> 00:22:33,110

 

602
00:22:33,110 --> 00:22:35,350

 

603
00:22:35,350 --> 00:22:37,430

 

604
00:22:37,430 --> 00:22:39,190

 

605
00:22:39,190 --> 00:22:42,230

 

606
00:22:42,230 --> 00:22:44,549

 

607
00:22:44,549 --> 00:22:46,710

 

608
00:22:46,710 --> 00:22:49,190

 

609
00:22:49,190 --> 00:22:50,789

 

610
00:22:50,789 --> 00:22:52,870

 

611
00:22:52,870 --> 00:22:54,789

 

612
00:22:54,789 --> 00:22:57,510

 

613
00:22:57,510 --> 00:22:59,190

 

614
00:22:59,190 --> 00:23:01,270

 

615
00:23:01,270 --> 00:23:03,110

 

616
00:23:03,110 --> 00:23:04,710

 

617
00:23:04,710 --> 00:23:06,390

 

618
00:23:06,390 --> 00:23:08,549

 

619
00:23:08,549 --> 00:23:09,669

 

620
00:23:09,669 --> 00:23:09,679

 

621
00:23:09,679 --> 00:23:10,710


622
00:23:10,710 --> 00:23:12,710

 

623
00:23:12,710 --> 00:23:14,390

 

624
00:23:14,390 --> 00:23:16,070

 

625
00:23:16,070 --> 00:23:16,080

 

626
00:23:16,080 --> 00:23:16,870


627
00:23:16,870 --> 00:23:19,110

 

628
00:23:19,110 --> 00:23:19,120

 

629
00:23:19,120 --> 00:23:20,710


630
00:23:20,710 --> 00:23:22,710

 

631
00:23:22,710 --> 00:23:22,720

 

632
00:23:22,720 --> 00:23:25,270


633
00:23:25,270 --> 00:23:26,950

 

634
00:23:26,950 --> 00:23:28,789

 

635
00:23:28,789 --> 00:23:31,510

 

636
00:23:31,510 --> 00:23:32,710

 

637
00:23:32,710 --> 00:23:34,950

 

638
00:23:34,950 --> 00:23:36,789

 

639
00:23:36,789 --> 00:23:38,710

 

640
00:23:38,710 --> 00:23:40,549

 

641
00:23:40,549 --> 00:23:40,559

 

642
00:23:40,559 --> 00:23:44,149


643
00:23:44,149 --> 00:23:47,590

 

644
00:23:47,590 --> 00:23:47,600

 

645
00:23:47,600 --> 00:23:48,830


646
00:23:48,830 --> 00:23:54,710

 

647
00:23:54,710 --> 00:23:57,430

 

648
00:23:57,430 --> 00:23:57,440

 

649
00:23:57,440 --> 00:23:59,669


650
00:23:59,669 --> 00:24:02,549

 

651
00:24:02,549 --> 00:24:05,590

 

652
00:24:05,590 --> 00:24:08,070

 

653
00:24:08,070 --> 00:24:10,630

 

654
00:24:10,630 --> 00:24:12,470

 

655
00:24:12,470 --> 00:24:14,070

 

656
00:24:14,070 --> 00:24:17,430

 

657
00:24:17,430 --> 00:24:17,440

 

658
00:24:17,440 --> 00:24:19,510


659
00:24:19,510 --> 00:24:22,789

 

660
00:24:22,789 --> 00:24:24,950

 

661
00:24:24,950 --> 00:24:26,630

 

662
00:24:26,630 --> 00:24:27,990

 

663
00:24:27,990 --> 00:24:30,070

 

664
00:24:30,070 --> 00:24:34,230

 

665
00:24:34,230 --> 00:24:36,470

 

666
00:24:36,470 --> 00:24:37,590

 

667
00:24:37,590 --> 00:24:38,950

 

668
00:24:38,950 --> 00:24:41,029

 

669
00:24:41,029 --> 00:24:43,430

 

670
00:24:43,430 --> 00:24:46,230

 

671
00:24:46,230 --> 00:24:47,909

 

672
00:24:47,909 --> 00:24:49,669

 

673
00:24:49,669 --> 00:24:51,430

 

674
00:24:51,430 --> 00:24:53,350

 

675
00:24:53,350 --> 00:24:55,990

 

676
00:24:55,990 --> 00:24:58,549

 

677
00:24:58,549 --> 00:25:00,870

 

678
00:25:00,870 --> 00:25:02,630

 

679
00:25:02,630 --> 00:25:04,310

 

680
00:25:04,310 --> 00:25:05,669

 

681
00:25:05,669 --> 00:25:08,070

 

682
00:25:08,070 --> 00:25:10,390

 

683
00:25:10,390 --> 00:25:12,149

 

684
00:25:12,149 --> 00:25:14,870

 

685
00:25:14,870 --> 00:25:16,630

 

686
00:25:16,630 --> 00:25:18,549

 

687
00:25:18,549 --> 00:25:20,710

 

688
00:25:20,710 --> 00:25:23,029

 

689
00:25:23,029 --> 00:25:24,870

 

690
00:25:24,870 --> 00:25:27,669

 

691
00:25:27,669 --> 00:25:30,549

 

692
00:25:30,549 --> 00:25:32,549

 

693
00:25:32,549 --> 00:25:37,029

 

694
00:25:37,029 --> 00:25:38,549

 

695
00:25:38,549 --> 00:25:40,390

 

696
00:25:40,390 --> 00:25:41,510

 

697
00:25:41,510 --> 00:25:43,110

 

698
00:25:43,110 --> 00:25:45,750

 

699
00:25:45,750 --> 00:25:47,669

 

700
00:25:47,669 --> 00:25:47,679

 

701
00:25:47,679 --> 00:25:48,630


702
00:25:48,630 --> 00:25:50,549

 

703
00:25:50,549 --> 00:25:52,230

 

704
00:25:52,230 --> 00:25:54,470

 

705
00:25:54,470 --> 00:25:57,269

 

706
00:25:57,269 --> 00:25:58,789

 

707
00:25:58,789 --> 00:26:00,470

 

708
00:26:00,470 --> 00:26:02,789

 

709
00:26:02,789 --> 00:26:02,799

 

710
00:26:02,799 --> 00:26:04,070


711
00:26:04,070 --> 00:26:06,870

 

712
00:26:06,870 --> 00:26:09,430

 

713
00:26:09,430 --> 00:26:12,310

 

714
00:26:12,310 --> 00:26:14,310

 

715
00:26:14,310 --> 00:26:16,149

 

716
00:26:16,149 --> 00:26:19,990

 

717
00:26:19,990 --> 00:26:21,909

 

718
00:26:21,909 --> 00:26:24,310

 

719
00:26:24,310 --> 00:26:24,320

 

720
00:26:24,320 --> 00:26:25,669


721
00:26:25,669 --> 00:26:28,230

 

722
00:26:28,230 --> 00:26:30,950

 

723
00:26:30,950 --> 00:26:33,110

 

724
00:26:33,110 --> 00:26:38,310

 

725
00:26:38,310 --> 00:26:39,990

 

726
00:26:39,990 --> 00:26:41,830

 

727
00:26:41,830 --> 00:26:44,630

 

728
00:26:44,630 --> 00:26:46,630

 

729
00:26:46,630 --> 00:26:47,909

 

730
00:26:47,909 --> 00:26:50,710

 

731
00:26:50,710 --> 00:26:53,669

 

732
00:26:53,669 --> 00:26:55,110

 

733
00:26:55,110 --> 00:26:58,230

 

734
00:26:58,230 --> 00:26:59,430

 

735
00:26:59,430 --> 00:27:02,230

 

736
00:27:02,230 --> 00:27:03,750

 

737
00:27:03,750 --> 00:27:03,760

 

738
00:27:03,760 --> 00:27:05,029


739
00:27:05,029 --> 00:27:08,630

 

740
00:27:08,630 --> 00:27:10,870

 

741
00:27:10,870 --> 00:27:11,990

 

742
00:27:11,990 --> 00:27:13,669

 

743
00:27:13,669 --> 00:27:15,909

 

744
00:27:15,909 --> 00:27:15,919

 

745
00:27:15,919 --> 00:27:16,870


746
00:27:16,870 --> 00:27:18,789

 

747
00:27:18,789 --> 00:27:21,510

 

748
00:27:21,510 --> 00:27:23,510

 

749
00:27:23,510 --> 00:27:25,190

 

750
00:27:25,190 --> 00:27:27,430

 

751
00:27:27,430 --> 00:27:29,430

 

752
00:27:29,430 --> 00:27:32,549

 

753
00:27:32,549 --> 00:27:35,110

 

754
00:27:35,110 --> 00:27:37,029

 

755
00:27:37,029 --> 00:27:39,669

 

756
00:27:39,669 --> 00:27:41,909

 

757
00:27:41,909 --> 00:27:44,710

 

758
00:27:44,710 --> 00:27:46,830

 

759
00:27:46,830 --> 00:27:46,840

 

760
00:27:46,840 --> 00:27:48,710


761
00:27:48,710 --> 00:27:50,149

 

762
00:27:50,149 --> 00:27:51,990

 

763
00:27:51,990 --> 00:27:55,190

 

764
00:27:55,190 --> 00:27:57,350

 

765
00:27:57,350 --> 00:27:59,269

 

766
00:27:59,269 --> 00:28:00,789

 

767
00:28:00,789 --> 00:28:02,950

 

768
00:28:02,950 --> 00:28:04,310

 

769
00:28:04,310 --> 00:28:06,630

 

770
00:28:06,630 --> 00:28:08,070

 

771
00:28:08,070 --> 00:28:10,389

 

772
00:28:10,389 --> 00:28:12,549

 

773
00:28:12,549 --> 00:28:16,230

 

774
00:28:16,230 --> 00:28:17,190

 

775
00:28:17,190 --> 00:28:24,470

 

776
00:28:24,470 --> 00:28:27,029

 

777
00:28:27,029 --> 00:28:31,110

 

778
00:28:31,110 --> 00:28:32,549

 

779
00:28:32,549 --> 00:28:32,559

 

780
00:28:32,559 --> 00:28:33,750


781
00:28:33,750 --> 00:28:36,870

 

782
00:28:36,870 --> 00:28:36,880

 

783
00:28:36,880 --> 00:28:37,750


784
00:28:37,750 --> 00:28:37,760

 

785
00:28:37,760 --> 00:28:38,789


786
00:28:38,789 --> 00:28:40,950

 

787
00:28:40,950 --> 00:28:41,990

 

788
00:28:41,990 --> 00:28:44,310

 

789
00:28:44,310 --> 00:28:46,389

 

790
00:28:46,389 --> 00:28:47,909

 

791
00:28:47,909 --> 00:28:50,070

 

792
00:28:50,070 --> 00:28:50,080

 

793
00:28:50,080 --> 00:28:52,310


794
00:28:52,310 --> 00:28:53,909

 

795
00:28:53,909 --> 00:28:56,070

 

796
00:28:56,070 --> 00:28:58,470

 

797
00:28:58,470 --> 00:29:00,070

 

798
00:29:00,070 --> 00:29:01,830

 

799
00:29:01,830 --> 00:29:03,430

 

800
00:29:03,430 --> 00:29:05,110

 

801
00:29:05,110 --> 00:29:07,190

 

802
00:29:07,190 --> 00:29:08,950

 

803
00:29:08,950 --> 00:29:08,960

 

804
00:29:08,960 --> 00:29:09,830


805
00:29:09,830 --> 00:29:12,149

 

806
00:29:12,149 --> 00:29:14,870

 

807
00:29:14,870 --> 00:29:16,710

 

808
00:29:16,710 --> 00:29:19,110

 

809
00:29:19,110 --> 00:29:20,310

 

810
00:29:20,310 --> 00:29:22,389

 

811
00:29:22,389 --> 00:29:24,549

 

812
00:29:24,549 --> 00:29:26,070

 

813
00:29:26,070 --> 00:29:27,990

 

814
00:29:27,990 --> 00:29:28,000

 

815
00:29:28,000 --> 00:29:28,789


816
00:29:28,789 --> 00:29:28,799

 

817
00:29:28,799 --> 00:29:30,389


818
00:29:30,389 --> 00:29:31,669

 

819
00:29:31,669 --> 00:29:33,830

 

820
00:29:33,830 --> 00:29:33,840

 

821
00:29:33,840 --> 00:29:34,549


822
00:29:34,549 --> 00:29:35,990

 

823
00:29:35,990 --> 00:29:38,549

 

824
00:29:38,549 --> 00:29:40,149

 

825
00:29:40,149 --> 00:29:41,669

 

826
00:29:41,669 --> 00:29:43,350

 

827
00:29:43,350 --> 00:29:44,710

 

828
00:29:44,710 --> 00:29:44,720

 

829
00:29:44,720 --> 00:29:46,310


830
00:29:46,310 --> 00:29:48,950

 

831
00:29:48,950 --> 00:29:48,960

 

832
00:29:48,960 --> 00:29:50,630


833
00:29:50,630 --> 00:29:51,909

 

834
00:29:51,909 --> 00:29:53,909

 

835
00:29:53,909 --> 00:29:55,510

 

836
00:29:55,510 --> 00:29:57,510

 

837
00:29:57,510 --> 00:29:59,590

 

838
00:29:59,590 --> 00:30:01,510

 

839
00:30:01,510 --> 00:30:03,190

 

840
00:30:03,190 --> 00:30:05,669

 

841
00:30:05,669 --> 00:30:07,750

 

842
00:30:07,750 --> 00:30:11,830

 

843
00:30:11,830 --> 00:30:13,190

 

844
00:30:13,190 --> 00:30:14,549

 

845
00:30:14,549 --> 00:30:15,990

 

846
00:30:15,990 --> 00:30:16,000

 

847
00:30:16,000 --> 00:30:17,190


848
00:30:17,190 --> 00:30:20,710

 

849
00:30:20,710 --> 00:30:22,389

 

850
00:30:22,389 --> 00:30:24,710

 

851
00:30:24,710 --> 00:30:26,149

 

852
00:30:26,149 --> 00:30:27,750

 

853
00:30:27,750 --> 00:30:29,190

 

854
00:30:29,190 --> 00:30:29,200

 

855
00:30:29,200 --> 00:30:30,070


856
00:30:30,070 --> 00:30:31,430

 

857
00:30:31,430 --> 00:30:33,190

 

858
00:30:33,190 --> 00:30:34,310

 

859
00:30:34,310 --> 00:30:35,510

 

860
00:30:35,510 --> 00:30:37,029

 

861
00:30:37,029 --> 00:30:38,710

 

862
00:30:38,710 --> 00:30:40,070

 

863
00:30:40,070 --> 00:30:41,510

 

864
00:30:41,510 --> 00:30:41,520

 

865
00:30:41,520 --> 00:30:43,830


866
00:30:43,830 --> 00:30:45,269

 

867
00:30:45,269 --> 00:30:47,269

 

868
00:30:47,269 --> 00:30:49,909

 

869
00:30:49,909 --> 00:30:52,549

 

870
00:30:52,549 --> 00:30:53,990

 

871
00:30:53,990 --> 00:30:55,830

 

872
00:30:55,830 --> 00:30:57,590

 

873
00:30:57,590 --> 00:30:59,909

 

874
00:30:59,909 --> 00:31:00,950

 

875
00:31:00,950 --> 00:31:03,430

 

876
00:31:03,430 --> 00:31:04,870

 

877
00:31:04,870 --> 00:31:06,789

 

878
00:31:06,789 --> 00:31:07,830

 

879
00:31:07,830 --> 00:31:09,269

 

880
00:31:09,269 --> 00:31:11,269

 

881
00:31:11,269 --> 00:31:12,870

 

882
00:31:12,870 --> 00:31:15,350

 

883
00:31:15,350 --> 00:31:16,789

 

884
00:31:16,789 --> 00:31:19,830

 

885
00:31:19,830 --> 00:31:21,350

 

886
00:31:21,350 --> 00:31:21,360

 

887
00:31:21,360 --> 00:31:22,710


888
00:31:22,710 --> 00:31:25,669

 

889
00:31:25,669 --> 00:31:28,389

 

890
00:31:28,389 --> 00:31:30,630

 

891
00:31:30,630 --> 00:31:33,269

 

892
00:31:33,269 --> 00:31:35,909

 

893
00:31:35,909 --> 00:31:35,919

 

894
00:31:35,919 --> 00:31:37,509


895
00:31:37,509 --> 00:31:39,190

 

896
00:31:39,190 --> 00:31:43,669

 

897
00:31:43,669 --> 00:31:43,679

 

898
00:31:43,679 --> 00:31:44,549


899
00:31:44,549 --> 00:31:46,870

 

900
00:31:46,870 --> 00:31:49,509

 

901
00:31:49,509 --> 00:31:51,430

 

902
00:31:51,430 --> 00:31:53,590

 

903
00:31:53,590 --> 00:31:55,269

 

904
00:31:55,269 --> 00:31:57,269

 

905
00:31:57,269 --> 00:31:57,279

 

906
00:31:57,279 --> 00:31:58,230


907
00:31:58,230 --> 00:31:59,750

 

908
00:31:59,750 --> 00:32:01,990

 

909
00:32:01,990 --> 00:32:03,669

 

910
00:32:03,669 --> 00:32:05,990

 

911
00:32:05,990 --> 00:32:07,350

 

912
00:32:07,350 --> 00:32:08,710

 

913
00:32:08,710 --> 00:32:09,990

 

914
00:32:09,990 --> 00:32:11,750

 

915
00:32:11,750 --> 00:32:14,470

 

916
00:32:14,470 --> 00:32:14,480

 

917
00:32:14,480 --> 00:32:15,990


918
00:32:15,990 --> 00:32:17,350

 

919
00:32:17,350 --> 00:32:19,190

 

920
00:32:19,190 --> 00:32:22,149

 

921
00:32:22,149 --> 00:32:22,159

 

922
00:32:22,159 --> 00:32:23,350


923
00:32:23,350 --> 00:32:31,750

 

924
00:32:31,750 --> 00:32:31,760

 

925
00:32:31,760 --> 00:32:33,269


926
00:32:33,269 --> 00:32:35,669

 

927
00:32:35,669 --> 00:32:35,679

 

928
00:32:35,679 --> 00:32:37,029


929
00:32:37,029 --> 00:32:42,630

 

930
00:32:42,630 --> 00:32:52,789

 

931
00:32:52,789 --> 00:32:54,549

 

932
00:32:54,549 --> 00:32:56,389

 

933
00:32:56,389 --> 00:32:58,310

 

934
00:32:58,310 --> 00:33:06,310

 

935
00:33:06,310 --> 00:33:18,070

 

936
00:33:18,070 --> 00:33:18,080

 

937
00:33:18,080 --> 00:33:19,830


938
00:33:19,830 --> 00:33:21,830

 

939
00:33:21,830 --> 00:33:21,840

 

940
00:33:21,840 --> 00:33:23,750


941
00:33:23,750 --> 00:33:25,990

 

942
00:33:25,990 --> 00:33:27,590

 

943
00:33:27,590 --> 00:33:29,269

 

944
00:33:29,269 --> 00:33:30,789

 

945
00:33:30,789 --> 00:33:32,070

 

946
00:33:32,070 --> 00:33:35,190

 

947
00:33:35,190 --> 00:33:36,950

 

948
00:33:36,950 --> 00:33:39,909

 

949
00:33:39,909 --> 00:33:42,710

 

950
00:33:42,710 --> 00:33:44,149

 

951
00:33:44,149 --> 00:33:47,430

 

952
00:33:47,430 --> 00:33:48,710

 

953
00:33:48,710 --> 00:33:50,070

 

954
00:33:50,070 --> 00:33:51,750

 

955
00:33:51,750 --> 00:33:53,430

 

956
00:33:53,430 --> 00:33:54,470

 

957
00:33:54,470 --> 00:33:56,149

 

958
00:33:56,149 --> 00:33:56,159

 

959
00:33:56,159 --> 00:33:57,509


960
00:33:57,509 --> 00:33:58,870

 

961
00:33:58,870 --> 00:34:00,149

 

962
00:34:00,149 --> 00:34:01,990

 

963
00:34:01,990 --> 00:34:04,549

 

964
00:34:04,549 --> 00:34:05,990

 

965
00:34:05,990 --> 00:34:07,509

 

966
00:34:07,509 --> 00:34:09,270

 

967
00:34:09,270 --> 00:34:11,109

 

968
00:34:11,109 --> 00:34:12,629

 

969
00:34:12,629 --> 00:34:15,109

 

970
00:34:15,109 --> 00:34:18,710

 

971
00:34:18,710 --> 00:34:20,950

 

972
00:34:20,950 --> 00:34:23,750

 

973
00:34:23,750 --> 00:34:26,470

 

974
00:34:26,470 --> 00:34:27,909

 

975
00:34:27,909 --> 00:34:29,990

 

976
00:34:29,990 --> 00:34:32,310

 

977
00:34:32,310 --> 00:34:34,389

 

978
00:34:34,389 --> 00:34:34,399

 

979
00:34:34,399 --> 00:34:36,230


980
00:34:36,230 --> 00:34:37,510

 

981
00:34:37,510 --> 00:34:39,669

 

982
00:34:39,669 --> 00:34:41,270

 

983
00:34:41,270 --> 00:34:42,629

 

984
00:34:42,629 --> 00:34:44,950

 

985
00:34:44,950 --> 00:34:46,149

 

986
00:34:46,149 --> 00:34:47,270

 

987
00:34:47,270 --> 00:34:48,950

 

988
00:34:48,950 --> 00:34:50,869

 

989
00:34:50,869 --> 00:34:53,430

 

990
00:34:53,430 --> 00:34:55,589

 

991
00:34:55,589 --> 00:34:57,910

 

992
00:34:57,910 --> 00:34:59,430

 

993
00:34:59,430 --> 00:35:01,270

 

994
00:35:01,270 --> 00:35:03,750

 

995
00:35:03,750 --> 00:35:05,990

 

996
00:35:05,990 --> 00:35:08,470

 

997
00:35:08,470 --> 00:35:09,990

 

998
00:35:09,990 --> 00:35:12,069

 

999
00:35:12,069 --> 00:35:12,079

 

1000
00:35:12,079 --> 00:35:13,109


1001
00:35:13,109 --> 00:35:15,109

 

1002
00:35:15,109 --> 00:35:16,870

 

1003
00:35:16,870 --> 00:35:18,710

 

1004
00:35:18,710 --> 00:35:21,030

 

1005
00:35:21,030 --> 00:35:23,430

 

1006
00:35:23,430 --> 00:35:25,510

 

1007
00:35:25,510 --> 00:35:25,520

 

1008
00:35:25,520 --> 00:35:26,630


1009
00:35:26,630 --> 00:35:26,640

 

1010
00:35:26,640 --> 00:35:27,750


1011
00:35:27,750 --> 00:35:30,390

 

1012
00:35:30,390 --> 00:35:32,470

 

1013
00:35:32,470 --> 00:35:33,829

 

1014
00:35:33,829 --> 00:35:35,990

 

1015
00:35:35,990 --> 00:35:39,510

 

1016
00:35:39,510 --> 00:35:41,109

 

1017
00:35:41,109 --> 00:35:42,829

 

1018
00:35:42,829 --> 00:35:45,030

 

1019
00:35:45,030 --> 00:35:45,040

 

1020
00:35:45,040 --> 00:35:46,470


1021
00:35:46,470 --> 00:35:48,230

 

1022
00:35:48,230 --> 00:35:50,069

 

1023
00:35:50,069 --> 00:35:54,470

 

1024
00:35:54,470 --> 00:36:00,150

 

1025
00:36:00,150 --> 00:36:03,109

 

1026
00:36:03,109 --> 00:36:05,430

 

1027
00:36:05,430 --> 00:36:07,670

 

1028
00:36:07,670 --> 00:36:08,950

 

1029
00:36:08,950 --> 00:36:10,710

 

1030
00:36:10,710 --> 00:36:13,109

 

1031
00:36:13,109 --> 00:36:13,119

 

1032
00:36:13,119 --> 00:36:14,069


1033
00:36:14,069 --> 00:36:16,230

 

1034
00:36:16,230 --> 00:36:18,390

 

1035
00:36:18,390 --> 00:36:20,390

 

1036
00:36:20,390 --> 00:36:24,230

 

1037
00:36:24,230 --> 00:36:25,910

 

1038
00:36:25,910 --> 00:36:26,950

 

1039
00:36:26,950 --> 00:36:28,470

 

1040
00:36:28,470 --> 00:36:29,910

 

1041
00:36:29,910 --> 00:36:31,190

 

1042
00:36:31,190 --> 00:36:31,200

 

1043
00:36:31,200 --> 00:36:32,870


1044
00:36:32,870 --> 00:36:34,310

 

1045
00:36:34,310 --> 00:36:35,750

 

1046
00:36:35,750 --> 00:36:37,270

 

1047
00:36:37,270 --> 00:36:39,270

 

1048
00:36:39,270 --> 00:36:41,670

 

1049
00:36:41,670 --> 00:36:49,430

 

1050
00:36:49,430 --> 00:36:51,910

 

1051
00:36:51,910 --> 00:36:54,069

 

1052
00:36:54,069 --> 00:36:55,430

 

1053
00:36:55,430 --> 00:36:57,030

 

1054
00:36:57,030 --> 00:36:59,750

 

1055
00:36:59,750 --> 00:37:01,510

 

1056
00:37:01,510 --> 00:37:01,520

 

1057
00:37:01,520 --> 00:37:02,310


1058
00:37:02,310 --> 00:37:05,109

 

1059
00:37:05,109 --> 00:37:06,870

 

1060
00:37:06,870 --> 00:37:09,109

 

1061
00:37:09,109 --> 00:37:11,109

 

1062
00:37:11,109 --> 00:37:13,190

 

1063
00:37:13,190 --> 00:37:15,750

 

1064
00:37:15,750 --> 00:37:17,589

 

1065
00:37:17,589 --> 00:37:19,190

 

1066
00:37:19,190 --> 00:37:20,710

 

1067
00:37:20,710 --> 00:37:23,510

 

1068
00:37:23,510 --> 00:37:25,829

 

1069
00:37:25,829 --> 00:37:27,910

 

1070
00:37:27,910 --> 00:37:30,710

 

1071
00:37:30,710 --> 00:37:32,630

 

1072
00:37:32,630 --> 00:37:34,069

 

1073
00:37:34,069 --> 00:37:36,630

 

1074
00:37:36,630 --> 00:37:39,270

 

1075
00:37:39,270 --> 00:37:41,430

 

1076
00:37:41,430 --> 00:37:43,990

 

1077
00:37:43,990 --> 00:37:46,230

 

1078
00:37:46,230 --> 00:37:47,990

 

1079
00:37:47,990 --> 00:37:49,750

 

1080
00:37:49,750 --> 00:37:51,349

 

1081
00:37:51,349 --> 00:37:53,190

 

1082
00:37:53,190 --> 00:37:54,790

 

1083
00:37:54,790 --> 00:37:57,030

 

1084
00:37:57,030 --> 00:37:58,630

 

1085
00:37:58,630 --> 00:38:00,710

 

1086
00:38:00,710 --> 00:38:02,230

 

1087
00:38:02,230 --> 00:38:04,550

 

1088
00:38:04,550 --> 00:38:06,630

 

1089
00:38:06,630 --> 00:38:09,030

 

1090
00:38:09,030 --> 00:38:11,190

 

1091
00:38:11,190 --> 00:38:13,430

 

1092
00:38:13,430 --> 00:38:15,750

 

1093
00:38:15,750 --> 00:38:17,430

 

1094
00:38:17,430 --> 00:38:19,990

 

1095
00:38:19,990 --> 00:38:22,390

 

1096
00:38:22,390 --> 00:38:24,710

 

1097
00:38:24,710 --> 00:38:26,390

 

1098
00:38:26,390 --> 00:38:28,230

 

1099
00:38:28,230 --> 00:38:31,190

 

1100
00:38:31,190 --> 00:38:32,470

 

1101
00:38:32,470 --> 00:38:34,710

 

1102
00:38:34,710 --> 00:38:36,870

 

1103
00:38:36,870 --> 00:38:38,310

 

1104
00:38:38,310 --> 00:38:42,600

 

1105
00:38:42,600 --> 00:38:42,610

 

1106
00:38:42,610 --> 00:38:46,150


1107
00:38:46,150 --> 00:38:48,550

 

1108
00:38:48,550 --> 00:38:50,310

 

1109
00:38:50,310 --> 00:38:51,670

 

1110
00:38:51,670 --> 00:38:53,349

 

1111
00:38:53,349 --> 00:38:55,270

 

1112
00:38:55,270 --> 00:38:57,589

 

1113
00:38:57,589 --> 00:38:58,950

 

1114
00:38:58,950 --> 00:39:01,270

 

1115
00:39:01,270 --> 00:39:02,870

 

1116
00:39:02,870 --> 00:39:02,880

 

1117
00:39:02,880 --> 00:39:05,349


1118
00:39:05,349 --> 00:39:05,359

 

1119
00:39:05,359 --> 00:39:07,430


1120
00:39:07,430 --> 00:39:07,440

 

1121
00:39:07,440 --> 00:39:09,109


1122
00:39:09,109 --> 00:39:09,119

 

1123
00:39:09,119 --> 00:39:11,030


1124
00:39:11,030 --> 00:39:12,950

 

1125
00:39:12,950 --> 00:39:15,589

 

1126
00:39:15,589 --> 00:39:17,270

 

1127
00:39:17,270 --> 00:39:18,630

 

1128
00:39:18,630 --> 00:39:20,710

 

1129
00:39:20,710 --> 00:39:22,390

 

1130
00:39:22,390 --> 00:39:25,109

 

1131
00:39:25,109 --> 00:39:26,710

 

1132
00:39:26,710 --> 00:39:29,030

 

1133
00:39:29,030 --> 00:39:30,390

 

1134
00:39:30,390 --> 00:39:31,589

 

1135
00:39:31,589 --> 00:39:34,310

 

1136
00:39:34,310 --> 00:39:36,310

 

1137
00:39:36,310 --> 00:39:37,750

 

1138
00:39:37,750 --> 00:39:39,589

 

1139
00:39:39,589 --> 00:39:42,470

 

1140
00:39:42,470 --> 00:39:43,670

 

1141
00:39:43,670 --> 00:39:47,349

 

1142
00:39:47,349 --> 00:39:49,510

 

1143
00:39:49,510 --> 00:39:51,430

 

1144
00:39:51,430 --> 00:39:53,829

 

1145
00:39:53,829 --> 00:39:55,589

 

1146
00:39:55,589 --> 00:39:57,430

 

1147
00:39:57,430 --> 00:39:59,990

 

1148
00:39:59,990 --> 00:40:00,000

 

1149
00:40:00,000 --> 00:40:00,950


1150
00:40:00,950 --> 00:40:02,710

 

1151
00:40:02,710 --> 00:40:04,150

 

1152
00:40:04,150 --> 00:40:04,160

 

1153
00:40:04,160 --> 00:40:05,750


1154
00:40:05,750 --> 00:40:07,030

 

1155
00:40:07,030 --> 00:40:09,990

 

1156
00:40:09,990 --> 00:40:11,190

 

1157
00:40:11,190 --> 00:40:13,589

 

1158
00:40:13,589 --> 00:40:14,870

 

1159
00:40:14,870 --> 00:40:16,870

 

1160
00:40:16,870 --> 00:40:19,589

 

1161
00:40:19,589 --> 00:40:21,430

 

1162
00:40:21,430 --> 00:40:21,440

 

1163
00:40:21,440 --> 00:40:22,470


1164
00:40:22,470 --> 00:40:23,990

 

1165
00:40:23,990 --> 00:40:25,589

 

1166
00:40:25,589 --> 00:40:27,109

 

1167
00:40:27,109 --> 00:40:29,109

 

1168
00:40:29,109 --> 00:40:31,349

 

1169
00:40:31,349 --> 00:40:32,470

 

1170
00:40:32,470 --> 00:40:34,069

 

1171
00:40:34,069 --> 00:40:36,309

 

1172
00:40:36,309 --> 00:40:37,829

 

1173
00:40:37,829 --> 00:40:39,430

 

1174
00:40:39,430 --> 00:40:42,230

 

1175
00:40:42,230 --> 00:40:44,790

 

1176
00:40:44,790 --> 00:40:46,710

 

1177
00:40:46,710 --> 00:40:48,710

 

1178
00:40:48,710 --> 00:40:51,190

 

1179
00:40:51,190 --> 00:40:52,870

 

1180
00:40:52,870 --> 00:40:55,349

 

1181
00:40:55,349 --> 00:41:00,069

 

1182
00:41:00,069 --> 00:41:00,079

 

1183
00:41:00,079 --> 00:41:01,589


1184
00:41:01,589 --> 00:41:04,150

 

1185
00:41:04,150 --> 00:41:05,510

 

1186
00:41:05,510 --> 00:41:08,069

 

1187
00:41:08,069 --> 00:41:09,430

 

1188
00:41:09,430 --> 00:41:12,069

 

1189
00:41:12,069 --> 00:41:12,079

 

1190
00:41:12,079 --> 00:41:13,190


1191
00:41:13,190 --> 00:41:14,790

 

1192
00:41:14,790 --> 00:41:16,950

 

1193
00:41:16,950 --> 00:41:16,960

 

1194
00:41:16,960 --> 00:41:17,910


1195
00:41:17,910 --> 00:41:21,670

 

1196
00:41:21,670 --> 00:41:23,430

 

1197
00:41:23,430 --> 00:41:25,670

 

1198
00:41:25,670 --> 00:41:25,680

 

1199
00:41:25,680 --> 00:41:26,390


1200
00:41:26,390 --> 00:41:27,990

 

1201
00:41:27,990 --> 00:41:29,910

 

1202
00:41:29,910 --> 00:41:31,349

 

1203
00:41:31,349 --> 00:41:33,829

 

1204
00:41:33,829 --> 00:41:35,750

 

1205
00:41:35,750 --> 00:41:37,670

 

1206
00:41:37,670 --> 00:41:39,589

 

1207
00:41:39,589 --> 00:41:42,230

 

1208
00:41:42,230 --> 00:41:43,750

 

1209
00:41:43,750 --> 00:41:46,870

 

1210
00:41:46,870 --> 00:41:49,190

 

1211
00:41:49,190 --> 00:41:50,870

 

1212
00:41:50,870 --> 00:41:52,150

 

1213
00:41:52,150 --> 00:41:53,670

 

1214
00:41:53,670 --> 00:41:55,109

 

1215
00:41:55,109 --> 00:41:56,390

 

1216
00:41:56,390 --> 00:41:59,829

 

1217
00:41:59,829 --> 00:42:02,230

 

1218
00:42:02,230 --> 00:42:05,030

 

1219
00:42:05,030 --> 00:42:06,790

 

1220
00:42:06,790 --> 00:42:09,270

 

1221
00:42:09,270 --> 00:42:10,950

 

1222
00:42:10,950 --> 00:42:12,790

 

1223
00:42:12,790 --> 00:42:14,470

 

1224
00:42:14,470 --> 00:42:17,589

 

1225
00:42:17,589 --> 00:42:17,599

 

1226
00:42:17,599 --> 00:42:18,870


1227
00:42:18,870 --> 00:42:18,880

 

1228
00:42:18,880 --> 00:42:19,670


1229
00:42:19,670 --> 00:42:21,670

 

1230
00:42:21,670 --> 00:42:24,790

 

1231
00:42:24,790 --> 00:42:28,790

 

1232
00:42:28,790 --> 00:42:31,190

 

1233
00:42:31,190 --> 00:42:32,710

 

1234
00:42:32,710 --> 00:42:35,990

 

1235
00:42:35,990 --> 00:42:38,069

 

1236
00:42:38,069 --> 00:42:39,430

 

1237
00:42:39,430 --> 00:42:41,190

 

1238
00:42:41,190 --> 00:42:43,829

 

1239
00:42:43,829 --> 00:42:47,349

 

1240
00:42:47,349 --> 00:42:49,589

 

1241
00:42:49,589 --> 00:42:51,589

 

1242
00:42:51,589 --> 00:42:53,510

 

1243
00:42:53,510 --> 00:42:54,950

 

1244
00:42:54,950 --> 00:42:54,960

 

1245
00:42:54,960 --> 00:42:56,309


1246
00:42:56,309 --> 00:42:58,710

 

1247
00:42:58,710 --> 00:43:00,390

 

1248
00:43:00,390 --> 00:43:02,470

 

1249
00:43:02,470 --> 00:43:03,910

 

1250
00:43:03,910 --> 00:43:06,309

 

1251
00:43:06,309 --> 00:43:08,470

 

1252
00:43:08,470 --> 00:43:10,630

 

1253
00:43:10,630 --> 00:43:13,109

 

1254
00:43:13,109 --> 00:43:15,510

 

1255
00:43:15,510 --> 00:43:17,270

 

1256
00:43:17,270 --> 00:43:18,950

 

1257
00:43:18,950 --> 00:43:20,230

 

1258
00:43:20,230 --> 00:43:21,910

 

1259
00:43:21,910 --> 00:43:23,589

 

1260
00:43:23,589 --> 00:43:25,990

 

1261
00:43:25,990 --> 00:43:26,000

 

1262
00:43:26,000 --> 00:43:27,190


1263
00:43:27,190 --> 00:43:29,109

 

1264
00:43:29,109 --> 00:43:31,109

 

1265
00:43:31,109 --> 00:43:33,109

 

1266
00:43:33,109 --> 00:43:34,470

 

1267
00:43:34,470 --> 00:43:36,309

 

1268
00:43:36,309 --> 00:43:43,349

 

1269
00:43:43,349 --> 00:43:43,359

 

1270
00:43:43,359 --> 00:43:45,440