1
00:00:03,280 --> 00:00:04,630

today we're talking about  the local behavior of a crease pattern  so you take some crease pattern for some  flat folding we're thinking about flat  foldability  is a foldability question i give you a  crease pattern like this i want to know  does it fold flat like this one does  and we're studying what happens locally  right around a single vertex  so i didn't mention sort of graph  terminology it's probably useful to  mention that  these corners wherever all the edges  come together those are vertices these  triangles or in general some regions  divided by the creases  we call faces all right so  uh if we just sort of imagine cutting a  little  disk around that vertex and seeing how  it behaves  we get a nice circular piece of paper  with some  crease pattern and we want to understand  when those things fold flat  and when they don't sometimes they do  sometimes they don't this  is kind of like a one-dimensional  folding problem so let me write some  stuff  about that  so we get we call single vertex crease  patterns  so something like this is the example i  have  made something like that that should be  flat foldable  so we think of uh having  a disc of paper  it doesn't really matter but it's it's  going to be easier to reason about and  think about a disc of paper  really we just mean some small region  around that vertex  there's obviously one vertex in the  crease pattern  let's say we have n creases  emanating from one vertex  uh this pattern is going to be defined  by  some sequence of angles  in general theta 1 up to theta n if  there are increases there will be n  angles between them say in clockwise  order  and let's see  all right when we fold this thing flat  like in this picture we take  this disc we fold it along all of those  creases  what i'd like to focus on is what  happens to the boundary  of the paper so there's this outer  circle the boundary is a circle  and locally if you look at one of these  creases it's like folding the circle in  half  onto itself and when you make all of  these creases if you're successful  you end up folding the circle onto  a portion of the circle so if you look  at this boundary which is a little hard  to see  why don't i trace it it's gonna do  something like  this  okay i'm gonna  so this is what the 3d thing looks like  blown up a little bit and i changed the  angles a little bit  but this is what a flat folding would  look like and i'm really just focusing  on  how the circle maps around this circle  so one of the a flat folding of a disc  also folds a circle onto a circle  so we get  we have a flat folding in one of these  single vertex crease patterns  we get a folding  of a circle  onto a circle this is nice because  circles are one-dimensional and last  class we talked about  folding one-dimensional line segments a  circle is just a little bit more  complicated  it's a different topology we can use a  lot of the same  mindset at least as we did for folding  line segments onto  the line in particular you can see  yeah this is kind of circular but you  could also imagine  unrolling it a little bit and making it  straight  something like this so that it lies  on a straight line if you just unroll  that circle  so you also get a folding of a circle in  some sense  onto a line  by unrolling  now at this point i want to mention a  slight issue  here i mean this  for this unrolling to work  if you're taking a circle like this  in particular you must make at least one  fold for this to be possible  because right now if i have if i haven't  folded anything i just have a big circle  i can't unroll a circle onto a line i  haven't  collapsed it yet here i've made it small  enough  that it only occupies a portion of the  entire circle and so i can unroll it  so this is true as long as i make at  least one fold  okay technically i need to assume  another thing  which is that this thing came from a  flat piece of paper  uh i said there are these angles and  what i intended to mean  is that the sum of those angles  is 360 degrees as it is in a flat piece  of paper  but i'm not always going to require this  assumption it's  it's what we care about for thinking  about a crease pattern of a flat piece  of paper  and we look at each vertex this will be  true  but when we reason about these single  vertex crease patterns it's really  useful  to think about sort of in the middle  this could get smaller  so for example you have a pattern like  this let's say i just take  these two creases in the bottom and i  fold a crimp  because i know a crimp isn't a kind of a  good thing to do  so now i have what's called a cone of  paper  the sum of the angles there's two angles  here this one and this one each one  looks like 270 no not 270 135  in total it's 270. so the total angle  there is now less than 360. this is what  we call a convex cone  so if you relax this constraint  to allow less than or equal to 360  then we get a convex cone  and today i only want to talk about flat  paper and convex cones we need this  for the induction essentially for  proving things about the flat case  in the textbook if you look at this  chapter on single vertex  crease patterns they als we also prove  or study the case where the sum of the  angles is greater than 360 which you  could never make from flat paper  but hey it's fun to think about it's a  natural generalization  and you can characterize all the things  that we do here you can generalize to  that situation  it's just a little bit harder  all right  so what does it take to fold a circle  onto a line in this situation of a  convex cone  where we make at least one fold our  folding is going to lie along portion of  the circle we can  unroll it a little bit get something on  lying on a straight line  it's kind of like  folding a line segment onto a straight  line  that was flat folding of one-dimensional  pieces of paper  but there's a lot of differences but  they are similar  but a lot of the things we proved last  time do not hold in the case of circular  paper  so i have a bunch of them here  so one problem you may recall before if  we had a line segment  and we put an arbitrary crease pattern  then we could just assign a mountain  valley assignment alternating mountain  valley  and that would always fold flat no  collisions  okay in this world of having a circular  piece of paper  that's no longer true for example  if you have this kind of crease pattern  on a circle or  in the disk it would look like this  those two creases  you can't fold that thing flat it's not  going to work  okay we'll see exactly what what you  need to forbid for that to make sense  another example  is something like this  so if you're all right so  if i try to draw this in the the line  what's happening  is i have a short segment and then i  have a long segment  and so the problem is those ends don't  meet but even when the ends meet  like in this picture imagine these are  vertically aligned  if i do this alternating mountain valley  pattern mountain valley assignment uh  i can't join these ends and make a  circle because that would collide with  everything  so this thing actually does have a flat  folding  starting from a circle and i'm not going  to try to unmap it to a circle  it does fold but not like that  not with that alternating mountain  valley pattern  there are other annoying things let me  tell you  sort of one more on the proof side  last time we had  a lemma that said if your crease pattern  is mingling which still is a meaningful  notion here then you have a crimp  or an end fold in that case here of  course we don't have any end folds  here we also do not get this implication  mingling does not imply the existence of  a crimp and the  where the proof breaks down is if you  have  so if you remember our parentheses from  last time  we had this kind of pattern and we said  well  you keep going and either on the left  side you have a  beginning open paren or on the right  side you have a closing  round parenthesis in either case you've  got an end fold now this can actually  happen in a circle  and so you have nothing you can do so  you could have this pattern  there would be no crimp and probably in  that case you would not be flat-foldable  but in particular this implication fails  so  we have to do more work and but we're  going to get the same kind of result  that in linear time  we can tell whether one of these things  folds flat  so that's good news  so uh before last time  telling whether a crease pattern folded  flat was trivial the answer was always  yes  now it's not so trivial so let's start  with characterizing crease patterns that  fold flat  then we will go to mountain valley  patterns that fold flat  so  be great if i could just say that but i  mean here of course single vertex  so i give you one of these sequence of  angles  let's say it sums to 360 or less  when is it going to be flat foldable  and the answer is very simple  let me write it down  this thing is going to be flat foldable  if and only if  the sum of the odd angles  is equal to the sum of the even angles  and yeah i'm implicitly assuming and  requiring here  that n is even that's something we'll  prove so  n is actually even and minus one is  going to be odd  and if you happen to have started with a  flat piece of paper  uh for the sum of all of these things is  360 then of course  if you have them evenly divided this sum  will be 180 degrees  but if you started with the convex cone  that it might it'll be smaller  but for this reason a lot of people or  some people call this the 180 degree  property  or the pi property if you'd like to  think in radians instead of degrees  and we can do a little example here this  guy  this crease pattern uh has  a 90 degree angle here and it has a 90  degree angle down here  so the sum of those two is 180 those are  alternating angles  different parity classes this is 45  this is the complement 135  and that was also something 180 of  course one does it  only if the other does so it's flat  foldable we're golden  something like this is bad because  i mean it's an even number but you the  odd angles do not equal the sum of the  even angles  one's clearly bigger okay and this is a  general property this is  also called kawasaki's theorem or  kawasaki jost  jostan's theorem they proved it in 1989  so this is pretty much the beginning of  origami mathematics  so let's prove it there's two things to  prove one is that uh  any flat foldable crease pattern has  this property which is pretty easy  and the other which is a little bit  harder is that if you have this property  you really are flat foldable that's the  full characterization  so let's start with the easy direction  if you're flat foldable you must have  that sum  okay so it's all about  thinking what happens at a crease  so we're thinking about the boundary of  this piece of paper thinking about the  circle  and what's happening on the circle  is you travel uh  say counterclockwise for some amount of  time that would be  first angle then you make a fold it's  either a mountain or valley at this  point we don't care  either way you turn around maybe you  turn around this way or you turn around  this way  but in terms of your travel along a  circle  so we went theta one counterclockwise  now we're going to go theta two  clockwise then we're gonna go theta  three  counterclockwise and so on we keep  alternating back and forth  it's much easier to think draw on a line  so we go  to the right theta 1 then we go to the  left  theta 2 then we go to the right  theta 3 and i don't know about the  layering here  i'm just drawing it in terms of  horizontal travel the y-coordinate means  nothing  okay in order for this thing to be a  flat folding  to be a valid folding of a circle of  paper  these guys have to have the same x  coordinate  so i don't know i mean the y coordinates  they also have to work out but we're not  thinking about the y coordinates we're  just thinking about x travel  we've got to get back to where we  started if we're folding a circle  that's it so how much trout how much  total travel do we do  in a sine sense how much do we go right  well we go theta 1 to the right  then we go theta 2 to the left which is  like going negative theta two  to the right and then we go theta three  to the right  then we go negative theta four to the  right and you add up  this alternating summation that must  equal  zero okay the other thing  uh that i should say is that n has to be  even  um this is maybe obvious  at this point but we're supposed to  alternate every time we hit a crease we  have to change direction  and so in the end it's important that  after theta 4 we change direction to  visit theta 1.  if they were aligned like if theta 4  went over here  and then theta 5 went like that  and there was an odd number of creases  this would be bad even though they're  lined up there's no crease here you just  went straight from theta 5  to theta 1. you're supposed to change  direction every crease  so also n is even  to alternate directions  question all right all right good  clear this is pretty easy once you  realize the angles correspond to x  coordinates in this linear space  so why is this enough in order to show  that  this property is enough for flat  foldability  we need to actually worry about the  stacking order of these edges we need to  worry about the y coordinates we need to  make sure that they can avoid collision  with some mountain valley assignment the  good news is we're free to use whatever  mountains and valleys we want  we can stack things however we want we  just need to find some valid state  and our intuition from the  one-dimensional case was  we should try alternating mountains and  valleys that seemed like a good thing it  avoided collisions  for an open piece of paper for a line  segment of course we know that's not  enough  here somewhere i drew  this example so it's alternating in the  middle  and then that last crease is a problem  but that's all we need to fix  turns out it's not so hard  so we have a nice circle  what i'd like to do is cut that circle  so here i sort of cut it and thought  about it as a line segment and then i  wanted to rejoin  okay i'm going to be careful about where  i cut it  okay i want to cut at an extreme left or  extreme right it doesn't matter  you can see here the cut ended up being  in the middle that's bad  so how do i fix it  what does extreme mean how do i tell  given a circle it's a little hard to say  what extreme means because it's  circular but if i take a picture like  this i've already drawn it with nice x  coordinates i don't know about the y  coordinates yet  say oh that's bad my cut point ended up  being non-extreme  well this is extreme so cut here instead  okay so what that means is redraw this  picture  with this being the cut point so maybe  this is the beginning  so we go like that again and now we wrap  around  so now we have this segment and now we  have the segment  and lo and behold it remained extreme  and this is always true you can do this  sort of  cut and rejoin the ends that were messed  up before  and you'll preserve the x coordinates x  coordinates aren't changing when you do  this kind of transformation  so there's a clear leftmost and there's  a clear rightmost you pick either one  you cut there instead  and now we have this nice property that  it's easy to join the ends  because it's that as far left as you can  go there can't be anything that  penetrates because that would be  farther left  all right let me write down some words  corresponding to that  so once we cut at this extreme crease  you can fold this 1d  segment just like we used to be able to  so we can fold it flat  using the accordion fold alternating  mountain and valley  we know that works and then all we need  to show  is that the two ends can join  those ends are the two copies  of the cut crease  that we did in this first step  uh we need two things one is that they  are aligned  and that's where we need this condition  that the sum of the odd angles equals  the sum of the even angles which is the  same as saying the alternating sum  is equal to zero that tells us that  whatever we do the x coordinates are  lined up at the end  so they're aligned by the by the  assumption that we made here  and  you can join without crossing  so this is a statement about  y-coordinates and we know it's okay  because it's at the left extreme  so by this choice of the proper cut  there could be other things that come  right to the boundary here but that's  considered okay that's not crossing  that's just touching  nothing can go farther left because it  was the left extreme  that's the end of the proof any  questions about that  it's kawasaki's theorem  i'll mention just for fun this sort of  classic  if you allow one of these  a non-convex cone where you have more  than 360 degrees  of material this statement is not true  and what can there's one other situation  which can happen  which is that the alternating sum of  angles is not zero but it's uh plus or  minus 360 degrees  which is fun and you can read it you can  see examples of that in the book  all right so obviously if you want to  tell whether a crease pattern  or a single vertex is flat foldable you  just compute this thing in linear time  and you know  yes or no uh so crease patterns are  pretty easy just have one extra  condition  what about mountain valley assignments  last time  we looked at mountain valley assignments  and we showed that crimps and end folds  were always enough to fold any mountain  valley assignment that's out there  that will continue to be true here  except now we don't have end folds now  it's crimps all the way  what does it mean  so  that is the flat foldable mountain  valley patterns  we're going to prove that crimps are  enough but before we get there i need a  bunch of sort of warm-up  facts about valid mountain valley  patterns  the first thing is just counting  mountains and valleys  so even if you play with a simple  example like this one  you see it has three mountains and one  valley  or three valleys and one mountain in  fact if  if you have a degree four vertex with  four creases emanating from that point  those are your only choices you could  try for example making it all valleys  and it i mean you can't do it  uh it's hard to demonstrate uh you could  try  making it two mountains and two valleys  so maybe i try to do mountain mountain  here  and then somehow it's gonna be valley  valley there it's  no good okay you can that what's  happening here is that the number of  mountains  and the number of valleys must differ by  exactly two it doesn't  matter who's bigger so the difference  could be plus two or minus two  because you can always flip over and  negate that difference  swapping mountains for valleys but they  always have to differ by exactly two  why is that well  this is uh yeah we're again going to  think of a picture like this we have  horizontal  travel according to the theta eyes we're  also going to think about the vertical  picture  if this thing is flat foldable so i'm  assuming it's flat foldable  then this must happen this is called  maekawa's theorem by the way  it's proved by june mayakawa in  1986 a little earlier  also proved by jacques christine around  the same time  back before we were good at internet and  communication and whatnot  and there were little pockets of  mathematical origamis  and they found each other basically in  89  when there was the 89 as the first  origami science math and education  conference  started bringing these people together  the last one was this summer  right so why is this true  i don't know flat folding looks like  something okay it has no crossings  it has horizontal travel we understand  it ends up back where it started there's  also some notion of layers and vertical  travel which is a little tricky to think  about  but the one thing that's easy to think  about i mean this forms essentially a  polygon  it's like a really squash polygon and  these vertical segments are actually  really just  points okay imagine  that's just where you turn around so  it's like a polygon just  stretched or squashed down onto a line  we know some things about polygons  okay it's an n vertex polygon you know  that like the sum of the interior angles  is  whatever it is i always get confused  with that formula so i don't like to  think about it  one thing we an easier way to think  about that same statement  is just think about how much turning the  polygon does so you start here  you turn right 180 degrees you turn left  180 degrees  you turn left 180 right 180 right 180  right 180. how much is it in total oh  gosh  do i have to add no it's so simple it's  360.  right just if you think about you're  going around in a circle  any polygon not just flat but anything  in in two dimensions  uh the total amount of turning you do is  360. or if you count backwards it's  negative 360.  okay so this is the key something you  should all know  about polygons if you look at the sum of  the turn angles how much turning you do  at each vertex  that will always be plus or minus 360  in any polygon  this is equivalent to the sum of the  interior angles being whatever pi times  n minus 2  which i don't remember but this is much  easier to remember  and it's useful because we can map  mountains and valleys  to left and right turns as i said every  time this was a valley  it was a left turn by 180.  every time it was a mountain it was a  right turn by 180.  valleys  mountains  right turns so i'm going to think of  that as negative 180.  now of course technically it could be  the other way around could be this is  negative this is positive but  i already have a plus and minus here so  it's symmetric  so if we sum up the turn angles which we  know is supposed to be plus or minus 360  some of the turn angles  is going to be we're going to have 180  for each valley so 180 times the number  of valleys  and we're going to have negative 180 for  each mountain  and so the 180s factor out  and this thing is supposed to equal 360  plus or minus 360. how could that be  well number of valleys minus number of  mountains should be either two for plus  360 or negative two for minus 360.  so that proves the theorem  and that's why in a degree four vertex  one of them has to be three the other  one has to be one  there's no other option because that  total number is four  it's the only way to get the difference  to be plus or minus two  so that's kind of nice but it's not  enough  unfortunately if i gave you a mountain  valley pattern that satisfies this  condition  it still might not be flat foldable  this may be no surprise it's not like we  had such a simple test  for the one dimensional case should be  easier we had to give an algorithm we  said repeatedly crimp and enfold  if you get stuck the answer is no if you  don't get stuck  you folded it great  i love this eraser increases entropy  all right let me tell you one case that  is easy  to think about and because it it's nice  it reduces directly to the one  dimensional case  and that's what i call the generic case  generic is this very convenient term we  use in mathematics it's actually really  tricky to define  so i'd like to not define it if i can  but  maybe i should  yeah all right uh  the simple version which is not quite  enough is to say that all of the angles  in the crease pattern are different  okay here for example two of them are  the same there's two 90 degree angles  this is not generic  it's just easier to draw but in general  you  i mean you imagine you just draw some  random thing  none of those lengths are going to be  the same yeah the alternating  sum is 0 but that's it  and that's what i the generic case is  that is the only thing  that holds true about these angles you  can take the alternating sum  that equals zero if you take some  alternating sum of some subset of the  angles that will not equal zero  if you take a random example this is  going to be true okay just  bear with me suppose never any two  angles are the same  then look at the globally smallest  crease  it's the globally smallest theta  say it's theta i so the picture is  theta i theta i minus 1  theta i plus 1. and what do we know if  it's globally smallest and none of the  values are equal  then we know that these two are bigger  no big surprise uh think about what  happens  if this is small these are bigger and  you try to make these both valleys  or both mountains that's of course bad  this is one of the  situations we had in the one dimensional  case i mean really this is a circle  but i can think just locally about  what's happening here so  in this situation one of these must be a  mountain  and the other must be a i don't know  which order they are  but i know they're different and then i  know i can apply  one of these crimps  incidentally the word crimp jason kuh is  asking about like why do you call it  crimp and not pleat  probably i should call it pleat but it's  crimp in my mind it's been crimped in my  mind since 1998 when we wrote this the  1d paper  before i knew the word pleat and so  that's my excuse  so pleat or crimp take your pick crimp  is usually mini pleats together  that's crimping  cool so crimps we kind of like why do we  like  crimps  we proved in the one-dimensional case  last time that  if you make a crimp and your original  thing was flat-foldable the new thing  will be flat-foldable  good news is that is still true i told  you all these things were no longer true  in the circular case  it's still true in the circular case uh  should i tell you let me just remind you  the same proof works  remind you what the proof looked like we  had a crimp  and then and we're hoping to fold that  first  we know at some point these creases get  folded but there may be other junk in  here  maybe other junk in here we'd like to  get rid of that junk  and so what we did is move this junk up  to here which was always safe we moved  this junk down to here  and that was still a folded state a flat  folded state  okay i remember all this stuff  therefore we could have folded the crimp  first and then we have a flat folding  in this situation where this paper is  effectively glued together  which means it was safe to fold this  crimp and glue these together and never  touch them again  so that is still true in the circular  case that proof didn't really assume  anything about the paper would even work  for like tree-shaped paper or something  weird  so if we can find a crimp which we can  in the generic case  then you just make a crimp make a crimp  repeat if you ever get stuck you know  that the original thing was not flat  foldable because the thing you have is  not flat foldable  why if there's no crimp in the generic  case that means you have two valleys  at around the smallest angle and that's  clearly bad  um so i'm not writing a lot of details  here is that clear  if i have an angle that's surrounded by  strictly larger angles i know  there must be one mount in one valley  then i can safely make the crimp and  repeat  the trouble comes in when these angles  are equal  we never had to think about angles being  equal in the one-dimensional case it  didn't matter much  we just said oh you know it's safe to  make a crimp  as long as this was greater than or  equal to this  and this was greater than or equal to  this that is still true it's safe to  make a crimp  but how do we know that there is a crimp  how do we know that there's a mountain  followed by a valley  we don't i mean if they're equal this  this would be all right  that's two valleys is valid if these  three angles are equal  but i claim still somewhere there's got  to be a crimp there's no longer a notion  of the globally smallest angle because  some of them are equal  we've got to find that crimp but it's  out there  so we just need to prove that it exists  so  so to do that we're going to generalize  this  theorem and kawa just in that number of  mountains minus number of valleys is  plus or minus two that's true  and it's a statement about the sum of  all of the or you know  all of the angles all the creases what  i'd like is something a little bit more  localized  and in particular we're going to think  about when i have a whole bunch of  angles that are equal how many mountains  and valleys can there be in there  so what i call local counts  so while everything i've said so far is  pretty classic this result  was proved in 2001 by tom hall so much  more recent  over a decade later  so let's think about k equal angles  and i want that to be a maximal sequence  of equal angles so the ones right after  and right before  are different and furthermore i'm going  to assume that they're bigger  strictly larger  angles okay so  something like that a bunch of equal  angles k of them  bigger angles on either side this almost  always  exists um i should mention this it's so  i'm going to say something about this  situation but the situation should exist  because  i take the smallest angle out there and  then i see how many friends it has that  are all equal  if it's the smallest angle or it's one  of the smallest angles  then the ones are surrounding it are  going to be bigger  unless all the angles are equal  so there's one case which we'll worry  about later it's very easy  all the angles are equal in fact i could  tell you how to worry about that  if all the angles are equal  then you know everything is crimpable as  long as we can find a switch between  mountain and valley  somewhere and because the number of  mountains and valleys is plus or minus  two  there's got to be so they're almost in  equal balance  there's got to be a valley it can't be  all valleys  can't be all mountains so somewhere  there's a transition between mountain  and valley  so you get a crimp out of that  okay but otherwise if they're not all  equal let's think about k  angle equal angles surrounded by two  larger guys  forgot how technical this is all right  then  i want to look at the number of  mountains and the number of valleys  among these uh k plus one creases  so within that range  it's going to be zero if k is odd  it's going to be plus or minus one if k  is even  okay it's not plus or minus two  intuitively plus or minus  two is when you go all the way around  here we're just looking at  a one you know segment portion of the  entire circle  remember this is actually a circle  somewhere  and when you this corresponds to  in the odd case you're going to be going  in the same direction afterwards  and the even case you're going to have  turned around but not a full circle so  it's going to be plus or minus 1.  there's a lot of ways to prove this one  way is to think about convex  cones so  if k is even and you fold this thing  see one two three four five six  equal angles and then there's the two  longer guys they may not be the same  length  but that's what you get in the even case  so what i'd like to do is just  in order to just reduce the things that  we've already thought about i mean you  could talk about turn angles again  or you could just say hey i know  mayakawa's theorem  so as long as i can make this into a  circle which i can do  by adding that stuff now i have a nice  circular  it's a flat folded state there's no  crossings here  presumably because this did something i  don't actually know that it's zigzagging  it might do other stuff  but it's not going to cross what i just  added  so the number of mountains minus number  of valleys in total here  must be plus or minus 2. i only added  one crease this one  so that's going to make it either plus  or minus 1 or plus or minus 3.  if you think about it a little bit it  has to be plus or minus 1 and hopefully  that's what i wrote here yes  because i mean really we're thinking  about turn angles again i'm afraid  can't totally reduce it we just turned  around here  and we're about to finish the circle  which would make it plus or minus 2  before we finish the circle is plus or  minus 1.  k is odd it's  again this might go like that whatever  but because it's an odd number these  guys  are going in the same direction and so  the  total turn angle here must be zero and  one way to see that  is to extend it into a full polygon  you know that it's plus or minus two  for this whole thing and i added two  guys  in the same direction if i remove them  it's going to go down to plus or minus  zero  that was a bit hand wavy but do you buy  that  yeah plus or minus zero is zero  yeah you went from two down to zero  because you removed two identical guys  so let me let me tie this all together  because there's so many  little cases  so i want to know what mountain what  mountain valley patterns are flat  foldable  okay single vertex  and the patterns  okay i claim  there is a crimp if it's flat foldable  there has to be a crimp that you can  find to do remember crimp was  where this length  was less than or equal to this length  and less than or equal to  this length and one of these was m one  of them was v that was the definition of  a crimp  claims such a thing always exists the  proof is  two parts if all the angles are equal  all the angles are equal we know that  these conditions are always satisfied  because everything is equal  so we just need to find an m followed by  a v or v followed by an m  and by maekalo's theorem  we know it's not all m's or all v's so  somewhere there has to be a transition  from m to v's  so just use that lemma theorem count  property otherwise they're not all  equal then i can apply this local counts  result  take the smallest angle take one  instance of the smallest angle there  could be many of them  find all of its neighboring friends that  are equal  then i know that the next angles before  and after have to be strictly bigger  then i know that the number of mountains  and valleys in here  if i look at the difference and it's  zero  that means the number of mountains  equals the number of valleys therefore  there's at least one of each  therefore there's a transition for  mountain valley in there there's a  transition from mountain to valley  like right here that's a valid crimp  this guy's bigger  this guy's equal this would also be a  fine transition because everybody's  equal  this would also be a fine transition any  any transition from mountain valley in  here  i've got a crimp the even case is a  little  i've got to think a little bit more k is  even  k plus 1 which is the number of creases  here is  odd and the difference between mountains  and valleys among those k plus one  creases is plus or minus one  think about it for a second that means  there's again at least one mountain and  at least one valley  and uh because cave to be even has to be  at least  two you've gotta have at least three  things so there's gonna be at least one  of  one of one and two of the other in fact  and i just need one mountain in one  valley somewhere in there and then i  know there has to be a transition from  mountain to valley  otherwise use local counts  and in either case  we get either a mountain to valley  transition  or a valley to mountain  transition and i get it in such a way  that these  these inequalities and the lengths hold  so in other words that it's a crimp  therefore any flat foldable single  vertex mountain valley pattern  has a crimp make it and as i was arguing  here  just like last time that crimp will  still  after you do the crimp you'll still be  flat foldable so repeat  cook until done is the i like to put it  question what about the trivial case of  just floating  trivial case of folding a circle in half  all right that's a good question  it's kind of a technicality  uh one version would be to say that's  zero vertices  that doesn't count we're thinking about  one vertex and as soon as you think  about it as one vertex which  is fine then you actually have two  uh creases there so it's two creases and  hopefully this works  uh oh i see you're saying it's not a  crimp that's a good point  it is flat foldable and yet there's no  crimp damn it  all right sorry my  college right yeah so it's i made a  slight mistake here  i said if all the angles are equal then  meikawa says there's at least one of  each  that's not true when n equals two then  there's  two of one and zero of the other okay as  long as you have more than two  creases then this is true  that's enough so you have to handle this  as a special case  uh that's true this this this is an  important special case to remember  because this will be the final picture  after you do a sequence of crimps  you this is the good case uh  if your thing is flat foldable you will  always end up with that that's a  case where everything is excuse me the  same  smallest it's it's another example where  all the angles are equal  yeah so that's why this is the situation  it's either maekawa and you get a crimp  here because n is greater than 2 and  here i should say  if n is greater than 2  then this is true of course the n equals  2 case  it looks like that it better be both  valleys or both mountains otherwise it's  not going to be flat foldable i forgot  about that  i should add in the notes but it's in  the textbook right  that's true it won't actually look like  this it's going to at the end of the  algorithm it's going to be a cone  the two angles will be equal because of  kawasaki's theorem  and they should still be both mountains  or both valleys but it's going to be  we can see it right here so i apply my  crimp  and now i have a cone and there's two  angles there's one  here which has been fused together  there's one here  which is original and they're equal and  it's two mountains  or two valleys if i turn it upside down  and then it finishes  so that's the algorithm in action two  steps first you find this crimp  here this is actually the globally  smallest angle it's surrounded by bigger  angles so it's really easy  then i have a cone with two equal angles  which is what has to happen at the end  and then i i'm done  good  this is what always happens oh  is this the only way to flat fold these  things no  there are other ah yes  one of the two um  i think it depends what you're counting  if you're counting mountain valley  assignments i mean if you just want to  know  can every mountain valley assignment be  folded by crimping then the answer is  yes that's what we proved  but if you want to get every possible  folded state  crimps are not going to be enough the  reason they're not enough is because  we're using this thing  said hey here's some hypothetical folded  state we can rip it out and make the  crimp have been done but then you're  there's a folded state you're not able  to reach by crimping  i mean we need to work that into an  actual example where crimping forbids  you from reaching some folded state but  i'm pretty sure one exists because of  this  yeah maybe some kind of spiraling thing  right what we're missing here is getting  all the possible layer stacking  orders so we're just trying to match the  mountain valley assignment  we're not going to match a target layer  ordering because  we simplified the layer orders in order  to make crimps possible  so crimps won't get you some some of the  folded states  but one thing you can play with  over here that has been played with by  tom hall  is if i give you  a crease pattern  how many flat foldable mountain valley  assignments does it have okay we have  this nice linear time algorithm to tell  you whether a particular one is doable  how many are there and  there is if you work through all of the  things i've shown you in a little bit  more detail  you can recover this was finding a crimp  you could actually look at what are all  the crimps that are possible  and actually count how many different  ways  there are to fold how many crimps you  can do and then given those crimps how  many crimps you can do after that  and in same linear time kind of  algorithm you can figure out how many  different mountain valley patterns are  flat foldable that takes a little bit  more care  i will tell you the extremes  how small could the  number of flat foldable mountain valley  assignments be for a given crease  pattern  well do you have any guesses when when  do i have the least  choice of where to make crimps  yeah uh  for general n sorry but yeah three  three angles are i guess two angles i  really can't do any i don't have any  choice  but for general and how should i  disperse the theta eyes  in order to make there be very few  possible crimps  when there are only two transitions  between mountains and valleys all the  mountains are together all the valleys  are together  that's right if i was asking the  question about mountains and valleys but  i was asking a question about theta eyes  so i want to count how many mountain  valley patterns there are  mountain valley assignments that are  flat foldable  for a given crease pattern  all the angles are consecutive  increasing yeah that pretty much works  in fact what i need is the generic case  where all the angles are different  and they never become equal by folding  which is what generic means so then  there was only really one crimp i could  do  which is the globally smallest angle  so actually i still have two choices i  could do mountain then valley or valley  than mountain  but i only have two choices for that  crimp then i will have two choices for  the next crimp  two choices for the next crimp in  general i get two to the n  possible mountain valley assignments for  the generic case  okay what about what's the opposite  where i get as much choice as possible  yeah two to the n no uh  two to the n over two thank you right  they're two to the end conceivable  mountain valley assignments can't be  that big  uh every time i do a crimp i eat two  creases not one that's what i forgot  good still pretty big though  it's like square root of all the  possible things you could imagine  are indeed feasible it went from 2 to  the n to 2n over 2.  when would i get the most choice in  crimping  all angles are equal that's the opposite  extreme and in this case  a little messier  when all the angles are equal the only  property you need and  you can see that from the proof the only  property you need is that the number of  mountains minus number of valleys is  plus or minus two  all the a and the n is even as long as  you have that  uh we showed here you're going to have  an alternation from mountain valley  that's a valid crimp because all the  angles are equal and you can keep going  so it's just how do you disperse that  many  how do i disperse n over two minus one  mountains among n different positions  that's what this represents and  positions how do i pick n over two minus  one of them to be mountains  or could be n over two minus one of them  are valleys  and that's this factor of two so it  could have more mountains or more  valleys  and then you somehow place those among  those and this is if you don't know this  notation  that's just what it means if you want to  know  with using other notation it's like this  n over 2 minus 1 factorial over 2  plus 1 factorial if you don't know  factorials  i'll tell you about them later  cool so that's kind of the end of the  single  vertex situation um  yeah i think i'll mention one  interesting open question here which i  would love to explore maybe in our  problem session  um which looks like it'll be mondays at  five i think um  this is one vertex and we went through  all this work to solve single vertex  situation and it's interesting  as we'll see to think about locally how  each vertex behaves  but you would think if you have a crease  pattern with two vertices  shouldn't be that much harder so here we  have linear time  how quickly can you tell whether a two  vertex crease pattern  is flat foldable as far as i know no one  has looked at that problem  surely we can do it in like quadratic  time but maybe even linear time i think  can't be that hard i think  in general if i have a small number of  vertices say k vertices  much smaller than n creases  is there the algorithms people is there  fixed parameter retractable algorithm  and k  or can i get something even getting  something like n to the  some function of k would be progress i  think this should be doable  but ideally we get a running time that's  exponential in k and linear in n  that would be my hope why do i say it  has to be exponential because in general  if i give you crease pattern with n  vertices lots of vertices  this problem is np complete which is  there's not going to be a polynomial  time algorithm for it  nothing good we will prove that next  wednesday  so i'll hold off on that a little bit  but for two two vertices how hard could  it be  all right  i want to talk about one related topic  and then we will go to origami design  and do a little bit on the tree method  but before we get there  i want to talk about local foldability  which is a cool topic people tend to  forget about it i really like it i think  it would make a cool project also  it's a nice algorithm it goes back to  vernon hayes 1996  so it's also right at the beginning of  origami mathematics  and it's this idea all right i give you  a crease pattern now  arbitrarily many vertices and i ask you  if i ask you does it fold flat that's  np-complete intractable  same paper but what if i ask you just to  give me a mountain valley assignment  that might fold flat well  that's also mp complete but if if you  actually want it to fall flat this is  really the same problem  but if i ask you give me a mountain  valley assignment  so that if i checked every vertex  according to this algorithm  at least every vertex folds flat that  would seem nice  you know definitely i have to find a  mountain valley assignment that  satisfies  these conditions that as i do successive  crimps  every vertex if i cut out the vertex  separately it would fold flat this is  the notion of local foldability  and there's a linear time algorithm to  tell you to give you a mountain valley  assignment  that ought to work in that each vertex  it works it still may not work globally  for some other reason  but it's pretty good i think this would  be actually pretty practical  i think in a lot of real world origami  settings when you're doing  flat folding anyway locally valid  locally  foldable is going to be enough to be  globally foldable  that's a guess i don't know if it's true  consistent mountain valley assignment  if there is one  so that each vertex  locally folds flat  let me uh  try to concoct a small example that's  relevant here  gotta think  yeah i think that works all right here  we go  i think this is in the textbook i  remember drawing it in the past  three years ago or whatever all right so  here's a crease pattern  on a square whatever yes  i should have this saying thank you  uh so this satisfies kawasaki's theorem  that was the hard part  because these angles sum to 180 and it's  symmetric  all around okay this is uh these angles  are the smallest they're  60 degrees this is an equilateral  triangle  so we have this not quite the generic  case but we have a the smallest angle is  surrounded  by two larger angles therefore one of  these is a mountain  the other is a valley that means that  these two creases  have to have different assignments i'm  going to write a not equal sign  one of these is not in the other's value  they can't be the same also  these two cannot be the same also these  two cannot be the same  that's not possible because i've got  three  it has to alternate you can't alternate  three times you can only alternate an  even number of times  okay two two of these mountains one's  valley and then  you've got a problem okay so this thing  is not flat foldable  and this algorithm will tell you that  because it'll say hey i can't even find  a mountain valley assignment  that could possibly fold each vertex  flat  so this is a nice algorithm  at least it will detect annoying  situations like that  so in order to solve this we're going to  use  and i'm just going to sketch how this  algorithm works we're going to use this  characterization  and this idea that we can really find  all possible mountain valley assignments  just by trying all the possible crimp  sequences now there are exponentially  many so it's not like i'm actually going  to try them all  but i need to explore that space of  candidate crimps  and see what happens all right  uh so  the idea is kind of crazy  the beginning of the algorithm is fold  each vertex flat  somehow i don't care how just  pick a crimp do it pick a crimp do it  okay separately for each vertex they're  not going to be compatible  and don't look at the mountain valley  assignment you get but look at the  crimping sequence you get so here let's  do a little  example a  non-trivial example so here we have the  peace sign  and these two angles are equal they're  smallest i could crimp this angle first  or i could crimp this angle first i have  a choice so i'll draw both of them  if i crimp this angle first i know  these guys are paired up in the sense  that they must be not equal ones  mountain ones valley  after i do that crimp just like in the  real example i had  you know this angle will equal that  angle and these two guys  must have equal assignments they must be  both mountain or both valley  in this situation these two guys are not  equal  but if i crimp the this pair first and  these two guys must be equal equal now  in the sense of  being mountain or valley okay  so there's still you know exponentially  many possibilities in how to do this but  just pick one pick one of these ways of  pairing up you're going to pair up  each of the increases into n over two  pairs  and they're going to have some not  equals and equal signs  okay so now if you imagine that the  general picture like here  i for example i might get in this  pattern i would be forced to get not  equals  and these guys paired up and in general  i want to look at  these kinds of cycles  you know if i come if i come into a  vertex  here's a vertex it's paired up with  somebody so if i come in here i can go  out  somewhere and i come to this vertex it's  paired with somebody  so i'm gonna just i can keep wandering  around  and in general there will be a bunch of  these  paths that you can follow what could the  paths do they  either close up on themselves maybe  things are paired up  in such a way that you make a return  trip  or it could be some other path let me  draw a dotted path  it could come in here and maybe it gets  paired up with this guy  maybe it goes off to infinity it reaches  the boundary of the paper  those are the two possibilities you get  paths which go off to the edge  off to infinity on either end or you  could get cycles  the cycles are the problem because  whenever i have a cycle  i have a parity constraint for example  when they're all not equals uh the  length of the cycle must be even if they  were  yeah what's the general statement it's  like the parity of the cycle which is  whether it's even or odd should  be equal to the parity of the number of  not equal signs  something like that i'm going to cheat  here  great i just said parody problems it's  something like that  it's either the parity of the cycle  should equal the parity of the number of  equals  or the variety of the number not equals  i think number of  equals anyway one of the two  and you can just check i mean you're  forced if i say okay let's make this  mountain  then this is either equals or not equals  it'll tell me whether this is mountain  or valley just walk around the cycle  either you get a contradiction or you  don't  if you get a contradiction we have a  problem  how do we how could we possibly fix the  problem  well  when we made you look at each of these  crimps i mean in fact you could look at  each of these vertices separately but  you think about one of these crimps and  say well could i have done it another  way  sometimes there are crimps that have  other equal choices  maybe they're a bunch of equal angles  and i could have done a different  pairing  what happens when i try a different  pairing well instead of  this being in one cycle and let's say  this being in another cycle  if i pair these guys up instead i'll end  up merging those two cycles into one  bigger thing  could be a path or a cycle and the  algorithm says  just keep doing those cycle merges and  if you get stuck your thing is not  locally foldable  that's the hard part to prove otherwise  you will find  a one of these patterns that you can  actually  resolve mountains and valleys all the  way through so  let's say  so start with sum  folding say local folding  whatever i'm going to say with some  pairing  of creases at vertices  and  merge  two pads or cycles  whenever possible  and when i say merge i mean  whenever you have what are the possible  things you could possibly merge  when you have at some point during the  algorithm a bunch of equal angles  you have a choice which of these you  crimp obviously  the mountain valley sign is not fixed  you can crimp any of them  you picked one of them and it's sort of  merging  whatever this thing is attached to to  whatever this thing is attached to  if there's something else that's in  that's disconnected from that thing  i want you to instead merge two of them  that combines two different connected  components two paths or cycles  merging means i decrease the total  number of pads or cycles i combine two  into one  whenever that's possible do it you can  prove that if you have  a parity problem in the merge thing you  had to have had a parity problem  originally and merging can only fix  parity problems that's the claim that's  what i will not prove  once you know that and it doesn't matter  in what order or how you choose to merge  you just merge  as much as possible and either the  resulting thing is okay or not  and accordingly you will tell whether  this thing is locally foldable  sorry i want to move on to other things  but i think this would be a fun thing to  actually implement it's an easy  algorithm  and it's  i think a pretty good test for problems  like this to prevent flat foldability  all right  so let's move on from foldability to  origami design  so a bit of a big transition and we're  gonna  talk about origami design a lot more  next  class also but uh  just started off today and the  particular  algorithm for origami design i want to  talk about is called tree method  this is probably the oldest algorithm  for origami design  uh in that it's been around and people  have been thinking about it and  developing it  for many years uh through this period  called the bug wars and people are  trying to design  more and more complicated insects it's  like well i can make an insect  with six legs oh yeah well i can make  you know a spider with eight legs oh  yeah well i can make  an insect that has you know a beetle it  has wings and horns and there's thorns  on the horns and  all these crazy things during that time  there are a lot of people thinking about  how do i make more and more complicated  especially  uh sort of more more limbs in my  creatures  more and very precise uh arrangements of  those limbs let's say  and that is what the tree method deals  with and is really formalized by  robert lang who published a paper in  96 describing it as a as a  sort of complete algorithm and  it's still not known for sure that that  algorithm always works  but uh that's what we've we've been  working on me and  marty and rob lang for the last  four years or so and soon we will  publish that thing  and prove that this thing always works  but i'm going to describe to the  algorithm without the proof that it  works  and what it does tell you its goal  from a mathematical perspective  so it's interested in practical origami  design so we're going to start from a  square piece of paper  it would also work for triangular pieces  of paper or anything convex  but squares are what people usually care  about rectangles are  sometimes it's used for rectangles also  okay the idea is i give to you  a stick figure  so that's formally it's a tree a graph  without any cycles  it's a metric tree meaning that i put  lengths on the edges  i know this length this edge length is  maybe twice as long as this one  so i really draw it with edge lengths in  mind  then what i want you to do is find  some folding of a piece of paper  i should really be looking at what i'm  trying to match uh  so here here  maybe this goes down  such that i want to find one some  folding of a square paper in fact the  smallest square possible  so that when i project  like this vertically  the projection of that uh flat  folding is exactly that metric tree  okay and this is called a uniaxial this  thing is called a uniaxial base  let me tell you a little bit why it's  called any axial base  we're thinking about what are called  origami bases these are  like the beginning of origami models and  most classic origami models like more  than  60 years ago start from one of these  bases  you've got water bomb base in the top  left  preliminary base fish base  bird base windmill base  and frog base i remember them all i have  a little example here this is the water  bomb base  so it's just very simple crease pattern  and you know why this is useful is it  gives you sort of four  flaps of paper to work with you know  maybe you make one of them  uh the head and the other two wings  and the back one a tail if you're making  a crane it doesn't actually starts  not from the space but from the other  one  but same idea if you're folding a crane  one of these would be  head other tail and two wings this is  great if you're making a four-flap  animal and if you think about its  projection and it's  easier to think about in this other  picture  the projection of this thing is a  four-limbed  star right you can see it's a plus sign  all of them are the same length and so  this is actually something you can get  out of the tree method you just give  that as your input  you will get this 3d thing and this  crease pattern as your  uh  origami as the output let me show you  the program in action  uh so robert lang implemented this thing  it's called tree maker  it's freely available open source all  that good stuff  and uh yeah good  and i'm not an expert at using it so  bear with me  but if we wanted to say i would like  that  star okay now i i drew it obviously not  with all the lengths equal  but it's ignoring the lengths that i  drew and they're actually specified here  so all the lengths here are supposed to  be one  and then i say okay optimize and then  make a crease pattern and then show me  the crease pattern  there it is exactly the crease pattern i  made although actually  i can see from the mountain valley  assignment because this is not really  this is not flat origami  it's it made it this way so it's flat of  course the projection is the same  still for limbs okay  and so you know stash is for valleys and  solid lines for mountains  but you can make anything you want so  let's say  we want to make uh i don't know a lizard  or something  so the blue lines are the tree  uh  so here's can i do this yeah there we go  uh  so here i have a forearm head  four leg another forearm body segment  tail and two hind legs maybe i want to  make that  so i say optimize and then make a crease  pattern  and then boom you fold that it will have  exactly that projection  and assuming it did a reasonable job at  computation this will be the smaller the  best way to fold this thing and it's the  smallest square that matches exactly  that shape you get the best scale factor  between the size of your piece of paper  and the target shape  but actually doing that optimization the  first step i did  is mp complete so it's not going to do  it perfectly we'll prove that  wednesday but it the heuristics are  pretty good  it finds a local minimum and often it  finds a pretty good one  and sometimes you can coax it to find  better ones but yeah  it's not perfect but hey it's mp  complete so you can't do it  this actually shows you what it would  look like in x-ray view  and then you can say oh that was nice  but let's where's my  yeah that was good but maybe  i really wanted the head segment to be  shorter uh you know length 0.5  and then i wanted the tail to be really  long  and then you can optimize that and find  a crease  pattern and it'll complain because it's  having trouble oh dear  demo effect i should have tried this  example before  it's not oh gosh it's one of these  annoying ones  i should say  i should add some feature like  maybe strain split  something add a little bit  maybe do that and yeah there we go  uh i cheated and i'll explain how i  cheated last time but  if sometimes the particular method fails  but you can fix it by adding in another  tiny limb off the edge somewhere and  of course you can then get rid of that  at the end when you're folding  it only makes the problem slightly  harder and it'll still find a folding  but we'll talk about that  next time and i'm way out of time  so we will stop there

2
00:00:04,630 --> 00:00:08,950
today we're talking about  the local behavior of a crease pattern  so you take some crease pattern for some  flat folding we're thinking about flat  foldability  is a foldability question i give you a  crease pattern like this i want to know  does it fold flat like this one does  and we're studying what happens locally  right around a single vertex  so i didn't mention sort of graph  terminology it's probably useful to  mention that  these corners wherever all the edges  come together those are vertices these  triangles or in general some regions  divided by the creases  we call faces all right so  uh if we just sort of imagine cutting a  little  disk around that vertex and seeing how  it behaves  we get a nice circular piece of paper  with some  crease pattern and we want to understand  when those things fold flat  and when they don't sometimes they do  sometimes they don't this  is kind of like a one-dimensional  folding problem so let me write some  stuff  about that  so we get we call single vertex crease  patterns  so something like this is the example i  have  made something like that that should be  flat foldable  so we think of uh having  a disc of paper  it doesn't really matter but it's it's  going to be easier to reason about and  think about a disc of paper  really we just mean some small region  around that vertex  there's obviously one vertex in the  crease pattern  let's say we have n creases  emanating from one vertex  uh this pattern is going to be defined  by  some sequence of angles  in general theta 1 up to theta n if  there are increases there will be n  angles between them say in clockwise  order  and let's see  all right when we fold this thing flat  like in this picture we take  this disc we fold it along all of those  creases  what i'd like to focus on is what  happens to the boundary  of the paper so there's this outer  circle the boundary is a circle  and locally if you look at one of these  creases it's like folding the circle in  half  onto itself and when you make all of  these creases if you're successful  you end up folding the circle onto  a portion of the circle so if you look  at this boundary which is a little hard  to see  why don't i trace it it's gonna do  something like  this  okay i'm gonna  so this is what the 3d thing looks like  blown up a little bit and i changed the  angles a little bit  but this is what a flat folding would  look like and i'm really just focusing  on  how the circle maps around this circle  so one of the a flat folding of a disc  also folds a circle onto a circle  so we get  we have a flat folding in one of these  single vertex crease patterns  we get a folding  of a circle  onto a circle this is nice because  circles are one-dimensional and last  class we talked about  folding one-dimensional line segments a  circle is just a little bit more  complicated  it's a different topology we can use a  lot of the same  mindset at least as we did for folding  line segments onto  the line in particular you can see  yeah this is kind of circular but you  could also imagine  unrolling it a little bit and making it  straight  something like this so that it lies  on a straight line if you just unroll  that circle  so you also get a folding of a circle in  some sense  onto a line  by unrolling  now at this point i want to mention a  slight issue  here i mean this  for this unrolling to work  if you're taking a circle like this  in particular you must make at least one  fold for this to be possible  because right now if i have if i haven't  folded anything i just have a big circle  i can't unroll a circle onto a line i  haven't  collapsed it yet here i've made it small  enough  that it only occupies a portion of the  entire circle and so i can unroll it  so this is true as long as i make at  least one fold  okay technically i need to assume  another thing  which is that this thing came from a  flat piece of paper  uh i said there are these angles and  what i intended to mean  is that the sum of those angles  is 360 degrees as it is in a flat piece  of paper  but i'm not always going to require this  assumption it's  it's what we care about for thinking  about a crease pattern of a flat piece  of paper  and we look at each vertex this will be  true  but when we reason about these single  vertex crease patterns it's really  useful  to think about sort of in the middle  this could get smaller  so for example you have a pattern like  this let's say i just take  these two creases in the bottom and i  fold a crimp  because i know a crimp isn't a kind of a  good thing to do  so now i have what's called a cone of  paper  the sum of the angles there's two angles  here this one and this one each one  looks like 270 no not 270 135  in total it's 270. so the total angle  there is now less than 360. this is what  we call a convex cone  so if you relax this constraint  to allow less than or equal to 360  then we get a convex cone  and today i only want to talk about flat  paper and convex cones we need this  for the induction essentially for  proving things about the flat case  in the textbook if you look at this  chapter on single vertex  crease patterns they als we also prove  or study the case where the sum of the  angles is greater than 360 which you  could never make from flat paper  but hey it's fun to think about it's a  natural generalization  and you can characterize all the things  that we do here you can generalize to  that situation  it's just a little bit harder  all right  so what does it take to fold a circle  onto a line in this situation of a  convex cone  where we make at least one fold our  folding is going to lie along portion of  the circle we can  unroll it a little bit get something on  lying on a straight line  it's kind of like  folding a line segment onto a straight  line  that was flat folding of one-dimensional  pieces of paper  but there's a lot of differences but  they are similar  but a lot of the things we proved last  time do not hold in the case of circular  paper  so i have a bunch of them here  so one problem you may recall before if  we had a line segment  and we put an arbitrary crease pattern  then we could just assign a mountain  valley assignment alternating mountain  valley  and that would always fold flat no  collisions  okay in this world of having a circular  piece of paper  that's no longer true for example  if you have this kind of crease pattern  on a circle or  in the disk it would look like this  those two creases  you can't fold that thing flat it's not  going to work  okay we'll see exactly what what you  need to forbid for that to make sense  another example  is something like this  so if you're all right so  if i try to draw this in the the line  what's happening  is i have a short segment and then i  have a long segment  and so the problem is those ends don't  meet but even when the ends meet  like in this picture imagine these are  vertically aligned  if i do this alternating mountain valley  pattern mountain valley assignment uh  i can't join these ends and make a  circle because that would collide with  everything  so this thing actually does have a flat  folding  starting from a circle and i'm not going  to try to unmap it to a circle  it does fold but not like that  not with that alternating mountain  valley pattern  there are other annoying things let me  tell you  sort of one more on the proof side  last time we had  a lemma that said if your crease pattern  is mingling which still is a meaningful  notion here then you have a crimp  or an end fold in that case here of  course we don't have any end folds  here we also do not get this implication  mingling does not imply the existence of  a crimp and the  where the proof breaks down is if you  have  so if you remember our parentheses from  last time  we had this kind of pattern and we said  well  you keep going and either on the left  side you have a  beginning open paren or on the right  side you have a closing  round parenthesis in either case you've  got an end fold now this can actually  happen in a circle  and so you have nothing you can do so  you could have this pattern  there would be no crimp and probably in  that case you would not be flat-foldable  but in particular this implication fails  so  we have to do more work and but we're  going to get the same kind of result  that in linear time  we can tell whether one of these things  folds flat  so that's good news  so uh before last time  telling whether a crease pattern folded  flat was trivial the answer was always  yes  now it's not so trivial so let's start  with characterizing crease patterns that  fold flat  then we will go to mountain valley  patterns that fold flat  so  be great if i could just say that but i  mean here of course single vertex  so i give you one of these sequence of  angles  let's say it sums to 360 or less  when is it going to be flat foldable  and the answer is very simple  let me write it down  this thing is going to be flat foldable  if and only if  the sum of the odd angles  is equal to the sum of the even angles  and yeah i'm implicitly assuming and  requiring here  that n is even that's something we'll  prove so  n is actually even and minus one is  going to be odd  and if you happen to have started with a  flat piece of paper  uh for the sum of all of these things is  360 then of course  if you have them evenly divided this sum  will be 180 degrees  but if you started with the convex cone  that it might it'll be smaller  but for this reason a lot of people or  some people call this the 180 degree  property  or the pi property if you'd like to  think in radians instead of degrees  and we can do a little example here this  guy  this crease pattern uh has  a 90 degree angle here and it has a 90  degree angle down here  so the sum of those two is 180 those are  alternating angles  different parity classes this is 45  this is the complement 135  and that was also something 180 of  course one does it  only if the other does so it's flat  foldable we're golden  something like this is bad because  i mean it's an even number but you the  odd angles do not equal the sum of the  even angles  one's clearly bigger okay and this is a  general property this is  also called kawasaki's theorem or  kawasaki jost  jostan's theorem they proved it in 1989  so this is pretty much the beginning of  origami mathematics  so let's prove it there's two things to  prove one is that uh  any flat foldable crease pattern has  this property which is pretty easy  and the other which is a little bit  harder is that if you have this property  you really are flat foldable that's the  full characterization  so let's start with the easy direction  if you're flat foldable you must have  that sum  okay so it's all about  thinking what happens at a crease  so we're thinking about the boundary of  this piece of paper thinking about the  circle  and what's happening on the circle  is you travel uh  say counterclockwise for some amount of  time that would be  first angle then you make a fold it's  either a mountain or valley at this  point we don't care  either way you turn around maybe you  turn around this way or you turn around  this way  but in terms of your travel along a  circle  so we went theta one counterclockwise  now we're going to go theta two  clockwise then we're gonna go theta  three  counterclockwise and so on we keep  alternating back and forth  it's much easier to think draw on a line  so we go  to the right theta 1 then we go to the  left  theta 2 then we go to the right  theta 3 and i don't know about the  layering here  i'm just drawing it in terms of  horizontal travel the y-coordinate means  nothing  okay in order for this thing to be a  flat folding  to be a valid folding of a circle of  paper  these guys have to have the same x  coordinate  so i don't know i mean the y coordinates  they also have to work out but we're not  thinking about the y coordinates we're  just thinking about x travel  we've got to get back to where we  started if we're folding a circle  that's it so how much trout how much  total travel do we do  in a sine sense how much do we go right  well we go theta 1 to the right  then we go theta 2 to the left which is  like going negative theta two  to the right and then we go theta three  to the right  then we go negative theta four to the  right and you add up  this alternating summation that must  equal  zero okay the other thing  uh that i should say is that n has to be  even  um this is maybe obvious  at this point but we're supposed to  alternate every time we hit a crease we  have to change direction  and so in the end it's important that  after theta 4 we change direction to  visit theta 1.  if they were aligned like if theta 4  went over here  and then theta 5 went like that  and there was an odd number of creases  this would be bad even though they're  lined up there's no crease here you just  went straight from theta 5  to theta 1. you're supposed to change  direction every crease  so also n is even  to alternate directions  question all right all right good  clear this is pretty easy once you  realize the angles correspond to x  coordinates in this linear space  so why is this enough in order to show  that  this property is enough for flat  foldability  we need to actually worry about the  stacking order of these edges we need to  worry about the y coordinates we need to  make sure that they can avoid collision  with some mountain valley assignment the  good news is we're free to use whatever  mountains and valleys we want  we can stack things however we want we  just need to find some valid state  and our intuition from the  one-dimensional case was  we should try alternating mountains and  valleys that seemed like a good thing it  avoided collisions  for an open piece of paper for a line  segment of course we know that's not  enough  here somewhere i drew  this example so it's alternating in the  middle  and then that last crease is a problem  but that's all we need to fix  turns out it's not so hard  so we have a nice circle  what i'd like to do is cut that circle  so here i sort of cut it and thought  about it as a line segment and then i  wanted to rejoin  okay i'm going to be careful about where  i cut it  okay i want to cut at an extreme left or  extreme right it doesn't matter  you can see here the cut ended up being  in the middle that's bad  so how do i fix it  what does extreme mean how do i tell  given a circle it's a little hard to say  what extreme means because it's  circular but if i take a picture like  this i've already drawn it with nice x  coordinates i don't know about the y  coordinates yet  say oh that's bad my cut point ended up  being non-extreme  well this is extreme so cut here instead  okay so what that means is redraw this  picture  with this being the cut point so maybe  this is the beginning  so we go like that again and now we wrap  around  so now we have this segment and now we  have the segment  and lo and behold it remained extreme  and this is always true you can do this  sort of  cut and rejoin the ends that were messed  up before  and you'll preserve the x coordinates x  coordinates aren't changing when you do  this kind of transformation  so there's a clear leftmost and there's  a clear rightmost you pick either one  you cut there instead  and now we have this nice property that  it's easy to join the ends  because it's that as far left as you can  go there can't be anything that  penetrates because that would be  farther left  all right let me write down some words  corresponding to that  so once we cut at this extreme crease  you can fold this 1d  segment just like we used to be able to  so we can fold it flat  using the accordion fold alternating  mountain and valley  we know that works and then all we need  to show  is that the two ends can join  those ends are the two copies  of the cut crease  that we did in this first step  uh we need two things one is that they  are aligned  and that's where we need this condition  that the sum of the odd angles equals  the sum of the even angles which is the  same as saying the alternating sum  is equal to zero that tells us that  whatever we do the x coordinates are  lined up at the end  so they're aligned by the by the  assumption that we made here  and  you can join without crossing  so this is a statement about  y-coordinates and we know it's okay  because it's at the left extreme  so by this choice of the proper cut  there could be other things that come  right to the boundary here but that's  considered okay that's not crossing  that's just touching  nothing can go farther left because it  was the left extreme  that's the end of the proof any  questions about that  it's kawasaki's theorem  i'll mention just for fun this sort of  classic  if you allow one of these  a non-convex cone where you have more  than 360 degrees  of material this statement is not true  and what can there's one other situation  which can happen  which is that the alternating sum of  angles is not zero but it's uh plus or  minus 360 degrees  which is fun and you can read it you can  see examples of that in the book  all right so obviously if you want to  tell whether a crease pattern  or a single vertex is flat foldable you  just compute this thing in linear time  and you know  yes or no uh so crease patterns are  pretty easy just have one extra  condition  what about mountain valley assignments  last time  we looked at mountain valley assignments  and we showed that crimps and end folds  were always enough to fold any mountain  valley assignment that's out there  that will continue to be true here  except now we don't have end folds now  it's crimps all the way  what does it mean  so  that is the flat foldable mountain  valley patterns  we're going to prove that crimps are  enough but before we get there i need a  bunch of sort of warm-up  facts about valid mountain valley  patterns  the first thing is just counting  mountains and valleys  so even if you play with a simple  example like this one  you see it has three mountains and one  valley  or three valleys and one mountain in  fact if  if you have a degree four vertex with  four creases emanating from that point  those are your only choices you could  try for example making it all valleys  and it i mean you can't do it  uh it's hard to demonstrate uh you could  try  making it two mountains and two valleys  so maybe i try to do mountain mountain  here  and then somehow it's gonna be valley  valley there it's  no good okay you can that what's  happening here is that the number of  mountains  and the number of valleys must differ by  exactly two it doesn't  matter who's bigger so the difference  could be plus two or minus two  because you can always flip over and  negate that difference  swapping mountains for valleys but they  always have to differ by exactly two  why is that well  this is uh yeah we're again going to  think of a picture like this we have  horizontal  travel according to the theta eyes we're  also going to think about the vertical  picture  if this thing is flat foldable so i'm  assuming it's flat foldable  then this must happen this is called  maekawa's theorem by the way  it's proved by june mayakawa in  1986 a little earlier  also proved by jacques christine around  the same time  back before we were good at internet and  communication and whatnot  and there were little pockets of  mathematical origamis  and they found each other basically in  89  when there was the 89 as the first  origami science math and education  conference  started bringing these people together  the last one was this summer  right so why is this true  i don't know flat folding looks like  something okay it has no crossings  it has horizontal travel we understand  it ends up back where it started there's  also some notion of layers and vertical  travel which is a little tricky to think  about  but the one thing that's easy to think  about i mean this forms essentially a  polygon  it's like a really squash polygon and  these vertical segments are actually  really just  points okay imagine  that's just where you turn around so  it's like a polygon just  stretched or squashed down onto a line  we know some things about polygons  okay it's an n vertex polygon you know  that like the sum of the interior angles  is  whatever it is i always get confused  with that formula so i don't like to  think about it  one thing we an easier way to think  about that same statement  is just think about how much turning the  polygon does so you start here  you turn right 180 degrees you turn left  180 degrees  you turn left 180 right 180 right 180  right 180. how much is it in total oh  gosh  do i have to add no it's so simple it's  360.  right just if you think about you're  going around in a circle  any polygon not just flat but anything  in in two dimensions  uh the total amount of turning you do is  360. or if you count backwards it's  negative 360.  okay so this is the key something you  should all know  about polygons if you look at the sum of  the turn angles how much turning you do  at each vertex  that will always be plus or minus 360  in any polygon  this is equivalent to the sum of the  interior angles being whatever pi times  n minus 2  which i don't remember but this is much  easier to remember  and it's useful because we can map  mountains and valleys  to left and right turns as i said every  time this was a valley  it was a left turn by 180.  every time it was a mountain it was a  right turn by 180.  valleys  mountains  right turns so i'm going to think of  that as negative 180.  now of course technically it could be  the other way around could be this is  negative this is positive but  i already have a plus and minus here so  it's symmetric  so if we sum up the turn angles which we  know is supposed to be plus or minus 360  some of the turn angles  is going to be we're going to have 180  for each valley so 180 times the number  of valleys  and we're going to have negative 180 for  each mountain  and so the 180s factor out  and this thing is supposed to equal 360  plus or minus 360. how could that be  well number of valleys minus number of  mountains should be either two for plus  360 or negative two for minus 360.  so that proves the theorem  and that's why in a degree four vertex  one of them has to be three the other  one has to be one  there's no other option because that  total number is four  it's the only way to get the difference  to be plus or minus two  so that's kind of nice but it's not  enough  unfortunately if i gave you a mountain  valley pattern that satisfies this  condition  it still might not be flat foldable  this may be no surprise it's not like we  had such a simple test  for the one dimensional case should be  easier we had to give an algorithm we  said repeatedly crimp and enfold  if you get stuck the answer is no if you  don't get stuck  you folded it great  i love this eraser increases entropy  all right let me tell you one case that  is easy  to think about and because it it's nice  it reduces directly to the one  dimensional case  and that's what i call the generic case  generic is this very convenient term we  use in mathematics it's actually really  tricky to define  so i'd like to not define it if i can  but  maybe i should  yeah all right uh  the simple version which is not quite  enough is to say that all of the angles  in the crease pattern are different  okay here for example two of them are  the same there's two 90 degree angles  this is not generic  it's just easier to draw but in general  you  i mean you imagine you just draw some  random thing  none of those lengths are going to be  the same yeah the alternating  sum is 0 but that's it  and that's what i the generic case is  that is the only thing  that holds true about these angles you  can take the alternating sum  that equals zero if you take some  alternating sum of some subset of the  angles that will not equal zero  if you take a random example this is  going to be true okay just  bear with me suppose never any two  angles are the same  then look at the globally smallest  crease  it's the globally smallest theta  say it's theta i so the picture is  theta i theta i minus 1  theta i plus 1. and what do we know if  it's globally smallest and none of the  values are equal  then we know that these two are bigger  no big surprise uh think about what  happens  if this is small these are bigger and  you try to make these both valleys  or both mountains that's of course bad  this is one of the  situations we had in the one dimensional  case i mean really this is a circle  but i can think just locally about  what's happening here so  in this situation one of these must be a  mountain  and the other must be a i don't know  which order they are  but i know they're different and then i  know i can apply  one of these crimps  incidentally the word crimp jason kuh is  asking about like why do you call it  crimp and not pleat  probably i should call it pleat but it's  crimp in my mind it's been crimped in my  mind since 1998 when we wrote this the  1d paper  before i knew the word pleat and so  that's my excuse  so pleat or crimp take your pick crimp  is usually mini pleats together  that's crimping  cool so crimps we kind of like why do we  like  crimps  we proved in the one-dimensional case  last time that  if you make a crimp and your original  thing was flat-foldable the new thing  will be flat-foldable  good news is that is still true i told  you all these things were no longer true  in the circular case  it's still true in the circular case uh  should i tell you let me just remind you  the same proof works  remind you what the proof looked like we  had a crimp  and then and we're hoping to fold that  first  we know at some point these creases get  folded but there may be other junk in  here  maybe other junk in here we'd like to  get rid of that junk  and so what we did is move this junk up  to here which was always safe we moved  this junk down to here  and that was still a folded state a flat  folded state  okay i remember all this stuff  therefore we could have folded the crimp  first and then we have a flat folding  in this situation where this paper is  effectively glued together  which means it was safe to fold this  crimp and glue these together and never  touch them again  so that is still true in the circular  case that proof didn't really assume  anything about the paper would even work  for like tree-shaped paper or something  weird  so if we can find a crimp which we can  in the generic case  then you just make a crimp make a crimp  repeat if you ever get stuck you know  that the original thing was not flat  foldable because the thing you have is  not flat foldable  why if there's no crimp in the generic  case that means you have two valleys  at around the smallest angle and that's  clearly bad  um so i'm not writing a lot of details  here is that clear  if i have an angle that's surrounded by  strictly larger angles i know  there must be one mount in one valley  then i can safely make the crimp and  repeat  the trouble comes in when these angles  are equal  we never had to think about angles being  equal in the one-dimensional case it  didn't matter much  we just said oh you know it's safe to  make a crimp  as long as this was greater than or  equal to this  and this was greater than or equal to  this that is still true it's safe to  make a crimp  but how do we know that there is a crimp  how do we know that there's a mountain  followed by a valley  we don't i mean if they're equal this  this would be all right  that's two valleys is valid if these  three angles are equal  but i claim still somewhere there's got  to be a crimp there's no longer a notion  of the globally smallest angle because  some of them are equal  we've got to find that crimp but it's  out there  so we just need to prove that it exists  so  so to do that we're going to generalize  this  theorem and kawa just in that number of  mountains minus number of valleys is  plus or minus two that's true  and it's a statement about the sum of  all of the or you know  all of the angles all the creases what  i'd like is something a little bit more  localized  and in particular we're going to think  about when i have a whole bunch of  angles that are equal how many mountains  and valleys can there be in there  so what i call local counts  so while everything i've said so far is  pretty classic this result  was proved in 2001 by tom hall so much  more recent  over a decade later  so let's think about k equal angles  and i want that to be a maximal sequence  of equal angles so the ones right after  and right before  are different and furthermore i'm going  to assume that they're bigger  strictly larger  angles okay so  something like that a bunch of equal  angles k of them  bigger angles on either side this almost  always  exists um i should mention this it's so  i'm going to say something about this  situation but the situation should exist  because  i take the smallest angle out there and  then i see how many friends it has that  are all equal  if it's the smallest angle or it's one  of the smallest angles  then the ones are surrounding it are  going to be bigger  unless all the angles are equal  so there's one case which we'll worry  about later it's very easy  all the angles are equal in fact i could  tell you how to worry about that  if all the angles are equal  then you know everything is crimpable as  long as we can find a switch between  mountain and valley  somewhere and because the number of  mountains and valleys is plus or minus  two  there's got to be so they're almost in  equal balance  there's got to be a valley it can't be  all valleys  can't be all mountains so somewhere  there's a transition between mountain  and valley  so you get a crimp out of that  okay but otherwise if they're not all  equal let's think about k  angle equal angles surrounded by two  larger guys  forgot how technical this is all right  then  i want to look at the number of  mountains and the number of valleys  among these uh k plus one creases  so within that range  it's going to be zero if k is odd  it's going to be plus or minus one if k  is even  okay it's not plus or minus two  intuitively plus or minus  two is when you go all the way around  here we're just looking at  a one you know segment portion of the  entire circle  remember this is actually a circle  somewhere  and when you this corresponds to  in the odd case you're going to be going  in the same direction afterwards  and the even case you're going to have  turned around but not a full circle so  it's going to be plus or minus 1.  there's a lot of ways to prove this one  way is to think about convex  cones so  if k is even and you fold this thing  see one two three four five six  equal angles and then there's the two  longer guys they may not be the same  length  but that's what you get in the even case  so what i'd like to do is just  in order to just reduce the things that  we've already thought about i mean you  could talk about turn angles again  or you could just say hey i know  mayakawa's theorem  so as long as i can make this into a  circle which i can do  by adding that stuff now i have a nice  circular  it's a flat folded state there's no  crossings here  presumably because this did something i  don't actually know that it's zigzagging  it might do other stuff  but it's not going to cross what i just  added  so the number of mountains minus number  of valleys in total here  must be plus or minus 2. i only added  one crease this one  so that's going to make it either plus  or minus 1 or plus or minus 3.  if you think about it a little bit it  has to be plus or minus 1 and hopefully  that's what i wrote here yes  because i mean really we're thinking  about turn angles again i'm afraid  can't totally reduce it we just turned  around here  and we're about to finish the circle  which would make it plus or minus 2  before we finish the circle is plus or  minus 1.  k is odd it's  again this might go like that whatever  but because it's an odd number these  guys  are going in the same direction and so  the  total turn angle here must be zero and  one way to see that  is to extend it into a full polygon  you know that it's plus or minus two  for this whole thing and i added two  guys  in the same direction if i remove them  it's going to go down to plus or minus  zero  that was a bit hand wavy but do you buy  that  yeah plus or minus zero is zero  yeah you went from two down to zero  because you removed two identical guys  so let me let me tie this all together  because there's so many  little cases  so i want to know what mountain what  mountain valley patterns are flat  foldable  okay single vertex  and the patterns  okay i claim  there is a crimp if it's flat foldable  there has to be a crimp that you can  find to do remember crimp was  where this length  was less than or equal to this length  and less than or equal to  this length and one of these was m one  of them was v that was the definition of  a crimp  claims such a thing always exists the  proof is  two parts if all the angles are equal  all the angles are equal we know that  these conditions are always satisfied  because everything is equal  so we just need to find an m followed by  a v or v followed by an m  and by maekalo's theorem  we know it's not all m's or all v's so  somewhere there has to be a transition  from m to v's  so just use that lemma theorem count  property otherwise they're not all  equal then i can apply this local counts  result  take the smallest angle take one  instance of the smallest angle there  could be many of them  find all of its neighboring friends that  are equal  then i know that the next angles before  and after have to be strictly bigger  then i know that the number of mountains  and valleys in here  if i look at the difference and it's  zero  that means the number of mountains  equals the number of valleys therefore  there's at least one of each  therefore there's a transition for  mountain valley in there there's a  transition from mountain to valley  like right here that's a valid crimp  this guy's bigger  this guy's equal this would also be a  fine transition because everybody's  equal  this would also be a fine transition any  any transition from mountain valley in  here  i've got a crimp the even case is a  little  i've got to think a little bit more k is  even  k plus 1 which is the number of creases  here is  odd and the difference between mountains  and valleys among those k plus one  creases is plus or minus one  think about it for a second that means  there's again at least one mountain and  at least one valley  and uh because cave to be even has to be  at least  two you've gotta have at least three  things so there's gonna be at least one  of  one of one and two of the other in fact  and i just need one mountain in one  valley somewhere in there and then i  know there has to be a transition from  mountain to valley  otherwise use local counts  and in either case  we get either a mountain to valley  transition  or a valley to mountain  transition and i get it in such a way  that these  these inequalities and the lengths hold  so in other words that it's a crimp  therefore any flat foldable single  vertex mountain valley pattern  has a crimp make it and as i was arguing  here  just like last time that crimp will  still  after you do the crimp you'll still be  flat foldable so repeat  cook until done is the i like to put it  question what about the trivial case of  just floating  trivial case of folding a circle in half  all right that's a good question  it's kind of a technicality  uh one version would be to say that's  zero vertices  that doesn't count we're thinking about  one vertex and as soon as you think  about it as one vertex which  is fine then you actually have two  uh creases there so it's two creases and  hopefully this works  uh oh i see you're saying it's not a  crimp that's a good point  it is flat foldable and yet there's no  crimp damn it  all right sorry my  college right yeah so it's i made a  slight mistake here  i said if all the angles are equal then  meikawa says there's at least one of  each  that's not true when n equals two then  there's  two of one and zero of the other okay as  long as you have more than two  creases then this is true  that's enough so you have to handle this  as a special case  uh that's true this this this is an  important special case to remember  because this will be the final picture  after you do a sequence of crimps  you this is the good case uh  if your thing is flat foldable you will  always end up with that that's a  case where everything is excuse me the  same  smallest it's it's another example where  all the angles are equal  yeah so that's why this is the situation  it's either maekawa and you get a crimp  here because n is greater than 2 and  here i should say  if n is greater than 2  then this is true of course the n equals  2 case  it looks like that it better be both  valleys or both mountains otherwise it's  not going to be flat foldable i forgot  about that  i should add in the notes but it's in  the textbook right  that's true it won't actually look like  this it's going to at the end of the  algorithm it's going to be a cone  the two angles will be equal because of  kawasaki's theorem  and they should still be both mountains  or both valleys but it's going to be  we can see it right here so i apply my  crimp  and now i have a cone and there's two  angles there's one  here which has been fused together  there's one here  which is original and they're equal and  it's two mountains  or two valleys if i turn it upside down  and then it finishes  so that's the algorithm in action two  steps first you find this crimp  here this is actually the globally  smallest angle it's surrounded by bigger  angles so it's really easy  then i have a cone with two equal angles  which is what has to happen at the end  and then i i'm done  good  this is what always happens oh  is this the only way to flat fold these  things no  there are other ah yes  one of the two um  i think it depends what you're counting  if you're counting mountain valley  assignments i mean if you just want to  know  can every mountain valley assignment be  folded by crimping then the answer is  yes that's what we proved  but if you want to get every possible  folded state  crimps are not going to be enough the  reason they're not enough is because  we're using this thing  said hey here's some hypothetical folded  state we can rip it out and make the  crimp have been done but then you're  there's a folded state you're not able  to reach by crimping  i mean we need to work that into an  actual example where crimping forbids  you from reaching some folded state but  i'm pretty sure one exists because of  this  yeah maybe some kind of spiraling thing  right what we're missing here is getting  all the possible layer stacking  orders so we're just trying to match the  mountain valley assignment  we're not going to match a target layer  ordering because  we simplified the layer orders in order  to make crimps possible  so crimps won't get you some some of the  folded states  but one thing you can play with  over here that has been played with by  tom hall  is if i give you  a crease pattern  how many flat foldable mountain valley  assignments does it have okay we have  this nice linear time algorithm to tell  you whether a particular one is doable  how many are there and  there is if you work through all of the  things i've shown you in a little bit  more detail  you can recover this was finding a crimp  you could actually look at what are all  the crimps that are possible  and actually count how many different  ways  there are to fold how many crimps you  can do and then given those crimps how  many crimps you can do after that  and in same linear time kind of  algorithm you can figure out how many  different mountain valley patterns are  flat foldable that takes a little bit  more care  i will tell you the extremes  how small could the  number of flat foldable mountain valley  assignments be for a given crease  pattern  well do you have any guesses when when  do i have the least  choice of where to make crimps  yeah uh  for general n sorry but yeah three  three angles are i guess two angles i  really can't do any i don't have any  choice  but for general and how should i  disperse the theta eyes  in order to make there be very few  possible crimps  when there are only two transitions  between mountains and valleys all the  mountains are together all the valleys  are together  that's right if i was asking the  question about mountains and valleys but  i was asking a question about theta eyes  so i want to count how many mountain  valley patterns there are  mountain valley assignments that are  flat foldable  for a given crease pattern  all the angles are consecutive  increasing yeah that pretty much works  in fact what i need is the generic case  where all the angles are different  and they never become equal by folding  which is what generic means so then  there was only really one crimp i could  do  which is the globally smallest angle  so actually i still have two choices i  could do mountain then valley or valley  than mountain  but i only have two choices for that  crimp then i will have two choices for  the next crimp  two choices for the next crimp in  general i get two to the n  possible mountain valley assignments for  the generic case  okay what about what's the opposite  where i get as much choice as possible  yeah two to the n no uh  two to the n over two thank you right  they're two to the end conceivable  mountain valley assignments can't be  that big  uh every time i do a crimp i eat two  creases not one that's what i forgot  good still pretty big though  it's like square root of all the  possible things you could imagine  are indeed feasible it went from 2 to  the n to 2n over 2.  when would i get the most choice in  crimping  all angles are equal that's the opposite  extreme and in this case  a little messier  when all the angles are equal the only  property you need and  you can see that from the proof the only  property you need is that the number of  mountains minus number of valleys is  plus or minus two  all the a and the n is even as long as  you have that  uh we showed here you're going to have  an alternation from mountain valley  that's a valid crimp because all the  angles are equal and you can keep going  so it's just how do you disperse that  many  how do i disperse n over two minus one  mountains among n different positions  that's what this represents and  positions how do i pick n over two minus  one of them to be mountains  or could be n over two minus one of them  are valleys  and that's this factor of two so it  could have more mountains or more  valleys  and then you somehow place those among  those and this is if you don't know this  notation  that's just what it means if you want to  know  with using other notation it's like this  n over 2 minus 1 factorial over 2  plus 1 factorial if you don't know  factorials  i'll tell you about them later  cool so that's kind of the end of the  single  vertex situation um  yeah i think i'll mention one  interesting open question here which i  would love to explore maybe in our  problem session  um which looks like it'll be mondays at  five i think um  this is one vertex and we went through  all this work to solve single vertex  situation and it's interesting  as we'll see to think about locally how  each vertex behaves  but you would think if you have a crease  pattern with two vertices  shouldn't be that much harder so here we  have linear time  how quickly can you tell whether a two  vertex crease pattern  is flat foldable as far as i know no one  has looked at that problem  surely we can do it in like quadratic  time but maybe even linear time i think  can't be that hard i think  in general if i have a small number of  vertices say k vertices  much smaller than n creases  is there the algorithms people is there  fixed parameter retractable algorithm  and k  or can i get something even getting  something like n to the  some function of k would be progress i  think this should be doable  but ideally we get a running time that's  exponential in k and linear in n  that would be my hope why do i say it  has to be exponential because in general  if i give you crease pattern with n  vertices lots of vertices  this problem is np complete which is  there's not going to be a polynomial  time algorithm for it  nothing good we will prove that next  wednesday  so i'll hold off on that a little bit  but for two two vertices how hard could  it be  all right  i want to talk about one related topic  and then we will go to origami design  and do a little bit on the tree method  but before we get there  i want to talk about local foldability  which is a cool topic people tend to  forget about it i really like it i think  it would make a cool project also  it's a nice algorithm it goes back to  vernon hayes 1996  so it's also right at the beginning of  origami mathematics  and it's this idea all right i give you  a crease pattern now  arbitrarily many vertices and i ask you  if i ask you does it fold flat that's  np-complete intractable  same paper but what if i ask you just to  give me a mountain valley assignment  that might fold flat well  that's also mp complete but if if you  actually want it to fall flat this is  really the same problem  but if i ask you give me a mountain  valley assignment  so that if i checked every vertex  according to this algorithm  at least every vertex folds flat that  would seem nice  you know definitely i have to find a  mountain valley assignment that  satisfies  these conditions that as i do successive  crimps  every vertex if i cut out the vertex  separately it would fold flat this is  the notion of local foldability  and there's a linear time algorithm to  tell you to give you a mountain valley  assignment  that ought to work in that each vertex  it works it still may not work globally  for some other reason  but it's pretty good i think this would  be actually pretty practical  i think in a lot of real world origami  settings when you're doing  flat folding anyway locally valid  locally  foldable is going to be enough to be  globally foldable  that's a guess i don't know if it's true  consistent mountain valley assignment  if there is one  so that each vertex  locally folds flat  let me uh  try to concoct a small example that's  relevant here  gotta think  yeah i think that works all right here  we go  i think this is in the textbook i  remember drawing it in the past  three years ago or whatever all right so  here's a crease pattern  on a square whatever yes  i should have this saying thank you  uh so this satisfies kawasaki's theorem  that was the hard part  because these angles sum to 180 and it's  symmetric  all around okay this is uh these angles  are the smallest they're  60 degrees this is an equilateral  triangle  so we have this not quite the generic  case but we have a the smallest angle is  surrounded  by two larger angles therefore one of  these is a mountain  the other is a valley that means that  these two creases  have to have different assignments i'm  going to write a not equal sign  one of these is not in the other's value  they can't be the same also  these two cannot be the same also these  two cannot be the same  that's not possible because i've got  three  it has to alternate you can't alternate  three times you can only alternate an  even number of times  okay two two of these mountains one's  valley and then  you've got a problem okay so this thing  is not flat foldable  and this algorithm will tell you that  because it'll say hey i can't even find  a mountain valley assignment  that could possibly fold each vertex  flat  so this is a nice algorithm  at least it will detect annoying  situations like that  so in order to solve this we're going to  use  and i'm just going to sketch how this  algorithm works we're going to use this  characterization  and this idea that we can really find  all possible mountain valley assignments  just by trying all the possible crimp  sequences now there are exponentially  many so it's not like i'm actually going  to try them all  but i need to explore that space of  candidate crimps  and see what happens all right  uh so  the idea is kind of crazy  the beginning of the algorithm is fold  each vertex flat  somehow i don't care how just  pick a crimp do it pick a crimp do it  okay separately for each vertex they're  not going to be compatible  and don't look at the mountain valley  assignment you get but look at the  crimping sequence you get so here let's  do a little  example a  non-trivial example so here we have the  peace sign  and these two angles are equal they're  smallest i could crimp this angle first  or i could crimp this angle first i have  a choice so i'll draw both of them  if i crimp this angle first i know  these guys are paired up in the sense  that they must be not equal ones  mountain ones valley  after i do that crimp just like in the  real example i had  you know this angle will equal that  angle and these two guys  must have equal assignments they must be  both mountain or both valley  in this situation these two guys are not  equal  but if i crimp the this pair first and  these two guys must be equal equal now  in the sense of  being mountain or valley okay  so there's still you know exponentially  many possibilities in how to do this but  just pick one pick one of these ways of  pairing up you're going to pair up  each of the increases into n over two  pairs  and they're going to have some not  equals and equal signs  okay so now if you imagine that the  general picture like here  i for example i might get in this  pattern i would be forced to get not  equals  and these guys paired up and in general  i want to look at  these kinds of cycles  you know if i come if i come into a  vertex  here's a vertex it's paired up with  somebody so if i come in here i can go  out  somewhere and i come to this vertex it's  paired with somebody  so i'm gonna just i can keep wandering  around  and in general there will be a bunch of  these  paths that you can follow what could the  paths do they  either close up on themselves maybe  things are paired up  in such a way that you make a return  trip  or it could be some other path let me  draw a dotted path  it could come in here and maybe it gets  paired up with this guy  maybe it goes off to infinity it reaches  the boundary of the paper  those are the two possibilities you get  paths which go off to the edge  off to infinity on either end or you  could get cycles  the cycles are the problem because  whenever i have a cycle  i have a parity constraint for example  when they're all not equals uh the  length of the cycle must be even if they  were  yeah what's the general statement it's  like the parity of the cycle which is  whether it's even or odd should  be equal to the parity of the number of  not equal signs  something like that i'm going to cheat  here  great i just said parody problems it's  something like that  it's either the parity of the cycle  should equal the parity of the number of  equals  or the variety of the number not equals  i think number of  equals anyway one of the two  and you can just check i mean you're  forced if i say okay let's make this  mountain  then this is either equals or not equals  it'll tell me whether this is mountain  or valley just walk around the cycle  either you get a contradiction or you  don't  if you get a contradiction we have a  problem  how do we how could we possibly fix the  problem  well  when we made you look at each of these  crimps i mean in fact you could look at  each of these vertices separately but  you think about one of these crimps and  say well could i have done it another  way  sometimes there are crimps that have  other equal choices  maybe they're a bunch of equal angles  and i could have done a different  pairing  what happens when i try a different  pairing well instead of  this being in one cycle and let's say  this being in another cycle  if i pair these guys up instead i'll end  up merging those two cycles into one  bigger thing  could be a path or a cycle and the  algorithm says  just keep doing those cycle merges and  if you get stuck your thing is not  locally foldable  that's the hard part to prove otherwise  you will find  a one of these patterns that you can  actually  resolve mountains and valleys all the  way through so  let's say  so start with sum  folding say local folding  whatever i'm going to say with some  pairing  of creases at vertices  and  merge  two pads or cycles  whenever possible  and when i say merge i mean  whenever you have what are the possible  things you could possibly merge  when you have at some point during the  algorithm a bunch of equal angles  you have a choice which of these you  crimp obviously  the mountain valley sign is not fixed  you can crimp any of them  you picked one of them and it's sort of  merging  whatever this thing is attached to to  whatever this thing is attached to  if there's something else that's in  that's disconnected from that thing  i want you to instead merge two of them  that combines two different connected  components two paths or cycles  merging means i decrease the total  number of pads or cycles i combine two  into one  whenever that's possible do it you can  prove that if you have  a parity problem in the merge thing you  had to have had a parity problem  originally and merging can only fix  parity problems that's the claim that's  what i will not prove  once you know that and it doesn't matter  in what order or how you choose to merge  you just merge  as much as possible and either the  resulting thing is okay or not  and accordingly you will tell whether  this thing is locally foldable  sorry i want to move on to other things  but i think this would be a fun thing to  actually implement it's an easy  algorithm  and it's  i think a pretty good test for problems  like this to prevent flat foldability  all right  so let's move on from foldability to  origami design  so a bit of a big transition and we're  gonna  talk about origami design a lot more  next  class also but uh  just started off today and the  particular  algorithm for origami design i want to  talk about is called tree method  this is probably the oldest algorithm  for origami design  uh in that it's been around and people  have been thinking about it and  developing it  for many years uh through this period  called the bug wars and people are  trying to design  more and more complicated insects it's  like well i can make an insect  with six legs oh yeah well i can make  you know a spider with eight legs oh  yeah well i can make  an insect that has you know a beetle it  has wings and horns and there's thorns  on the horns and  all these crazy things during that time  there are a lot of people thinking about  how do i make more and more complicated  especially  uh sort of more more limbs in my  creatures  more and very precise uh arrangements of  those limbs let's say  and that is what the tree method deals  with and is really formalized by  robert lang who published a paper in  96 describing it as a as a  sort of complete algorithm and  it's still not known for sure that that  algorithm always works  but uh that's what we've we've been  working on me and  marty and rob lang for the last  four years or so and soon we will  publish that thing  and prove that this thing always works  but i'm going to describe to the  algorithm without the proof that it  works  and what it does tell you its goal  from a mathematical perspective  so it's interested in practical origami  design so we're going to start from a  square piece of paper  it would also work for triangular pieces  of paper or anything convex  but squares are what people usually care  about rectangles are  sometimes it's used for rectangles also  okay the idea is i give to you  a stick figure  so that's formally it's a tree a graph  without any cycles  it's a metric tree meaning that i put  lengths on the edges  i know this length this edge length is  maybe twice as long as this one  so i really draw it with edge lengths in  mind  then what i want you to do is find  some folding of a piece of paper  i should really be looking at what i'm  trying to match uh  so here here  maybe this goes down  such that i want to find one some  folding of a square paper in fact the  smallest square possible  so that when i project  like this vertically  the projection of that uh flat  folding is exactly that metric tree  okay and this is called a uniaxial this  thing is called a uniaxial base  let me tell you a little bit why it's  called any axial base  we're thinking about what are called  origami bases these are  like the beginning of origami models and  most classic origami models like more  than  60 years ago start from one of these  bases  you've got water bomb base in the top  left  preliminary base fish base  bird base windmill base  and frog base i remember them all i have  a little example here this is the water  bomb base  so it's just very simple crease pattern  and you know why this is useful is it  gives you sort of four  flaps of paper to work with you know  maybe you make one of them  uh the head and the other two wings  and the back one a tail if you're making  a crane it doesn't actually starts  not from the space but from the other  one  but same idea if you're folding a crane  one of these would be  head other tail and two wings this is  great if you're making a four-flap  animal and if you think about its  projection and it's  easier to think about in this other  picture  the projection of this thing is a  four-limbed  star right you can see it's a plus sign  all of them are the same length and so  this is actually something you can get  out of the tree method you just give  that as your input  you will get this 3d thing and this  crease pattern as your  uh  origami as the output let me show you  the program in action  uh so robert lang implemented this thing  it's called tree maker  it's freely available open source all  that good stuff  and uh yeah good  and i'm not an expert at using it so  bear with me  but if we wanted to say i would like  that  star okay now i i drew it obviously not  with all the lengths equal  but it's ignoring the lengths that i  drew and they're actually specified here  so all the lengths here are supposed to  be one  and then i say okay optimize and then  make a crease pattern and then show me  the crease pattern  there it is exactly the crease pattern i  made although actually  i can see from the mountain valley  assignment because this is not really  this is not flat origami  it's it made it this way so it's flat of  course the projection is the same  still for limbs okay  and so you know stash is for valleys and  solid lines for mountains  but you can make anything you want so  let's say  we want to make uh i don't know a lizard  or something  so the blue lines are the tree  uh  so here's can i do this yeah there we go  uh  so here i have a forearm head  four leg another forearm body segment  tail and two hind legs maybe i want to  make that  so i say optimize and then make a crease  pattern  and then boom you fold that it will have  exactly that projection  and assuming it did a reasonable job at  computation this will be the smaller the  best way to fold this thing and it's the  smallest square that matches exactly  that shape you get the best scale factor  between the size of your piece of paper  and the target shape  but actually doing that optimization the  first step i did  is mp complete so it's not going to do  it perfectly we'll prove that  wednesday but it the heuristics are  pretty good  it finds a local minimum and often it  finds a pretty good one  and sometimes you can coax it to find  better ones but yeah  it's not perfect but hey it's mp  complete so you can't do it  this actually shows you what it would  look like in x-ray view  and then you can say oh that was nice  but let's where's my  yeah that was good but maybe  i really wanted the head segment to be  shorter uh you know length 0.5  and then i wanted the tail to be really  long  and then you can optimize that and find  a crease  pattern and it'll complain because it's  having trouble oh dear  demo effect i should have tried this  example before  it's not oh gosh it's one of these  annoying ones  i should say  i should add some feature like  maybe strain split  something add a little bit  maybe do that and yeah there we go  uh i cheated and i'll explain how i  cheated last time but  if sometimes the particular method fails  but you can fix it by adding in another  tiny limb off the edge somewhere and  of course you can then get rid of that  at the end when you're folding  it only makes the problem slightly  harder and it'll still find a folding  but we'll talk about that  next time and i'm way out of time  so we will stop there
 

3
00:00:08,950 --> 00:00:12,470

 

4
00:00:12,470 --> 00:00:13,990

 

5
00:00:13,990 --> 00:00:14,000

 

6
00:00:14,000 --> 00:00:15,589


7
00:00:15,589 --> 00:00:16,870

 

8
00:00:16,870 --> 00:00:18,310

 

9
00:00:18,310 --> 00:00:22,230

 

10
00:00:22,230 --> 00:00:25,589

 

11
00:00:25,589 --> 00:00:28,070

 

12
00:00:28,070 --> 00:00:30,150

 

13
00:00:30,150 --> 00:00:31,589

 

14
00:00:31,589 --> 00:00:32,310

 

15
00:00:32,310 --> 00:00:34,310

 

16
00:00:34,310 --> 00:00:36,549

 

17
00:00:36,549 --> 00:00:38,869

 

18
00:00:38,869 --> 00:00:40,470

 

19
00:00:40,470 --> 00:00:43,670

 

20
00:00:43,670 --> 00:00:46,470

 

21
00:00:46,470 --> 00:00:46,480

 

22
00:00:46,480 --> 00:00:47,510


23
00:00:47,510 --> 00:00:49,670

 

24
00:00:49,670 --> 00:00:50,630

 

25
00:00:50,630 --> 00:00:52,950

 

26
00:00:52,950 --> 00:00:53,670

 

27
00:00:53,670 --> 00:00:55,910

 

28
00:00:55,910 --> 00:00:57,590

 

29
00:00:57,590 --> 00:00:59,029

 

30
00:00:59,029 --> 00:01:00,950

 

31
00:01:00,950 --> 00:01:04,789

 

32
00:01:04,789 --> 00:01:07,350

 

33
00:01:07,350 --> 00:01:07,360

 

34
00:01:07,360 --> 00:01:08,630


35
00:01:08,630 --> 00:01:16,070

 

36
00:01:16,070 --> 00:01:24,830

 

37
00:01:24,830 --> 00:01:24,840

 

38
00:01:24,840 --> 00:01:30,789


39
00:01:30,789 --> 00:01:34,789

 

40
00:01:34,789 --> 00:01:34,799

 

41
00:01:34,799 --> 00:01:36,390


42
00:01:36,390 --> 00:01:37,990

 

43
00:01:37,990 --> 00:01:41,510

 

44
00:01:41,510 --> 00:01:44,710

 

45
00:01:44,710 --> 00:01:48,469

 

46
00:01:48,469 --> 00:01:50,630

 

47
00:01:50,630 --> 00:01:51,910

 

48
00:01:51,910 --> 00:01:53,590

 

49
00:01:53,590 --> 00:01:55,510

 

50
00:01:55,510 --> 00:01:56,630

 

51
00:01:56,630 --> 00:01:57,990

 

52
00:01:57,990 --> 00:01:59,670

 

53
00:01:59,670 --> 00:02:03,670

 

54
00:02:03,670 --> 00:02:15,350

 

55
00:02:15,350 --> 00:02:18,470

 

56
00:02:18,470 --> 00:02:18,480

 

57
00:02:18,480 --> 00:02:19,830


58
00:02:19,830 --> 00:02:38,390

 

59
00:02:38,390 --> 00:02:41,910

 

60
00:02:41,910 --> 00:02:43,350

 

61
00:02:43,350 --> 00:02:46,869

 

62
00:02:46,869 --> 00:02:46,879

 

63
00:02:46,879 --> 00:02:48,830


64
00:02:48,830 --> 00:02:52,470

 

65
00:02:52,470 --> 00:02:57,830

 

66
00:02:57,830 --> 00:03:01,110

 

67
00:03:01,110 --> 00:03:04,470

 

68
00:03:04,470 --> 00:03:04,480

 

69
00:03:04,480 --> 00:03:06,229


70
00:03:06,229 --> 00:03:07,830

 

71
00:03:07,830 --> 00:03:09,589

 

72
00:03:09,589 --> 00:03:11,350

 

73
00:03:11,350 --> 00:03:13,589

 

74
00:03:13,589 --> 00:03:15,190

 

75
00:03:15,190 --> 00:03:17,030

 

76
00:03:17,030 --> 00:03:17,040

 

77
00:03:17,040 --> 00:03:18,070


78
00:03:18,070 --> 00:03:20,710

 

79
00:03:20,710 --> 00:03:22,710

 

80
00:03:22,710 --> 00:03:25,750

 

81
00:03:25,750 --> 00:03:28,390

 

82
00:03:28,390 --> 00:03:29,670

 

83
00:03:29,670 --> 00:03:31,509

 

84
00:03:31,509 --> 00:03:33,350

 

85
00:03:33,350 --> 00:03:35,589

 

86
00:03:35,589 --> 00:03:35,599

 

87
00:03:35,599 --> 00:03:39,910


88
00:03:39,910 --> 00:03:43,190

 

89
00:03:43,190 --> 00:03:47,110

 

90
00:03:47,110 --> 00:03:49,990

 

91
00:03:49,990 --> 00:03:51,030

 

92
00:03:51,030 --> 00:03:52,390

 

93
00:03:52,390 --> 00:03:53,910

 

94
00:03:53,910 --> 00:03:53,920

 

95
00:03:53,920 --> 00:03:54,470


96
00:03:54,470 --> 00:04:00,149

 

97
00:04:00,149 --> 00:04:04,630

 

98
00:04:04,630 --> 00:04:08,710

 

99
00:04:08,710 --> 00:04:12,229

 

100
00:04:12,229 --> 00:04:13,670

 

101
00:04:13,670 --> 00:04:15,910

 

102
00:04:15,910 --> 00:04:21,189

 

103
00:04:21,189 --> 00:04:26,550

 

104
00:04:26,550 --> 00:04:30,070

 

105
00:04:30,070 --> 00:04:32,390

 

106
00:04:32,390 --> 00:04:33,430

 

107
00:04:33,430 --> 00:04:35,189

 

108
00:04:35,189 --> 00:04:36,469

 

109
00:04:36,469 --> 00:04:36,479

 

110
00:04:36,479 --> 00:04:37,590


111
00:04:37,590 --> 00:04:40,310

 

112
00:04:40,310 --> 00:04:41,430

 

113
00:04:41,430 --> 00:04:44,150

 

114
00:04:44,150 --> 00:04:45,510

 

115
00:04:45,510 --> 00:04:48,790

 

116
00:04:48,790 --> 00:04:51,030

 

117
00:04:51,030 --> 00:04:52,310

 

118
00:04:52,310 --> 00:04:54,390

 

119
00:04:54,390 --> 00:04:54,400

 

120
00:04:54,400 --> 00:04:55,510


121
00:04:55,510 --> 00:04:59,189

 

122
00:04:59,189 --> 00:05:00,950

 

123
00:05:00,950 --> 00:05:03,029

 

124
00:05:03,029 --> 00:05:04,950

 

125
00:05:04,950 --> 00:05:06,710

 

126
00:05:06,710 --> 00:05:10,870

 

127
00:05:10,870 --> 00:05:14,629

 

128
00:05:14,629 --> 00:05:16,070

 

129
00:05:16,070 --> 00:05:18,150

 

130
00:05:18,150 --> 00:05:22,469

 

131
00:05:22,469 --> 00:05:25,510

 

132
00:05:25,510 --> 00:05:28,710

 

133
00:05:28,710 --> 00:05:30,550

 

134
00:05:30,550 --> 00:05:32,230

 

135
00:05:32,230 --> 00:05:34,150

 

136
00:05:34,150 --> 00:05:35,909

 

137
00:05:35,909 --> 00:05:37,990

 

138
00:05:37,990 --> 00:05:38,000

 

139
00:05:38,000 --> 00:05:38,870


140
00:05:38,870 --> 00:05:41,350

 

141
00:05:41,350 --> 00:05:41,360

 

142
00:05:41,360 --> 00:05:41,990


143
00:05:41,990 --> 00:05:43,749

 

144
00:05:43,749 --> 00:05:45,830

 

145
00:05:45,830 --> 00:05:48,150

 

146
00:05:48,150 --> 00:05:52,390

 

147
00:05:52,390 --> 00:05:54,790

 

148
00:05:54,790 --> 00:05:55,510

 

149
00:05:55,510 --> 00:05:57,510

 

150
00:05:57,510 --> 00:05:59,510

 

151
00:05:59,510 --> 00:06:01,670

 

152
00:06:01,670 --> 00:06:03,590

 

153
00:06:03,590 --> 00:06:06,710

 

154
00:06:06,710 --> 00:06:10,309

 

155
00:06:10,309 --> 00:06:13,029

 

156
00:06:13,029 --> 00:06:15,110

 

157
00:06:15,110 --> 00:06:16,230

 

158
00:06:16,230 --> 00:06:17,830

 

159
00:06:17,830 --> 00:06:19,670

 

160
00:06:19,670 --> 00:06:20,550

 

161
00:06:20,550 --> 00:06:22,070

 

162
00:06:22,070 --> 00:06:22,080

 

163
00:06:22,080 --> 00:06:23,830


164
00:06:23,830 --> 00:06:26,710

 

165
00:06:26,710 --> 00:06:28,070

 

166
00:06:28,070 --> 00:06:28,080

 

167
00:06:28,080 --> 00:06:28,870


168
00:06:28,870 --> 00:06:31,670

 

169
00:06:31,670 --> 00:06:32,870

 

170
00:06:32,870 --> 00:06:35,189

 

171
00:06:35,189 --> 00:06:37,029

 

172
00:06:37,029 --> 00:06:38,950

 

173
00:06:38,950 --> 00:06:40,309

 

174
00:06:40,309 --> 00:06:42,150

 

175
00:06:42,150 --> 00:06:43,350

 

176
00:06:43,350 --> 00:06:45,189

 

177
00:06:45,189 --> 00:06:45,199

 

178
00:06:45,199 --> 00:06:46,390


179
00:06:46,390 --> 00:06:48,390

 

180
00:06:48,390 --> 00:06:50,710

 

181
00:06:50,710 --> 00:06:55,830

 

182
00:06:55,830 --> 00:06:58,150

 

183
00:06:58,150 --> 00:06:59,909

 

184
00:06:59,909 --> 00:07:03,589

 

185
00:07:03,589 --> 00:07:07,830

 

186
00:07:07,830 --> 00:07:11,430

 

187
00:07:11,430 --> 00:07:16,629

 

188
00:07:16,629 --> 00:07:18,629

 

189
00:07:18,629 --> 00:07:20,550

 

190
00:07:20,550 --> 00:07:22,070

 

191
00:07:22,070 --> 00:07:24,469

 

192
00:07:24,469 --> 00:07:26,309

 

193
00:07:26,309 --> 00:07:27,589

 

194
00:07:27,589 --> 00:07:29,909

 

195
00:07:29,909 --> 00:07:31,589

 

196
00:07:31,589 --> 00:07:33,270

 

197
00:07:33,270 --> 00:07:35,110

 

198
00:07:35,110 --> 00:07:37,110

 

199
00:07:37,110 --> 00:07:38,550

 

200
00:07:38,550 --> 00:07:40,150

 

201
00:07:40,150 --> 00:07:41,749

 

202
00:07:41,749 --> 00:07:42,790

 

203
00:07:42,790 --> 00:07:46,790

 

204
00:07:46,790 --> 00:07:51,749

 

205
00:07:51,749 --> 00:07:56,390

 

206
00:07:56,390 --> 00:07:58,390

 

207
00:07:58,390 --> 00:07:59,510

 

208
00:07:59,510 --> 00:08:02,150

 

209
00:08:02,150 --> 00:08:03,749

 

210
00:08:03,749 --> 00:08:04,790

 

211
00:08:04,790 --> 00:08:07,189

 

212
00:08:07,189 --> 00:08:10,790

 

213
00:08:10,790 --> 00:08:13,990

 

214
00:08:13,990 --> 00:08:16,070

 

215
00:08:16,070 --> 00:08:16,080

 

216
00:08:16,080 --> 00:08:17,029


217
00:08:17,029 --> 00:08:20,230

 

218
00:08:20,230 --> 00:08:27,909

 

219
00:08:27,909 --> 00:08:30,550

 

220
00:08:30,550 --> 00:08:32,790

 

221
00:08:32,790 --> 00:08:34,149

 

222
00:08:34,149 --> 00:08:35,909

 

223
00:08:35,909 --> 00:08:35,919

 

224
00:08:35,919 --> 00:08:36,709


225
00:08:36,709 --> 00:08:40,230

 

226
00:08:40,230 --> 00:08:43,110

 

227
00:08:43,110 --> 00:08:44,310

 

228
00:08:44,310 --> 00:08:47,030

 

229
00:08:47,030 --> 00:08:48,710

 

230
00:08:48,710 --> 00:08:50,150

 

231
00:08:50,150 --> 00:08:50,160

 

232
00:08:50,160 --> 00:08:50,949


233
00:08:50,949 --> 00:08:52,829

 

234
00:08:52,829 --> 00:08:52,839

 

235
00:08:52,839 --> 00:08:54,389


236
00:08:54,389 --> 00:08:56,949

 

237
00:08:56,949 --> 00:08:58,310

 

238
00:08:58,310 --> 00:09:02,790

 

239
00:09:02,790 --> 00:09:04,550

 

240
00:09:04,550 --> 00:09:05,990

 

241
00:09:05,990 --> 00:09:08,070

 

242
00:09:08,070 --> 00:09:09,750

 

243
00:09:09,750 --> 00:09:11,590

 

244
00:09:11,590 --> 00:09:12,870

 

245
00:09:12,870 --> 00:09:15,590

 

246
00:09:15,590 --> 00:09:18,790

 

247
00:09:18,790 --> 00:09:22,550

 

248
00:09:22,550 --> 00:09:28,310

 

249
00:09:28,310 --> 00:09:31,910

 

250
00:09:31,910 --> 00:09:34,070

 

251
00:09:34,070 --> 00:09:35,110

 

252
00:09:35,110 --> 00:09:37,190

 

253
00:09:37,190 --> 00:09:38,230

 

254
00:09:38,230 --> 00:09:39,829

 

255
00:09:39,829 --> 00:09:42,310

 

256
00:09:42,310 --> 00:09:43,750

 

257
00:09:43,750 --> 00:09:45,509

 

258
00:09:45,509 --> 00:09:47,269

 

259
00:09:47,269 --> 00:09:50,470

 

260
00:09:50,470 --> 00:09:52,230

 

261
00:09:52,230 --> 00:09:54,470

 

262
00:09:54,470 --> 00:09:54,480

 

263
00:09:54,480 --> 00:09:55,430


264
00:09:55,430 --> 00:09:57,829

 

265
00:09:57,829 --> 00:09:57,839

 

266
00:09:57,839 --> 00:09:59,110


267
00:09:59,110 --> 00:10:01,030

 

268
00:10:01,030 --> 00:10:03,269

 

269
00:10:03,269 --> 00:10:07,990

 

270
00:10:07,990 --> 00:10:09,190

 

271
00:10:09,190 --> 00:10:11,509

 

272
00:10:11,509 --> 00:10:13,590

 

273
00:10:13,590 --> 00:10:16,310

 

274
00:10:16,310 --> 00:10:19,590

 

275
00:10:19,590 --> 00:10:24,790

 

276
00:10:24,790 --> 00:10:27,190

 

277
00:10:27,190 --> 00:10:29,030

 

278
00:10:29,030 --> 00:10:32,870

 

279
00:10:32,870 --> 00:10:34,389

 

280
00:10:34,389 --> 00:10:36,069

 

281
00:10:36,069 --> 00:10:38,470

 

282
00:10:38,470 --> 00:10:39,990

 

283
00:10:39,990 --> 00:10:40,790

 

284
00:10:40,790 --> 00:10:43,190

 

285
00:10:43,190 --> 00:10:43,200

 

286
00:10:43,200 --> 00:10:44,389


287
00:10:44,389 --> 00:10:47,190

 

288
00:10:47,190 --> 00:10:48,470

 

289
00:10:48,470 --> 00:10:50,630

 

290
00:10:50,630 --> 00:10:50,640

 

291
00:10:50,640 --> 00:10:54,069


292
00:10:54,069 --> 00:10:56,710

 

293
00:10:56,710 --> 00:10:57,590

 

294
00:10:57,590 --> 00:10:59,190

 

295
00:10:59,190 --> 00:11:01,110

 

296
00:11:01,110 --> 00:11:02,949

 

297
00:11:02,949 --> 00:11:04,710

 

298
00:11:04,710 --> 00:11:05,910

 

299
00:11:05,910 --> 00:11:08,150

 

300
00:11:08,150 --> 00:11:09,269

 

301
00:11:09,269 --> 00:11:12,150

 

302
00:11:12,150 --> 00:11:14,550

 

303
00:11:14,550 --> 00:11:17,430

 

304
00:11:17,430 --> 00:11:17,440

 

305
00:11:17,440 --> 00:11:17,829


306
00:11:17,829 --> 00:11:19,829

 

307
00:11:19,829 --> 00:11:21,030

 

308
00:11:21,030 --> 00:11:22,550

 

309
00:11:22,550 --> 00:11:24,230

 

310
00:11:24,230 --> 00:11:28,150

 

311
00:11:28,150 --> 00:11:32,389

 

312
00:11:32,389 --> 00:11:36,150

 

313
00:11:36,150 --> 00:11:37,509

 

314
00:11:37,509 --> 00:11:39,030

 

315
00:11:39,030 --> 00:11:39,040

 

316
00:11:39,040 --> 00:11:40,069


317
00:11:40,069 --> 00:11:42,069

 

318
00:11:42,069 --> 00:11:43,590

 

319
00:11:43,590 --> 00:11:44,389

 

320
00:11:44,389 --> 00:11:46,230

 

321
00:11:46,230 --> 00:11:54,829

 

322
00:11:54,829 --> 00:11:54,839

 

323
00:11:54,839 --> 00:12:16,870


324
00:12:16,870 --> 00:12:18,550

 

325
00:12:18,550 --> 00:12:25,829

 

326
00:12:25,829 --> 00:12:27,750

 

327
00:12:27,750 --> 00:12:27,760

 

328
00:12:27,760 --> 00:12:28,949


329
00:12:28,949 --> 00:12:31,990

 

330
00:12:31,990 --> 00:12:35,509

 

331
00:12:35,509 --> 00:12:39,430

 

332
00:12:39,430 --> 00:12:53,750

 

333
00:12:53,750 --> 00:12:56,470

 

334
00:12:56,470 --> 00:12:59,590

 

335
00:12:59,590 --> 00:13:07,110

 

336
00:13:07,110 --> 00:13:16,069

 

337
00:13:16,069 --> 00:13:19,829

 

338
00:13:19,829 --> 00:13:20,949

 

339
00:13:20,949 --> 00:13:22,870

 

340
00:13:22,870 --> 00:13:24,150

 

341
00:13:24,150 --> 00:13:25,990

 

342
00:13:25,990 --> 00:13:27,670

 

343
00:13:27,670 --> 00:13:29,670

 

344
00:13:29,670 --> 00:13:31,350

 

345
00:13:31,350 --> 00:13:33,750

 

346
00:13:33,750 --> 00:13:34,949

 

347
00:13:34,949 --> 00:13:37,269

 

348
00:13:37,269 --> 00:13:40,629

 

349
00:13:40,629 --> 00:13:42,310

 

350
00:13:42,310 --> 00:13:48,069

 

351
00:13:48,069 --> 00:13:50,150

 

352
00:13:50,150 --> 00:13:51,829

 

353
00:13:51,829 --> 00:13:51,839

 

354
00:13:51,839 --> 00:13:52,550


355
00:13:52,550 --> 00:13:54,230

 

356
00:13:54,230 --> 00:13:57,670

 

357
00:13:57,670 --> 00:14:00,949

 

358
00:14:00,949 --> 00:14:00,959

 

359
00:14:00,959 --> 00:14:02,150


360
00:14:02,150 --> 00:14:06,069

 

361
00:14:06,069 --> 00:14:09,189

 

362
00:14:09,189 --> 00:14:10,710

 

363
00:14:10,710 --> 00:14:13,030

 

364
00:14:13,030 --> 00:14:14,550

 

365
00:14:14,550 --> 00:14:17,829

 

366
00:14:17,829 --> 00:14:21,670

 

367
00:14:21,670 --> 00:14:24,150

 

368
00:14:24,150 --> 00:14:25,189

 

369
00:14:25,189 --> 00:14:26,389

 

370
00:14:26,389 --> 00:14:28,389

 

371
00:14:28,389 --> 00:14:32,870

 

372
00:14:32,870 --> 00:14:35,750

 

373
00:14:35,750 --> 00:14:37,509

 

374
00:14:37,509 --> 00:14:39,110

 

375
00:14:39,110 --> 00:14:41,750

 

376
00:14:41,750 --> 00:14:42,949

 

377
00:14:42,949 --> 00:14:45,269

 

378
00:14:45,269 --> 00:14:46,629

 

379
00:14:46,629 --> 00:14:50,629

 

380
00:14:50,629 --> 00:14:53,030

 

381
00:14:53,030 --> 00:14:55,910

 

382
00:14:55,910 --> 00:14:59,030

 

383
00:14:59,030 --> 00:15:00,790

 

384
00:15:00,790 --> 00:15:02,550

 

385
00:15:02,550 --> 00:15:04,389

 

386
00:15:04,389 --> 00:15:05,509

 

387
00:15:05,509 --> 00:15:07,269

 

388
00:15:07,269 --> 00:15:08,949

 

389
00:15:08,949 --> 00:15:13,990

 

390
00:15:13,990 --> 00:15:17,030

 

391
00:15:17,030 --> 00:15:19,269

 

392
00:15:19,269 --> 00:15:22,710

 

393
00:15:22,710 --> 00:15:25,990

 

394
00:15:25,990 --> 00:15:29,269

 

395
00:15:29,269 --> 00:15:30,870

 

396
00:15:30,870 --> 00:15:32,150

 

397
00:15:32,150 --> 00:15:32,160

 

398
00:15:32,160 --> 00:15:33,509


399
00:15:33,509 --> 00:15:37,110

 

400
00:15:37,110 --> 00:15:40,150

 

401
00:15:40,150 --> 00:15:42,870

 

402
00:15:42,870 --> 00:15:44,150

 

403
00:15:44,150 --> 00:15:46,550

 

404
00:15:46,550 --> 00:15:47,749

 

405
00:15:47,749 --> 00:15:48,949

 

406
00:15:48,949 --> 00:15:51,350

 

407
00:15:51,350 --> 00:15:52,710

 

408
00:15:52,710 --> 00:15:53,670

 

409
00:15:53,670 --> 00:15:55,749

 

410
00:15:55,749 --> 00:15:55,759

 

411
00:15:55,759 --> 00:15:56,870


412
00:15:56,870 --> 00:15:59,030

 

413
00:15:59,030 --> 00:16:00,790

 

414
00:16:00,790 --> 00:16:03,189

 

415
00:16:03,189 --> 00:16:03,199

 

416
00:16:03,199 --> 00:16:04,230


417
00:16:04,230 --> 00:16:05,749

 

418
00:16:05,749 --> 00:16:07,749

 

419
00:16:07,749 --> 00:16:10,150

 

420
00:16:10,150 --> 00:16:10,949

 

421
00:16:10,949 --> 00:16:14,150

 

422
00:16:14,150 --> 00:16:14,160

 

423
00:16:14,160 --> 00:16:14,710


424
00:16:14,710 --> 00:16:18,150

 

425
00:16:18,150 --> 00:16:21,670

 

426
00:16:21,670 --> 00:16:22,790

 

427
00:16:22,790 --> 00:16:24,470

 

428
00:16:24,470 --> 00:16:26,310

 

429
00:16:26,310 --> 00:16:26,320

 

430
00:16:26,320 --> 00:16:29,749


431
00:16:29,749 --> 00:16:31,829

 

432
00:16:31,829 --> 00:16:33,910

 

433
00:16:33,910 --> 00:16:35,829

 

434
00:16:35,829 --> 00:16:35,839

 

435
00:16:35,839 --> 00:16:37,189


436
00:16:37,189 --> 00:16:39,509

 

437
00:16:39,509 --> 00:16:39,519

 

438
00:16:39,519 --> 00:16:41,110


439
00:16:41,110 --> 00:16:42,870

 

440
00:16:42,870 --> 00:16:44,470

 

441
00:16:44,470 --> 00:16:45,590

 

442
00:16:45,590 --> 00:16:48,389

 

443
00:16:48,389 --> 00:16:49,509

 

444
00:16:49,509 --> 00:16:51,590

 

445
00:16:51,590 --> 00:16:55,030

 

446
00:16:55,030 --> 00:16:56,310

 

447
00:16:56,310 --> 00:16:58,150

 

448
00:16:58,150 --> 00:17:00,150

 

449
00:17:00,150 --> 00:17:01,990

 

450
00:17:01,990 --> 00:17:03,670

 

451
00:17:03,670 --> 00:17:06,390

 

452
00:17:06,390 --> 00:17:07,429

 

453
00:17:07,429 --> 00:17:09,270

 

454
00:17:09,270 --> 00:17:11,189

 

455
00:17:11,189 --> 00:17:13,990

 

456
00:17:13,990 --> 00:17:14,000

 

457
00:17:14,000 --> 00:17:14,630


458
00:17:14,630 --> 00:17:18,870

 

459
00:17:18,870 --> 00:17:20,949

 

460
00:17:20,949 --> 00:17:20,959

 

461
00:17:20,959 --> 00:17:22,549


462
00:17:22,549 --> 00:17:26,230

 

463
00:17:26,230 --> 00:17:27,909

 

464
00:17:27,909 --> 00:17:29,990

 

465
00:17:29,990 --> 00:17:32,150

 

466
00:17:32,150 --> 00:17:35,190

 

467
00:17:35,190 --> 00:17:37,510

 

468
00:17:37,510 --> 00:17:39,029

 

469
00:17:39,029 --> 00:17:42,230

 

470
00:17:42,230 --> 00:17:43,430

 

471
00:17:43,430 --> 00:17:47,669

 

472
00:17:47,669 --> 00:17:49,430

 

473
00:17:49,430 --> 00:17:50,870

 

474
00:17:50,870 --> 00:17:52,630

 

475
00:17:52,630 --> 00:17:54,549

 

476
00:17:54,549 --> 00:17:56,230

 

477
00:17:56,230 --> 00:17:57,669

 

478
00:17:57,669 --> 00:18:01,270

 

479
00:18:01,270 --> 00:18:09,510

 

480
00:18:09,510 --> 00:18:14,549

 

481
00:18:14,549 --> 00:18:17,510

 

482
00:18:17,510 --> 00:18:19,110

 

483
00:18:19,110 --> 00:18:26,470

 

484
00:18:26,470 --> 00:18:29,350

 

485
00:18:29,350 --> 00:18:29,360

 

486
00:18:29,360 --> 00:18:29,909


487
00:18:29,909 --> 00:18:31,270

 

488
00:18:31,270 --> 00:18:31,280

 

489
00:18:31,280 --> 00:18:33,110


490
00:18:33,110 --> 00:18:34,870

 

491
00:18:34,870 --> 00:18:36,789

 

492
00:18:36,789 --> 00:18:38,310

 

493
00:18:38,310 --> 00:18:40,230

 

494
00:18:40,230 --> 00:18:41,830

 

495
00:18:41,830 --> 00:18:43,270

 

496
00:18:43,270 --> 00:18:44,470

 

497
00:18:44,470 --> 00:18:45,909

 

498
00:18:45,909 --> 00:18:48,390

 

499
00:18:48,390 --> 00:18:49,590

 

500
00:18:49,590 --> 00:18:51,510

 

501
00:18:51,510 --> 00:18:53,110

 

502
00:18:53,110 --> 00:18:55,029

 

503
00:18:55,029 --> 00:18:57,270

 

504
00:18:57,270 --> 00:18:59,430

 

505
00:18:59,430 --> 00:19:01,110

 

506
00:19:01,110 --> 00:19:01,120

 

507
00:19:01,120 --> 00:19:02,070


508
00:19:02,070 --> 00:19:05,190

 

509
00:19:05,190 --> 00:19:07,669

 

510
00:19:07,669 --> 00:19:07,679

 

511
00:19:07,679 --> 00:19:09,350


512
00:19:09,350 --> 00:19:12,150

 

513
00:19:12,150 --> 00:19:13,750

 

514
00:19:13,750 --> 00:19:17,029

 

515
00:19:17,029 --> 00:19:20,390

 

516
00:19:20,390 --> 00:19:23,350

 

517
00:19:23,350 --> 00:19:24,870

 

518
00:19:24,870 --> 00:19:26,230

 

519
00:19:26,230 --> 00:19:27,909

 

520
00:19:27,909 --> 00:19:29,750

 

521
00:19:29,750 --> 00:19:49,430

 

522
00:19:49,430 --> 00:19:51,669

 

523
00:19:51,669 --> 00:19:53,029

 

524
00:19:53,029 --> 00:19:54,630

 

525
00:19:54,630 --> 00:19:57,110

 

526
00:19:57,110 --> 00:20:01,350

 

527
00:20:01,350 --> 00:20:03,510

 

528
00:20:03,510 --> 00:20:04,870

 

529
00:20:04,870 --> 00:20:06,310

 

530
00:20:06,310 --> 00:20:08,789

 

531
00:20:08,789 --> 00:20:10,549

 

532
00:20:10,549 --> 00:20:11,750

 

533
00:20:11,750 --> 00:20:13,029

 

534
00:20:13,029 --> 00:20:14,870

 

535
00:20:14,870 --> 00:20:16,710

 

536
00:20:16,710 --> 00:20:21,350

 

537
00:20:21,350 --> 00:20:23,909

 

538
00:20:23,909 --> 00:20:23,919

 

539
00:20:23,919 --> 00:20:24,710


540
00:20:24,710 --> 00:20:27,270

 

541
00:20:27,270 --> 00:20:29,270

 

542
00:20:29,270 --> 00:20:32,310

 

543
00:20:32,310 --> 00:20:32,320

 

544
00:20:32,320 --> 00:20:33,270


545
00:20:33,270 --> 00:20:36,310

 

546
00:20:36,310 --> 00:20:38,149

 

547
00:20:38,149 --> 00:20:40,230

 

548
00:20:40,230 --> 00:20:41,830

 

549
00:20:41,830 --> 00:20:42,630

 

550
00:20:42,630 --> 00:20:45,590

 

551
00:20:45,590 --> 00:20:46,789

 

552
00:20:46,789 --> 00:20:48,310

 

553
00:20:48,310 --> 00:20:49,669

 

554
00:20:49,669 --> 00:20:51,110

 

555
00:20:51,110 --> 00:20:52,710

 

556
00:20:52,710 --> 00:20:54,710

 

557
00:20:54,710 --> 00:20:55,990

 

558
00:20:55,990 --> 00:20:58,230

 

559
00:20:58,230 --> 00:20:59,830

 

560
00:20:59,830 --> 00:21:02,149

 

561
00:21:02,149 --> 00:21:03,270

 

562
00:21:03,270 --> 00:21:05,029

 

563
00:21:05,029 --> 00:21:08,390

 

564
00:21:08,390 --> 00:21:10,070

 

565
00:21:10,070 --> 00:21:12,470

 

566
00:21:12,470 --> 00:21:15,590

 

567
00:21:15,590 --> 00:21:20,310

 

568
00:21:20,310 --> 00:21:26,789

 

569
00:21:26,789 --> 00:21:29,990

 

570
00:21:29,990 --> 00:21:31,909

 

571
00:21:31,909 --> 00:21:34,950

 

572
00:21:34,950 --> 00:21:38,070

 

573
00:21:38,070 --> 00:21:38,549

 

574
00:21:38,549 --> 00:21:44,390

 

575
00:21:44,390 --> 00:21:47,510

 

576
00:21:47,510 --> 00:21:54,549

 

577
00:21:54,549 --> 00:21:59,909

 

578
00:21:59,909 --> 00:22:02,549

 

579
00:22:02,549 --> 00:22:04,710

 

580
00:22:04,710 --> 00:22:07,830

 

581
00:22:07,830 --> 00:22:10,470

 

582
00:22:10,470 --> 00:22:12,310

 

583
00:22:12,310 --> 00:22:14,310

 

584
00:22:14,310 --> 00:22:16,390

 

585
00:22:16,390 --> 00:22:17,909

 

586
00:22:17,909 --> 00:22:20,390

 

587
00:22:20,390 --> 00:22:22,710

 

588
00:22:22,710 --> 00:22:24,830

 

589
00:22:24,830 --> 00:22:24,840

 

590
00:22:24,840 --> 00:22:27,750


591
00:22:27,750 --> 00:22:34,549

 

592
00:22:34,549 --> 00:22:35,750

 

593
00:22:35,750 --> 00:22:38,149

 

594
00:22:38,149 --> 00:22:40,230

 

595
00:22:40,230 --> 00:22:43,270

 

596
00:22:43,270 --> 00:22:44,789

 

597
00:22:44,789 --> 00:22:46,549

 

598
00:22:46,549 --> 00:22:48,310

 

599
00:22:48,310 --> 00:22:50,230

 

600
00:22:50,230 --> 00:22:52,789

 

601
00:22:52,789 --> 00:22:56,549

 

602
00:22:56,549 --> 00:22:57,750

 

603
00:22:57,750 --> 00:23:00,549

 

604
00:23:00,549 --> 00:23:05,110

 

605
00:23:05,110 --> 00:23:07,350

 

606
00:23:07,350 --> 00:23:07,360

 

607
00:23:07,360 --> 00:23:09,750


608
00:23:09,750 --> 00:23:13,270

 

609
00:23:13,270 --> 00:23:15,830

 

610
00:23:15,830 --> 00:23:17,190

 

611
00:23:17,190 --> 00:23:20,549

 

612
00:23:20,549 --> 00:23:23,029

 

613
00:23:23,029 --> 00:23:23,990

 

614
00:23:23,990 --> 00:23:25,430

 

615
00:23:25,430 --> 00:23:27,510

 

616
00:23:27,510 --> 00:23:29,750

 

617
00:23:29,750 --> 00:23:31,110

 

618
00:23:31,110 --> 00:23:36,789

 

619
00:23:36,789 --> 00:23:38,870

 

620
00:23:38,870 --> 00:23:40,230

 

621
00:23:40,230 --> 00:23:42,789

 

622
00:23:42,789 --> 00:23:44,310

 

623
00:23:44,310 --> 00:23:45,350

 

624
00:23:45,350 --> 00:23:49,350

 

625
00:23:49,350 --> 00:23:51,830

 

626
00:23:51,830 --> 00:23:51,840

 

627
00:23:51,840 --> 00:23:53,669


628
00:23:53,669 --> 00:23:55,510

 

629
00:23:55,510 --> 00:23:56,710

 

630
00:23:56,710 --> 00:23:58,149

 

631
00:23:58,149 --> 00:24:00,789

 

632
00:24:00,789 --> 00:24:02,470

 

633
00:24:02,470 --> 00:24:05,669

 

634
00:24:05,669 --> 00:24:08,070

 

635
00:24:08,070 --> 00:24:09,830

 

636
00:24:09,830 --> 00:24:13,110

 

637
00:24:13,110 --> 00:24:16,830

 

638
00:24:16,830 --> 00:24:16,840

 

639
00:24:16,840 --> 00:24:23,190


640
00:24:23,190 --> 00:24:24,950

 

641
00:24:24,950 --> 00:24:34,630

 

642
00:24:34,630 --> 00:24:35,669

 

643
00:24:35,669 --> 00:24:37,269

 

644
00:24:37,269 --> 00:24:38,950

 

645
00:24:38,950 --> 00:24:41,430

 

646
00:24:41,430 --> 00:24:41,440

 

647
00:24:41,440 --> 00:24:43,430


648
00:24:43,430 --> 00:24:46,149

 

649
00:24:46,149 --> 00:24:49,269

 

650
00:24:49,269 --> 00:24:50,710

 

651
00:24:50,710 --> 00:24:52,470

 

652
00:24:52,470 --> 00:24:55,909

 

653
00:24:55,909 --> 00:24:55,919

 

654
00:24:55,919 --> 00:24:56,390


655
00:24:56,390 --> 00:24:58,870

 

656
00:24:58,870 --> 00:25:01,029

 

657
00:25:01,029 --> 00:25:02,710

 

658
00:25:02,710 --> 00:25:04,549

 

659
00:25:04,549 --> 00:25:07,029

 

660
00:25:07,029 --> 00:25:09,590

 

661
00:25:09,590 --> 00:25:13,510

 

662
00:25:13,510 --> 00:25:16,390

 

663
00:25:16,390 --> 00:25:16,400

 

664
00:25:16,400 --> 00:25:17,029


665
00:25:17,029 --> 00:25:18,950

 

666
00:25:18,950 --> 00:25:20,390

 

667
00:25:20,390 --> 00:25:20,400

 

668
00:25:20,400 --> 00:25:21,190


669
00:25:21,190 --> 00:25:22,950

 

670
00:25:22,950 --> 00:25:24,390

 

671
00:25:24,390 --> 00:25:27,750

 

672
00:25:27,750 --> 00:25:30,149

 

673
00:25:30,149 --> 00:25:30,159

 

674
00:25:30,159 --> 00:25:33,510


675
00:25:33,510 --> 00:25:37,110

 

676
00:25:37,110 --> 00:25:40,390

 

677
00:25:40,390 --> 00:25:42,310

 

678
00:25:42,310 --> 00:25:43,990

 

679
00:25:43,990 --> 00:25:45,190

 

680
00:25:45,190 --> 00:25:47,350

 

681
00:25:47,350 --> 00:25:50,310

 

682
00:25:50,310 --> 00:25:54,470

 

683
00:25:54,470 --> 00:25:58,390

 

684
00:25:58,390 --> 00:26:01,990

 

685
00:26:01,990 --> 00:26:04,070

 

686
00:26:04,070 --> 00:26:04,080

 

687
00:26:04,080 --> 00:26:05,190


688
00:26:05,190 --> 00:26:07,190

 

689
00:26:07,190 --> 00:26:08,830

 

690
00:26:08,830 --> 00:26:08,840

 

691
00:26:08,840 --> 00:26:10,230


692
00:26:10,230 --> 00:26:11,990

 

693
00:26:11,990 --> 00:26:13,830

 

694
00:26:13,830 --> 00:26:17,190

 

695
00:26:17,190 --> 00:26:19,350

 

696
00:26:19,350 --> 00:26:24,470

 

697
00:26:24,470 --> 00:26:29,590

 

698
00:26:29,590 --> 00:26:31,990

 

699
00:26:31,990 --> 00:26:34,390

 

700
00:26:34,390 --> 00:26:36,710

 

701
00:26:36,710 --> 00:26:39,269

 

702
00:26:39,269 --> 00:26:41,190

 

703
00:26:41,190 --> 00:26:42,630

 

704
00:26:42,630 --> 00:26:45,269

 

705
00:26:45,269 --> 00:26:45,279

 

706
00:26:45,279 --> 00:26:46,549


707
00:26:46,549 --> 00:26:50,710

 

708
00:26:50,710 --> 00:26:52,390

 

709
00:26:52,390 --> 00:26:52,400

 

710
00:26:52,400 --> 00:26:53,269


711
00:26:53,269 --> 00:26:55,830

 

712
00:26:55,830 --> 00:26:59,750

 

713
00:26:59,750 --> 00:27:06,950

 

714
00:27:06,950 --> 00:27:08,390

 

715
00:27:08,390 --> 00:27:11,909

 

716
00:27:11,909 --> 00:27:15,830

 

717
00:27:15,830 --> 00:27:17,990

 

718
00:27:17,990 --> 00:27:20,549

 

719
00:27:20,549 --> 00:27:22,549

 

720
00:27:22,549 --> 00:27:22,559

 

721
00:27:22,559 --> 00:27:23,590


722
00:27:23,590 --> 00:27:25,029

 

723
00:27:25,029 --> 00:27:26,549

 

724
00:27:26,549 --> 00:27:26,559

 

725
00:27:26,559 --> 00:27:27,190


726
00:27:27,190 --> 00:27:29,029

 

727
00:27:29,029 --> 00:27:30,870

 

728
00:27:30,870 --> 00:27:31,669

 

729
00:27:31,669 --> 00:27:35,029

 

730
00:27:35,029 --> 00:27:37,190

 

731
00:27:37,190 --> 00:27:39,350

 

732
00:27:39,350 --> 00:27:42,950

 

733
00:27:42,950 --> 00:27:46,149

 

734
00:27:46,149 --> 00:27:47,990

 

735
00:27:47,990 --> 00:27:49,990

 

736
00:27:49,990 --> 00:27:50,000

 

737
00:27:50,000 --> 00:27:50,710


738
00:27:50,710 --> 00:27:52,389

 

739
00:27:52,389 --> 00:27:53,750

 

740
00:27:53,750 --> 00:27:54,789

 

741
00:27:54,789 --> 00:27:56,710

 

742
00:27:56,710 --> 00:27:58,630

 

743
00:27:58,630 --> 00:28:00,149

 

744
00:28:00,149 --> 00:28:02,789

 

745
00:28:02,789 --> 00:28:05,190

 

746
00:28:05,190 --> 00:28:06,230

 

747
00:28:06,230 --> 00:28:09,590

 

748
00:28:09,590 --> 00:28:12,149

 

749
00:28:12,149 --> 00:28:12,159

 

750
00:28:12,159 --> 00:28:12,950


751
00:28:12,950 --> 00:28:16,310

 

752
00:28:16,310 --> 00:28:16,320

 

753
00:28:16,320 --> 00:28:17,269


754
00:28:17,269 --> 00:28:19,029

 

755
00:28:19,029 --> 00:28:20,870

 

756
00:28:20,870 --> 00:28:22,789

 

757
00:28:22,789 --> 00:28:24,950

 

758
00:28:24,950 --> 00:28:26,789

 

759
00:28:26,789 --> 00:28:28,710

 

760
00:28:28,710 --> 00:28:30,630

 

761
00:28:30,630 --> 00:28:32,870

 

762
00:28:32,870 --> 00:28:33,990

 

763
00:28:33,990 --> 00:28:36,070

 

764
00:28:36,070 --> 00:28:37,750

 

765
00:28:37,750 --> 00:28:38,950

 

766
00:28:38,950 --> 00:28:42,149

 

767
00:28:42,149 --> 00:28:47,190

 

768
00:28:47,190 --> 00:28:48,630

 

769
00:28:48,630 --> 00:28:50,630

 

770
00:28:50,630 --> 00:28:51,510

 

771
00:28:51,510 --> 00:28:54,149

 

772
00:28:54,149 --> 00:28:56,710

 

773
00:28:56,710 --> 00:28:59,590

 

774
00:28:59,590 --> 00:29:00,870

 

775
00:29:00,870 --> 00:29:02,630

 

776
00:29:02,630 --> 00:29:04,389

 

777
00:29:04,389 --> 00:29:08,630

 

778
00:29:08,630 --> 00:29:09,909

 

779
00:29:09,909 --> 00:29:12,310

 

780
00:29:12,310 --> 00:29:12,320

 

781
00:29:12,320 --> 00:29:24,830


782
00:29:24,830 --> 00:29:24,840

 

783
00:29:24,840 --> 00:29:31,510


784
00:29:31,510 --> 00:29:33,190

 

785
00:29:33,190 --> 00:29:35,190

 

786
00:29:35,190 --> 00:29:36,549

 

787
00:29:36,549 --> 00:29:37,830

 

788
00:29:37,830 --> 00:29:40,070

 

789
00:29:40,070 --> 00:29:41,750

 

790
00:29:41,750 --> 00:29:44,310

 

791
00:29:44,310 --> 00:29:46,389

 

792
00:29:46,389 --> 00:29:49,430

 

793
00:29:49,430 --> 00:29:55,510

 

794
00:29:55,510 --> 00:30:00,070

 

795
00:30:00,070 --> 00:30:03,510

 

796
00:30:03,510 --> 00:30:05,830

 

797
00:30:05,830 --> 00:30:08,710

 

798
00:30:08,710 --> 00:30:15,750

 

799
00:30:15,750 --> 00:30:30,549

 

800
00:30:30,549 --> 00:30:34,230

 

801
00:30:34,230 --> 00:30:37,350

 

802
00:30:37,350 --> 00:30:38,710

 

803
00:30:38,710 --> 00:30:40,549

 

804
00:30:40,549 --> 00:30:43,909

 

805
00:30:43,909 --> 00:30:49,110

 

806
00:30:49,110 --> 00:30:51,190

 

807
00:30:51,190 --> 00:30:52,389

 

808
00:30:52,389 --> 00:30:54,630

 

809
00:30:54,630 --> 00:30:56,549

 

810
00:30:56,549 --> 00:30:58,070

 

811
00:30:58,070 --> 00:30:59,350

 

812
00:30:59,350 --> 00:31:02,149

 

813
00:31:02,149 --> 00:31:05,190

 

814
00:31:05,190 --> 00:31:05,200

 

815
00:31:05,200 --> 00:31:05,590


816
00:31:05,590 --> 00:31:07,430

 

817
00:31:07,430 --> 00:31:08,950

 

818
00:31:08,950 --> 00:31:08,960

 

819
00:31:08,960 --> 00:31:09,990


820
00:31:09,990 --> 00:31:13,909

 

821
00:31:13,909 --> 00:31:15,830

 

822
00:31:15,830 --> 00:31:17,750

 

823
00:31:17,750 --> 00:31:19,750

 

824
00:31:19,750 --> 00:31:21,830

 

825
00:31:21,830 --> 00:31:23,669

 

826
00:31:23,669 --> 00:31:25,990

 

827
00:31:25,990 --> 00:31:26,789

 

828
00:31:26,789 --> 00:31:42,870

 

829
00:31:42,870 --> 00:31:49,110

 

830
00:31:49,110 --> 00:31:52,230

 

831
00:31:52,230 --> 00:31:53,750

 

832
00:31:53,750 --> 00:31:56,230

 

833
00:31:56,230 --> 00:31:57,590

 

834
00:31:57,590 --> 00:31:58,950

 

835
00:31:58,950 --> 00:32:06,310

 

836
00:32:06,310 --> 00:32:08,070

 

837
00:32:08,070 --> 00:32:09,590

 

838
00:32:09,590 --> 00:32:12,310

 

839
00:32:12,310 --> 00:32:15,190

 

840
00:32:15,190 --> 00:32:15,200

 

841
00:32:15,200 --> 00:32:15,669


842
00:32:15,669 --> 00:32:18,710

 

843
00:32:18,710 --> 00:32:22,149

 

844
00:32:22,149 --> 00:32:23,990

 

845
00:32:23,990 --> 00:32:26,070

 

846
00:32:26,070 --> 00:32:28,950

 

847
00:32:28,950 --> 00:32:30,470

 

848
00:32:30,470 --> 00:32:31,830

 

849
00:32:31,830 --> 00:32:32,789

 

850
00:32:32,789 --> 00:32:35,509

 

851
00:32:35,509 --> 00:32:35,519

 

852
00:32:35,519 --> 00:32:35,909


853
00:32:35,909 --> 00:32:37,350

 

854
00:32:37,350 --> 00:32:38,950

 

855
00:32:38,950 --> 00:32:40,149

 

856
00:32:40,149 --> 00:32:41,990

 

857
00:32:41,990 --> 00:32:45,190

 

858
00:32:45,190 --> 00:32:46,870

 

859
00:32:46,870 --> 00:32:48,710

 

860
00:32:48,710 --> 00:32:50,950

 

861
00:32:50,950 --> 00:32:52,549

 

862
00:32:52,549 --> 00:32:54,710

 

863
00:32:54,710 --> 00:32:56,230

 

864
00:32:56,230 --> 00:32:57,990

 

865
00:32:57,990 --> 00:32:59,669

 

866
00:32:59,669 --> 00:33:01,669

 

867
00:33:01,669 --> 00:33:04,230

 

868
00:33:04,230 --> 00:33:06,630

 

869
00:33:06,630 --> 00:33:10,149

 

870
00:33:10,149 --> 00:33:10,159

 

871
00:33:10,159 --> 00:33:11,269


872
00:33:11,269 --> 00:33:18,230

 

873
00:33:18,230 --> 00:33:22,149

 

874
00:33:22,149 --> 00:33:25,750

 

875
00:33:25,750 --> 00:33:28,789

 

876
00:33:28,789 --> 00:33:30,230

 

877
00:33:30,230 --> 00:33:31,830

 

878
00:33:31,830 --> 00:33:38,070

 

879
00:33:38,070 --> 00:33:40,870

 

880
00:33:40,870 --> 00:33:40,880

 

881
00:33:40,880 --> 00:33:42,070


882
00:33:42,070 --> 00:33:44,789

 

883
00:33:44,789 --> 00:33:47,430

 

884
00:33:47,430 --> 00:33:50,070

 

885
00:33:50,070 --> 00:33:53,909

 

886
00:33:53,909 --> 00:33:55,509

 

887
00:33:55,509 --> 00:33:57,669

 

888
00:33:57,669 --> 00:33:58,950

 

889
00:33:58,950 --> 00:34:00,710

 

890
00:34:00,710 --> 00:34:03,190

 

891
00:34:03,190 --> 00:34:03,200

 

892
00:34:03,200 --> 00:34:03,830


893
00:34:03,830 --> 00:34:06,070

 

894
00:34:06,070 --> 00:34:08,149

 

895
00:34:08,149 --> 00:34:10,389

 

896
00:34:10,389 --> 00:34:12,310

 

897
00:34:12,310 --> 00:34:15,589

 

898
00:34:15,589 --> 00:34:18,629

 

899
00:34:18,629 --> 00:34:19,990

 

900
00:34:19,990 --> 00:34:21,990

 

901
00:34:21,990 --> 00:34:24,950

 

902
00:34:24,950 --> 00:34:26,629

 

903
00:34:26,629 --> 00:34:28,790

 

904
00:34:28,790 --> 00:34:30,069

 

905
00:34:30,069 --> 00:34:31,909

 

906
00:34:31,909 --> 00:34:33,349

 

907
00:34:33,349 --> 00:34:35,829

 

908
00:34:35,829 --> 00:34:37,430

 

909
00:34:37,430 --> 00:34:41,109

 

910
00:34:41,109 --> 00:34:44,470

 

911
00:34:44,470 --> 00:34:44,480

 

912
00:34:44,480 --> 00:34:46,829


913
00:34:46,829 --> 00:34:46,839

 

914
00:34:46,839 --> 00:34:48,710


915
00:34:48,710 --> 00:34:50,790

 

916
00:34:50,790 --> 00:34:52,230

 

917
00:34:52,230 --> 00:34:53,990

 

918
00:34:53,990 --> 00:34:55,589

 

919
00:34:55,589 --> 00:34:57,510

 

920
00:34:57,510 --> 00:35:00,150

 

921
00:35:00,150 --> 00:35:01,510

 

922
00:35:01,510 --> 00:35:02,710

 

923
00:35:02,710 --> 00:35:06,069

 

924
00:35:06,069 --> 00:35:08,310

 

925
00:35:08,310 --> 00:35:09,589

 

926
00:35:09,589 --> 00:35:11,589

 

927
00:35:11,589 --> 00:35:13,510

 

928
00:35:13,510 --> 00:35:15,910

 

929
00:35:15,910 --> 00:35:15,920

 

930
00:35:15,920 --> 00:35:16,710


931
00:35:16,710 --> 00:35:18,230

 

932
00:35:18,230 --> 00:35:20,390

 

933
00:35:20,390 --> 00:35:20,400

 

934
00:35:20,400 --> 00:35:21,190


935
00:35:21,190 --> 00:35:23,510

 

936
00:35:23,510 --> 00:35:24,710

 

937
00:35:24,710 --> 00:35:26,870

 

938
00:35:26,870 --> 00:35:28,550

 

939
00:35:28,550 --> 00:35:30,310

 

940
00:35:30,310 --> 00:35:33,030

 

941
00:35:33,030 --> 00:35:35,510

 

942
00:35:35,510 --> 00:35:38,870

 

943
00:35:38,870 --> 00:35:40,310

 

944
00:35:40,310 --> 00:35:42,870

 

945
00:35:42,870 --> 00:35:44,870

 

946
00:35:44,870 --> 00:35:47,430

 

947
00:35:47,430 --> 00:35:49,270

 

948
00:35:49,270 --> 00:35:51,109

 

949
00:35:51,109 --> 00:35:52,790

 

950
00:35:52,790 --> 00:35:54,310

 

951
00:35:54,310 --> 00:35:55,829

 

952
00:35:55,829 --> 00:35:58,150

 

953
00:35:58,150 --> 00:35:59,910

 

954
00:35:59,910 --> 00:35:59,920

 

955
00:35:59,920 --> 00:36:02,550


956
00:36:02,550 --> 00:36:04,870

 

957
00:36:04,870 --> 00:36:06,470

 

958
00:36:06,470 --> 00:36:08,069

 

959
00:36:08,069 --> 00:36:09,750

 

960
00:36:09,750 --> 00:36:11,109

 

961
00:36:11,109 --> 00:36:12,310

 

962
00:36:12,310 --> 00:36:15,750

 

963
00:36:15,750 --> 00:36:17,750

 

964
00:36:17,750 --> 00:36:19,750

 

965
00:36:19,750 --> 00:36:22,069

 

966
00:36:22,069 --> 00:36:24,829

 

967
00:36:24,829 --> 00:36:28,230

 

968
00:36:28,230 --> 00:36:30,950

 

969
00:36:30,950 --> 00:36:32,950

 

970
00:36:32,950 --> 00:36:34,950

 

971
00:36:34,950 --> 00:36:36,550

 

972
00:36:36,550 --> 00:36:38,069

 

973
00:36:38,069 --> 00:36:38,079

 

974
00:36:38,079 --> 00:36:39,430


975
00:36:39,430 --> 00:36:41,589

 

976
00:36:41,589 --> 00:36:43,349

 

977
00:36:43,349 --> 00:36:45,430

 

978
00:36:45,430 --> 00:36:46,790

 

979
00:36:46,790 --> 00:36:48,950

 

980
00:36:48,950 --> 00:36:51,270

 

981
00:36:51,270 --> 00:36:52,310

 

982
00:36:52,310 --> 00:36:54,310

 

983
00:36:54,310 --> 00:36:55,430

 

984
00:36:55,430 --> 00:36:56,710

 

985
00:36:56,710 --> 00:36:58,790

 

986
00:36:58,790 --> 00:36:59,910

 

987
00:36:59,910 --> 00:37:01,589

 

988
00:37:01,589 --> 00:37:02,710

 

989
00:37:02,710 --> 00:37:05,670

 

990
00:37:05,670 --> 00:37:07,990

 

991
00:37:07,990 --> 00:37:11,349

 

992
00:37:11,349 --> 00:37:13,510

 

993
00:37:13,510 --> 00:37:14,710

 

994
00:37:14,710 --> 00:37:16,390

 

995
00:37:16,390 --> 00:37:18,310

 

996
00:37:18,310 --> 00:37:19,670

 

997
00:37:19,670 --> 00:37:21,190

 

998
00:37:21,190 --> 00:37:22,790

 

999
00:37:22,790 --> 00:37:24,710

 

1000
00:37:24,710 --> 00:37:48,829

 

1001
00:37:48,829 --> 00:37:48,839

 

1002
00:37:48,839 --> 00:37:57,829


1003
00:37:57,829 --> 00:38:00,470

 

1004
00:38:00,470 --> 00:38:00,480

 

1005
00:38:00,480 --> 00:38:01,270


1006
00:38:01,270 --> 00:38:04,630

 

1007
00:38:04,630 --> 00:38:05,910

 

1008
00:38:05,910 --> 00:38:08,150

 

1009
00:38:08,150 --> 00:38:09,670

 

1010
00:38:09,670 --> 00:38:11,190

 

1011
00:38:11,190 --> 00:38:13,750

 

1012
00:38:13,750 --> 00:38:15,270

 

1013
00:38:15,270 --> 00:38:15,280

 

1014
00:38:15,280 --> 00:38:16,710


1015
00:38:16,710 --> 00:38:17,990

 

1016
00:38:17,990 --> 00:38:20,390

 

1017
00:38:20,390 --> 00:38:23,109

 

1018
00:38:23,109 --> 00:38:27,430

 

1019
00:38:27,430 --> 00:38:35,030

 

1020
00:38:35,030 --> 00:38:36,630

 

1021
00:38:36,630 --> 00:38:38,630

 

1022
00:38:38,630 --> 00:38:41,990

 

1023
00:38:41,990 --> 00:38:44,710

 

1024
00:38:44,710 --> 00:38:48,470

 

1025
00:38:48,470 --> 00:38:58,550

 

1026
00:38:58,550 --> 00:39:00,710

 

1027
00:39:00,710 --> 00:39:02,710

 

1028
00:39:02,710 --> 00:39:03,910

 

1029
00:39:03,910 --> 00:39:07,109

 

1030
00:39:07,109 --> 00:39:16,069

 

1031
00:39:16,069 --> 00:39:20,150

 

1032
00:39:20,150 --> 00:39:27,510

 

1033
00:39:27,510 --> 00:39:29,589

 

1034
00:39:29,589 --> 00:39:31,109

 

1035
00:39:31,109 --> 00:39:34,550

 

1036
00:39:34,550 --> 00:39:34,560

 

1037
00:39:34,560 --> 00:39:35,030


1038
00:39:35,030 --> 00:39:37,990

 

1039
00:39:37,990 --> 00:39:39,190

 

1040
00:39:39,190 --> 00:39:41,190

 

1041
00:39:41,190 --> 00:39:41,200

 

1042
00:39:41,200 --> 00:39:41,829


1043
00:39:41,829 --> 00:39:44,069

 

1044
00:39:44,069 --> 00:39:45,910

 

1045
00:39:45,910 --> 00:39:47,109

 

1046
00:39:47,109 --> 00:39:49,349

 

1047
00:39:49,349 --> 00:39:50,790

 

1048
00:39:50,790 --> 00:39:53,030

 

1049
00:39:53,030 --> 00:39:54,829

 

1050
00:39:54,829 --> 00:39:58,390

 

1051
00:39:58,390 --> 00:39:59,990

 

1052
00:39:59,990 --> 00:40:01,589

 

1053
00:40:01,589 --> 00:40:04,790

 

1054
00:40:04,790 --> 00:40:06,150

 

1055
00:40:06,150 --> 00:40:09,270

 

1056
00:40:09,270 --> 00:40:11,910

 

1057
00:40:11,910 --> 00:40:13,430

 

1058
00:40:13,430 --> 00:40:14,710

 

1059
00:40:14,710 --> 00:40:17,190

 

1060
00:40:17,190 --> 00:40:18,710

 

1061
00:40:18,710 --> 00:40:18,720

 

1062
00:40:18,720 --> 00:40:19,349


1063
00:40:19,349 --> 00:40:21,750

 

1064
00:40:21,750 --> 00:40:22,950

 

1065
00:40:22,950 --> 00:40:26,309

 

1066
00:40:26,309 --> 00:40:27,270

 

1067
00:40:27,270 --> 00:40:28,630

 

1068
00:40:28,630 --> 00:40:29,910

 

1069
00:40:29,910 --> 00:40:31,030

 

1070
00:40:31,030 --> 00:40:34,069

 

1071
00:40:34,069 --> 00:40:36,069

 

1072
00:40:36,069 --> 00:40:38,309

 

1073
00:40:38,309 --> 00:40:40,470

 

1074
00:40:40,470 --> 00:40:42,870

 

1075
00:40:42,870 --> 00:40:45,750

 

1076
00:40:45,750 --> 00:40:45,760

 

1077
00:40:45,760 --> 00:40:50,790


1078
00:40:50,790 --> 00:40:52,150

 

1079
00:40:52,150 --> 00:40:54,470

 

1080
00:40:54,470 --> 00:40:58,150

 

1081
00:40:58,150 --> 00:41:04,069

 

1082
00:41:04,069 --> 00:41:09,990

 

1083
00:41:09,990 --> 00:41:12,710

 

1084
00:41:12,710 --> 00:41:16,550

 

1085
00:41:16,550 --> 00:41:18,550

 

1086
00:41:18,550 --> 00:41:20,069

 

1087
00:41:20,069 --> 00:41:21,990

 

1088
00:41:21,990 --> 00:41:23,270

 

1089
00:41:23,270 --> 00:41:25,430

 

1090
00:41:25,430 --> 00:41:27,190

 

1091
00:41:27,190 --> 00:41:28,630

 

1092
00:41:28,630 --> 00:41:28,640

 

1093
00:41:28,640 --> 00:41:30,470


1094
00:41:30,470 --> 00:41:34,150

 

1095
00:41:34,150 --> 00:41:36,069

 

1096
00:41:36,069 --> 00:41:38,150

 

1097
00:41:38,150 --> 00:41:39,829

 

1098
00:41:39,829 --> 00:41:41,750

 

1099
00:41:41,750 --> 00:41:43,750

 

1100
00:41:43,750 --> 00:41:46,790

 

1101
00:41:46,790 --> 00:41:48,150

 

1102
00:41:48,150 --> 00:41:55,190

 

1103
00:41:55,190 --> 00:42:00,069

 

1104
00:42:00,069 --> 00:42:05,430

 

1105
00:42:05,430 --> 00:42:07,270

 

1106
00:42:07,270 --> 00:42:08,630

 

1107
00:42:08,630 --> 00:42:08,640

 

1108
00:42:08,640 --> 00:42:09,430


1109
00:42:09,430 --> 00:42:13,589

 

1110
00:42:13,589 --> 00:42:17,589

 

1111
00:42:17,589 --> 00:42:19,109

 

1112
00:42:19,109 --> 00:42:20,230

 

1113
00:42:20,230 --> 00:42:21,829

 

1114
00:42:21,829 --> 00:42:23,510

 

1115
00:42:23,510 --> 00:42:24,870

 

1116
00:42:24,870 --> 00:42:26,470

 

1117
00:42:26,470 --> 00:42:28,550

 

1118
00:42:28,550 --> 00:42:32,470

 

1119
00:42:32,470 --> 00:42:32,480

 

1120
00:42:32,480 --> 00:42:33,589


1121
00:42:33,589 --> 00:42:34,790

 

1122
00:42:34,790 --> 00:42:37,270

 

1123
00:42:37,270 --> 00:42:39,109

 

1124
00:42:39,109 --> 00:42:40,470

 

1125
00:42:40,470 --> 00:42:41,990

 

1126
00:42:41,990 --> 00:42:43,510

 

1127
00:42:43,510 --> 00:42:43,520

 

1128
00:42:43,520 --> 00:42:45,190


1129
00:42:45,190 --> 00:42:46,790

 

1130
00:42:46,790 --> 00:42:48,390

 

1131
00:42:48,390 --> 00:42:51,990

 

1132
00:42:51,990 --> 00:42:55,190

 

1133
00:42:55,190 --> 00:42:56,790

 

1134
00:42:56,790 --> 00:42:59,510

 

1135
00:42:59,510 --> 00:43:00,790

 

1136
00:43:00,790 --> 00:43:02,470

 

1137
00:43:02,470 --> 00:43:06,630

 

1138
00:43:06,630 --> 00:43:09,829

 

1139
00:43:09,829 --> 00:43:11,430

 

1140
00:43:11,430 --> 00:43:13,910

 

1141
00:43:13,910 --> 00:43:15,670

 

1142
00:43:15,670 --> 00:43:17,270

 

1143
00:43:17,270 --> 00:43:18,790

 

1144
00:43:18,790 --> 00:43:20,470

 

1145
00:43:20,470 --> 00:43:23,430

 

1146
00:43:23,430 --> 00:43:28,470

 

1147
00:43:28,470 --> 00:43:31,030

 

1148
00:43:31,030 --> 00:43:32,550

 

1149
00:43:32,550 --> 00:43:32,560

 

1150
00:43:32,560 --> 00:43:33,109


1151
00:43:33,109 --> 00:43:36,309

 

1152
00:43:36,309 --> 00:43:36,319

 

1153
00:43:36,319 --> 00:43:36,790


1154
00:43:36,790 --> 00:43:39,109

 

1155
00:43:39,109 --> 00:43:40,390

 

1156
00:43:40,390 --> 00:43:44,069

 

1157
00:43:44,069 --> 00:43:47,109

 

1158
00:43:47,109 --> 00:43:49,750

 

1159
00:43:49,750 --> 00:43:49,760

 

1160
00:43:49,760 --> 00:43:50,309


1161
00:43:50,309 --> 00:43:52,870

 

1162
00:43:52,870 --> 00:43:54,309

 

1163
00:43:54,309 --> 00:43:54,319

 

1164
00:43:54,319 --> 00:43:59,829


1165
00:43:59,829 --> 00:44:01,750

 

1166
00:44:01,750 --> 00:44:01,760

 

1167
00:44:01,760 --> 00:44:06,829


1168
00:44:06,829 --> 00:44:10,470

 

1169
00:44:10,470 --> 00:44:12,790

 

1170
00:44:12,790 --> 00:44:18,710

 

1171
00:44:18,710 --> 00:44:22,309

 

1172
00:44:22,309 --> 00:44:23,589

 

1173
00:44:23,589 --> 00:44:34,470

 

1174
00:44:34,470 --> 00:44:36,630

 

1175
00:44:36,630 --> 00:44:37,910

 

1176
00:44:37,910 --> 00:44:37,920

 

1177
00:44:37,920 --> 00:44:40,550


1178
00:44:40,550 --> 00:44:47,910

 

1179
00:44:47,910 --> 00:44:56,150

 

1180
00:44:56,150 --> 00:45:01,670

 

1181
00:45:01,670 --> 00:45:05,829

 

1182
00:45:05,829 --> 00:45:07,190

 

1183
00:45:07,190 --> 00:45:10,069

 

1184
00:45:10,069 --> 00:45:14,309

 

1185
00:45:14,309 --> 00:45:17,430

 

1186
00:45:17,430 --> 00:45:19,270

 

1187
00:45:19,270 --> 00:45:21,190

 

1188
00:45:21,190 --> 00:45:22,870

 

1189
00:45:22,870 --> 00:45:23,670

 

1190
00:45:23,670 --> 00:45:26,150

 

1191
00:45:26,150 --> 00:45:28,870

 

1192
00:45:28,870 --> 00:45:37,109

 

1193
00:45:37,109 --> 00:45:38,470

 

1194
00:45:38,470 --> 00:45:40,230

 

1195
00:45:40,230 --> 00:45:42,950

 

1196
00:45:42,950 --> 00:45:44,710

 

1197
00:45:44,710 --> 00:45:46,950

 

1198
00:45:46,950 --> 00:45:50,390

 

1199
00:45:50,390 --> 00:45:53,270

 

1200
00:45:53,270 --> 00:45:54,790

 

1201
00:45:54,790 --> 00:46:03,990

 

1202
00:46:03,990 --> 00:46:08,309

 

1203
00:46:08,309 --> 00:46:11,589

 

1204
00:46:11,589 --> 00:46:14,790

 

1205
00:46:14,790 --> 00:46:14,800

 

1206
00:46:14,800 --> 00:46:16,150


1207
00:46:16,150 --> 00:46:19,270

 

1208
00:46:19,270 --> 00:46:20,550

 

1209
00:46:20,550 --> 00:46:21,750

 

1210
00:46:21,750 --> 00:46:24,069

 

1211
00:46:24,069 --> 00:46:25,270

 

1212
00:46:25,270 --> 00:46:27,109

 

1213
00:46:27,109 --> 00:46:29,589

 

1214
00:46:29,589 --> 00:46:31,030

 

1215
00:46:31,030 --> 00:46:33,430

 

1216
00:46:33,430 --> 00:46:35,349

 

1217
00:46:35,349 --> 00:46:35,359

 

1218
00:46:35,359 --> 00:46:36,630


1219
00:46:36,630 --> 00:46:37,990

 

1220
00:46:37,990 --> 00:46:39,190

 

1221
00:46:39,190 --> 00:46:41,270

 

1222
00:46:41,270 --> 00:46:42,470

 

1223
00:46:42,470 --> 00:46:44,390

 

1224
00:46:44,390 --> 00:46:46,230

 

1225
00:46:46,230 --> 00:46:48,870

 

1226
00:46:48,870 --> 00:46:49,750

 

1227
00:46:49,750 --> 00:46:53,190

 

1228
00:46:53,190 --> 00:46:54,550

 

1229
00:46:54,550 --> 00:46:54,560

 

1230
00:46:54,560 --> 00:46:55,030


1231
00:46:55,030 --> 00:46:56,710

 

1232
00:46:56,710 --> 00:46:58,230

 

1233
00:46:58,230 --> 00:46:58,240

 

1234
00:46:58,240 --> 00:46:58,710


1235
00:46:58,710 --> 00:47:02,150

 

1236
00:47:02,150 --> 00:47:02,160

 

1237
00:47:02,160 --> 00:47:02,630


1238
00:47:02,630 --> 00:47:04,790

 

1239
00:47:04,790 --> 00:47:04,800

 

1240
00:47:04,800 --> 00:47:07,510


1241
00:47:07,510 --> 00:47:09,430

 

1242
00:47:09,430 --> 00:47:10,630

 

1243
00:47:10,630 --> 00:47:13,750

 

1244
00:47:13,750 --> 00:47:15,190

 

1245
00:47:15,190 --> 00:47:16,870

 

1246
00:47:16,870 --> 00:47:18,069

 

1247
00:47:18,069 --> 00:47:19,510

 

1248
00:47:19,510 --> 00:47:21,670

 

1249
00:47:21,670 --> 00:47:24,470

 

1250
00:47:24,470 --> 00:47:25,109

 

1251
00:47:25,109 --> 00:47:27,109

 

1252
00:47:27,109 --> 00:47:28,790

 

1253
00:47:28,790 --> 00:47:28,800

 

1254
00:47:28,800 --> 00:47:29,109


1255
00:47:29,109 --> 00:47:31,750

 

1256
00:47:31,750 --> 00:47:33,030

 

1257
00:47:33,030 --> 00:47:34,230

 

1258
00:47:34,230 --> 00:47:35,589

 

1259
00:47:35,589 --> 00:47:39,109

 

1260
00:47:39,109 --> 00:47:45,829

 

1261
00:47:45,829 --> 00:47:52,790

 

1262
00:47:52,790 --> 00:47:56,549

 

1263
00:47:56,549 --> 00:47:56,559

 

1264
00:47:56,559 --> 00:47:57,750


1265
00:47:57,750 --> 00:48:01,750

 

1266
00:48:01,750 --> 00:48:05,190

 

1267
00:48:05,190 --> 00:48:05,990

 

1268
00:48:05,990 --> 00:48:09,109

 

1269
00:48:09,109 --> 00:48:15,430

 

1270
00:48:15,430 --> 00:48:17,430

 

1271
00:48:17,430 --> 00:48:19,030

 

1272
00:48:19,030 --> 00:48:22,549

 

1273
00:48:22,549 --> 00:48:22,559

 

1274
00:48:22,559 --> 00:48:23,430


1275
00:48:23,430 --> 00:48:26,230

 

1276
00:48:26,230 --> 00:48:26,240

 

1277
00:48:26,240 --> 00:48:26,870


1278
00:48:26,870 --> 00:48:28,069

 

1279
00:48:28,069 --> 00:48:29,990

 

1280
00:48:29,990 --> 00:48:34,069

 

1281
00:48:34,069 --> 00:48:35,910

 

1282
00:48:35,910 --> 00:48:38,230

 

1283
00:48:38,230 --> 00:48:39,829

 

1284
00:48:39,829 --> 00:48:41,270

 

1285
00:48:41,270 --> 00:48:44,549

 

1286
00:48:44,549 --> 00:48:47,270

 

1287
00:48:47,270 --> 00:48:48,230

 

1288
00:48:48,230 --> 00:48:49,510

 

1289
00:48:49,510 --> 00:48:51,349

 

1290
00:48:51,349 --> 00:48:52,790

 

1291
00:48:52,790 --> 00:48:56,630

 

1292
00:48:56,630 --> 00:48:59,589

 

1293
00:48:59,589 --> 00:49:02,069

 

1294
00:49:02,069 --> 00:49:03,510

 

1295
00:49:03,510 --> 00:49:08,870

 

1296
00:49:08,870 --> 00:49:10,309

 

1297
00:49:10,309 --> 00:49:13,030

 

1298
00:49:13,030 --> 00:49:16,230

 

1299
00:49:16,230 --> 00:49:19,670

 

1300
00:49:19,670 --> 00:49:21,109

 

1301
00:49:21,109 --> 00:49:22,630

 

1302
00:49:22,630 --> 00:49:24,150

 

1303
00:49:24,150 --> 00:49:24,160

 

1304
00:49:24,160 --> 00:49:24,630


1305
00:49:24,630 --> 00:49:27,349

 

1306
00:49:27,349 --> 00:49:27,359

 

1307
00:49:27,359 --> 00:49:27,829


1308
00:49:27,829 --> 00:49:30,630

 

1309
00:49:30,630 --> 00:49:32,069

 

1310
00:49:32,069 --> 00:49:35,910

 

1311
00:49:35,910 --> 00:49:37,670

 

1312
00:49:37,670 --> 00:49:43,349

 

1313
00:49:43,349 --> 00:49:45,670

 

1314
00:49:45,670 --> 00:49:47,349

 

1315
00:49:47,349 --> 00:49:50,150

 

1316
00:49:50,150 --> 00:49:51,910

 

1317
00:49:51,910 --> 00:49:55,109

 

1318
00:49:55,109 --> 00:49:56,390

 

1319
00:49:56,390 --> 00:49:58,150

 

1320
00:49:58,150 --> 00:50:01,270

 

1321
00:50:01,270 --> 00:50:01,280

 

1322
00:50:01,280 --> 00:50:01,910


1323
00:50:01,910 --> 00:50:04,710

 

1324
00:50:04,710 --> 00:50:05,910

 

1325
00:50:05,910 --> 00:50:09,030

 

1326
00:50:09,030 --> 00:50:13,670

 

1327
00:50:13,670 --> 00:50:16,470

 

1328
00:50:16,470 --> 00:50:17,430

 

1329
00:50:17,430 --> 00:50:20,470

 

1330
00:50:20,470 --> 00:50:22,630

 

1331
00:50:22,630 --> 00:50:24,309

 

1332
00:50:24,309 --> 00:50:25,589

 

1333
00:50:25,589 --> 00:50:26,950

 

1334
00:50:26,950 --> 00:50:28,390

 

1335
00:50:28,390 --> 00:50:29,109

 

1336
00:50:29,109 --> 00:50:30,390

 

1337
00:50:30,390 --> 00:50:35,589

 

1338
00:50:35,589 --> 00:50:37,349

 

1339
00:50:37,349 --> 00:50:39,109

 

1340
00:50:39,109 --> 00:50:41,109

 

1341
00:50:41,109 --> 00:50:43,270

 

1342
00:50:43,270 --> 00:50:45,510

 

1343
00:50:45,510 --> 00:50:47,349

 

1344
00:50:47,349 --> 00:50:49,990

 

1345
00:50:49,990 --> 00:50:51,829

 

1346
00:50:51,829 --> 00:50:51,839

 

1347
00:50:51,839 --> 00:50:53,109


1348
00:50:53,109 --> 00:50:55,030

 

1349
00:50:55,030 --> 00:50:56,230

 

1350
00:50:56,230 --> 00:50:58,710

 

1351
00:50:58,710 --> 00:51:00,150

 

1352
00:51:00,150 --> 00:51:03,109

 

1353
00:51:03,109 --> 00:51:04,630

 

1354
00:51:04,630 --> 00:51:07,190

 

1355
00:51:07,190 --> 00:51:08,710

 

1356
00:51:08,710 --> 00:51:10,549

 

1357
00:51:10,549 --> 00:51:13,030

 

1358
00:51:13,030 --> 00:51:14,309

 

1359
00:51:14,309 --> 00:51:15,670

 

1360
00:51:15,670 --> 00:51:17,589

 

1361
00:51:17,589 --> 00:51:20,950

 

1362
00:51:20,950 --> 00:51:23,190

 

1363
00:51:23,190 --> 00:51:27,270

 

1364
00:51:27,270 --> 00:51:27,280

 

1365
00:51:27,280 --> 00:51:31,510


1366
00:51:31,510 --> 00:51:34,870

 

1367
00:51:34,870 --> 00:51:36,630

 

1368
00:51:36,630 --> 00:51:38,390

 

1369
00:51:38,390 --> 00:51:41,510

 

1370
00:51:41,510 --> 00:51:52,950

 

1371
00:51:52,950 --> 00:51:56,470

 

1372
00:51:56,470 --> 00:51:57,670

 

1373
00:51:57,670 --> 00:51:59,910

 

1374
00:51:59,910 --> 00:51:59,920

 

1375
00:51:59,920 --> 00:52:00,710


1376
00:52:00,710 --> 00:52:03,030

 

1377
00:52:03,030 --> 00:52:04,390

 

1378
00:52:04,390 --> 00:52:05,670

 

1379
00:52:05,670 --> 00:52:07,510

 

1380
00:52:07,510 --> 00:52:08,870

 

1381
00:52:08,870 --> 00:52:10,150

 

1382
00:52:10,150 --> 00:52:11,510

 

1383
00:52:11,510 --> 00:52:12,950

 

1384
00:52:12,950 --> 00:52:14,790

 

1385
00:52:14,790 --> 00:52:16,790

 

1386
00:52:16,790 --> 00:52:18,549

 

1387
00:52:18,549 --> 00:52:19,829

 

1388
00:52:19,829 --> 00:52:21,670

 

1389
00:52:21,670 --> 00:52:22,870

 

1390
00:52:22,870 --> 00:52:25,349

 

1391
00:52:25,349 --> 00:52:26,870

 

1392
00:52:26,870 --> 00:52:28,390

 

1393
00:52:28,390 --> 00:52:28,400

 

1394
00:52:28,400 --> 00:52:30,870


1395
00:52:30,870 --> 00:52:35,430

 

1396
00:52:35,430 --> 00:52:36,950

 

1397
00:52:36,950 --> 00:52:39,030

 

1398
00:52:39,030 --> 00:52:41,589

 

1399
00:52:41,589 --> 00:52:43,109

 

1400
00:52:43,109 --> 00:52:44,870

 

1401
00:52:44,870 --> 00:52:46,630

 

1402
00:52:46,630 --> 00:52:48,309

 

1403
00:52:48,309 --> 00:52:50,549

 

1404
00:52:50,549 --> 00:52:52,950

 

1405
00:52:52,950 --> 00:52:55,829

 

1406
00:52:55,829 --> 00:53:02,069

 

1407
00:53:02,069 --> 00:53:06,069

 

1408
00:53:06,069 --> 00:53:08,309

 

1409
00:53:08,309 --> 00:53:12,150

 

1410
00:53:12,150 --> 00:53:15,349

 

1411
00:53:15,349 --> 00:53:18,710

 

1412
00:53:18,710 --> 00:53:20,870

 

1413
00:53:20,870 --> 00:53:22,470

 

1414
00:53:22,470 --> 00:53:24,309

 

1415
00:53:24,309 --> 00:53:27,349

 

1416
00:53:27,349 --> 00:53:30,549

 

1417
00:53:30,549 --> 00:53:31,829

 

1418
00:53:31,829 --> 00:53:33,030

 

1419
00:53:33,030 --> 00:53:36,309

 

1420
00:53:36,309 --> 00:53:37,510

 

1421
00:53:37,510 --> 00:53:39,510

 

1422
00:53:39,510 --> 00:53:42,069

 

1423
00:53:42,069 --> 00:53:42,079

 

1424
00:53:42,079 --> 00:53:43,190


1425
00:53:43,190 --> 00:53:45,190

 

1426
00:53:45,190 --> 00:53:46,790

 

1427
00:53:46,790 --> 00:53:48,150

 

1428
00:53:48,150 --> 00:53:50,150

 

1429
00:53:50,150 --> 00:53:52,549

 

1430
00:53:52,549 --> 00:53:53,910

 

1431
00:53:53,910 --> 00:53:55,990

 

1432
00:53:55,990 --> 00:53:56,950

 

1433
00:53:56,950 --> 00:54:02,790

 

1434
00:54:02,790 --> 00:54:05,829

 

1435
00:54:05,829 --> 00:54:08,230

 

1436
00:54:08,230 --> 00:54:09,829

 

1437
00:54:09,829 --> 00:54:09,839

 

1438
00:54:09,839 --> 00:54:12,230


1439
00:54:12,230 --> 00:54:14,470

 

1440
00:54:14,470 --> 00:54:15,430

 

1441
00:54:15,430 --> 00:54:23,990

 

1442
00:54:23,990 --> 00:54:27,109

 

1443
00:54:27,109 --> 00:54:30,309

 

1444
00:54:30,309 --> 00:54:32,309

 

1445
00:54:32,309 --> 00:54:33,750

 

1446
00:54:33,750 --> 00:54:33,760

 

1447
00:54:33,760 --> 00:54:34,870


1448
00:54:34,870 --> 00:54:37,589

 

1449
00:54:37,589 --> 00:54:39,510

 

1450
00:54:39,510 --> 00:54:41,349

 

1451
00:54:41,349 --> 00:54:45,030

 

1452
00:54:45,030 --> 00:54:46,390

 

1453
00:54:46,390 --> 00:54:48,230

 

1454
00:54:48,230 --> 00:54:49,589

 

1455
00:54:49,589 --> 00:54:51,510

 

1456
00:54:51,510 --> 00:54:52,630

 

1457
00:54:52,630 --> 00:54:53,910

 

1458
00:54:53,910 --> 00:54:56,069

 

1459
00:54:56,069 --> 00:54:58,549

 

1460
00:54:58,549 --> 00:55:00,950

 

1461
00:55:00,950 --> 00:55:02,950

 

1462
00:55:02,950 --> 00:55:07,670

 

1463
00:55:07,670 --> 00:55:15,270

 

1464
00:55:15,270 --> 00:55:16,710

 

1465
00:55:16,710 --> 00:55:18,630

 

1466
00:55:18,630 --> 00:55:21,190

 

1467
00:55:21,190 --> 00:55:23,190

 

1468
00:55:23,190 --> 00:55:26,789

 

1469
00:55:26,789 --> 00:55:29,030

 

1470
00:55:29,030 --> 00:55:30,789

 

1471
00:55:30,789 --> 00:55:30,799

 

1472
00:55:30,799 --> 00:55:31,190


1473
00:55:31,190 --> 00:55:34,789

 

1474
00:55:34,789 --> 00:55:36,549

 

1475
00:55:36,549 --> 00:55:38,069

 

1476
00:55:38,069 --> 00:55:39,349

 

1477
00:55:39,349 --> 00:55:41,430

 

1478
00:55:41,430 --> 00:55:43,349

 

1479
00:55:43,349 --> 00:55:44,309

 

1480
00:55:44,309 --> 00:55:45,589

 

1481
00:55:45,589 --> 00:55:50,630

 

1482
00:55:50,630 --> 00:55:52,789

 

1483
00:55:52,789 --> 00:55:59,190

 

1484
00:55:59,190 --> 00:56:02,390

 

1485
00:56:02,390 --> 00:56:09,430

 

1486
00:56:09,430 --> 00:56:13,430

 

1487
00:56:13,430 --> 00:56:15,109

 

1488
00:56:15,109 --> 00:56:16,870

 

1489
00:56:16,870 --> 00:56:18,150

 

1490
00:56:18,150 --> 00:56:19,030

 

1491
00:56:19,030 --> 00:56:20,710

 

1492
00:56:20,710 --> 00:56:23,270

 

1493
00:56:23,270 --> 00:56:26,390

 

1494
00:56:26,390 --> 00:56:28,230

 

1495
00:56:28,230 --> 00:56:29,589

 

1496
00:56:29,589 --> 00:56:32,470

 

1497
00:56:32,470 --> 00:56:34,150

 

1498
00:56:34,150 --> 00:56:36,069

 

1499
00:56:36,069 --> 00:56:36,079

 

1500
00:56:36,079 --> 00:56:38,950


1501
00:56:38,950 --> 00:56:41,030

 

1502
00:56:41,030 --> 00:56:43,589

 

1503
00:56:43,589 --> 00:57:05,109

 

1504
00:57:05,109 --> 00:57:07,670

 

1505
00:57:07,670 --> 00:57:08,630

 

1506
00:57:08,630 --> 00:57:10,470

 

1507
00:57:10,470 --> 00:57:11,670

 

1508
00:57:11,670 --> 00:57:13,030

 

1509
00:57:13,030 --> 00:57:14,710

 

1510
00:57:14,710 --> 00:57:17,589

 

1511
00:57:17,589 --> 00:57:19,190

 

1512
00:57:19,190 --> 00:57:22,150

 

1513
00:57:22,150 --> 00:57:23,510

 

1514
00:57:23,510 --> 00:57:24,710

 

1515
00:57:24,710 --> 00:57:27,109

 

1516
00:57:27,109 --> 00:57:30,309

 

1517
00:57:30,309 --> 00:57:30,319

 

1518
00:57:30,319 --> 00:57:31,589


1519
00:57:31,589 --> 00:57:35,349

 

1520
00:57:35,349 --> 00:57:39,270

 

1521
00:57:39,270 --> 00:57:40,950

 

1522
00:57:40,950 --> 00:57:42,950

 

1523
00:57:42,950 --> 00:57:44,549

 

1524
00:57:44,549 --> 00:57:46,470

 

1525
00:57:46,470 --> 00:57:47,670

 

1526
00:57:47,670 --> 00:57:49,430

 

1527
00:57:49,430 --> 00:57:50,549

 

1528
00:57:50,549 --> 00:57:50,559

 

1529
00:57:50,559 --> 00:57:51,270


1530
00:57:51,270 --> 00:57:53,430

 

1531
00:57:53,430 --> 00:57:55,030

 

1532
00:57:55,030 --> 00:57:55,040

 

1533
00:57:55,040 --> 00:57:56,309


1534
00:57:56,309 --> 00:57:58,789

 

1535
00:57:58,789 --> 00:57:58,799

 

1536
00:57:58,799 --> 00:57:59,589


1537
00:57:59,589 --> 00:58:03,430

 

1538
00:58:03,430 --> 00:58:06,870

 

1539
00:58:06,870 --> 00:58:09,030

 

1540
00:58:09,030 --> 00:58:09,040

 

1541
00:58:09,040 --> 00:58:10,470


1542
00:58:10,470 --> 00:58:14,069

 

1543
00:58:14,069 --> 00:58:16,870

 

1544
00:58:16,870 --> 00:58:16,880

 

1545
00:58:16,880 --> 00:58:17,510


1546
00:58:17,510 --> 00:58:21,990

 

1547
00:58:21,990 --> 00:58:23,670

 

1548
00:58:23,670 --> 00:58:25,030

 

1549
00:58:25,030 --> 00:58:26,470

 

1550
00:58:26,470 --> 00:58:27,750

 

1551
00:58:27,750 --> 00:58:30,789

 

1552
00:58:30,789 --> 00:58:34,950

 

1553
00:58:34,950 --> 00:58:36,870

 

1554
00:58:36,870 --> 00:58:38,390

 

1555
00:58:38,390 --> 00:58:40,230

 

1556
00:58:40,230 --> 00:58:42,789

 

1557
00:58:42,789 --> 00:58:45,349

 

1558
00:58:45,349 --> 00:58:47,349

 

1559
00:58:47,349 --> 00:58:49,510

 

1560
00:58:49,510 --> 00:58:51,430

 

1561
00:58:51,430 --> 00:58:52,710

 

1562
00:58:52,710 --> 00:58:54,630

 

1563
00:58:54,630 --> 00:58:55,910

 

1564
00:58:55,910 --> 00:58:57,670

 

1565
00:58:57,670 --> 00:58:59,750

 

1566
00:58:59,750 --> 00:59:01,829

 

1567
00:59:01,829 --> 00:59:04,390

 

1568
00:59:04,390 --> 00:59:08,150

 

1569
00:59:08,150 --> 00:59:10,309

 

1570
00:59:10,309 --> 00:59:12,630

 

1571
00:59:12,630 --> 00:59:16,390

 

1572
00:59:16,390 --> 00:59:18,710

 

1573
00:59:18,710 --> 00:59:20,230

 

1574
00:59:20,230 --> 00:59:21,190

 

1575
00:59:21,190 --> 00:59:23,030

 

1576
00:59:23,030 --> 00:59:24,789

 

1577
00:59:24,789 --> 00:59:27,510

 

1578
00:59:27,510 --> 00:59:29,030

 

1579
00:59:29,030 --> 00:59:32,470

 

1580
00:59:32,470 --> 00:59:35,829

 

1581
00:59:35,829 --> 00:59:38,870

 

1582
00:59:38,870 --> 00:59:40,549

 

1583
00:59:40,549 --> 00:59:42,150

 

1584
00:59:42,150 --> 00:59:43,750

 

1585
00:59:43,750 --> 00:59:46,549

 

1586
00:59:46,549 --> 00:59:48,069

 

1587
00:59:48,069 --> 00:59:49,270

 

1588
00:59:49,270 --> 00:59:52,470

 

1589
00:59:52,470 --> 00:59:52,480

 

1590
00:59:52,480 --> 00:59:53,510


1591
00:59:53,510 --> 00:59:56,549

 

1592
00:59:56,549 --> 00:59:58,549

 

1593
00:59:58,549 --> 01:00:01,349

 

1594
01:00:01,349 --> 01:00:23,670

 

1595
01:00:23,670 --> 01:00:25,510

 

1596
01:00:25,510 --> 01:00:27,990

 

1597
01:00:27,990 --> 01:00:30,870

 

1598
01:00:30,870 --> 01:00:32,870

 

1599
01:00:32,870 --> 01:00:40,549

 

1600
01:00:40,549 --> 01:00:42,150

 

1601
01:00:42,150 --> 01:00:44,230

 

1602
01:00:44,230 --> 01:00:46,549

 

1603
01:00:46,549 --> 01:00:48,789

 

1604
01:00:48,789 --> 01:00:50,549

 

1605
01:00:50,549 --> 01:00:53,030

 

1606
01:00:53,030 --> 01:00:55,750

 

1607
01:00:55,750 --> 01:00:57,750

 

1608
01:00:57,750 --> 01:00:59,109

 

1609
01:00:59,109 --> 01:01:02,309

 

1610
01:01:02,309 --> 01:01:03,990

 

1611
01:01:03,990 --> 01:01:05,750

 

1612
01:01:05,750 --> 01:01:08,549

 

1613
01:01:08,549 --> 01:01:10,710

 

1614
01:01:10,710 --> 01:01:14,309

 

1615
01:01:14,309 --> 01:01:17,190

 

1616
01:01:17,190 --> 01:01:18,630

 

1617
01:01:18,630 --> 01:01:19,990

 

1618
01:01:19,990 --> 01:01:22,150

 

1619
01:01:22,150 --> 01:01:23,270

 

1620
01:01:23,270 --> 01:01:24,950

 

1621
01:01:24,950 --> 01:01:26,390

 

1622
01:01:26,390 --> 01:01:29,510

 

1623
01:01:29,510 --> 01:01:32,230

 

1624
01:01:32,230 --> 01:01:33,510

 

1625
01:01:33,510 --> 01:01:34,470

 

1626
01:01:34,470 --> 01:01:34,480

 

1627
01:01:34,480 --> 01:01:35,270


1628
01:01:35,270 --> 01:01:37,190

 

1629
01:01:37,190 --> 01:01:37,200

 

1630
01:01:37,200 --> 01:01:38,309


1631
01:01:38,309 --> 01:01:40,230

 

1632
01:01:40,230 --> 01:01:41,829

 

1633
01:01:41,829 --> 01:01:43,670

 

1634
01:01:43,670 --> 01:01:45,430

 

1635
01:01:45,430 --> 01:01:47,589

 

1636
01:01:47,589 --> 01:01:47,599

 

1637
01:01:47,599 --> 01:01:48,630


1638
01:01:48,630 --> 01:01:50,789

 

1639
01:01:50,789 --> 01:01:52,710

 

1640
01:01:52,710 --> 01:01:56,230

 

1641
01:01:56,230 --> 01:01:59,750

 

1642
01:01:59,750 --> 01:02:01,270

 

1643
01:02:01,270 --> 01:02:04,950

 

1644
01:02:04,950 --> 01:02:06,390

 

1645
01:02:06,390 --> 01:02:09,510

 

1646
01:02:09,510 --> 01:02:09,520

 

1647
01:02:09,520 --> 01:02:10,230


1648
01:02:10,230 --> 01:02:11,670

 

1649
01:02:11,670 --> 01:02:13,270

 

1650
01:02:13,270 --> 01:02:22,470

 

1651
01:02:22,470 --> 01:02:28,230

 

1652
01:02:28,230 --> 01:02:36,789

 

1653
01:02:36,789 --> 01:02:43,349

 

1654
01:02:43,349 --> 01:03:05,430

 

1655
01:03:05,430 --> 01:03:08,950

 

1656
01:03:08,950 --> 01:03:10,710

 

1657
01:03:10,710 --> 01:03:20,710

 

1658
01:03:20,710 --> 01:03:31,750

 

1659
01:03:31,750 --> 01:03:33,109

 

1660
01:03:33,109 --> 01:03:35,349

 

1661
01:03:35,349 --> 01:03:36,630

 

1662
01:03:36,630 --> 01:03:40,549

 

1663
01:03:40,549 --> 01:03:43,589

 

1664
01:03:43,589 --> 01:03:45,990

 

1665
01:03:45,990 --> 01:03:49,349

 

1666
01:03:49,349 --> 01:03:53,270

 

1667
01:03:53,270 --> 01:03:55,829

 

1668
01:03:55,829 --> 01:03:56,870

 

1669
01:03:56,870 --> 01:03:59,510

 

1670
01:03:59,510 --> 01:03:59,520

 

1671
01:03:59,520 --> 01:04:00,470


1672
01:04:00,470 --> 01:04:04,069

 

1673
01:04:04,069 --> 01:04:05,430

 

1674
01:04:05,430 --> 01:04:07,990

 

1675
01:04:07,990 --> 01:04:08,000

 

1676
01:04:08,000 --> 01:04:09,349


1677
01:04:09,349 --> 01:04:11,190

 

1678
01:04:11,190 --> 01:04:13,109

 

1679
01:04:13,109 --> 01:04:13,119

 

1680
01:04:13,119 --> 01:04:14,230


1681
01:04:14,230 --> 01:04:16,710

 

1682
01:04:16,710 --> 01:04:17,670

 

1683
01:04:17,670 --> 01:04:20,710

 

1684
01:04:20,710 --> 01:04:22,230

 

1685
01:04:22,230 --> 01:04:23,990

 

1686
01:04:23,990 --> 01:04:26,230

 

1687
01:04:26,230 --> 01:04:27,510

 

1688
01:04:27,510 --> 01:04:29,510

 

1689
01:04:29,510 --> 01:04:31,589

 

1690
01:04:31,589 --> 01:04:33,270

 

1691
01:04:33,270 --> 01:04:35,910

 

1692
01:04:35,910 --> 01:04:35,920

 

1693
01:04:35,920 --> 01:04:36,549


1694
01:04:36,549 --> 01:04:38,309

 

1695
01:04:38,309 --> 01:04:40,309

 

1696
01:04:40,309 --> 01:04:41,990

 

1697
01:04:41,990 --> 01:04:44,150

 

1698
01:04:44,150 --> 01:04:45,270

 

1699
01:04:45,270 --> 01:04:47,029

 

1700
01:04:47,029 --> 01:04:48,470

 

1701
01:04:48,470 --> 01:04:50,309

 

1702
01:04:50,309 --> 01:04:51,910

 

1703
01:04:51,910 --> 01:04:53,270

 

1704
01:04:53,270 --> 01:04:55,109

 

1705
01:04:55,109 --> 01:04:55,119

 

1706
01:04:55,119 --> 01:04:57,349


1707
01:04:57,349 --> 01:05:01,990

 

1708
01:05:01,990 --> 01:05:03,270

 

1709
01:05:03,270 --> 01:05:06,829

 

1710
01:05:06,829 --> 01:05:10,390

 

1711
01:05:10,390 --> 01:05:10,400

 

1712
01:05:10,400 --> 01:05:11,510


1713
01:05:11,510 --> 01:05:12,630

 

1714
01:05:12,630 --> 01:05:14,390

 

1715
01:05:14,390 --> 01:05:14,400

 

1716
01:05:14,400 --> 01:05:15,670


1717
01:05:15,670 --> 01:05:17,510

 

1718
01:05:17,510 --> 01:05:19,750

 

1719
01:05:19,750 --> 01:05:21,670

 

1720
01:05:21,670 --> 01:05:23,670

 

1721
01:05:23,670 --> 01:05:25,029

 

1722
01:05:25,029 --> 01:05:26,069

 

1723
01:05:26,069 --> 01:05:28,309

 

1724
01:05:28,309 --> 01:05:30,069

 

1725
01:05:30,069 --> 01:05:34,829

 

1726
01:05:34,829 --> 01:05:38,870

 

1727
01:05:38,870 --> 01:05:43,109

 

1728
01:05:43,109 --> 01:05:45,510

 

1729
01:05:45,510 --> 01:05:47,029

 

1730
01:05:47,029 --> 01:05:50,390

 

1731
01:05:50,390 --> 01:05:52,549

 

1732
01:05:52,549 --> 01:05:54,390

 

1733
01:05:54,390 --> 01:05:56,309

 

1734
01:05:56,309 --> 01:05:57,750

 

1735
01:05:57,750 --> 01:06:00,150

 

1736
01:06:00,150 --> 01:06:03,430

 

1737
01:06:03,430 --> 01:06:03,990

 

1738
01:06:03,990 --> 01:06:07,190

 

1739
01:06:07,190 --> 01:06:09,510

 

1740
01:06:09,510 --> 01:06:10,549

 

1741
01:06:10,549 --> 01:06:12,309

 

1742
01:06:12,309 --> 01:06:14,950

 

1743
01:06:14,950 --> 01:06:16,710

 

1744
01:06:16,710 --> 01:06:22,950

 

1745
01:06:22,950 --> 01:06:26,710

 

1746
01:06:26,710 --> 01:06:28,390

 

1747
01:06:28,390 --> 01:06:29,910

 

1748
01:06:29,910 --> 01:06:31,589

 

1749
01:06:31,589 --> 01:06:33,910

 

1750
01:06:33,910 --> 01:06:36,309

 

1751
01:06:36,309 --> 01:06:37,430

 

1752
01:06:37,430 --> 01:06:39,430

 

1753
01:06:39,430 --> 01:06:42,390

 

1754
01:06:42,390 --> 01:06:44,390

 

1755
01:06:44,390 --> 01:06:47,670

 

1756
01:06:47,670 --> 01:06:47,680

 

1757
01:06:47,680 --> 01:06:48,549


1758
01:06:48,549 --> 01:06:51,109

 

1759
01:06:51,109 --> 01:06:52,950

 

1760
01:06:52,950 --> 01:06:53,670

 

1761
01:06:53,670 --> 01:06:56,950

 

1762
01:06:56,950 --> 01:06:59,829

 

1763
01:06:59,829 --> 01:07:01,910

 

1764
01:07:01,910 --> 01:07:04,309

 

1765
01:07:04,309 --> 01:07:06,150

 

1766
01:07:06,150 --> 01:07:08,230

 

1767
01:07:08,230 --> 01:07:08,240

 

1768
01:07:08,240 --> 01:07:09,750


1769
01:07:09,750 --> 01:07:11,510

 

1770
01:07:11,510 --> 01:07:13,589

 

1771
01:07:13,589 --> 01:07:16,069

 

1772
01:07:16,069 --> 01:07:17,990

 

1773
01:07:17,990 --> 01:07:20,150

 

1774
01:07:20,150 --> 01:07:21,750

 

1775
01:07:21,750 --> 01:07:21,760

 

1776
01:07:21,760 --> 01:07:22,630


1777
01:07:22,630 --> 01:07:25,670

 

1778
01:07:25,670 --> 01:07:27,349

 

1779
01:07:27,349 --> 01:07:31,270

 

1780
01:07:31,270 --> 01:07:33,510

 

1781
01:07:33,510 --> 01:07:33,520

 

1782
01:07:33,520 --> 01:07:35,910


1783
01:07:35,910 --> 01:07:37,670

 

1784
01:07:37,670 --> 01:07:40,230

 

1785
01:07:40,230 --> 01:07:40,240

 

1786
01:07:40,240 --> 01:07:40,870


1787
01:07:40,870 --> 01:07:43,670

 

1788
01:07:43,670 --> 01:07:45,109

 

1789
01:07:45,109 --> 01:07:47,190

 

1790
01:07:47,190 --> 01:07:47,200

 

1791
01:07:47,200 --> 01:07:48,230


1792
01:07:48,230 --> 01:07:50,069

 

1793
01:07:50,069 --> 01:07:50,079

 

1794
01:07:50,079 --> 01:07:51,430


1795
01:07:51,430 --> 01:07:53,910

 

1796
01:07:53,910 --> 01:07:54,950

 

1797
01:07:54,950 --> 01:07:57,990

 

1798
01:07:57,990 --> 01:07:59,510

 

1799
01:07:59,510 --> 01:08:02,390

 

1800
01:08:02,390 --> 01:08:02,400

 

1801
01:08:02,400 --> 01:08:03,510


1802
01:08:03,510 --> 01:08:06,390

 

1803
01:08:06,390 --> 01:08:08,470

 

1804
01:08:08,470 --> 01:08:10,230

 

1805
01:08:10,230 --> 01:08:12,069

 

1806
01:08:12,069 --> 01:08:13,829

 

1807
01:08:13,829 --> 01:08:15,270

 

1808
01:08:15,270 --> 01:08:16,950

 

1809
01:08:16,950 --> 01:08:19,110

 

1810
01:08:19,110 --> 01:08:21,030

 

1811
01:08:21,030 --> 01:08:22,789

 

1812
01:08:22,789 --> 01:08:25,030

 

1813
01:08:25,030 --> 01:08:27,269

 

1814
01:08:27,269 --> 01:08:30,309

 

1815
01:08:30,309 --> 01:08:33,430

 

1816
01:08:33,430 --> 01:08:36,829

 

1817
01:08:36,829 --> 01:08:36,839

 

1818
01:08:36,839 --> 01:08:39,110


1819
01:08:39,110 --> 01:08:40,789

 

1820
01:08:40,789 --> 01:08:43,269

 

1821
01:08:43,269 --> 01:08:44,550

 

1822
01:08:44,550 --> 01:08:47,829

 

1823
01:08:47,829 --> 01:08:52,149

 

1824
01:08:52,149 --> 01:08:55,110

 

1825
01:08:55,110 --> 01:08:55,120

 

1826
01:08:55,120 --> 01:09:03,269


1827
01:09:03,269 --> 01:09:05,829

 

1828
01:09:05,829 --> 01:09:06,709

 

1829
01:09:06,709 --> 01:09:08,870

 

1830
01:09:08,870 --> 01:09:10,149

 

1831
01:09:10,149 --> 01:09:10,159

 

1832
01:09:10,159 --> 01:09:10,789


1833
01:09:10,789 --> 01:09:12,870

 

1834
01:09:12,870 --> 01:09:14,229

 

1835
01:09:14,229 --> 01:09:18,070

 

1836
01:09:18,070 --> 01:09:19,669

 

1837
01:09:19,669 --> 01:09:21,269

 

1838
01:09:21,269 --> 01:09:21,279

 

1839
01:09:21,279 --> 01:09:22,070


1840
01:09:22,070 --> 01:09:23,910

 

1841
01:09:23,910 --> 01:09:25,110

 

1842
01:09:25,110 --> 01:09:26,630

 

1843
01:09:26,630 --> 01:09:27,910

 

1844
01:09:27,910 --> 01:09:27,920

 

1845
01:09:27,920 --> 01:09:28,630


1846
01:09:28,630 --> 01:09:29,990

 

1847
01:09:29,990 --> 01:09:30,000

 

1848
01:09:30,000 --> 01:09:31,910


1849
01:09:31,910 --> 01:09:33,669

 

1850
01:09:33,669 --> 01:09:33,679

 

1851
01:09:33,679 --> 01:09:35,030


1852
01:09:35,030 --> 01:09:35,040

 

1853
01:09:35,040 --> 01:09:38,630


1854
01:09:38,630 --> 01:09:40,789

 

1855
01:09:40,789 --> 01:09:42,630

 

1856
01:09:42,630 --> 01:09:44,709

 

1857
01:09:44,709 --> 01:09:45,910

 

1858
01:09:45,910 --> 01:09:47,349

 

1859
01:09:47,349 --> 01:09:47,359

 

1860
01:09:47,359 --> 01:09:47,990


1861
01:09:47,990 --> 01:09:49,910

 

1862
01:09:49,910 --> 01:09:51,749

 

1863
01:09:51,749 --> 01:09:53,669

 

1864
01:09:53,669 --> 01:09:54,790

 

1865
01:09:54,790 --> 01:09:54,800

 

1866
01:09:54,800 --> 01:09:56,630


1867
01:09:56,630 --> 01:09:57,990

 

1868
01:09:57,990 --> 01:10:01,590

 

1869
01:10:01,590 --> 01:10:04,310

 

1870
01:10:04,310 --> 01:10:07,110

 

1871
01:10:07,110 --> 01:10:09,510

 

1872
01:10:09,510 --> 01:10:11,270

 

1873
01:10:11,270 --> 01:10:13,030

 

1874
01:10:13,030 --> 01:10:15,910

 

1875
01:10:15,910 --> 01:10:16,790

 

1876
01:10:16,790 --> 01:10:20,149

 

1877
01:10:20,149 --> 01:10:22,709

 

1878
01:10:22,709 --> 01:10:23,669

 

1879
01:10:23,669 --> 01:10:26,149

 

1880
01:10:26,149 --> 01:10:26,950

 

1881
01:10:26,950 --> 01:10:29,590

 

1882
01:10:29,590 --> 01:10:29,600

 

1883
01:10:29,600 --> 01:10:30,149


1884
01:10:30,149 --> 01:10:31,510

 

1885
01:10:31,510 --> 01:10:34,870

 

1886
01:10:34,870 --> 01:10:39,750

 

1887
01:10:39,750 --> 01:10:46,310

 

1888
01:10:46,310 --> 01:10:51,110

 

1889
01:10:51,110 --> 01:10:53,110

 

1890
01:10:53,110 --> 01:10:53,120

 

1891
01:10:53,120 --> 01:10:56,630


1892
01:10:56,630 --> 01:11:10,830

 

1893
01:11:10,830 --> 01:11:10,840

 

1894
01:11:10,840 --> 01:11:14,790


1895
01:11:14,790 --> 01:11:14,800

 

1896
01:11:14,800 --> 01:11:19,350


1897
01:11:19,350 --> 01:11:28,550

 

1898
01:11:28,550 --> 01:11:32,790

 

1899
01:11:32,790 --> 01:11:37,189

 

1900
01:11:37,189 --> 01:11:39,990

 

1901
01:11:39,990 --> 01:11:41,590

 

1902
01:11:41,590 --> 01:11:43,189

 

1903
01:11:43,189 --> 01:11:45,430

 

1904
01:11:45,430 --> 01:11:46,709

 

1905
01:11:46,709 --> 01:11:48,790

 

1906
01:11:48,790 --> 01:11:50,229

 

1907
01:11:50,229 --> 01:11:53,270

 

1908
01:11:53,270 --> 01:11:55,910

 

1909
01:11:55,910 --> 01:11:55,920

 

1910
01:11:55,920 --> 01:11:56,630


1911
01:11:56,630 --> 01:11:58,229

 

1912
01:11:58,229 --> 01:12:01,189

 

1913
01:12:01,189 --> 01:12:03,030

 

1914
01:12:03,030 --> 01:12:05,510

 

1915
01:12:05,510 --> 01:12:08,790

 

1916
01:12:08,790 --> 01:12:10,470

 

1917
01:12:10,470 --> 01:12:12,790

 

1918
01:12:12,790 --> 01:12:14,550

 

1919
01:12:14,550 --> 01:12:16,310

 

1920
01:12:16,310 --> 01:12:17,030

 

1921
01:12:17,030 --> 01:12:19,510

 

1922
01:12:19,510 --> 01:12:20,950

 

1923
01:12:20,950 --> 01:12:23,590

 

1924
01:12:23,590 --> 01:12:24,790

 

1925
01:12:24,790 --> 01:12:27,430

 

1926
01:12:27,430 --> 01:12:30,070

 

1927
01:12:30,070 --> 01:12:31,750

 

1928
01:12:31,750 --> 01:12:33,270

 

1929
01:12:33,270 --> 01:12:35,430

 

1930
01:12:35,430 --> 01:12:35,990

 

1931
01:12:35,990 --> 01:12:39,030

 

1932
01:12:39,030 --> 01:12:42,390

 

1933
01:12:42,390 --> 01:12:43,830

 

1934
01:12:43,830 --> 01:12:46,470

 

1935
01:12:46,470 --> 01:12:48,550

 

1936
01:12:48,550 --> 01:12:50,550

 

1937
01:12:50,550 --> 01:12:51,750

 

1938
01:12:51,750 --> 01:12:51,760

 

1939
01:12:51,760 --> 01:12:53,270


1940
01:12:53,270 --> 01:12:57,669

 

1941
01:12:57,669 --> 01:13:00,470

 

1942
01:13:00,470 --> 01:13:04,470

 

1943
01:13:04,470 --> 01:13:13,510

 

1944
01:13:13,510 --> 01:13:16,630

 

1945
01:13:16,630 --> 01:13:21,669

 

1946
01:13:21,669 --> 01:13:23,990

 

1947
01:13:23,990 --> 01:13:24,000

 

1948
01:13:24,000 --> 01:13:24,790


1949
01:13:24,790 --> 01:13:27,510

 

1950
01:13:27,510 --> 01:13:27,520

 

1951
01:13:27,520 --> 01:13:28,310


1952
01:13:28,310 --> 01:13:33,270

 

1953
01:13:33,270 --> 01:13:35,350

 

1954
01:13:35,350 --> 01:13:35,360

 

1955
01:13:35,360 --> 01:13:37,030


1956
01:13:37,030 --> 01:13:38,630

 

1957
01:13:38,630 --> 01:13:41,669

 

1958
01:13:41,669 --> 01:13:44,470

 

1959
01:13:44,470 --> 01:13:45,990

 

1960
01:13:45,990 --> 01:13:47,750

 

1961
01:13:47,750 --> 01:13:48,709

 

1962
01:13:48,709 --> 01:13:49,590

 

1963
01:13:49,590 --> 01:13:52,550

 

1964
01:13:52,550 --> 01:13:54,070

 

1965
01:13:54,070 --> 01:13:55,030

 

1966
01:13:55,030 --> 01:13:57,030

 

1967
01:13:57,030 --> 01:13:59,030

 

1968
01:13:59,030 --> 01:14:01,910

 

1969
01:14:01,910 --> 01:14:03,590

 

1970
01:14:03,590 --> 01:14:05,030

 

1971
01:14:05,030 --> 01:14:07,510

 

1972
01:14:07,510 --> 01:14:09,669

 

1973
01:14:09,669 --> 01:14:10,550

 

1974
01:14:10,550 --> 01:14:13,750

 

1975
01:14:13,750 --> 01:14:14,870

 

1976
01:14:14,870 --> 01:14:16,709

 

1977
01:14:16,709 --> 01:14:16,719

 

1978
01:14:16,719 --> 01:14:17,910


1979
01:14:17,910 --> 01:14:20,709

 

1980
01:14:20,709 --> 01:14:20,719

 

1981
01:14:20,719 --> 01:14:21,669


1982
01:14:21,669 --> 01:14:24,470

 

1983
01:14:24,470 --> 01:14:25,830

 

1984
01:14:25,830 --> 01:14:27,590

 

1985
01:14:27,590 --> 01:14:29,669

 

1986
01:14:29,669 --> 01:14:34,149

 

1987
01:14:34,149 --> 01:14:37,189

 

1988
01:14:37,189 --> 01:14:41,270

 

1989
01:14:41,270 --> 01:14:43,110

 

1990
01:14:43,110 --> 01:14:44,390

 

1991
01:14:44,390 --> 01:14:46,470

 

1992
01:14:46,470 --> 01:14:47,990

 

1993
01:14:47,990 --> 01:14:51,270

 

1994
01:14:51,270 --> 01:14:53,350

 

1995
01:14:53,350 --> 01:14:54,470

 

1996
01:14:54,470 --> 01:14:56,709

 

1997
01:14:56,709 --> 01:14:57,669

 

1998
01:14:57,669 --> 01:14:59,030

 

1999
01:14:59,030 --> 01:14:59,040

 

2000
01:14:59,040 --> 01:14:59,990


2001
01:14:59,990 --> 01:15:05,750

 

2002
01:15:05,750 --> 01:15:18,229

 

2003
01:15:18,229 --> 01:15:20,390

 

2004
01:15:20,390 --> 01:15:21,510

 

2005
01:15:21,510 --> 01:15:22,630

 

2006
01:15:22,630 --> 01:15:24,550

 

2007
01:15:24,550 --> 01:15:27,510

 

2008
01:15:27,510 --> 01:15:29,910

 

2009
01:15:29,910 --> 01:15:31,030

 

2010
01:15:31,030 --> 01:16:04,950

 

2011
01:16:04,950 --> 01:16:09,350

 

2012
01:16:09,350 --> 01:16:13,030

 

2013
01:16:13,030 --> 01:16:15,110

 

2014
01:16:15,110 --> 01:16:16,950

 

2015
01:16:16,950 --> 01:16:19,910

 

2016
01:16:19,910 --> 01:16:21,270

 

2017
01:16:21,270 --> 01:16:23,350

 

2018
01:16:23,350 --> 01:16:24,709

 

2019
01:16:24,709 --> 01:16:26,950

 

2020
01:16:26,950 --> 01:16:26,960

 

2021
01:16:26,960 --> 01:16:28,229


2022
01:16:28,229 --> 01:16:32,709

 

2023
01:16:32,709 --> 01:16:38,630

 

2024
01:16:38,630 --> 01:16:39,910

 

2025
01:16:39,910 --> 01:16:41,910

 

2026
01:16:41,910 --> 01:16:50,550

 

2027
01:16:50,550 --> 01:17:05,669

 

2028
01:17:05,669 --> 01:17:07,430

 

2029
01:17:07,430 --> 01:17:09,350

 

2030
01:17:09,350 --> 01:17:11,750

 

2031
01:17:11,750 --> 01:17:15,030

 

2032
01:17:15,030 --> 01:17:18,950

 

2033
01:17:18,950 --> 01:17:22,229

 

2034
01:17:22,229 --> 01:17:26,070

 

2035
01:17:26,070 --> 01:17:28,470

 

2036
01:17:28,470 --> 01:17:30,630

 

2037
01:17:30,630 --> 01:17:31,830

 

2038
01:17:31,830 --> 01:17:37,110

 

2039
01:17:37,110 --> 01:17:38,630

 

2040
01:17:38,630 --> 01:17:40,709

 

2041
01:17:40,709 --> 01:17:43,990

 

2042
01:17:43,990 --> 01:17:46,950

 

2043
01:17:46,950 --> 01:17:46,960

 

2044
01:17:46,960 --> 01:17:48,149


2045
01:17:48,149 --> 01:17:50,870

 

2046
01:17:50,870 --> 01:17:50,880

 

2047
01:17:50,880 --> 01:17:51,669


2048
01:17:51,669 --> 01:17:55,030

 

2049
01:17:55,030 --> 01:17:55,040

 

2050
01:17:55,040 --> 01:17:56,310


2051
01:17:56,310 --> 01:18:00,229

 

2052
01:18:00,229 --> 01:18:03,430

 

2053
01:18:03,430 --> 01:18:07,189

 

2054
01:18:07,189 --> 01:18:09,270

 

2055
01:18:09,270 --> 01:18:10,709

 

2056
01:18:10,709 --> 01:18:14,709

 

2057
01:18:14,709 --> 01:18:17,270

 

2058
01:18:17,270 --> 01:18:18,390

 

2059
01:18:18,390 --> 01:18:20,390

 

2060
01:18:20,390 --> 01:18:22,070

 

2061
01:18:22,070 --> 01:18:25,110

 

2062
01:18:25,110 --> 01:18:27,189

 

2063
01:18:27,189 --> 01:18:28,950

 

2064
01:18:28,950 --> 01:18:31,430

 

2065
01:18:31,430 --> 01:18:31,440

 

2066
01:18:31,440 --> 01:18:32,229


2067
01:18:32,229 --> 01:18:35,030

 

2068
01:18:35,030 --> 01:18:36,310

 

2069
01:18:36,310 --> 01:18:38,870

 

2070
01:18:38,870 --> 01:18:40,550

 

2071
01:18:40,550 --> 01:18:43,590

 

2072
01:18:43,590 --> 01:18:44,630

 

2073
01:18:44,630 --> 01:18:46,550

 

2074
01:18:46,550 --> 01:18:46,560

 

2075
01:18:46,560 --> 01:18:49,350


2076
01:18:49,350 --> 01:18:51,350

 

2077
01:18:51,350 --> 01:18:51,360

 

2078
01:18:51,360 --> 01:18:52,830


2079
01:18:52,830 --> 01:18:56,390

 

2080
01:18:56,390 --> 01:18:58,630

 

2081
01:18:58,630 --> 01:19:00,149

 

2082
01:19:00,149 --> 01:19:03,110

 

2083
01:19:03,110 --> 01:19:04,229

 

2084
01:19:04,229 --> 01:19:06,630

 

2085
01:19:06,630 --> 01:19:08,950

 

2086
01:19:08,950 --> 01:19:08,960

 

2087
01:19:08,960 --> 01:19:12,390


2088
01:19:12,390 --> 01:19:14,709

 

2089
01:19:14,709 --> 01:19:16,550

 

2090
01:19:16,550 --> 01:19:18,709

 

2091
01:19:18,709 --> 01:19:19,910

 

2092
01:19:19,910 --> 01:19:22,310

 

2093
01:19:22,310 --> 01:19:23,750

 

2094
01:19:23,750 --> 01:19:28,310

 

2095
01:19:28,310 --> 01:19:30,390

 

2096
01:19:30,390 --> 01:19:31,830

 

2097
01:19:31,830 --> 01:19:34,830

 

2098
01:19:34,830 --> 01:19:34,840

 

2099
01:19:34,840 --> 01:19:38,550


2100
01:19:38,550 --> 01:19:41,990

 

2101
01:19:41,990 --> 01:19:43,110

 

2102
01:19:43,110 --> 01:19:44,550

 

2103
01:19:44,550 --> 01:19:46,390

 

2104
01:19:46,390 --> 01:19:47,590

 

2105
01:19:47,590 --> 01:19:48,470

 

2106
01:19:48,470 --> 01:19:52,229

 

2107
01:19:52,229 --> 01:19:54,470

 

2108
01:19:54,470 --> 01:19:55,830

 

2109
01:19:55,830 --> 01:19:58,070

 

2110
01:19:58,070 --> 01:19:59,510

 

2111
01:19:59,510 --> 01:20:00,709

 

2112
01:20:00,709 --> 01:20:02,229

 

2113
01:20:02,229 --> 01:20:04,390

 

2114
01:20:04,390 --> 01:20:06,790

 

2115
01:20:06,790 --> 01:20:08,229

 

2116
01:20:08,229 --> 01:20:12,550

 

2117
01:20:12,550 --> 01:20:15,110

 

2118
01:20:15,110 --> 01:20:17,189

 

2119
01:20:17,189 --> 01:20:19,990

 

2120
01:20:19,990 --> 01:20:21,270

 

2121
01:20:21,270 --> 01:20:25,030

 

2122
01:20:25,030 --> 01:20:26,310

 

2123
01:20:26,310 --> 01:20:30,830

 

2124
01:20:30,830 --> 01:20:30,840

 

2125
01:20:30,840 --> 01:20:37,590


2126
01:20:37,590 --> 01:20:40,229

 

2127
01:20:40,229 --> 01:20:40,239

 

2128
01:20:40,239 --> 01:20:40,870


2129
01:20:40,870 --> 01:20:44,709

 

2130
01:20:44,709 --> 01:20:49,189

 

2131
01:20:49,189 --> 01:20:51,990

 

2132
01:20:51,990 --> 01:20:53,830

 

2133
01:20:53,830 --> 01:20:57,030

 

2134
01:20:57,030 --> 01:20:57,040

 

2135
01:20:57,040 --> 01:20:58,070


2136
01:20:58,070 --> 01:21:00,149

 

2137
01:21:00,149 --> 01:21:01,910

 

2138
01:21:01,910 --> 01:21:04,070

 

2139
01:21:04,070 --> 01:21:05,910

 

2140
01:21:05,910 --> 01:21:07,430

 

2141
01:21:07,430 --> 01:21:09,830

 

2142
01:21:09,830 --> 01:21:14,390

 

2143
01:21:14,390 --> 01:21:15,910

 

2144
01:21:15,910 --> 01:21:17,510

 

2145
01:21:17,510 --> 01:21:19,750

 

2146
01:21:19,750 --> 01:21:20,870

 

2147
01:21:20,870 --> 01:21:22,550

 

2148
01:21:22,550 --> 01:21:24,390

 

2149
01:21:24,390 --> 01:21:27,030

 

2150
01:21:27,030 --> 01:21:28,950

 

2151
01:21:28,950 --> 01:21:31,430

 

2152
01:21:31,430 --> 01:21:32,629

 

2153
01:21:32,629 --> 01:21:34,310

 

2154
01:21:34,310 --> 01:21:35,830

 

2155
01:21:35,830 --> 01:21:37,590

 

2156
01:21:37,590 --> 01:21:39,510

 

2157
01:21:39,510 --> 01:21:41,030

 

2158
01:21:41,030 --> 01:21:46,550

 

2159
01:21:46,550 --> 01:21:49,910

 

2160
01:21:49,910 --> 01:21:53,350

 

2161
01:21:53,350 --> 01:21:56,550

 

2162
01:21:56,550 --> 01:22:00,149

 

2163
01:22:00,149 --> 01:22:03,510

 

2164
01:22:03,510 --> 01:22:06,470

 

2165
01:22:06,470 --> 01:22:06,480

 

2166
01:22:06,480 --> 01:22:07,830


2167
01:22:07,830 --> 01:22:10,870

 

2168
01:22:10,870 --> 01:22:11,750

 

2169
01:22:11,750 --> 01:22:13,510

 

2170
01:22:13,510 --> 01:22:17,990

 

2171
01:22:17,990 --> 01:22:19,430

 

2172
01:22:19,430 --> 01:22:21,350

 

2173
01:22:21,350 --> 01:22:24,629

 

2174
01:22:24,629 --> 01:22:25,430

 

2175
01:22:25,430 --> 01:22:31,830

 

2176
01:22:31,830 --> 01:22:39,910

 

2177
01:22:39,910 --> 01:22:43,350

 

2178
01:22:43,350 --> 01:22:46,629

 

2179
01:22:46,629 --> 01:22:53,510

 

2180
01:22:53,510 --> 01:22:56,149

 

2181
01:22:56,149 --> 01:22:58,070

 

2182
01:22:58,070 --> 01:23:00,950

 

2183
01:23:00,950 --> 01:23:04,149

 

2184
01:23:04,149 --> 01:23:07,350

 

2185
01:23:07,350 --> 01:23:08,629

 

2186
01:23:08,629 --> 01:23:10,390

 

2187
01:23:10,390 --> 01:23:11,669

 

2188
01:23:11,669 --> 01:23:13,830

 

2189
01:23:13,830 --> 01:23:15,110

 

2190
01:23:15,110 --> 01:23:18,310

 

2191
01:23:18,310 --> 01:23:21,840