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all right  um today i think is the last lecture  at least for the while uh about origami  and i'm gonna talk about where i got  started thinking about origami  mathematics  which is the folding cut problem  and this is sort of motivated by a magic  trick  the idea is you take a piece of paper  you fold  it flat  you make one complete straight cut  so you cut along a line  and you unfold the pieces  and the question is what shapes can you  get by that process  so this is like a magic trick i showed  you  making a swan which i have here just for  jog your memory you have a rectangle  paper and you can see the swan on there  and you can see a bunch of creases  you fold along all the creases not the  swan lines  and you end up with all the edges of the  swan lying right along that line  you cut along the line  and you get your swan  as we did before and you also get the  anti-swan  the other piece i didn't show that last  time but it's really it's not like we're  making we're not allowed to make any  extra creases we really want this one  all right so we cut we cut along exactly  the edges of the swan by  lining them up onto a line so really you  can think of this as a magic trick in  cutting  but you can also think of it as an  origami problem which is i want to line  up all these edges by folding  how do i do it and that way it connects  to a lot of origami design  problems this problem has an old history  uh it goes back to 1721 this is the  oldest reference  we know this is a japanese puzzle book  and i think this is kind of like it's  called mathematical contests  is the translation and it's sort of like  uh the old version of  uh like these big problem solving  sessions  that kids do these days for to get  better at math  and one of the pages poses this problem  i have this japanese emblem shape  can i make it from a piece of paper by  folding in one straight cut  and the answer is yes and this is a  solution if you cheat and look in the  back of the book  so i'll let you read that for a minute  and  you're making folds along lines that end  up lining up other parts of the  the shape so that in the end everything  lies along the line then you cut along  the line  uh we learned about this a bit later  than the original our we learned about  this problem from martin gardner  originally  and he knew about the magic world so  houdini before he was an escape artist  he was a general magician  and in 1922 he he wrote or probably had  ghost written  this book houdini's paper magic and  one of the pages is about folding  and it says here you can take a square  paper fold it flat make one straight cut  and get a regular five-pointed star  and that was pretty cool uh and then  other magicians picked it up in  particular this guy  gerald lowe wrote this book paper capers  or some more like  it's a very small book magic book and he  can make all sorts of different  things and he would incorporate them  into magic tricks and he was primarily  using simple folds he would just fold  along one line at a time  and make all these make one straight cut  and make all these cool patterns like  here i have  one of his examples redone  to start from a rectangle i fold i fold  i fold i fold these are all simple folds  i take my  scissors i make one complete straight  cut  and usually when i perform this trick  i say look i made an isosceles triangle  wow i made five isosceles triangles  amazing  and then i made everything except the  five isosceles triangles right so  you're you saw that coming so and you  could make an arrangement of five of  these if you want  all that sort of very symmetric stuff is  easy to do by simple folds i'll talk  more about simple folds later  but we were really curious about the the  general challenge  so this is the sort of you can download  this from my webpage if you want to  make one it's pretty it's a it's a fun  trick  um good but i have some more interesting  examples  so here i have a rectangle folded  and i make one complete straight cut and  this one has actually a line of symmetry  so i fold it in half at the beginning  so i get an angelfish  all right you're not impressed keep  going there's another one  one straight cut can go all day here  i mean the point of today's lecture is  to see how this is done  in general here we have a butterfly  all right you guys are tough to impress  here uh here we go here's the  this one is is theme thematic it's  almost october  so it's sort of appropriate make one  straight cut  i haven't done this one in quite a while  and get tons of pieces  and if i'm lucky open it up the right  way around  so jack-o'-lantern  wow it's amazing how is it possible so  obviously you can make many shapes all  at once that's the general idea here i  i'll admit i cheated here because i  wanted kids to be able to fold this  the the outer octagon is cut initially  okay so it's not from a rectangle just  to make it easier to fold but you could  do it  all at once and now the big demo  all right in fact it's so big i think i  might want to use the x-acto knife  uh we'll try we'll try with scissors  it's got a lot of layers  oh yeah so i usually bring my own  scissors  from the other side  they're better scissors that's what's  special about my scissors  all right time for the x-acto knife make  sure i'm only cutting along the boundary  between red and white  and my lecture notes yeah who needs  those  it's flat getting there  a little more cutting  i believe all the pieces in this case  are rectangles  that's exciting all right straight cut  right  i don't remember which way this one  opens  just like this this should be  the mit logo  this one i encourage you to try at home  it's uh it's pretty crazy  it's definitely hard folding  i'm getting used to it now all right so  you can make anything  is the point some are a little more  difficult  because they have more layers and so on  if i did made it out of thinner paper  like some of the magicians do it is  super easy  onion skin paper or something i think i  have some pictures  of the crease patterns if you want to  see  what these look like  and these are all available for download  you want to impress your friends  go for it  all right they all use slightly  different color coatings but it's  mountain or valley  and so we want to see how to how to make  these  so let me state the theorem first we  have  our good friend the universality result  which is any  set of line segments on your piece of  paper  can be i'll phrase it as can be aligned  by flat folding  aligned means when you make the flat  folding  all those segments come onto a common  line nothing else comes to that line  and therefore you cut along the line you  get exactly those line segments  and there are two methods for solving  this problem  the first one is what i call the  straight skeleton method  and second one i'll call the disc  packing method  by slightly different authors this one  was me my dad and my advisor anna libu  this one was marshall byrne me david  epstein barry hayes  this one slightly after this one this is  sort of my first computational  origami paper and  they're quite different i mean at a high  level this one's practical  this one is theoretically good but  impractical  this one's actually theoretically bad in  a few situations which we'll get to  but it works very well almost always in  a  formal sense of almost always this one  always works  but it's a challenge to fold all of the  examples i showed you are made with the  straight skeleton method  so that's the idea that's we're going to  talk about both of them  and first i have a warm-up three  warm-ups  all right suppose you had a square paper  and you wanted to make a single cut what  folding do i do  nothing that was easy uh let's say i  have  two lines i want to make what should i  do  fold between them fold the lines onto  each other which is  angular bisector if i extend the lines  and i bisect the angle there  then i will fold one line to the other i  brought one just so you  totally obvious you have two lines you  fold along the angular bisector of their  extensions  it lines up the lines and nothing else  okay a little more exciting  what if i had a triangle  how would you line up the edges of the  triangle and nothing else  rabbit ear yes  this rabbit ear is you fold along the  three angular bisectors of the triangle  this is something we talked about in the  tree method  this is one of the sort of gadgets we  use in one of where these were active  paths  and so you fold along those angular  bisectors angular bisectors intuitively  are very good because locally they line  up edges  so if i fold along all three of them and  you may know they always meet at a point  uh then i kind of line up all those  edges if i just fold along those three  edges it won't be flat foldable it's  like this  floppy thing uh and so i've added in  these perpendicular the purple lines  to give me my hinges  so i can manipulate these arms you don't  have to use all of them but then you can  flatten it  and now along this line are all the  black guys  all the black lines you've got a bunch  of  flat foldings all right so  general idea and in the straight  skeleton method what we're going to use  are angular bisectors and perpendicular  folds perpendicular folds are also good  because they locally  locally folds a reflection if you fold  flat so this  line will fold on top of this line so  perpendiculars are good angular  bisectors are good if you fold along all  of them  you should line everything up that's the  intuition  and the question is  how do i fold along enough angular  bisectors in perpendicular so it  actually folds flat  and that is the straight skeleton or one  way to do that is a straight skeleton  and this actually was invented a few  years earlier  95 96 by a couple of austrians  i was an eye culture actually four  people sorry  so let me write down a definition and  then we will  i'll show you what it means  we've seen a thing like this very  briefly in the universal molecule  but this is going to be more general in  some sense  and actually why don't i put up an image  of one  while we're on defining it  all right so it is trajectory of the  vertices of the desired cut pattern  that's our input  the graph of edges we want to cut out  as we simultaneously shrink each region  that's every face  outlined by those cuts keeping edges  parallel  and a uniform perpendicular distance  from the original edges  so it's a bit of a mouthful let me uh  draw some pictures well let's maybe  start with a triangle  so the idea with triangle is you shrink  this is really the wrong order to do  things  i want to shrink there's two regions for  the triangle is the inside and the  outside  if i shrink the inside i get these  parallel lines all i want all of these  distances to be equal  those are the perpendicular distances i  keep shrinking  and at some point i can't shrink anymore  because i get a single point  the incenter of the triangle if i watch  where did the vertices go during that  time  it's along angular bisectors hey our  good friends  angular bisectors now i do the same  thing on the outside shrinking  the outside region is like expanding the  triangle  because as i expand the triangle the  outside gets smaller area  so actually it's just the same thing i  get concentric  triangles and i just keep going along  the angular bisectors  so that's it it's not going to give us  the perpendicular folds it's just the  angular bisector parts  you can still see the triangle in there  somewhere  okay that's a really simple example and  the only event that happened  is that the polygon disappeared when  that happens you stop  with that particular polygon  in general there are three things that  can happen  call these vents  three interesting things one is that an  edge  disappears  so for example if locally i have a  picture like this  and i shrink and i shrink  and i shrink at some point what's  happening is these  angular bisectors are meeting  and now i've i lost this edge shrank to  zero length  so i just forget about the edge pretend  it was never there just keep shrinking  now  what is these two edges so i shrink  and i shrink and i shrink  what happens is in some sense these  vertices merge  these edges merge and now i have one guy  going straight up there  and what is that edge doing it's not an  angular bisector of this or this  but it's an angular bisector of the  extension  of these two lines  right because if you if you look at this  one of these these two edges they are  parallel to the two originals so if you  bisect  those parallel offsets it's the same  thing as bisecting  the original edges in extension okay so  this looks good because  these two folds will line up those two  edges  or this fold will line up these two  edges this fold will line up those two  edges this fold will  line up these two edges and we're doing  kind of extra alignment but  everything looks kosher  good so when an edge disappears you just  forget about it  all right forget let's probably forget  about  something like that all right  then we have a region can disappear  that's what happened with the triangle  and then again you forget about it  there may be many regions you have to  keep shrinking the other regions but  when one disappears  you're done third thing that can happen  is that a region splits  so let's look at an interesting polygon  this one  with straight edges and when i shrink  this guy  see what's happening is this edge is  approaching that vertex  at some point they will meet  and what we're left with are two  triangles in this case  in general you split into two parts you  just keep shrinking the parts  so it's it's not really like you're  stopping at any point  it's just the same thing over and over  but if you're implementing this on a  computer you really have to realize that  that happens otherwise you'd shrink them  beyond each other and it would be  self-intersecting and ugly but you do  the obvious thing which is you cut  where they split and what will happen in  this case with straight skeleton  is you keep going along an angular  bisector until that  little triangle stops  there's one more edge here  so that's what the straight skeleton  will look like this edge is an angular  bisector of this one this one  this edge is an angular bisector of this  one and this one in general  if you look at an edge and you see  what original edge can i reach without  crossing another skeleton edge  those are the two that you bisect so  it's really easy to see you look at this  guy  only two i can reach are this one and  this one i look at this one  the only two i can reach for that one  and that one so it's an angular bisector  of those two  in fact in general if you finish the  outside here too it's the same deal  and then yeah all right enough here's a  bigger example  a little turtle drawn on a triangular  grid  and you can see there's angular  bisectors this is the straight skeleton  this guy for example bisects this  horizontal edge and this horizontal edge  that's a little bit of a boundary case  you have to think about but this is the  right interpretation  it's like an angle of 180 so you bisect  it to 90.  there are fun features here we get a  little bit of action on the outside of  the polygon  so far we haven't seen that so like  these guys meet  and there is some uh bigger  sorry there's a bigger turtle here  somewhere that's hard to draw  anyway what's happening is this edge is  shrinking to zero  while this one is offsetting down this  way and this one's offsetting down this  way  so the new turtle ends up being like  that  and so on  you're shrinking every face so in  general you have a whole bunch of  polygons  or i mean in general we're allowing  crazy things like  this as this pattern of cuts i want to  make  maybe you want to cut your square into  five pieces i'm going to shrink each of  them separately  or in parallel it doesn't actually  matter  so in this case it's just going to be  five angular bisectors  in general there are several regions you  shrink all of them a lot of the time we  think about polygons and then there's  two regions to shrink  and it looks like you're expanding the  turtle to go out but really you're  shrinking the outside region  just happens to be there's one infinite  region that one looks weird  a couple other special cases because i  want to do any graph  not just polygons you could have  something that just terminates  uh so a degree one vertex only has one  edge incident to it  in this case it's not quite well defined  what to do because you offset this  and you offset this but what happens  here and there you it's sort of  arbitrary you can do whatever you want  but the simplest thing you can do  is to make a imagine that there's a  little vertical segment here that  happens to be length zero  and it expands into the edge of a  rectangle  so you end up with these two 45 degree  angular bisect bisectors between this  vertical edge and the  horizontal one but it's  you have some flexibility there you can  design it how you want  the other case you can have is a degree  zero vertex there are no  edges here this is a little funny and  the way i defined it i just said i  wanted to align line segments  but you can also align points if you  really feel like it  and that would be represented by a dot  that has no cuts next to it if you want  to cut out just this point  i need to make it something you could  think of it as a tiny triangle for  consistency with this picture  we think of it as a little square and so  we when you expand  it or when you shrink the outside region  you get four  45-degree folds this is actually how  i culture it all defined it back in  96 and it's a fine definition but you  have flexibility here in your design  process  they'll all work and this would let you  take a whole bunch of points align them  onto a common line and nothing else is  on that line  because these folds are going to push  everything surrounding the point  away from the line  all right some fun facts straight  skeleton is nice and small if you have  an original points and line segments in  your  desired cut pattern the straight  skeleton has a linear and n  number of line segments linear number of  creases  so order n  uh other fun facts there's a one-to-one  correspondence between  the edges you want to cut along  like let me pick one over here maybe  like this this is an edge i want to cut  along and regions  of the straight skeleton so here's a  region a face of the straight skeleton  this guy there's exactly one cut edge  inside that this is always the case you  look everywhere here  every region of the straight skeleton  it's more obvious if i color them  different colors  there's one cut edge inside and all of  those guys  that surround that cut edge  bisect that edge and another one  and the other one is the one on the  other side and generally you take one of  the straight skeleton edges  there are two sides there's two faces of  the straight skeleton this one's crazy  non-convex  this one's just a little infinite  triangle down here  and that edge bisects those two cut  edges so it's very easy to walk around  the structure see what it bisects  lots of things get bisected but it's not  flat foldable  so we're not done and that's where we  need  the perpendiculars  so  so  all right down in definition and maybe  show  picture we're going for  there's a lot of structures here there's  what i call the cut graph the things  we're trying to make  and then there's this straight skeleton  which you can think of as a graph drawn  on the paper  there's vertices of straight skeleton  which you call skeleton vertices regions  of straight skeleton which we call  skeleton regions  edges of the cut graph which we call cut  edges and so on  we're going to add a new graph which is  the perpendicular graph which you can  think of as hinges from  the tree method of origami design  so what is this what does it mean  we started a straight skeleton vertex  usually there are three  skeleton edges coming together at a  vertex sometimes there are more like  this guy  there's four and there are three  if there's three edges coming together  there are three skeleton regions for  each one  each of those regions has one cut edge  in it so we try to walk  perpendicular and toward that  cut edge so here i walk perpendicular i  mean at right angles here  i just go off to infinity here  i walk perpendicular to this cut edge  and that's  that's cool but then i hit i leave the  skeleton region  at that point i enter a new skeleton  region which is this one  it's non-convex thing it contains one  cut edge and when i  where was i here i entered now i want to  move perpendicular to that  cut edge so that when i cross it i cross  it perpendicularly  now i enter a new region it contains  this cut edge  i move perpendicular to it  and i don't actually cross it but i  enter a new region now i'm in the region  of this cut edge so i move perpendicular  to that  and wow i hit another skeleton vertex i  stop  okay this example is because it's on a  triangular grid there's lots of  degeneracies like that  usually you'd eventually go off to  infinity or come around and meet  yourself  here i happen to hit a different vertex  you do that at all the vertices  all the skeleton vertices now there are  some weird ones  like this one notice there are no  no purple lines coming out from here and  that's because  every region you try to enter you  immediately leave  so if i tried there's four regions here  try each one of them  it's like this region has this cut edge  if i tried to go perpendicular to it i'd  enter a different region so i can't  actually go at all  it's like i i move and then i instantly  stop so you can think of there being a  very like a zero radius little uh thing  there  that's sort of the degenerate case of a  river being a disk river being a circle  same thing going on all of in general  when you have reflex vertices in their  angular bisectors meeting you're going  to lose some perpendiculars because you  can't  enter them here's another one where i  just have one perpendicular edge coming  out this  this one i can reach but if i try to be  perpendicular to either these i  enter the wrong region that's the  perpendicular folds  and that's pretty much the crease  pattern there are technically a few  other folds you have to deal with  but that is if you want to make  something right now  just apply those two algorithms and you  get your shape  just fold along that crease pattern it  will be flat foldable  almost always  why is it flat foldable so one thing we  can check  is local flat foldability at least  satisfies kawasaki's condition  because at a typical vertex you're going  to have three  skeleton edges coming together and so  there are three faces here each of them  has  a cut edge somewhere  hopefully i draw this reasonably well  and should have the property that when i  extend these  it didn't draw it so well when i extend  these  these are angular bisectors that's we  know these skeleton edges will  angularly bisect two cut edges the two  cut edges  that are defined by these guys so i  should get angular bisector here  pretend those meet angular bisector here  and then i also have perpendicular folds  so they may not actually meet  this guy but if they did they certainly  meet the extension  hey that's our good friend the rabbit  ear just regular triangle fold  and in particular you can see that  these angles are equal  i'll call this three prime  it's it's like 180 minus that i think  i'm mistaken and this is one prime  not the best notation and these are two  prime  and two prime and so i've got these nice  angle pairings  that means if i add the odd angles i get  the even angle same thing so i  definitely have kawasaki's theorem  everywhere you can check it works even  when you have these degenerate  situations where  uh three or more than three skeleton  edges come together  for the same reason you still get a  pairing just more than three of them  all right but some exciting things can  happen  so i'm going to look at proving  foldability but  one exciting thing that can happen is  you get a lot of  perpendicular folds at a very few  original cut lines so here i'm trying to  make this weird pinwheel shape i want  i want to cut out the bold lines of the  cut lines so i want to cut out the  square  and then these four squares arranged in  a pinwheel pattern around that one why  you'd want to do that i don't know but  we're mathematicians so we want to  consider all the cases  so the straight skeleton is the thin  black lines and that's linear size  that's nice  but the perpendiculars if this piece of  paper is infinite the number of  perpendiculars is infinite  if you have a finite piece of paper  which is what you usually buy in the  store  then it's a finite number of creases so  in any finite region this is a finite  number of creases but  it's a lot of them so that's one sad  thing you can't bound the number of  creases  as a function of the number of cut lines  but i think that's actually necessary i  don't think  it's possible to solve this problem  while bounding the number of creases in  terms of the number of cut lines  but that's one of the open problems  written down  on one of these pictures  one of these slot lecture notes  something else even more annoying though  it happens even in a finite piece of  paper  and it's even more obscure why you'd  want to make this but the bold blue  lines  are the cuts and then the thin black  lines are the straight skeleton  you can tell this spans many years  because i keep changing notational style  and this is from the textbook  and then you get these perpendicular  folds i haven't drawn all of them  but these dash lines the light blue  and this example is set up so that let  me get this right  this width  is an irrational multiple of this width  or this width one of those things are  irrational  uh so they're not very nice numbers and  and what i need to do to finish this  picture  is these guys go they enter a new  skeleton region they're actually going  to keep going straight because there's  two cut lines that are parallel to each  other  it's going to go up there and it's going  to cycle around let me do one round  uh who did i move this guy  so this guy i just extended down he's  going to turn around  make 180 degree turn and you can check  that each of these  you're set up to do 180 degree turn  around this axis around this axis on the  bottom  and on the top around this axis and  around this axis depending wherever you  enter it's like a race track  keep going around and around and if you  follow that one guy  a little bit farther it looks like that  and a little bit farther  it looks like that it never finishes  so in fact you completely fill this  region with creases  it's like a dense region of creases now  this would be a [ __ ] default  i don't recommend you try it and this is  really a situation where this algorithm  fails  the good news is if i move any of these  vertices  the cut vertices the tiniest amount this  will disappear  i really had to be very careful and get  lots of degeneracy for this to happen  we don't actually know how to prove that  it's a conjecture  that if you take any cut any graph of  cuts you want to make and you  perturb the vertices in a tiny epsilon  disk  then the resulting thing will not have  this density behavior i'm totally sure  it's true  but we don't have a proof so that's life  this is why i said skeleton method works  almost always  they're these in annoying situations  where  it doesn't really give you a crease  pattern so you feel like unless you  somehow think this is legitimate to make  infinitely many creases  but i don't think so  um all right  let me tell you a few more things to  make this practical for you  you wanna what you really want is a  mountain valley assignment before i  showed you lots of perpendiculars and  skeleton edges and  basically the way it works is if you  look at  any skeleton edge like this one  it's bisecting in this case a convex  angle  so i make it a mountain here red is  mountain blue is valley  dot dash is mountain dash is valley as  is standard  whenever i'm bisecting a reflex angle  then i make  the uh skeleton edge valley  and that's basically true convex angles  mountains  reflex angles valleys that's for the  straight skeleton edges yeah  uh like this fold or this one  oh this guy yeah all right this is a bit  of a special case here i'm really  bisecting an angle of zero  if you extend these guys out they meet  at infinity and they form a zero angle  because they're parallel  and i call zero a convex angle but i  just define it that way  and so this is a mountain uh whereas  like this guy  it's bisecting it's barely convex  is that really a mountain no  it's a typo good this one should be a  valley  pretty sure yeah  it should be about wow this is a weird  crease pattern  it's not a straight skeleton there never  mind this picture i know there's always  a bug  i think there's a there's a typo in the  book as we say  how did you  how did i make the initial pattern the  turtle i just draw some do something  look like a turtle  anything i i happen to draw this on a  grid but there's no reason i had to  because  the other one that you're explaining  that they have so many increases  happening  and by moving them around vertices i  mean then  the increments that you're creating  they're not uh  what's the what's the relationship  between those increments are you  so in this example i designed the ratios  to be really nasty  like a root 2 over 10 ratio or something  the whole thing will if i perturb these  vertices at all the whole thing will  fall apart i won't get these 180 degree  turns things will end up going off to  infinity  i the hard part here is actually the  irrational  ratio is quite common what's uncommon in  this picture  is that this thing is closed up and you  never escape  almost always there'll be a little gap  and you'll eventually  reach the gap and then go off to  infinity so that's what happens in the  typical case  oh if you drew a picture on the grid  this would never happen that you can  prove  yeah uh square grid  probably also the triangular grid you  need to be a little careful because you  want all these constructions to stay on  the grid  but i think something like that is true  okay  let's move on to how we construct a  folded state when this algorithm works  when it gives a valid crease pattern  you know that's locally flat foldable  because it satisfies kawasaki but how do  we actually know that it globally folds  flat  to do that you have to describe what it  looks like after it's folded  and our the idea here is to look at  what we call corridors but are  essentially discrete versions of rivers  from tree theory so you have these  constant width  strips that turn at skeleton edges  and they could go off to infinity on  both sides in general they could  loop around and meet themselves again  but in this case they actually all go to  infinity  and if you look at one of those strips  you can actually just like cut this out  of your your textbook just slice it up  and uh look at how it's folding well i  mean it meets  a skeleton edge and then maybe it meets  a cut edge usually you don't fold those  then it meets another skeleton edge it  just meets edges one at a time it's  never complicated  because we've divided along all these  perpendicular folds you really only meet  one edge at a time  which is good in fact if you fold  if you look at one of these skeleton  edges  it's creased along normally you think of  that as an angular bisector of let's see  these two cut edges  but you can also think of it as an  angular bisector of these two  perpendicular folds  because if you if you bisect these two  things  you also bisect two things that are  perpendicular to them  it's like two wrongs make a right  somehow okay so this guy bisects  those two creases so if you fold along  here  actually you align these creases it's  kind of it's a duality you're aligning  the perpendicular folds  i fold along here you line up this fold  with that one this fold with that one  i fold here i line up this this fold  with that one there's really there's  like a  zero length fold that's there you fold  along all these things you line up this  with itself  and so on so you follow along this thing  this  corridor folds down to basically a  rectangle  it's got some rough edges on the top and  the bottom but it lies in a  in a strip in 3d i think i have a  picture of that  so i took the one over here this  blue corridor and if you fold it up it  looks like this  okay now in particular you can check at  this point it's pretty easy to check  because of all this bisector-ness  bisector goodness that you bring into  alignment all the cut edges  so for example this guy because it  bisects that cut edge and that cut edge  brings them into alignment and you can  see that somewhere on this picture i  think it's these two guys  it can be a little more complicated like  over here i have a cut edge then i have  a bisector and then a bisector and then  a bisector and then another cut edge  but if you think about it right it's you  know i don't happen to meet  these uh cut edges but i'm effectively  bringing this into alignment with this  and then this into alignment with this  and then this into alignment with that  so in the end i aligned this with that  and that's what's happening  up here on the left where i don't quite  come all the way down but i still end up  lining everything up  so this is a solution to the fold and  cut problem as long as it actually folds  and to show that it actually folds you  just need to show that these  uh corridors or i forget i think we call  these accordions  it's been a long time this is 98 for  this paper  you take these accordions and you just  want to see how they fit together  and lo and behold they fit together in a  tree in this  picture it gets more complicated  in another picture which i will show you  in a moment  but in in this situation where every  quarter goes off to infinity on both  sides  you get a nice tree structure and as  long as you can fold this tree flat  then you can fold this thing flat  because each of these edges of the tree  is this very simple accordion structure  which is  trivi trivially folds flat  and if the other thing you have to check  here is you actually alternate mountain  valley  that's a little more subtle especially  when i don't draw the mountain valley  assignment  but it turns out you always alternate  between mountain and valley in this  picture  which is great that's the thing we  always we know always folds flat it's  like a 1d folding  so these are super easy to fold you can  fold each of them if you cut along all  the dashed lines  you can fold each separately then you  need to join them together and where  the edges here just like entry theory  the edges here correspond to these  rivers and now you need to somehow  attach them here  check that where you attach them there's  no crossings  that's i'm not going to describe but  it's pretty easy  it's planarity essentially of that  diagram and then you need to fold this  tree flat  folding a tree flat is actually kind of  a segue into next lecture which will be  about folding linkages  in this case it's really easy you just  pick up some root like the letter a over  there  and you hang the tree pull up  technically this is like a depth-first  search of the tree  so you just walk down always walk down  until you're finished then you go back  up  walk down some more branches you'll end  up drawing everything downward  away from a and it will be a flat  folding there won't be any crossings  here  and then this is a 1d representation of  what's really going on which is that  above each of these edges are is really  an accordion  and you need to join them together there  but we basically do this modular folding  where you fold each accordion separately  put them together according to this boom  you get your flat folding from this  picture you can read out the mountain  valley assignments for the perpendicular  folds  this looks like a valley this looks like  a valley this looks like a perpendicular  fold i didn't use  because there's no there's no crease  there that's flat  uh where's a mountain mountain's  probably at the top at  a what really happens  if you want to know really what we're  deciding is whether this starts mountain  or valley then it will actually  alternate back and forth  but as you move along perpendicular  that's basically how you construct a  folded state  in this situation of so-called linear  corridors  now there are a bunch of things i  haven't mentioned  but i think i don't want to talk about  them too much  so i get onto the other topics  so uh what i just talked about  is something called the linear corridor  case which is really where it's the most  beautiful  this construction of a folded state and  linear corridor intuitively something  like this it goes off to infinity on  both sides  has constant width all the way of course  it's really a discrete thing not a  smooth thing  uh let me say  conjecture  if your cut graph  has a maximum degree two  that means at most two edges at every  vertex  this is a very common scenario this is  if you want to make a polygon every  vertex has degree two  you might make several polygons every  vertex has degree two i'll even let you  have vertices of degree one  or zero for free but mainly we're  thinking about  degree 2 everywhere  then  we almost always  have a linear quarters  so this is why this situation is  interesting although unfortunately this  is still a conjecture  i'm sure this is true but proving it i  don't quite know the right techniques  so in this typical situation you take  any picture you want you slightly  perturb every vertex randomly  say then with probability one hundred  percent probability  you will get only linear quarters  and that's the situation where it turns  into a tree it's easy to fold  life is good the annoying case  is circular corridor case  this takes a lot more work to prove and  i'm not going to talk about it much  circular corridor looks like  this okay we had these in with rivers  also  so you just loop around still constant  width everywhere  but you meet yourself and you don't go  out to infinity  it's harder why is it harder well in  particular  if you look at how one quarter folds  it's no longer it's like the same  situation we had in like lectures two  and three  where we had on the one hand a 1d  folding was really easy  but then when you made it circular he's  folding a circle instead of folding a  line segment  now you had this wrap around issue so  like these guys would have to  line up it turns out they will line up  because everything is bisecting whatnot  these edges will line up but now you  have to join them together  and if this part went all the way down  here and came back up  then you'd get an intersection and  it turns out in general  i get i get a choice of who's first and  who's last i have a circular order of  things and i get to choose  where i break that circular order and  make it a linear order where i do the  wrap around  there are some circular corridors you  can't even  break them and make it work  it's kind of depressing so this is  definitely harder in that sometimes  it's not possible  but we can save a little bit  which is i don't have an example handy  i wish i did i'll have to reconstruct it  this is so long ago  uh here's a way to make it definitely  work  fold all the cut edges  okay so far in pictures i've been  drawing i didn't fold along the cut  edges  because i really wanted to separate the  green region here from the yellow region  if i folded this up this is quite a  complicated one to fold  you get the cut lines somewhere like  like over there  and then the green stuff will be always  above the cut line and the yellow stuff  will be below the cut line  in general this is called a side  assignment you have a bunch of regions  you decide which ones do you want to be  above and which ones do you want to be  below  and usually you have polygons and you  say the interiors are above and the  exteriors are below  but in general you could ask for  anything you want  you could say maybe i want both of these  to be above if you make both of them  above you have to valley fold  along all of the cut lines so if you do  that you say i want all regions to be  above the cut line you can still line  them up  you end up folding along all the cut  edges with valleys  and then wrap around is super easy you  take  you take this thing in fact everyone's  folding along the black lines  so really everybody comes down to the  floor  and then the wrap around is just  underneath the floor and  life is good so that's one way to deal  with this  case and you have to prove that that  works it's complicated  uh i would rather go on to other topics  i do think this would make a fun project  it's not easy to make these crease  patterns currently we draw them always  by hand meaning with a fancy drawing  program that knows how to do angular  bisectors  and it's it's an art to move around the  points  so that you get lots of alignments like  when you get  four straight skeleton edges coming  together that means you get fewer  creases that's a good thing whenever you  can make that happen  it's a win if you could give me software  to help do that  i would love you  yeah so let's move on  any questions about the straight  skeleton method now i'm going to switch  over  to disk packing  all right same problem  but now i'll give you a method that  always works  in theory just uh  difficult in practice  i think this is good chalk because it's  yellow  generally if it's yellow on the outside  it is a railroad shock which is the good  stuff  but what's the problem is we have bad  erasers  it's it's persistence of vision you just  get to remember what i used to write  uh i think i don't want to write down  the algorithm it's complicated  i want to give you a visual overview of  the main steps  because there's nice figures to do that  for us  so um we start with a very complicated  shape we want to make  like this quadrilateral and  you can see disc packing is involved the  very first thing we do and i'll tell you  why in a moment  is thicken the graph you want to build  so maybe it's a polygon maybe it's a  graph i'll think about the polygon case  for now because it's easier then i'll  come back to the general case i thicken  it by a tiny epsilon  just offset in both directions just like  as if you're starting the straight  skeleton method but then you stop after  epsilon  no events happen in epsilon time  okay then i have this picture  the purple stuff pink stuff whatever  magenta  50 magenta i happen to know drew this  figure  now i'm going to take some 50 cyan  discs and pack them to fill this region  now what i want  there's three properties i want one is  that every vertex  but you have to think about each region  separately because a little bit  confusing  let's think about the outside it's a  little easier bigger  every vertex of this graph on the  outside  should be the center of a disc so  there's a disc centered here  there's a disc centered here so just  centered here and just centered there  on the inside it's also true they're  just different disks  then also i want the edges of the graph  to be  covered by radii of disks  right so here's a radius of one disk  here's a diameter of a disk  here's a radius of a disk that covers  the edge  so in other words i want to fill along  the edge i want to have a bunch of  centers  so that i completely cover that edge  with blue we'll see why  later questions  uh the discs have to be non-overlapping  so these properties are actually quite  challenging to achieve  your question is why do we use small  discs  this because if i had a big disc here it  would intersect this disc  uh now i didn't have to make that disc  so big but if i made that one smaller  i'd have to have more discs here  so or here there's also two small discs  that one  i'm probably could have gotten away with  a bigger disc  uh oh no but then on the inside you have  a problem  so these guys actually have to match up  that's another constraint and the inside  and the outside have to match up here  there's a slight change in radii  to compensate for the epsilon along the  edges they're exactly the same  so if i made this one a big disc it  would overlap this one so i could make  that one smaller but then it's  other problems so you have to  simultaneously balance all these  constraints which is  a bit exciting um what else the distance  overlap  the last property is that the gaps  between the disks have always  three or four sides why because  i want it to it'll make life easier you  could try to deal with  more sides but three and four  is nice uh yeah i'll get to that  why do we care about number of sides  because i'm going to  draw a graph i'm going to subdivide my  original graph here  with these red lines to say whenever  disks  kiss i will draw connect the centers of  those disks  and because these gaps always have three  or four sides  uh that's not the best example here's  like three sides the  the red lines i draw will outline a  triangle whenever i have three sides  whenever i have four sides i'll have a  quadrilateral  so i've subdivided my regions into  triangles and quadrilaterals  okay for you have to believe that this  is possible i can sketch an algorithm  for you  uh which is you draw a tiny disk at each  of the corners  and then you draw lots of tiny discs  along the edges to fill the edges  and that will satisfy everything i mean  the discs will be non-overlapping  because they're super tiny they won't  get near any other discs from some other  side  and what other good things oh the but  the regions will be  ginormous they won't have three or four  sides they'll have hundred sides million  sides who knows  well uh whenever you have some crazy  region  outlined by disks might not be convex  whatever  just draw the biggest disc you can in  there  get turned into a disc eventually uh  that does not  intersect anybody but if it's the  biggest possible it will actually touch  at least three sides  if your degenerate it might touch four  but in general it will  touch three sides which will subdivide  that region into three pieces  and you can show that those pieces are  all a little bit smaller than what you  had before in terms of number of sides  except when you start with a  quadrilateral when there's four sides  you'll get  quadrilaterals and triangles so you  can't go below three and four be great  if we could always get down to  three sides in every region but we can  get down to three or four anything  bigger than three or four you can show  this will make it strictly smaller so  that is an algorithm it's not super  efficient  but it will find a disk packing with all  these properties then we do the  subdivision  now what do you think we're going to do  what do we do with the triangles  gravity here that's the key phrase for  today  so remember our good friend the rabbit  ear and then there was the  universal molecule length the universal  molecule for the quadrilateral we're  going to use that  for the quadrilaterals and it turns out  there's some nice properties here  which is uh the perpendicular folds of  the rabbit ear will always hit  right at the kissing point between the  two discs and same thing  in in here we've got these four discs  we've got this quadrilateral region  in between and the perpendicular folds  that come  out of these two points  you may not remember exactly what's  happening here is we  shrink and then in  in the tree method this became an active  path there's no notion of activepads  here  but we just make that so that  these perpendicular folds will end up  hitting the kissing discs  and we'll end up actually with a  four-armed starfish in in terms of the  tree you get  and the articulatable flaps here it  these guys will all meet at a point  that's just  the way this works with disc packing and  you can think of that you can think of  there being discs here and you're  actually applying the tree method  to that flap pattern and  that's that's probably the easiest way  to think about it but what's good for us  do i have a picture not yet but the  point is  i perpendicular is coming out of here i  have perpendiculars coming out of here  they will meet because these discs kiss  uh at the same point up from both viewed  from both sides perpendicular is meet  that's good that means i don't get  perpendiculars bouncing out all over the  place  so all this work is to make sure  perpendiculars are well behaved  it's a lot of work to do it but it does  it  now when you fold this thing uh what we  end up doing  is lining up remember there was there  was two copies of my polygon  there's the inner copy in the outer copy  i end up  lining up all of these guys  i got to go back to the other picture  i'm sorry so we have this inner copy  and what these molecules end up doing is  lining up all the edges of this quad all  the edges of this quad all the edges of  this triangle  all those edges on the inside will  become lined up on one  edge all the lines on the outside become  lined up on  another line turns out it will be  parallel to that line  but what we really wanted to line up  were these edges and you can see why we  had to do the offset at the beginning  because otherwise we get tons of extra  junk on our line  we only want these edges on our line so  we did the offset so that all this stuff  will come to one line all this stuff on  the outside will come to another line  and then we get this picture  so this is one line at the bottom  another line at the top we really wanted  to line up stuff  uh the the blue stuff there and there's  still some junk on our line these  these uh cyan triangles represent things  that come from down here  but we really don't want them to cross  our line they just happen to  so we have to sink them repeatedly do  lots of folds to make them underneath  that epsilon line  then we can cut along our line and we're  done easy  to prove that this works give you a  little  sketch this is kind of fun and it's it's  one piece of what we're  in the process of doing for tree maker  this is sort of like a special case of  tree maker you just have very simple  molecules  and relatively simple way in which they  are combined  so you have here i've done a simpler  example i want to make a square  and i end up decomposing in this case  into nine molecules nine quadrilateral  molecules very simple disc packing which  i have not shown the disc packing here  the idea is i'm gonna make some cuts in  my paper  to make my problem easier i'm gonna have  to pay for that because later i really  want those edges joined i have to glue  them back together  but to make it easier to think about i  cut  those four edges so that the  way in which my molecules are connected  to each other  is a tree because i like trees they're  easier to think about easier to do  induction over  so that's the blue lines are connecting  the centers in a tree  the other remaining edges in the grid  have been cut away  now the idea is it's kind of like  what i was drawing for the linear  corridor case you have a tree  you pick up the tree from some node and  just hang it down  and in this case we hang it from  this molecule the red edges are mountain  so these three of these are going to be  valley  one mountain so the idea is this thing  reaches around  the next guy juices around the next guy  which is around the next guy  and there's actually there's two valleys  here two little pockets  each of this guy goes in that pocket  this guy goes in that pocket and  recursively  it just works i think i have a picture  of  what's actually happening here yeah it's  it's hard to really draw  but each of these forearm starfish has  one mountain  and three valleys and you just nestle  inside your parent in the tree  and this is really easy to show that  there's no crossings here because you  just come you just join to your parent  and it's a nice nesting structure it's  just in the same way that trees can be  folded flat  uh you can fold all these molecules flat  and and join them together in a tree but  we didn't really have a tree  we had all those extra cuts that we have  to re-suture  so we have this picture and now we  really have to join up these edges and  think about what the mountain valley  assignment is there  and it works  uh this is this green thing  is the boundary and then i've connected  the dots each of these dots corresponds  to one green edge here i forget whether  it's this edge or  what that one i think it's just a single  edge  and so yeah for example  these two are  these two joints and then the joins  above  that nestle around it and then the other  branch at the top are these two joins  and then the leftmost cut are these two  joints you have to make these joins and  really all you need to check  is that these joints form a non-crossing  picture like they do here  and that's almost obvious because this  is a planar diagram  and we're cutting along a planar tree  and so this is again a depth first  search kind of thing  so it's there's one tree we call the  dual tree here  that works because it's a tree then  there's the cuts you make which are  different trees  primal tree if you want and that also  works because of planarity  and it all works that's the hand wavy  version  and you can read the textbook if you  want more details uh  oh gosh if you want to solve uh more  general graphs you can do it  in general you get you have to offset  all of those cut lines  and you get all these things and along  the uh  pink lines here you line things up  but you really wanted to line up the  these blue lines  purple lines and so you have to do more  sinking to get it to work  now i have all the things i want to line  up on this line and this line i fold in  the middle  and now they're lined up  ah if i fold in the middle yeah that  will work  good we might have to do more sinking  all right  questions about the disc packing method  so a bit of a whirlwind tour  but i wanted to get through it quickly  to tell you about a new result  just got accepted to a conference to  appear  in october pretty soon  and it's a project that started in this  class in the problem session three years  ago  and we just solved it took a while  took another co-author to join in  and it goes back to the early history of  folding cut which is  simple folds all the magicians were  using simple folds what can you make  with simple folds  now you've been wowed endowed that you  can make anything  with arbitrary folds but simple folds  you cannot make anything  because the first fold you make  say this one has to be a line of  symmetry  of your diagram  i've got to stop making my life hard  if you can fold something you can never  unfold it that's the usual simple fold  model  this has to line up the cuts you want to  line up over here better exactly be the  cuts you want to line up here so you can  only make symmetric diagrams the first  fold has to be a line of reflectional  symmetry  but is that the only property you need  no  you kind of have to have symmetry for a  while until you're done  how do you formalize that well we came  up with  an algorithm that in polynomial time  so an efficient algorithm will  tell you whether a given polygon can be  made  like this polygon  looks good i think  yeah i think this can be made so i think  i would fold along  angular bisector here and then this  basically disappears  folding over then i would fold along an  angular bisector here and then this  disappears into that  and then  maybe i can fold here does that work  barely i mean i got to make sure this  blank paper does not come onto that  but if that's a problem i can probably  make  well i can make a fold here for example  shrinks that up there's lots of things  you can do and this is a borderline  case whether it's yes or no i will give  you an algorithm that does it  for polygons with margin  a bit of a technical condition something  that  is pretty typical what i mean is the  thing you're trying to cut out  does not meet the boundary of the paper  there's no margin here it'd be hard to  print out  so i really want something that has  margin that's the typical case we care  about  but you actually need this for the proof  to work we also need that it's a single  polygon  uh it does not work with general graphs  this algorithm because more complicated  things can happen it might be np hard  for all we know  the general case so here's the algorithm  maybe i'll give you number form  first thing you do is guess the first  fold this is a  a powerful idea that even most  algorithmicists don't necessarily know  the idea is what could the first fold be  has to be a line of reflectional  symmetry turns out there's a linear  number of them at most you can find them  in linear time  with all these good algorithms for  finding them but which fold do i make  first  the answer is i don't care let's just  try them all one at a time  this is what i call guessing just  imagine from now on that we made the  right guess  but if you end up failing later on in  this algorithm just go back to here try  the next one  there's only n of them to try so you're  going to multiply the running time of  the rest of the algorithm by  n and if this is n to like n squared say  which i think the rest of the algorithm  is n squared  the whole algorithm will be n cubed  because you just run this over and over  for each possible first fold  we don't have a great theory to find the  first fold just try them all  that's step one step two  that's the only guess we're going to  make  fold down to convex hull  this is a central idea  so we have this this uh polygon we want  to make  there's all this extra blank paper i  don't like extra black paper just get  rid of  it make lots of folds that fold the  blank paper onto itself till it gets  so tiny it just goes slightly around the  convex hull  convex hall is the smallest convex  polygon that contains your shape  so be like that it'll reduce the paper  down to that  i'm going to do this a lot i might as  well  it makes the problem easier because i  have less blank space to worry about  blank space is a problem because if i  fold blank space onto a cut  it's bad it's not allowed  so the next thing we do is make  the best possible safe  fold  i need to define that but a safe fold is  just the folds we're trying to make  which is  line locally there are lines of symmetry  so like this one  was a good fold after i had made this  fold so this is all  all the right stuff here is gone it's a  good good fold because it folds this  onto this  and it folds blank space onto this blank  space so anything that comes into  on top of each other is identical that's  a safe fold  and repeat  that's the algorithm how does that work  why does this work so it's the obvious  algorithm basically it says  make safe folds until you're done if you  finish  then you've you're done and then the  answer is yes if you don't finish which  is a little hard to check because you  can always make  microscopic folds but you sort of take  the limit and uh  it's possible to do this in polynomial  time you find that oh  turns out i can't finish by making  say folds then you can show that  actually there was no way to make that  pattern  by simple folds so let me  give you an idea of why that's true  after i make a single fold the first  fold  my picture looks like this it's no  longer a polygon  it's it's like half a polygon it ends at  the  boundary of the paper it begins and ends  at the boundary paper and you have some  chain in the middle  call this a passage it's like path you  wander along  and i want to somehow bring all those  edges into alignment  this here i'm already using that it's a  polygon if it weren't a polygon there  might be more than one of these  now i want to make a fold like for  example this fold is safe because it  folds this onto this and it folds blank  space on the bank blank space if i  drew it right  and keep doing that now when i make a  fold like this  what happens is i can think of this  region  that i folded the smaller side as  disappearing right it just got absorbed  into the paper here so my the graph that  i was trying to line up got smaller  that's clearly a good thing makes my  problem easier  the piece of paper i was folding also  got smaller  that's a good thing but that's not  always true  suppose you had a piece of paper like  this which could happen  after a bunch of folding and then you  fold along a long line like that  because for example your passage looks  like that or something  when i make the fold this has to go off  when i make the folds i get you know  this  crazy thing not drawn to scale and this  polygon does not fit inside this polygon  so my paper got  bigger in some places and that's a worry  because now i have this stuff maybe it  happens to be blank space  maybe there's other junk that got out  here and now i have to  worry about collisions with this bigger  piece of paper and this is always our  sticking point but  there's some magic you can do  in fact the picture cannot look like  this  because look you've got some portion of  your passage  to the left of the crease you've got  some portion of the passage to the right  one of them has to be shorter plus this  is a line of symmetry  so wherever i have a  portion of my passage over here i will  have a portion of my passage over here  until i run out of length one of them is  shorter so  the shorter one like this one gets  totally absorbed by the larger one  so shorter side always disappears so in  this picture  i have the long side of my passage and  it's really a subset of the original  if i reduce this to the convex hull just  like this  this stuff disappears and in general  if you do this fold down a convex hull  this repeat  it goes back to step two if you fold  down to the convex hall you can show  that not only does your passage the  thing you want to cut out get smaller  but your piece of paper also gets  smaller guaranteed and once  once you know the paper gets smaller and  your the things you're trying to align  get smaller  you know that every move is safe so  you never get stuck by following this  algorithm this works for polygons  with margin but not in any other  situation as far as we can tell  um cool  the last thing i wanted to leave you on  is going out  a little a little away from regular 2d  flat sheets of paper  you can generalize and go up a dimension  to a folding polyhedra surface here a  surface of a polyhedron  you've probably done this you take a  cereal box you can collapse it flat  is that always possible this is called  the flattening problem  and the answer is yes and you can think  of it as a folding cut  problem because with the fold and cut  problem you have some polygon like this  diamond  you make some collection of folds that  brings the boundary of the diamond  to a line so if you forget about what's  happening on the inside of the paper you  just look at the boundary of the paper  you're folding that one dimensional  boundary  so that it collapses down to a single  line  what i want to do is this upper  dimension  i take a 3d cube of paper solid cube  i have would be embedded within it some  some  polygons that i want to bring to a  common plane  and i want to fold the solid cube  through four dimensions mind you  but flat so it ends up back in three  dimensions i get a different 3d solid  but there with multiple layers right on  top of each other a little bit of fourth  dimension  hanging out there but if i just look at  what's happening to the boundary of my  polyhedron say i start with a  dodecahedron or something embed it in  there  and i i want to fold this thing so that  all the sides of the dodecahedron come  to a common plane that is the 3d folding  cut problem  it remains unsolved i suspect it's  possible to solve even with straight  skeletons and perpendiculars but  it's really hard to draw the pictures so  we have not  resolved that one way or the other but  the boundary problem forget about what's  happening to the interior and the  exterior of the  kedron if you just look at the surface  of the dodecahedron that you can fold in  3d  we think and you can show  and burn in haze from the complexity  proof couple lectures ago  and also on this disc packing method  their co-authors  they proved just two years ago that if  you have any orientable manifold  which is things like polyhedra but no  mobius strips no klein bottles and other  ugly things they have to be manifolds so  you're not allowed to say  join three triangles together along a  single edge  that would be forbidden so it's locally  flat  in such a case you can flatten the thing  and the proof is very similar to the  disc packing  method of folding cut and in the in the  textbook we talk about how to apply that  to do  something that's just like a sphere a  regular polyhedron  that's pretty easy to do with the disc  packing method they generalize it to the  case where you have  polyhedral doughnuts and all sorts of  fun things  but there's tons of open problems here  so we know how to flatten surfaces and  that's useful for things like folding  airbags flat  but can you fold the 3d solid flat  uh you can think of where we're we have  here  1d edges which we are collapsing to a 1d  line there's also zero dimensional  points here  which we don't bring to a single point  it'd be nice if you could  the generalized fold and cut problem is  you take a d dimensional thing  and you have all of these uh there's  zero one two three up to d  dimensional parts to it or d minus one  dimensional parts  you wanna bring each of them down into  alignment so that  all the vertices come to a common point  all the edges come to a common line  all the faces two-dimensional faces come  to a common plane and so on up the  dimension hierarchy that is the ultimate  open problem i think we end the book  with it  and it's totally unsolved any questions  that's folding and cutting paper and  next time we'll start linkages

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all right  um today i think is the last lecture  at least for the while uh about origami  and i'm gonna talk about where i got  started thinking about origami  mathematics  which is the folding cut problem  and this is sort of motivated by a magic  trick  the idea is you take a piece of paper  you fold  it flat  you make one complete straight cut  so you cut along a line  and you unfold the pieces  and the question is what shapes can you  get by that process  so this is like a magic trick i showed  you  making a swan which i have here just for  jog your memory you have a rectangle  paper and you can see the swan on there  and you can see a bunch of creases  you fold along all the creases not the  swan lines  and you end up with all the edges of the  swan lying right along that line  you cut along the line  and you get your swan  as we did before and you also get the  anti-swan  the other piece i didn't show that last  time but it's really it's not like we're  making we're not allowed to make any  extra creases we really want this one  all right so we cut we cut along exactly  the edges of the swan by  lining them up onto a line so really you  can think of this as a magic trick in  cutting  but you can also think of it as an  origami problem which is i want to line  up all these edges by folding  how do i do it and that way it connects  to a lot of origami design  problems this problem has an old history  uh it goes back to 1721 this is the  oldest reference  we know this is a japanese puzzle book  and i think this is kind of like it's  called mathematical contests  is the translation and it's sort of like  uh the old version of  uh like these big problem solving  sessions  that kids do these days for to get  better at math  and one of the pages poses this problem  i have this japanese emblem shape  can i make it from a piece of paper by  folding in one straight cut  and the answer is yes and this is a  solution if you cheat and look in the  back of the book  so i'll let you read that for a minute  and  you're making folds along lines that end  up lining up other parts of the  the shape so that in the end everything  lies along the line then you cut along  the line  uh we learned about this a bit later  than the original our we learned about  this problem from martin gardner  originally  and he knew about the magic world so  houdini before he was an escape artist  he was a general magician  and in 1922 he he wrote or probably had  ghost written  this book houdini's paper magic and  one of the pages is about folding  and it says here you can take a square  paper fold it flat make one straight cut  and get a regular five-pointed star  and that was pretty cool uh and then  other magicians picked it up in  particular this guy  gerald lowe wrote this book paper capers  or some more like  it's a very small book magic book and he  can make all sorts of different  things and he would incorporate them  into magic tricks and he was primarily  using simple folds he would just fold  along one line at a time  and make all these make one straight cut  and make all these cool patterns like  here i have  one of his examples redone  to start from a rectangle i fold i fold  i fold i fold these are all simple folds  i take my  scissors i make one complete straight  cut  and usually when i perform this trick  i say look i made an isosceles triangle  wow i made five isosceles triangles  amazing  and then i made everything except the  five isosceles triangles right so  you're you saw that coming so and you  could make an arrangement of five of  these if you want  all that sort of very symmetric stuff is  easy to do by simple folds i'll talk  more about simple folds later  but we were really curious about the the  general challenge  so this is the sort of you can download  this from my webpage if you want to  make one it's pretty it's a it's a fun  trick  um good but i have some more interesting  examples  so here i have a rectangle folded  and i make one complete straight cut and  this one has actually a line of symmetry  so i fold it in half at the beginning  so i get an angelfish  all right you're not impressed keep  going there's another one  one straight cut can go all day here  i mean the point of today's lecture is  to see how this is done  in general here we have a butterfly  all right you guys are tough to impress  here uh here we go here's the  this one is is theme thematic it's  almost october  so it's sort of appropriate make one  straight cut  i haven't done this one in quite a while  and get tons of pieces  and if i'm lucky open it up the right  way around  so jack-o'-lantern  wow it's amazing how is it possible so  obviously you can make many shapes all  at once that's the general idea here i  i'll admit i cheated here because i  wanted kids to be able to fold this  the the outer octagon is cut initially  okay so it's not from a rectangle just  to make it easier to fold but you could  do it  all at once and now the big demo  all right in fact it's so big i think i  might want to use the x-acto knife  uh we'll try we'll try with scissors  it's got a lot of layers  oh yeah so i usually bring my own  scissors  from the other side  they're better scissors that's what's  special about my scissors  all right time for the x-acto knife make  sure i'm only cutting along the boundary  between red and white  and my lecture notes yeah who needs  those  it's flat getting there  a little more cutting  i believe all the pieces in this case  are rectangles  that's exciting all right straight cut  right  i don't remember which way this one  opens  just like this this should be  the mit logo  this one i encourage you to try at home  it's uh it's pretty crazy  it's definitely hard folding  i'm getting used to it now all right so  you can make anything  is the point some are a little more  difficult  because they have more layers and so on  if i did made it out of thinner paper  like some of the magicians do it is  super easy  onion skin paper or something i think i  have some pictures  of the crease patterns if you want to  see  what these look like  and these are all available for download  you want to impress your friends  go for it  all right they all use slightly  different color coatings but it's  mountain or valley  and so we want to see how to how to make  these  so let me state the theorem first we  have  our good friend the universality result  which is any  set of line segments on your piece of  paper  can be i'll phrase it as can be aligned  by flat folding  aligned means when you make the flat  folding  all those segments come onto a common  line nothing else comes to that line  and therefore you cut along the line you  get exactly those line segments  and there are two methods for solving  this problem  the first one is what i call the  straight skeleton method  and second one i'll call the disc  packing method  by slightly different authors this one  was me my dad and my advisor anna libu  this one was marshall byrne me david  epstein barry hayes  this one slightly after this one this is  sort of my first computational  origami paper and  they're quite different i mean at a high  level this one's practical  this one is theoretically good but  impractical  this one's actually theoretically bad in  a few situations which we'll get to  but it works very well almost always in  a  formal sense of almost always this one  always works  but it's a challenge to fold all of the  examples i showed you are made with the  straight skeleton method  so that's the idea that's we're going to  talk about both of them  and first i have a warm-up three  warm-ups  all right suppose you had a square paper  and you wanted to make a single cut what  folding do i do  nothing that was easy uh let's say i  have  two lines i want to make what should i  do  fold between them fold the lines onto  each other which is  angular bisector if i extend the lines  and i bisect the angle there  then i will fold one line to the other i  brought one just so you  totally obvious you have two lines you  fold along the angular bisector of their  extensions  it lines up the lines and nothing else  okay a little more exciting  what if i had a triangle  how would you line up the edges of the  triangle and nothing else  rabbit ear yes  this rabbit ear is you fold along the  three angular bisectors of the triangle  this is something we talked about in the  tree method  this is one of the sort of gadgets we  use in one of where these were active  paths  and so you fold along those angular  bisectors angular bisectors intuitively  are very good because locally they line  up edges  so if i fold along all three of them and  you may know they always meet at a point  uh then i kind of line up all those  edges if i just fold along those three  edges it won't be flat foldable it's  like this  floppy thing uh and so i've added in  these perpendicular the purple lines  to give me my hinges  so i can manipulate these arms you don't  have to use all of them but then you can  flatten it  and now along this line are all the  black guys  all the black lines you've got a bunch  of  flat foldings all right so  general idea and in the straight  skeleton method what we're going to use  are angular bisectors and perpendicular  folds perpendicular folds are also good  because they locally  locally folds a reflection if you fold  flat so this  line will fold on top of this line so  perpendiculars are good angular  bisectors are good if you fold along all  of them  you should line everything up that's the  intuition  and the question is  how do i fold along enough angular  bisectors in perpendicular so it  actually folds flat  and that is the straight skeleton or one  way to do that is a straight skeleton  and this actually was invented a few  years earlier  95 96 by a couple of austrians  i was an eye culture actually four  people sorry  so let me write down a definition and  then we will  i'll show you what it means  we've seen a thing like this very  briefly in the universal molecule  but this is going to be more general in  some sense  and actually why don't i put up an image  of one  while we're on defining it  all right so it is trajectory of the  vertices of the desired cut pattern  that's our input  the graph of edges we want to cut out  as we simultaneously shrink each region  that's every face  outlined by those cuts keeping edges  parallel  and a uniform perpendicular distance  from the original edges  so it's a bit of a mouthful let me uh  draw some pictures well let's maybe  start with a triangle  so the idea with triangle is you shrink  this is really the wrong order to do  things  i want to shrink there's two regions for  the triangle is the inside and the  outside  if i shrink the inside i get these  parallel lines all i want all of these  distances to be equal  those are the perpendicular distances i  keep shrinking  and at some point i can't shrink anymore  because i get a single point  the incenter of the triangle if i watch  where did the vertices go during that  time  it's along angular bisectors hey our  good friends  angular bisectors now i do the same  thing on the outside shrinking  the outside region is like expanding the  triangle  because as i expand the triangle the  outside gets smaller area  so actually it's just the same thing i  get concentric  triangles and i just keep going along  the angular bisectors  so that's it it's not going to give us  the perpendicular folds it's just the  angular bisector parts  you can still see the triangle in there  somewhere  okay that's a really simple example and  the only event that happened  is that the polygon disappeared when  that happens you stop  with that particular polygon  in general there are three things that  can happen  call these vents  three interesting things one is that an  edge  disappears  so for example if locally i have a  picture like this  and i shrink and i shrink  and i shrink at some point what's  happening is these  angular bisectors are meeting  and now i've i lost this edge shrank to  zero length  so i just forget about the edge pretend  it was never there just keep shrinking  now  what is these two edges so i shrink  and i shrink and i shrink  what happens is in some sense these  vertices merge  these edges merge and now i have one guy  going straight up there  and what is that edge doing it's not an  angular bisector of this or this  but it's an angular bisector of the  extension  of these two lines  right because if you if you look at this  one of these these two edges they are  parallel to the two originals so if you  bisect  those parallel offsets it's the same  thing as bisecting  the original edges in extension okay so  this looks good because  these two folds will line up those two  edges  or this fold will line up these two  edges this fold will line up those two  edges this fold will  line up these two edges and we're doing  kind of extra alignment but  everything looks kosher  good so when an edge disappears you just  forget about it  all right forget let's probably forget  about  something like that all right  then we have a region can disappear  that's what happened with the triangle  and then again you forget about it  there may be many regions you have to  keep shrinking the other regions but  when one disappears  you're done third thing that can happen  is that a region splits  so let's look at an interesting polygon  this one  with straight edges and when i shrink  this guy  see what's happening is this edge is  approaching that vertex  at some point they will meet  and what we're left with are two  triangles in this case  in general you split into two parts you  just keep shrinking the parts  so it's it's not really like you're  stopping at any point  it's just the same thing over and over  but if you're implementing this on a  computer you really have to realize that  that happens otherwise you'd shrink them  beyond each other and it would be  self-intersecting and ugly but you do  the obvious thing which is you cut  where they split and what will happen in  this case with straight skeleton  is you keep going along an angular  bisector until that  little triangle stops  there's one more edge here  so that's what the straight skeleton  will look like this edge is an angular  bisector of this one this one  this edge is an angular bisector of this  one and this one in general  if you look at an edge and you see  what original edge can i reach without  crossing another skeleton edge  those are the two that you bisect so  it's really easy to see you look at this  guy  only two i can reach are this one and  this one i look at this one  the only two i can reach for that one  and that one so it's an angular bisector  of those two  in fact in general if you finish the  outside here too it's the same deal  and then yeah all right enough here's a  bigger example  a little turtle drawn on a triangular  grid  and you can see there's angular  bisectors this is the straight skeleton  this guy for example bisects this  horizontal edge and this horizontal edge  that's a little bit of a boundary case  you have to think about but this is the  right interpretation  it's like an angle of 180 so you bisect  it to 90.  there are fun features here we get a  little bit of action on the outside of  the polygon  so far we haven't seen that so like  these guys meet  and there is some uh bigger  sorry there's a bigger turtle here  somewhere that's hard to draw  anyway what's happening is this edge is  shrinking to zero  while this one is offsetting down this  way and this one's offsetting down this  way  so the new turtle ends up being like  that  and so on  you're shrinking every face so in  general you have a whole bunch of  polygons  or i mean in general we're allowing  crazy things like  this as this pattern of cuts i want to  make  maybe you want to cut your square into  five pieces i'm going to shrink each of  them separately  or in parallel it doesn't actually  matter  so in this case it's just going to be  five angular bisectors  in general there are several regions you  shrink all of them a lot of the time we  think about polygons and then there's  two regions to shrink  and it looks like you're expanding the  turtle to go out but really you're  shrinking the outside region  just happens to be there's one infinite  region that one looks weird  a couple other special cases because i  want to do any graph  not just polygons you could have  something that just terminates  uh so a degree one vertex only has one  edge incident to it  in this case it's not quite well defined  what to do because you offset this  and you offset this but what happens  here and there you it's sort of  arbitrary you can do whatever you want  but the simplest thing you can do  is to make a imagine that there's a  little vertical segment here that  happens to be length zero  and it expands into the edge of a  rectangle  so you end up with these two 45 degree  angular bisect bisectors between this  vertical edge and the  horizontal one but it's  you have some flexibility there you can  design it how you want  the other case you can have is a degree  zero vertex there are no  edges here this is a little funny and  the way i defined it i just said i  wanted to align line segments  but you can also align points if you  really feel like it  and that would be represented by a dot  that has no cuts next to it if you want  to cut out just this point  i need to make it something you could  think of it as a tiny triangle for  consistency with this picture  we think of it as a little square and so  we when you expand  it or when you shrink the outside region  you get four  45-degree folds this is actually how  i culture it all defined it back in  96 and it's a fine definition but you  have flexibility here in your design  process  they'll all work and this would let you  take a whole bunch of points align them  onto a common line and nothing else is  on that line  because these folds are going to push  everything surrounding the point  away from the line  all right some fun facts straight  skeleton is nice and small if you have  an original points and line segments in  your  desired cut pattern the straight  skeleton has a linear and n  number of line segments linear number of  creases  so order n  uh other fun facts there's a one-to-one  correspondence between  the edges you want to cut along  like let me pick one over here maybe  like this this is an edge i want to cut  along and regions  of the straight skeleton so here's a  region a face of the straight skeleton  this guy there's exactly one cut edge  inside that this is always the case you  look everywhere here  every region of the straight skeleton  it's more obvious if i color them  different colors  there's one cut edge inside and all of  those guys  that surround that cut edge  bisect that edge and another one  and the other one is the one on the  other side and generally you take one of  the straight skeleton edges  there are two sides there's two faces of  the straight skeleton this one's crazy  non-convex  this one's just a little infinite  triangle down here  and that edge bisects those two cut  edges so it's very easy to walk around  the structure see what it bisects  lots of things get bisected but it's not  flat foldable  so we're not done and that's where we  need  the perpendiculars  so  so  all right down in definition and maybe  show  picture we're going for  there's a lot of structures here there's  what i call the cut graph the things  we're trying to make  and then there's this straight skeleton  which you can think of as a graph drawn  on the paper  there's vertices of straight skeleton  which you call skeleton vertices regions  of straight skeleton which we call  skeleton regions  edges of the cut graph which we call cut  edges and so on  we're going to add a new graph which is  the perpendicular graph which you can  think of as hinges from  the tree method of origami design  so what is this what does it mean  we started a straight skeleton vertex  usually there are three  skeleton edges coming together at a  vertex sometimes there are more like  this guy  there's four and there are three  if there's three edges coming together  there are three skeleton regions for  each one  each of those regions has one cut edge  in it so we try to walk  perpendicular and toward that  cut edge so here i walk perpendicular i  mean at right angles here  i just go off to infinity here  i walk perpendicular to this cut edge  and that's  that's cool but then i hit i leave the  skeleton region  at that point i enter a new skeleton  region which is this one  it's non-convex thing it contains one  cut edge and when i  where was i here i entered now i want to  move perpendicular to that  cut edge so that when i cross it i cross  it perpendicularly  now i enter a new region it contains  this cut edge  i move perpendicular to it  and i don't actually cross it but i  enter a new region now i'm in the region  of this cut edge so i move perpendicular  to that  and wow i hit another skeleton vertex i  stop  okay this example is because it's on a  triangular grid there's lots of  degeneracies like that  usually you'd eventually go off to  infinity or come around and meet  yourself  here i happen to hit a different vertex  you do that at all the vertices  all the skeleton vertices now there are  some weird ones  like this one notice there are no  no purple lines coming out from here and  that's because  every region you try to enter you  immediately leave  so if i tried there's four regions here  try each one of them  it's like this region has this cut edge  if i tried to go perpendicular to it i'd  enter a different region so i can't  actually go at all  it's like i i move and then i instantly  stop so you can think of there being a  very like a zero radius little uh thing  there  that's sort of the degenerate case of a  river being a disk river being a circle  same thing going on all of in general  when you have reflex vertices in their  angular bisectors meeting you're going  to lose some perpendiculars because you  can't  enter them here's another one where i  just have one perpendicular edge coming  out this  this one i can reach but if i try to be  perpendicular to either these i  enter the wrong region that's the  perpendicular folds  and that's pretty much the crease  pattern there are technically a few  other folds you have to deal with  but that is if you want to make  something right now  just apply those two algorithms and you  get your shape  just fold along that crease pattern it  will be flat foldable  almost always  why is it flat foldable so one thing we  can check  is local flat foldability at least  satisfies kawasaki's condition  because at a typical vertex you're going  to have three  skeleton edges coming together and so  there are three faces here each of them  has  a cut edge somewhere  hopefully i draw this reasonably well  and should have the property that when i  extend these  it didn't draw it so well when i extend  these  these are angular bisectors that's we  know these skeleton edges will  angularly bisect two cut edges the two  cut edges  that are defined by these guys so i  should get angular bisector here  pretend those meet angular bisector here  and then i also have perpendicular folds  so they may not actually meet  this guy but if they did they certainly  meet the extension  hey that's our good friend the rabbit  ear just regular triangle fold  and in particular you can see that  these angles are equal  i'll call this three prime  it's it's like 180 minus that i think  i'm mistaken and this is one prime  not the best notation and these are two  prime  and two prime and so i've got these nice  angle pairings  that means if i add the odd angles i get  the even angle same thing so i  definitely have kawasaki's theorem  everywhere you can check it works even  when you have these degenerate  situations where  uh three or more than three skeleton  edges come together  for the same reason you still get a  pairing just more than three of them  all right but some exciting things can  happen  so i'm going to look at proving  foldability but  one exciting thing that can happen is  you get a lot of  perpendicular folds at a very few  original cut lines so here i'm trying to  make this weird pinwheel shape i want  i want to cut out the bold lines of the  cut lines so i want to cut out the  square  and then these four squares arranged in  a pinwheel pattern around that one why  you'd want to do that i don't know but  we're mathematicians so we want to  consider all the cases  so the straight skeleton is the thin  black lines and that's linear size  that's nice  but the perpendiculars if this piece of  paper is infinite the number of  perpendiculars is infinite  if you have a finite piece of paper  which is what you usually buy in the  store  then it's a finite number of creases so  in any finite region this is a finite  number of creases but  it's a lot of them so that's one sad  thing you can't bound the number of  creases  as a function of the number of cut lines  but i think that's actually necessary i  don't think  it's possible to solve this problem  while bounding the number of creases in  terms of the number of cut lines  but that's one of the open problems  written down  on one of these pictures  one of these slot lecture notes  something else even more annoying though  it happens even in a finite piece of  paper  and it's even more obscure why you'd  want to make this but the bold blue  lines  are the cuts and then the thin black  lines are the straight skeleton  you can tell this spans many years  because i keep changing notational style  and this is from the textbook  and then you get these perpendicular  folds i haven't drawn all of them  but these dash lines the light blue  and this example is set up so that let  me get this right  this width  is an irrational multiple of this width  or this width one of those things are  irrational  uh so they're not very nice numbers and  and what i need to do to finish this  picture  is these guys go they enter a new  skeleton region they're actually going  to keep going straight because there's  two cut lines that are parallel to each  other  it's going to go up there and it's going  to cycle around let me do one round  uh who did i move this guy  so this guy i just extended down he's  going to turn around  make 180 degree turn and you can check  that each of these  you're set up to do 180 degree turn  around this axis around this axis on the  bottom  and on the top around this axis and  around this axis depending wherever you  enter it's like a race track  keep going around and around and if you  follow that one guy  a little bit farther it looks like that  and a little bit farther  it looks like that it never finishes  so in fact you completely fill this  region with creases  it's like a dense region of creases now  this would be a [ __ ] default  i don't recommend you try it and this is  really a situation where this algorithm  fails  the good news is if i move any of these  vertices  the cut vertices the tiniest amount this  will disappear  i really had to be very careful and get  lots of degeneracy for this to happen  we don't actually know how to prove that  it's a conjecture  that if you take any cut any graph of  cuts you want to make and you  perturb the vertices in a tiny epsilon  disk  then the resulting thing will not have  this density behavior i'm totally sure  it's true  but we don't have a proof so that's life  this is why i said skeleton method works  almost always  they're these in annoying situations  where  it doesn't really give you a crease  pattern so you feel like unless you  somehow think this is legitimate to make  infinitely many creases  but i don't think so  um all right  let me tell you a few more things to  make this practical for you  you wanna what you really want is a  mountain valley assignment before i  showed you lots of perpendiculars and  skeleton edges and  basically the way it works is if you  look at  any skeleton edge like this one  it's bisecting in this case a convex  angle  so i make it a mountain here red is  mountain blue is valley  dot dash is mountain dash is valley as  is standard  whenever i'm bisecting a reflex angle  then i make  the uh skeleton edge valley  and that's basically true convex angles  mountains  reflex angles valleys that's for the  straight skeleton edges yeah  uh like this fold or this one  oh this guy yeah all right this is a bit  of a special case here i'm really  bisecting an angle of zero  if you extend these guys out they meet  at infinity and they form a zero angle  because they're parallel  and i call zero a convex angle but i  just define it that way  and so this is a mountain uh whereas  like this guy  it's bisecting it's barely convex  is that really a mountain no  it's a typo good this one should be a  valley  pretty sure yeah  it should be about wow this is a weird  crease pattern  it's not a straight skeleton there never  mind this picture i know there's always  a bug  i think there's a there's a typo in the  book as we say  how did you  how did i make the initial pattern the  turtle i just draw some do something  look like a turtle  anything i i happen to draw this on a  grid but there's no reason i had to  because  the other one that you're explaining  that they have so many increases  happening  and by moving them around vertices i  mean then  the increments that you're creating  they're not uh  what's the what's the relationship  between those increments are you  so in this example i designed the ratios  to be really nasty  like a root 2 over 10 ratio or something  the whole thing will if i perturb these  vertices at all the whole thing will  fall apart i won't get these 180 degree  turns things will end up going off to  infinity  i the hard part here is actually the  irrational  ratio is quite common what's uncommon in  this picture  is that this thing is closed up and you  never escape  almost always there'll be a little gap  and you'll eventually  reach the gap and then go off to  infinity so that's what happens in the  typical case  oh if you drew a picture on the grid  this would never happen that you can  prove  yeah uh square grid  probably also the triangular grid you  need to be a little careful because you  want all these constructions to stay on  the grid  but i think something like that is true  okay  let's move on to how we construct a  folded state when this algorithm works  when it gives a valid crease pattern  you know that's locally flat foldable  because it satisfies kawasaki but how do  we actually know that it globally folds  flat  to do that you have to describe what it  looks like after it's folded  and our the idea here is to look at  what we call corridors but are  essentially discrete versions of rivers  from tree theory so you have these  constant width  strips that turn at skeleton edges  and they could go off to infinity on  both sides in general they could  loop around and meet themselves again  but in this case they actually all go to  infinity  and if you look at one of those strips  you can actually just like cut this out  of your your textbook just slice it up  and uh look at how it's folding well i  mean it meets  a skeleton edge and then maybe it meets  a cut edge usually you don't fold those  then it meets another skeleton edge it  just meets edges one at a time it's  never complicated  because we've divided along all these  perpendicular folds you really only meet  one edge at a time  which is good in fact if you fold  if you look at one of these skeleton  edges  it's creased along normally you think of  that as an angular bisector of let's see  these two cut edges  but you can also think of it as an  angular bisector of these two  perpendicular folds  because if you if you bisect these two  things  you also bisect two things that are  perpendicular to them  it's like two wrongs make a right  somehow okay so this guy bisects  those two creases so if you fold along  here  actually you align these creases it's  kind of it's a duality you're aligning  the perpendicular folds  i fold along here you line up this fold  with that one this fold with that one  i fold here i line up this this fold  with that one there's really there's  like a  zero length fold that's there you fold  along all these things you line up this  with itself  and so on so you follow along this thing  this  corridor folds down to basically a  rectangle  it's got some rough edges on the top and  the bottom but it lies in a  in a strip in 3d i think i have a  picture of that  so i took the one over here this  blue corridor and if you fold it up it  looks like this  okay now in particular you can check at  this point it's pretty easy to check  because of all this bisector-ness  bisector goodness that you bring into  alignment all the cut edges  so for example this guy because it  bisects that cut edge and that cut edge  brings them into alignment and you can  see that somewhere on this picture i  think it's these two guys  it can be a little more complicated like  over here i have a cut edge then i have  a bisector and then a bisector and then  a bisector and then another cut edge  but if you think about it right it's you  know i don't happen to meet  these uh cut edges but i'm effectively  bringing this into alignment with this  and then this into alignment with this  and then this into alignment with that  so in the end i aligned this with that  and that's what's happening  up here on the left where i don't quite  come all the way down but i still end up  lining everything up  so this is a solution to the fold and  cut problem as long as it actually folds  and to show that it actually folds you  just need to show that these  uh corridors or i forget i think we call  these accordions  it's been a long time this is 98 for  this paper  you take these accordions and you just  want to see how they fit together  and lo and behold they fit together in a  tree in this  picture it gets more complicated  in another picture which i will show you  in a moment  but in in this situation where every  quarter goes off to infinity on both  sides  you get a nice tree structure and as  long as you can fold this tree flat  then you can fold this thing flat  because each of these edges of the tree  is this very simple accordion structure  which is  trivi trivially folds flat  and if the other thing you have to check  here is you actually alternate mountain  valley  that's a little more subtle especially  when i don't draw the mountain valley  assignment  but it turns out you always alternate  between mountain and valley in this  picture  which is great that's the thing we  always we know always folds flat it's  like a 1d folding  so these are super easy to fold you can  fold each of them if you cut along all  the dashed lines  you can fold each separately then you  need to join them together and where  the edges here just like entry theory  the edges here correspond to these  rivers and now you need to somehow  attach them here  check that where you attach them there's  no crossings  that's i'm not going to describe but  it's pretty easy  it's planarity essentially of that  diagram and then you need to fold this  tree flat  folding a tree flat is actually kind of  a segue into next lecture which will be  about folding linkages  in this case it's really easy you just  pick up some root like the letter a over  there  and you hang the tree pull up  technically this is like a depth-first  search of the tree  so you just walk down always walk down  until you're finished then you go back  up  walk down some more branches you'll end  up drawing everything downward  away from a and it will be a flat  folding there won't be any crossings  here  and then this is a 1d representation of  what's really going on which is that  above each of these edges are is really  an accordion  and you need to join them together there  but we basically do this modular folding  where you fold each accordion separately  put them together according to this boom  you get your flat folding from this  picture you can read out the mountain  valley assignments for the perpendicular  folds  this looks like a valley this looks like  a valley this looks like a perpendicular  fold i didn't use  because there's no there's no crease  there that's flat  uh where's a mountain mountain's  probably at the top at  a what really happens  if you want to know really what we're  deciding is whether this starts mountain  or valley then it will actually  alternate back and forth  but as you move along perpendicular  that's basically how you construct a  folded state  in this situation of so-called linear  corridors  now there are a bunch of things i  haven't mentioned  but i think i don't want to talk about  them too much  so i get onto the other topics  so uh what i just talked about  is something called the linear corridor  case which is really where it's the most  beautiful  this construction of a folded state and  linear corridor intuitively something  like this it goes off to infinity on  both sides  has constant width all the way of course  it's really a discrete thing not a  smooth thing  uh let me say  conjecture  if your cut graph  has a maximum degree two  that means at most two edges at every  vertex  this is a very common scenario this is  if you want to make a polygon every  vertex has degree two  you might make several polygons every  vertex has degree two i'll even let you  have vertices of degree one  or zero for free but mainly we're  thinking about  degree 2 everywhere  then  we almost always  have a linear quarters  so this is why this situation is  interesting although unfortunately this  is still a conjecture  i'm sure this is true but proving it i  don't quite know the right techniques  so in this typical situation you take  any picture you want you slightly  perturb every vertex randomly  say then with probability one hundred  percent probability  you will get only linear quarters  and that's the situation where it turns  into a tree it's easy to fold  life is good the annoying case  is circular corridor case  this takes a lot more work to prove and  i'm not going to talk about it much  circular corridor looks like  this okay we had these in with rivers  also  so you just loop around still constant  width everywhere  but you meet yourself and you don't go  out to infinity  it's harder why is it harder well in  particular  if you look at how one quarter folds  it's no longer it's like the same  situation we had in like lectures two  and three  where we had on the one hand a 1d  folding was really easy  but then when you made it circular he's  folding a circle instead of folding a  line segment  now you had this wrap around issue so  like these guys would have to  line up it turns out they will line up  because everything is bisecting whatnot  these edges will line up but now you  have to join them together  and if this part went all the way down  here and came back up  then you'd get an intersection and  it turns out in general  i get i get a choice of who's first and  who's last i have a circular order of  things and i get to choose  where i break that circular order and  make it a linear order where i do the  wrap around  there are some circular corridors you  can't even  break them and make it work  it's kind of depressing so this is  definitely harder in that sometimes  it's not possible  but we can save a little bit  which is i don't have an example handy  i wish i did i'll have to reconstruct it  this is so long ago  uh here's a way to make it definitely  work  fold all the cut edges  okay so far in pictures i've been  drawing i didn't fold along the cut  edges  because i really wanted to separate the  green region here from the yellow region  if i folded this up this is quite a  complicated one to fold  you get the cut lines somewhere like  like over there  and then the green stuff will be always  above the cut line and the yellow stuff  will be below the cut line  in general this is called a side  assignment you have a bunch of regions  you decide which ones do you want to be  above and which ones do you want to be  below  and usually you have polygons and you  say the interiors are above and the  exteriors are below  but in general you could ask for  anything you want  you could say maybe i want both of these  to be above if you make both of them  above you have to valley fold  along all of the cut lines so if you do  that you say i want all regions to be  above the cut line you can still line  them up  you end up folding along all the cut  edges with valleys  and then wrap around is super easy you  take  you take this thing in fact everyone's  folding along the black lines  so really everybody comes down to the  floor  and then the wrap around is just  underneath the floor and  life is good so that's one way to deal  with this  case and you have to prove that that  works it's complicated  uh i would rather go on to other topics  i do think this would make a fun project  it's not easy to make these crease  patterns currently we draw them always  by hand meaning with a fancy drawing  program that knows how to do angular  bisectors  and it's it's an art to move around the  points  so that you get lots of alignments like  when you get  four straight skeleton edges coming  together that means you get fewer  creases that's a good thing whenever you  can make that happen  it's a win if you could give me software  to help do that  i would love you  yeah so let's move on  any questions about the straight  skeleton method now i'm going to switch  over  to disk packing  all right same problem  but now i'll give you a method that  always works  in theory just uh  difficult in practice  i think this is good chalk because it's  yellow  generally if it's yellow on the outside  it is a railroad shock which is the good  stuff  but what's the problem is we have bad  erasers  it's it's persistence of vision you just  get to remember what i used to write  uh i think i don't want to write down  the algorithm it's complicated  i want to give you a visual overview of  the main steps  because there's nice figures to do that  for us  so um we start with a very complicated  shape we want to make  like this quadrilateral and  you can see disc packing is involved the  very first thing we do and i'll tell you  why in a moment  is thicken the graph you want to build  so maybe it's a polygon maybe it's a  graph i'll think about the polygon case  for now because it's easier then i'll  come back to the general case i thicken  it by a tiny epsilon  just offset in both directions just like  as if you're starting the straight  skeleton method but then you stop after  epsilon  no events happen in epsilon time  okay then i have this picture  the purple stuff pink stuff whatever  magenta  50 magenta i happen to know drew this  figure  now i'm going to take some 50 cyan  discs and pack them to fill this region  now what i want  there's three properties i want one is  that every vertex  but you have to think about each region  separately because a little bit  confusing  let's think about the outside it's a  little easier bigger  every vertex of this graph on the  outside  should be the center of a disc so  there's a disc centered here  there's a disc centered here so just  centered here and just centered there  on the inside it's also true they're  just different disks  then also i want the edges of the graph  to be  covered by radii of disks  right so here's a radius of one disk  here's a diameter of a disk  here's a radius of a disk that covers  the edge  so in other words i want to fill along  the edge i want to have a bunch of  centers  so that i completely cover that edge  with blue we'll see why  later questions  uh the discs have to be non-overlapping  so these properties are actually quite  challenging to achieve  your question is why do we use small  discs  this because if i had a big disc here it  would intersect this disc  uh now i didn't have to make that disc  so big but if i made that one smaller  i'd have to have more discs here  so or here there's also two small discs  that one  i'm probably could have gotten away with  a bigger disc  uh oh no but then on the inside you have  a problem  so these guys actually have to match up  that's another constraint and the inside  and the outside have to match up here  there's a slight change in radii  to compensate for the epsilon along the  edges they're exactly the same  so if i made this one a big disc it  would overlap this one so i could make  that one smaller but then it's  other problems so you have to  simultaneously balance all these  constraints which is  a bit exciting um what else the distance  overlap  the last property is that the gaps  between the disks have always  three or four sides why because  i want it to it'll make life easier you  could try to deal with  more sides but three and four  is nice uh yeah i'll get to that  why do we care about number of sides  because i'm going to  draw a graph i'm going to subdivide my  original graph here  with these red lines to say whenever  disks  kiss i will draw connect the centers of  those disks  and because these gaps always have three  or four sides  uh that's not the best example here's  like three sides the  the red lines i draw will outline a  triangle whenever i have three sides  whenever i have four sides i'll have a  quadrilateral  so i've subdivided my regions into  triangles and quadrilaterals  okay for you have to believe that this  is possible i can sketch an algorithm  for you  uh which is you draw a tiny disk at each  of the corners  and then you draw lots of tiny discs  along the edges to fill the edges  and that will satisfy everything i mean  the discs will be non-overlapping  because they're super tiny they won't  get near any other discs from some other  side  and what other good things oh the but  the regions will be  ginormous they won't have three or four  sides they'll have hundred sides million  sides who knows  well uh whenever you have some crazy  region  outlined by disks might not be convex  whatever  just draw the biggest disc you can in  there  get turned into a disc eventually uh  that does not  intersect anybody but if it's the  biggest possible it will actually touch  at least three sides  if your degenerate it might touch four  but in general it will  touch three sides which will subdivide  that region into three pieces  and you can show that those pieces are  all a little bit smaller than what you  had before in terms of number of sides  except when you start with a  quadrilateral when there's four sides  you'll get  quadrilaterals and triangles so you  can't go below three and four be great  if we could always get down to  three sides in every region but we can  get down to three or four anything  bigger than three or four you can show  this will make it strictly smaller so  that is an algorithm it's not super  efficient  but it will find a disk packing with all  these properties then we do the  subdivision  now what do you think we're going to do  what do we do with the triangles  gravity here that's the key phrase for  today  so remember our good friend the rabbit  ear and then there was the  universal molecule length the universal  molecule for the quadrilateral we're  going to use that  for the quadrilaterals and it turns out  there's some nice properties here  which is uh the perpendicular folds of  the rabbit ear will always hit  right at the kissing point between the  two discs and same thing  in in here we've got these four discs  we've got this quadrilateral region  in between and the perpendicular folds  that come  out of these two points  you may not remember exactly what's  happening here is we  shrink and then in  in the tree method this became an active  path there's no notion of activepads  here  but we just make that so that  these perpendicular folds will end up  hitting the kissing discs  and we'll end up actually with a  four-armed starfish in in terms of the  tree you get  and the articulatable flaps here it  these guys will all meet at a point  that's just  the way this works with disc packing and  you can think of that you can think of  there being discs here and you're  actually applying the tree method  to that flap pattern and  that's that's probably the easiest way  to think about it but what's good for us  do i have a picture not yet but the  point is  i perpendicular is coming out of here i  have perpendiculars coming out of here  they will meet because these discs kiss  uh at the same point up from both viewed  from both sides perpendicular is meet  that's good that means i don't get  perpendiculars bouncing out all over the  place  so all this work is to make sure  perpendiculars are well behaved  it's a lot of work to do it but it does  it  now when you fold this thing uh what we  end up doing  is lining up remember there was there  was two copies of my polygon  there's the inner copy in the outer copy  i end up  lining up all of these guys  i got to go back to the other picture  i'm sorry so we have this inner copy  and what these molecules end up doing is  lining up all the edges of this quad all  the edges of this quad all the edges of  this triangle  all those edges on the inside will  become lined up on one  edge all the lines on the outside become  lined up on  another line turns out it will be  parallel to that line  but what we really wanted to line up  were these edges and you can see why we  had to do the offset at the beginning  because otherwise we get tons of extra  junk on our line  we only want these edges on our line so  we did the offset so that all this stuff  will come to one line all this stuff on  the outside will come to another line  and then we get this picture  so this is one line at the bottom  another line at the top we really wanted  to line up stuff  uh the the blue stuff there and there's  still some junk on our line these  these uh cyan triangles represent things  that come from down here  but we really don't want them to cross  our line they just happen to  so we have to sink them repeatedly do  lots of folds to make them underneath  that epsilon line  then we can cut along our line and we're  done easy  to prove that this works give you a  little  sketch this is kind of fun and it's it's  one piece of what we're  in the process of doing for tree maker  this is sort of like a special case of  tree maker you just have very simple  molecules  and relatively simple way in which they  are combined  so you have here i've done a simpler  example i want to make a square  and i end up decomposing in this case  into nine molecules nine quadrilateral  molecules very simple disc packing which  i have not shown the disc packing here  the idea is i'm gonna make some cuts in  my paper  to make my problem easier i'm gonna have  to pay for that because later i really  want those edges joined i have to glue  them back together  but to make it easier to think about i  cut  those four edges so that the  way in which my molecules are connected  to each other  is a tree because i like trees they're  easier to think about easier to do  induction over  so that's the blue lines are connecting  the centers in a tree  the other remaining edges in the grid  have been cut away  now the idea is it's kind of like  what i was drawing for the linear  corridor case you have a tree  you pick up the tree from some node and  just hang it down  and in this case we hang it from  this molecule the red edges are mountain  so these three of these are going to be  valley  one mountain so the idea is this thing  reaches around  the next guy juices around the next guy  which is around the next guy  and there's actually there's two valleys  here two little pockets  each of this guy goes in that pocket  this guy goes in that pocket and  recursively  it just works i think i have a picture  of  what's actually happening here yeah it's  it's hard to really draw  but each of these forearm starfish has  one mountain  and three valleys and you just nestle  inside your parent in the tree  and this is really easy to show that  there's no crossings here because you  just come you just join to your parent  and it's a nice nesting structure it's  just in the same way that trees can be  folded flat  uh you can fold all these molecules flat  and and join them together in a tree but  we didn't really have a tree  we had all those extra cuts that we have  to re-suture  so we have this picture and now we  really have to join up these edges and  think about what the mountain valley  assignment is there  and it works  uh this is this green thing  is the boundary and then i've connected  the dots each of these dots corresponds  to one green edge here i forget whether  it's this edge or  what that one i think it's just a single  edge  and so yeah for example  these two are  these two joints and then the joins  above  that nestle around it and then the other  branch at the top are these two joins  and then the leftmost cut are these two  joints you have to make these joins and  really all you need to check  is that these joints form a non-crossing  picture like they do here  and that's almost obvious because this  is a planar diagram  and we're cutting along a planar tree  and so this is again a depth first  search kind of thing  so it's there's one tree we call the  dual tree here  that works because it's a tree then  there's the cuts you make which are  different trees  primal tree if you want and that also  works because of planarity  and it all works that's the hand wavy  version  and you can read the textbook if you  want more details uh  oh gosh if you want to solve uh more  general graphs you can do it  in general you get you have to offset  all of those cut lines  and you get all these things and along  the uh  pink lines here you line things up  but you really wanted to line up the  these blue lines  purple lines and so you have to do more  sinking to get it to work  now i have all the things i want to line  up on this line and this line i fold in  the middle  and now they're lined up  ah if i fold in the middle yeah that  will work  good we might have to do more sinking  all right  questions about the disc packing method  so a bit of a whirlwind tour  but i wanted to get through it quickly  to tell you about a new result  just got accepted to a conference to  appear  in october pretty soon  and it's a project that started in this  class in the problem session three years  ago  and we just solved it took a while  took another co-author to join in  and it goes back to the early history of  folding cut which is  simple folds all the magicians were  using simple folds what can you make  with simple folds  now you've been wowed endowed that you  can make anything  with arbitrary folds but simple folds  you cannot make anything  because the first fold you make  say this one has to be a line of  symmetry  of your diagram  i've got to stop making my life hard  if you can fold something you can never  unfold it that's the usual simple fold  model  this has to line up the cuts you want to  line up over here better exactly be the  cuts you want to line up here so you can  only make symmetric diagrams the first  fold has to be a line of reflectional  symmetry  but is that the only property you need  no  you kind of have to have symmetry for a  while until you're done  how do you formalize that well we came  up with  an algorithm that in polynomial time  so an efficient algorithm will  tell you whether a given polygon can be  made  like this polygon  looks good i think  yeah i think this can be made so i think  i would fold along  angular bisector here and then this  basically disappears  folding over then i would fold along an  angular bisector here and then this  disappears into that  and then  maybe i can fold here does that work  barely i mean i got to make sure this  blank paper does not come onto that  but if that's a problem i can probably  make  well i can make a fold here for example  shrinks that up there's lots of things  you can do and this is a borderline  case whether it's yes or no i will give  you an algorithm that does it  for polygons with margin  a bit of a technical condition something  that  is pretty typical what i mean is the  thing you're trying to cut out  does not meet the boundary of the paper  there's no margin here it'd be hard to  print out  so i really want something that has  margin that's the typical case we care  about  but you actually need this for the proof  to work we also need that it's a single  polygon  uh it does not work with general graphs  this algorithm because more complicated  things can happen it might be np hard  for all we know  the general case so here's the algorithm  maybe i'll give you number form  first thing you do is guess the first  fold this is a  a powerful idea that even most  algorithmicists don't necessarily know  the idea is what could the first fold be  has to be a line of reflectional  symmetry turns out there's a linear  number of them at most you can find them  in linear time  with all these good algorithms for  finding them but which fold do i make  first  the answer is i don't care let's just  try them all one at a time  this is what i call guessing just  imagine from now on that we made the  right guess  but if you end up failing later on in  this algorithm just go back to here try  the next one  there's only n of them to try so you're  going to multiply the running time of  the rest of the algorithm by  n and if this is n to like n squared say  which i think the rest of the algorithm  is n squared  the whole algorithm will be n cubed  because you just run this over and over  for each possible first fold  we don't have a great theory to find the  first fold just try them all  that's step one step two  that's the only guess we're going to  make  fold down to convex hull  this is a central idea  so we have this this uh polygon we want  to make  there's all this extra blank paper i  don't like extra black paper just get  rid of  it make lots of folds that fold the  blank paper onto itself till it gets  so tiny it just goes slightly around the  convex hull  convex hall is the smallest convex  polygon that contains your shape  so be like that it'll reduce the paper  down to that  i'm going to do this a lot i might as  well  it makes the problem easier because i  have less blank space to worry about  blank space is a problem because if i  fold blank space onto a cut  it's bad it's not allowed  so the next thing we do is make  the best possible safe  fold  i need to define that but a safe fold is  just the folds we're trying to make  which is  line locally there are lines of symmetry  so like this one  was a good fold after i had made this  fold so this is all  all the right stuff here is gone it's a  good good fold because it folds this  onto this  and it folds blank space onto this blank  space so anything that comes into  on top of each other is identical that's  a safe fold  and repeat  that's the algorithm how does that work  why does this work so it's the obvious  algorithm basically it says  make safe folds until you're done if you  finish  then you've you're done and then the  answer is yes if you don't finish which  is a little hard to check because you  can always make  microscopic folds but you sort of take  the limit and uh  it's possible to do this in polynomial  time you find that oh  turns out i can't finish by making  say folds then you can show that  actually there was no way to make that  pattern  by simple folds so let me  give you an idea of why that's true  after i make a single fold the first  fold  my picture looks like this it's no  longer a polygon  it's it's like half a polygon it ends at  the  boundary of the paper it begins and ends  at the boundary paper and you have some  chain in the middle  call this a passage it's like path you  wander along  and i want to somehow bring all those  edges into alignment  this here i'm already using that it's a  polygon if it weren't a polygon there  might be more than one of these  now i want to make a fold like for  example this fold is safe because it  folds this onto this and it folds blank  space on the bank blank space if i  drew it right  and keep doing that now when i make a  fold like this  what happens is i can think of this  region  that i folded the smaller side as  disappearing right it just got absorbed  into the paper here so my the graph that  i was trying to line up got smaller  that's clearly a good thing makes my  problem easier  the piece of paper i was folding also  got smaller  that's a good thing but that's not  always true  suppose you had a piece of paper like  this which could happen  after a bunch of folding and then you  fold along a long line like that  because for example your passage looks  like that or something  when i make the fold this has to go off  when i make the folds i get you know  this  crazy thing not drawn to scale and this  polygon does not fit inside this polygon  so my paper got  bigger in some places and that's a worry  because now i have this stuff maybe it  happens to be blank space  maybe there's other junk that got out  here and now i have to  worry about collisions with this bigger  piece of paper and this is always our  sticking point but  there's some magic you can do  in fact the picture cannot look like  this  because look you've got some portion of  your passage  to the left of the crease you've got  some portion of the passage to the right  one of them has to be shorter plus this  is a line of symmetry  so wherever i have a  portion of my passage over here i will  have a portion of my passage over here  until i run out of length one of them is  shorter so  the shorter one like this one gets  totally absorbed by the larger one  so shorter side always disappears so in  this picture  i have the long side of my passage and  it's really a subset of the original  if i reduce this to the convex hull just  like this  this stuff disappears and in general  if you do this fold down a convex hull  this repeat  it goes back to step two if you fold  down to the convex hall you can show  that not only does your passage the  thing you want to cut out get smaller  but your piece of paper also gets  smaller guaranteed and once  once you know the paper gets smaller and  your the things you're trying to align  get smaller  you know that every move is safe so  you never get stuck by following this  algorithm this works for polygons  with margin but not in any other  situation as far as we can tell  um cool  the last thing i wanted to leave you on  is going out  a little a little away from regular 2d  flat sheets of paper  you can generalize and go up a dimension  to a folding polyhedra surface here a  surface of a polyhedron  you've probably done this you take a  cereal box you can collapse it flat  is that always possible this is called  the flattening problem  and the answer is yes and you can think  of it as a folding cut  problem because with the fold and cut  problem you have some polygon like this  diamond  you make some collection of folds that  brings the boundary of the diamond  to a line so if you forget about what's  happening on the inside of the paper you  just look at the boundary of the paper  you're folding that one dimensional  boundary  so that it collapses down to a single  line  what i want to do is this upper  dimension  i take a 3d cube of paper solid cube  i have would be embedded within it some  some  polygons that i want to bring to a  common plane  and i want to fold the solid cube  through four dimensions mind you  but flat so it ends up back in three  dimensions i get a different 3d solid  but there with multiple layers right on  top of each other a little bit of fourth  dimension  hanging out there but if i just look at  what's happening to the boundary of my  polyhedron say i start with a  dodecahedron or something embed it in  there  and i i want to fold this thing so that  all the sides of the dodecahedron come  to a common plane that is the 3d folding  cut problem  it remains unsolved i suspect it's  possible to solve even with straight  skeletons and perpendiculars but  it's really hard to draw the pictures so  we have not  resolved that one way or the other but  the boundary problem forget about what's  happening to the interior and the  exterior of the  kedron if you just look at the surface  of the dodecahedron that you can fold in  3d  we think and you can show  and burn in haze from the complexity  proof couple lectures ago  and also on this disc packing method  their co-authors  they proved just two years ago that if  you have any orientable manifold  which is things like polyhedra but no  mobius strips no klein bottles and other  ugly things they have to be manifolds so  you're not allowed to say  join three triangles together along a  single edge  that would be forbidden so it's locally  flat  in such a case you can flatten the thing  and the proof is very similar to the  disc packing  method of folding cut and in the in the  textbook we talk about how to apply that  to do  something that's just like a sphere a  regular polyhedron  that's pretty easy to do with the disc  packing method they generalize it to the  case where you have  polyhedral doughnuts and all sorts of  fun things  but there's tons of open problems here  so we know how to flatten surfaces and  that's useful for things like folding  airbags flat  but can you fold the 3d solid flat  uh you can think of where we're we have  here  1d edges which we are collapsing to a 1d  line there's also zero dimensional  points here  which we don't bring to a single point  it'd be nice if you could  the generalized fold and cut problem is  you take a d dimensional thing  and you have all of these uh there's  zero one two three up to d  dimensional parts to it or d minus one  dimensional parts  you wanna bring each of them down into  alignment so that  all the vertices come to a common point  all the edges come to a common line  all the faces two-dimensional faces come  to a common plane and so on up the  dimension hierarchy that is the ultimate  open problem i think we end the book  with it  and it's totally unsolved any questions  that's folding and cutting paper and  next time we'll start linkages
 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323
00:12:45,430 --> 00:12:46,230

 

324
00:12:46,230 --> 00:12:48,790

 

325
00:12:48,790 --> 00:12:50,069

 

326
00:12:50,069 --> 00:12:53,430

 

327
00:12:53,430 --> 00:12:53,440

 

328
00:12:53,440 --> 00:12:53,670


329
00:12:53,670 --> 00:12:57,110

 

330
00:12:57,110 --> 00:12:58,870

 

331
00:12:58,870 --> 00:13:00,870

 

332
00:13:00,870 --> 00:13:03,670

 

333
00:13:03,670 --> 00:13:05,269

 

334
00:13:05,269 --> 00:13:06,550

 

335
00:13:06,550 --> 00:13:08,310

 

336
00:13:08,310 --> 00:13:09,829

 

337
00:13:09,829 --> 00:13:12,550

 

338
00:13:12,550 --> 00:13:13,750

 

339
00:13:13,750 --> 00:13:15,190

 

340
00:13:15,190 --> 00:13:15,750

 

341
00:13:15,750 --> 00:13:17,829

 

342
00:13:17,829 --> 00:13:17,839

 

343
00:13:17,839 --> 00:13:19,190


344
00:13:19,190 --> 00:13:23,350

 

345
00:13:23,350 --> 00:13:24,790

 

346
00:13:24,790 --> 00:13:26,310

 

347
00:13:26,310 --> 00:13:28,949

 

348
00:13:28,949 --> 00:13:31,750

 

349
00:13:31,750 --> 00:13:44,069

 

350
00:13:44,069 --> 00:13:47,030

 

351
00:13:47,030 --> 00:13:47,829

 

352
00:13:47,829 --> 00:13:52,069

 

353
00:13:52,069 --> 00:13:54,310

 

354
00:13:54,310 --> 00:13:59,829

 

355
00:13:59,829 --> 00:14:01,509

 

356
00:14:01,509 --> 00:14:03,590

 

357
00:14:03,590 --> 00:14:06,870

 

358
00:14:06,870 --> 00:14:08,550

 

359
00:14:08,550 --> 00:14:11,509

 

360
00:14:11,509 --> 00:14:13,430

 

361
00:14:13,430 --> 00:14:31,509

 

362
00:14:31,509 --> 00:14:33,590

 

363
00:14:33,590 --> 00:14:35,110

 

364
00:14:35,110 --> 00:15:19,910

 

365
00:15:19,910 --> 00:15:21,590

 

366
00:15:21,590 --> 00:15:23,430

 

367
00:15:23,430 --> 00:15:24,550

 

368
00:15:24,550 --> 00:15:28,150

 

369
00:15:28,150 --> 00:15:30,150

 

370
00:15:30,150 --> 00:15:31,590

 

371
00:15:31,590 --> 00:15:34,069

 

372
00:15:34,069 --> 00:15:34,079

 

373
00:15:34,079 --> 00:15:34,629


374
00:15:34,629 --> 00:15:36,310

 

375
00:15:36,310 --> 00:15:38,629

 

376
00:15:38,629 --> 00:15:43,910

 

377
00:15:43,910 --> 00:15:46,790

 

378
00:15:46,790 --> 00:15:48,470

 

379
00:15:48,470 --> 00:15:52,230

 

380
00:15:52,230 --> 00:15:53,910

 

381
00:15:53,910 --> 00:15:53,920

 

382
00:15:53,920 --> 00:15:56,870


383
00:15:56,870 --> 00:15:58,470

 

384
00:15:58,470 --> 00:15:59,590

 

385
00:15:59,590 --> 00:15:59,600

 

386
00:15:59,600 --> 00:16:00,310


387
00:16:00,310 --> 00:16:03,749

 

388
00:16:03,749 --> 00:16:05,829

 

389
00:16:05,829 --> 00:16:07,030

 

390
00:16:07,030 --> 00:16:09,910

 

391
00:16:09,910 --> 00:16:11,990

 

392
00:16:11,990 --> 00:16:13,590

 

393
00:16:13,590 --> 00:16:15,189

 

394
00:16:15,189 --> 00:16:17,189

 

395
00:16:17,189 --> 00:16:18,629

 

396
00:16:18,629 --> 00:16:18,639

 

397
00:16:18,639 --> 00:16:19,350


398
00:16:19,350 --> 00:16:22,389

 

399
00:16:22,389 --> 00:16:23,430

 

400
00:16:23,430 --> 00:16:25,269

 

401
00:16:25,269 --> 00:16:27,110

 

402
00:16:27,110 --> 00:16:29,189

 

403
00:16:29,189 --> 00:16:29,199

 

404
00:16:29,199 --> 00:16:30,150


405
00:16:30,150 --> 00:16:31,509

 

406
00:16:31,509 --> 00:16:35,030

 

407
00:16:35,030 --> 00:16:37,430

 

408
00:16:37,430 --> 00:16:38,949

 

409
00:16:38,949 --> 00:16:40,629

 

410
00:16:40,629 --> 00:16:42,310

 

411
00:16:42,310 --> 00:16:43,990

 

412
00:16:43,990 --> 00:16:45,749

 

413
00:16:45,749 --> 00:16:47,430

 

414
00:16:47,430 --> 00:16:48,870

 

415
00:16:48,870 --> 00:16:48,880

 

416
00:16:48,880 --> 00:16:51,590


417
00:16:51,590 --> 00:16:53,749

 

418
00:16:53,749 --> 00:16:55,030

 

419
00:16:55,030 --> 00:16:57,670

 

420
00:16:57,670 --> 00:16:59,350

 

421
00:16:59,350 --> 00:17:03,829

 

422
00:17:03,829 --> 00:17:06,390

 

423
00:17:06,390 --> 00:17:08,230

 

424
00:17:08,230 --> 00:17:11,990

 

425
00:17:11,990 --> 00:17:15,270

 

426
00:17:15,270 --> 00:17:15,280

 

427
00:17:15,280 --> 00:17:16,829


428
00:17:16,829 --> 00:17:16,839

 

429
00:17:16,839 --> 00:17:21,429


430
00:17:21,429 --> 00:17:23,110

 

431
00:17:23,110 --> 00:17:24,870

 

432
00:17:24,870 --> 00:17:29,590

 

433
00:17:29,590 --> 00:17:32,549

 

434
00:17:32,549 --> 00:17:33,350

 

435
00:17:33,350 --> 00:17:36,630

 

436
00:17:36,630 --> 00:17:39,110

 

437
00:17:39,110 --> 00:17:40,230

 

438
00:17:40,230 --> 00:17:41,990

 

439
00:17:41,990 --> 00:17:43,430

 

440
00:17:43,430 --> 00:17:43,440

 

441
00:17:43,440 --> 00:17:43,830


442
00:17:43,830 --> 00:17:47,590

 

443
00:17:47,590 --> 00:17:50,789

 

444
00:17:50,789 --> 00:17:52,950

 

445
00:17:52,950 --> 00:17:55,270

 

446
00:17:55,270 --> 00:17:58,070

 

447
00:17:58,070 --> 00:18:00,310

 

448
00:18:00,310 --> 00:18:01,990

 

449
00:18:01,990 --> 00:18:04,070

 

450
00:18:04,070 --> 00:18:05,669

 

451
00:18:05,669 --> 00:18:05,679

 

452
00:18:05,679 --> 00:18:07,430


453
00:18:07,430 --> 00:18:11,270

 

454
00:18:11,270 --> 00:18:14,310

 

455
00:18:14,310 --> 00:18:16,150

 

456
00:18:16,150 --> 00:18:17,909

 

457
00:18:17,909 --> 00:18:17,919

 

458
00:18:17,919 --> 00:18:18,870


459
00:18:18,870 --> 00:18:20,630

 

460
00:18:20,630 --> 00:18:21,909

 

461
00:18:21,909 --> 00:18:24,549

 

462
00:18:24,549 --> 00:18:26,230

 

463
00:18:26,230 --> 00:18:27,909

 

464
00:18:27,909 --> 00:18:27,919

 

465
00:18:27,919 --> 00:18:29,430


466
00:18:29,430 --> 00:18:30,630

 

467
00:18:30,630 --> 00:18:31,990

 

468
00:18:31,990 --> 00:18:33,190

 

469
00:18:33,190 --> 00:18:34,950

 

470
00:18:34,950 --> 00:18:36,710

 

471
00:18:36,710 --> 00:18:40,830

 

472
00:18:40,830 --> 00:18:45,430

 

473
00:18:45,430 --> 00:18:50,150

 

474
00:18:50,150 --> 00:18:52,950

 

475
00:18:52,950 --> 00:18:52,960

 

476
00:18:52,960 --> 00:18:53,430


477
00:18:53,430 --> 00:18:57,830

 

478
00:18:57,830 --> 00:19:05,590

 

479
00:19:05,590 --> 00:19:08,870

 

480
00:19:08,870 --> 00:19:15,190

 

481
00:19:15,190 --> 00:19:16,549

 

482
00:19:16,549 --> 00:19:17,990

 

483
00:19:17,990 --> 00:19:19,029

 

484
00:19:19,029 --> 00:19:23,430

 

485
00:19:23,430 --> 00:19:29,909

 

486
00:19:29,909 --> 00:19:36,150

 

487
00:19:36,150 --> 00:19:40,950

 

488
00:19:40,950 --> 00:19:43,590

 

489
00:19:43,590 --> 00:19:47,270

 

490
00:19:47,270 --> 00:19:48,870

 

491
00:19:48,870 --> 00:19:53,990

 

492
00:19:53,990 --> 00:20:00,470

 

493
00:20:00,470 --> 00:20:02,149

 

494
00:20:02,149 --> 00:20:03,990

 

495
00:20:03,990 --> 00:20:05,990

 

496
00:20:05,990 --> 00:20:08,230

 

497
00:20:08,230 --> 00:20:09,909

 

498
00:20:09,909 --> 00:20:11,350

 

499
00:20:11,350 --> 00:20:13,669

 

500
00:20:13,669 --> 00:20:14,870

 

501
00:20:14,870 --> 00:20:16,310

 

502
00:20:16,310 --> 00:20:17,990

 

503
00:20:17,990 --> 00:20:19,510

 

504
00:20:19,510 --> 00:20:22,390

 

505
00:20:22,390 --> 00:20:24,549

 

506
00:20:24,549 --> 00:20:26,630

 

507
00:20:26,630 --> 00:20:28,230

 

508
00:20:28,230 --> 00:20:29,750

 

509
00:20:29,750 --> 00:20:31,750

 

510
00:20:31,750 --> 00:20:40,070

 

511
00:20:40,070 --> 00:20:43,830

 

512
00:20:43,830 --> 00:20:45,029

 

513
00:20:45,029 --> 00:20:46,549

 

514
00:20:46,549 --> 00:20:48,390

 

515
00:20:48,390 --> 00:20:49,909

 

516
00:20:49,909 --> 00:20:52,070

 

517
00:20:52,070 --> 00:20:55,830

 

518
00:20:55,830 --> 00:20:58,310

 

519
00:20:58,310 --> 00:21:00,390

 

520
00:21:00,390 --> 00:21:01,990

 

521
00:21:01,990 --> 00:21:03,590

 

522
00:21:03,590 --> 00:21:03,600

 

523
00:21:03,600 --> 00:21:04,070


524
00:21:04,070 --> 00:21:05,510

 

525
00:21:05,510 --> 00:21:07,270

 

526
00:21:07,270 --> 00:21:08,549

 

527
00:21:08,549 --> 00:21:10,070

 

528
00:21:10,070 --> 00:21:11,270

 

529
00:21:11,270 --> 00:21:12,630

 

530
00:21:12,630 --> 00:21:16,950

 

531
00:21:16,950 --> 00:21:19,990

 

532
00:21:19,990 --> 00:21:21,430

 

533
00:21:21,430 --> 00:21:24,149

 

534
00:21:24,149 --> 00:21:24,159

 

535
00:21:24,159 --> 00:21:25,110


536
00:21:25,110 --> 00:21:26,230

 

537
00:21:26,230 --> 00:21:28,230

 

538
00:21:28,230 --> 00:21:29,830

 

539
00:21:29,830 --> 00:21:31,430

 

540
00:21:31,430 --> 00:21:32,870

 

541
00:21:32,870 --> 00:21:34,230

 

542
00:21:34,230 --> 00:21:37,430

 

543
00:21:37,430 --> 00:21:39,669

 

544
00:21:39,669 --> 00:21:41,669

 

545
00:21:41,669 --> 00:21:43,750

 

546
00:21:43,750 --> 00:21:45,110

 

547
00:21:45,110 --> 00:21:46,070

 

548
00:21:46,070 --> 00:21:48,310

 

549
00:21:48,310 --> 00:21:50,230

 

550
00:21:50,230 --> 00:21:54,070

 

551
00:21:54,070 --> 00:21:55,669

 

552
00:21:55,669 --> 00:21:58,549

 

553
00:21:58,549 --> 00:22:01,750

 

554
00:22:01,750 --> 00:22:03,990

 

555
00:22:03,990 --> 00:22:05,669

 

556
00:22:05,669 --> 00:22:07,190

 

557
00:22:07,190 --> 00:22:07,200

 

558
00:22:07,200 --> 00:22:07,510


559
00:22:07,510 --> 00:22:10,830

 

560
00:22:10,830 --> 00:22:10,840

 

561
00:22:10,840 --> 00:22:12,149


562
00:22:12,149 --> 00:22:16,630

 

563
00:22:16,630 --> 00:22:19,110

 

564
00:22:19,110 --> 00:22:20,230

 

565
00:22:20,230 --> 00:22:20,240

 

566
00:22:20,240 --> 00:22:21,270


567
00:22:21,270 --> 00:22:23,990

 

568
00:22:23,990 --> 00:22:25,590

 

569
00:22:25,590 --> 00:22:27,909

 

570
00:22:27,909 --> 00:22:27,919

 

571
00:22:27,919 --> 00:22:28,950


572
00:22:28,950 --> 00:22:30,710

 

573
00:22:30,710 --> 00:22:32,630

 

574
00:22:32,630 --> 00:22:34,390

 

575
00:22:34,390 --> 00:22:35,750

 

576
00:22:35,750 --> 00:22:35,760

 

577
00:22:35,760 --> 00:22:37,750


578
00:22:37,750 --> 00:22:39,110

 

579
00:22:39,110 --> 00:22:41,270

 

580
00:22:41,270 --> 00:22:42,630

 

581
00:22:42,630 --> 00:22:44,710

 

582
00:22:44,710 --> 00:22:46,230

 

583
00:22:46,230 --> 00:22:47,750

 

584
00:22:47,750 --> 00:22:49,430

 

585
00:22:49,430 --> 00:22:50,870

 

586
00:22:50,870 --> 00:22:52,230

 

587
00:22:52,230 --> 00:22:54,230

 

588
00:22:54,230 --> 00:22:58,789

 

589
00:22:58,789 --> 00:23:02,070

 

590
00:23:02,070 --> 00:23:03,909

 

591
00:23:03,909 --> 00:23:06,390

 

592
00:23:06,390 --> 00:23:08,390

 

593
00:23:08,390 --> 00:23:11,029

 

594
00:23:11,029 --> 00:23:12,470

 

595
00:23:12,470 --> 00:23:14,149

 

596
00:23:14,149 --> 00:23:16,549

 

597
00:23:16,549 --> 00:23:18,149

 

598
00:23:18,149 --> 00:23:20,310

 

599
00:23:20,310 --> 00:23:21,590

 

600
00:23:21,590 --> 00:23:23,590

 

601
00:23:23,590 --> 00:23:25,909

 

602
00:23:25,909 --> 00:23:27,110

 

603
00:23:27,110 --> 00:23:28,630

 

604
00:23:28,630 --> 00:23:31,830

 

605
00:23:31,830 --> 00:23:31,840

 

606
00:23:31,840 --> 00:23:34,390


607
00:23:34,390 --> 00:23:38,870

 

608
00:23:38,870 --> 00:23:41,269

 

609
00:23:41,269 --> 00:23:42,390

 

610
00:23:42,390 --> 00:23:45,430

 

611
00:23:45,430 --> 00:23:46,950

 

612
00:23:46,950 --> 00:23:48,549

 

613
00:23:48,549 --> 00:23:50,070

 

614
00:23:50,070 --> 00:23:51,990

 

615
00:23:51,990 --> 00:23:54,710

 

616
00:23:54,710 --> 00:23:56,149

 

617
00:23:56,149 --> 00:23:57,830

 

618
00:23:57,830 --> 00:23:59,510

 

619
00:23:59,510 --> 00:24:00,870

 

620
00:24:00,870 --> 00:24:02,870

 

621
00:24:02,870 --> 00:24:04,870

 

622
00:24:04,870 --> 00:24:07,350

 

623
00:24:07,350 --> 00:24:09,110

 

624
00:24:09,110 --> 00:24:10,549

 

625
00:24:10,549 --> 00:24:11,990

 

626
00:24:11,990 --> 00:24:14,549

 

627
00:24:14,549 --> 00:24:16,310

 

628
00:24:16,310 --> 00:24:18,870

 

629
00:24:18,870 --> 00:24:19,669

 

630
00:24:19,669 --> 00:24:23,430

 

631
00:24:23,430 --> 00:24:27,350

 

632
00:24:27,350 --> 00:24:30,549

 

633
00:24:30,549 --> 00:24:32,230

 

634
00:24:32,230 --> 00:24:32,240

 

635
00:24:32,240 --> 00:24:33,590


636
00:24:33,590 --> 00:24:36,390

 

637
00:24:36,390 --> 00:24:37,909

 

638
00:24:37,909 --> 00:24:39,430

 

639
00:24:39,430 --> 00:24:41,190

 

640
00:24:41,190 --> 00:24:42,390

 

641
00:24:42,390 --> 00:24:44,710

 

642
00:24:44,710 --> 00:24:47,750

 

643
00:24:47,750 --> 00:24:50,950

 

644
00:24:50,950 --> 00:24:53,430

 

645
00:24:53,430 --> 00:24:56,789

 

646
00:24:56,789 --> 00:24:56,799

 

647
00:24:56,799 --> 00:24:57,350


648
00:24:57,350 --> 00:24:59,190

 

649
00:24:59,190 --> 00:25:01,110

 

650
00:25:01,110 --> 00:25:03,669

 

651
00:25:03,669 --> 00:25:03,679

 

652
00:25:03,679 --> 00:25:04,390


653
00:25:04,390 --> 00:25:11,110

 

654
00:25:11,110 --> 00:25:14,230

 

655
00:25:14,230 --> 00:25:16,149

 

656
00:25:16,149 --> 00:25:19,990

 

657
00:25:19,990 --> 00:25:23,190

 

658
00:25:23,190 --> 00:25:25,190

 

659
00:25:25,190 --> 00:25:26,870

 

660
00:25:26,870 --> 00:25:28,710

 

661
00:25:28,710 --> 00:25:31,029

 

662
00:25:31,029 --> 00:25:34,070

 

663
00:25:34,070 --> 00:25:36,070

 

664
00:25:36,070 --> 00:25:37,430

 

665
00:25:37,430 --> 00:25:39,029

 

666
00:25:39,029 --> 00:25:40,310

 

667
00:25:40,310 --> 00:25:41,350

 

668
00:25:41,350 --> 00:25:44,310

 

669
00:25:44,310 --> 00:25:45,190

 

670
00:25:45,190 --> 00:25:48,630

 

671
00:25:48,630 --> 00:25:52,310

 

672
00:25:52,310 --> 00:25:53,510

 

673
00:25:53,510 --> 00:25:54,789

 

674
00:25:54,789 --> 00:25:56,070

 

675
00:25:56,070 --> 00:25:57,990

 

676
00:25:57,990 --> 00:25:59,590

 

677
00:25:59,590 --> 00:25:59,600

 

678
00:25:59,600 --> 00:26:00,630


679
00:26:00,630 --> 00:26:02,149

 

680
00:26:02,149 --> 00:26:03,830

 

681
00:26:03,830 --> 00:26:05,909

 

682
00:26:05,909 --> 00:26:07,750

 

683
00:26:07,750 --> 00:26:10,310

 

684
00:26:10,310 --> 00:26:13,190

 

685
00:26:13,190 --> 00:26:14,310

 

686
00:26:14,310 --> 00:26:17,029

 

687
00:26:17,029 --> 00:26:17,039

 

688
00:26:17,039 --> 00:26:17,830


689
00:26:17,830 --> 00:26:22,830

 

690
00:26:22,830 --> 00:26:22,840

 

691
00:26:22,840 --> 00:26:36,830


692
00:26:36,830 --> 00:26:36,840

 

693
00:26:36,840 --> 00:26:39,350


694
00:26:39,350 --> 00:26:41,590

 

695
00:26:41,590 --> 00:26:41,600

 

696
00:26:41,600 --> 00:26:42,789


697
00:26:42,789 --> 00:27:44,070

 

698
00:27:44,070 --> 00:27:45,669

 

699
00:27:45,669 --> 00:27:47,110

 

700
00:27:47,110 --> 00:27:48,230

 

701
00:27:48,230 --> 00:27:49,909

 

702
00:27:49,909 --> 00:27:51,430

 

703
00:27:51,430 --> 00:27:52,230

 

704
00:27:52,230 --> 00:27:53,750

 

705
00:27:53,750 --> 00:27:56,070

 

706
00:27:56,070 --> 00:27:57,269

 

707
00:27:57,269 --> 00:27:58,470

 

708
00:27:58,470 --> 00:28:00,070

 

709
00:28:00,070 --> 00:28:01,669

 

710
00:28:01,669 --> 00:28:02,950

 

711
00:28:02,950 --> 00:28:05,029

 

712
00:28:05,029 --> 00:28:07,110

 

713
00:28:07,110 --> 00:28:10,310

 

714
00:28:10,310 --> 00:28:13,350

 

715
00:28:13,350 --> 00:28:15,510

 

716
00:28:15,510 --> 00:28:16,870

 

717
00:28:16,870 --> 00:28:18,470

 

718
00:28:18,470 --> 00:28:19,909

 

719
00:28:19,909 --> 00:28:20,789

 

720
00:28:20,789 --> 00:28:24,070

 

721
00:28:24,070 --> 00:28:25,430

 

722
00:28:25,430 --> 00:28:27,110

 

723
00:28:27,110 --> 00:28:28,470

 

724
00:28:28,470 --> 00:28:30,389

 

725
00:28:30,389 --> 00:28:32,389

 

726
00:28:32,389 --> 00:28:35,590

 

727
00:28:35,590 --> 00:28:37,669

 

728
00:28:37,669 --> 00:28:38,950

 

729
00:28:38,950 --> 00:28:42,230

 

730
00:28:42,230 --> 00:28:44,870

 

731
00:28:44,870 --> 00:28:45,430

 

732
00:28:45,430 --> 00:28:48,310

 

733
00:28:48,310 --> 00:28:49,830

 

734
00:28:49,830 --> 00:28:51,510

 

735
00:28:51,510 --> 00:28:53,590

 

736
00:28:53,590 --> 00:28:55,750

 

737
00:28:55,750 --> 00:28:57,110

 

738
00:28:57,110 --> 00:28:59,510

 

739
00:28:59,510 --> 00:29:01,269

 

740
00:29:01,269 --> 00:29:03,430

 

741
00:29:03,430 --> 00:29:04,549

 

742
00:29:04,549 --> 00:29:06,789

 

743
00:29:06,789 --> 00:29:07,830

 

744
00:29:07,830 --> 00:29:11,029

 

745
00:29:11,029 --> 00:29:12,549

 

746
00:29:12,549 --> 00:29:14,470

 

747
00:29:14,470 --> 00:29:16,070

 

748
00:29:16,070 --> 00:29:16,710

 

749
00:29:16,710 --> 00:29:19,110

 

750
00:29:19,110 --> 00:29:19,120

 

751
00:29:19,120 --> 00:29:20,310


752
00:29:20,310 --> 00:29:22,149

 

753
00:29:22,149 --> 00:29:23,190

 

754
00:29:23,190 --> 00:29:24,549

 

755
00:29:24,549 --> 00:29:25,750

 

756
00:29:25,750 --> 00:29:27,590

 

757
00:29:27,590 --> 00:29:27,600

 

758
00:29:27,600 --> 00:29:28,870


759
00:29:28,870 --> 00:29:31,269

 

760
00:29:31,269 --> 00:29:34,149

 

761
00:29:34,149 --> 00:29:35,669

 

762
00:29:35,669 --> 00:29:37,430

 

763
00:29:37,430 --> 00:29:40,710

 

764
00:29:40,710 --> 00:29:43,430

 

765
00:29:43,430 --> 00:29:44,149

 

766
00:29:44,149 --> 00:29:45,830

 

767
00:29:45,830 --> 00:29:47,750

 

768
00:29:47,750 --> 00:29:49,990

 

769
00:29:49,990 --> 00:29:50,950

 

770
00:29:50,950 --> 00:29:53,669

 

771
00:29:53,669 --> 00:29:55,430

 

772
00:29:55,430 --> 00:29:56,789

 

773
00:29:56,789 --> 00:29:57,990

 

774
00:29:57,990 --> 00:29:59,830

 

775
00:29:59,830 --> 00:30:02,230

 

776
00:30:02,230 --> 00:30:05,269

 

777
00:30:05,269 --> 00:30:05,279

 

778
00:30:05,279 --> 00:30:05,830


779
00:30:05,830 --> 00:30:07,350

 

780
00:30:07,350 --> 00:30:10,310

 

781
00:30:10,310 --> 00:30:12,870

 

782
00:30:12,870 --> 00:30:14,470

 

783
00:30:14,470 --> 00:30:15,830

 

784
00:30:15,830 --> 00:30:17,430

 

785
00:30:17,430 --> 00:30:17,440

 

786
00:30:17,440 --> 00:30:18,310


787
00:30:18,310 --> 00:30:21,350

 

788
00:30:21,350 --> 00:30:23,269

 

789
00:30:23,269 --> 00:30:24,070

 

790
00:30:24,070 --> 00:30:26,710

 

791
00:30:26,710 --> 00:30:28,149

 

792
00:30:28,149 --> 00:30:31,510

 

793
00:30:31,510 --> 00:30:33,029

 

794
00:30:33,029 --> 00:30:34,070

 

795
00:30:34,070 --> 00:30:35,430

 

796
00:30:35,430 --> 00:30:37,190

 

797
00:30:37,190 --> 00:30:38,710

 

798
00:30:38,710 --> 00:30:40,389

 

799
00:30:40,389 --> 00:30:43,669

 

800
00:30:43,669 --> 00:30:44,870

 

801
00:30:44,870 --> 00:30:46,310

 

802
00:30:46,310 --> 00:30:49,190

 

803
00:30:49,190 --> 00:30:52,470

 

804
00:30:52,470 --> 00:30:54,950

 

805
00:30:54,950 --> 00:30:55,510

 

806
00:30:55,510 --> 00:30:57,669

 

807
00:30:57,669 --> 00:31:00,950

 

808
00:31:00,950 --> 00:31:02,950

 

809
00:31:02,950 --> 00:31:04,230

 

810
00:31:04,230 --> 00:31:07,269

 

811
00:31:07,269 --> 00:31:09,110

 

812
00:31:09,110 --> 00:31:09,120

 

813
00:31:09,120 --> 00:31:10,230


814
00:31:10,230 --> 00:31:14,070

 

815
00:31:14,070 --> 00:31:15,909

 

816
00:31:15,909 --> 00:31:17,269

 

817
00:31:17,269 --> 00:31:19,430

 

818
00:31:19,430 --> 00:31:22,630

 

819
00:31:22,630 --> 00:31:22,640

 

820
00:31:22,640 --> 00:31:23,750


821
00:31:23,750 --> 00:31:26,630

 

822
00:31:26,630 --> 00:31:28,389

 

823
00:31:28,389 --> 00:31:31,430

 

824
00:31:31,430 --> 00:31:31,990

 

825
00:31:31,990 --> 00:31:34,310

 

826
00:31:34,310 --> 00:31:38,070

 

827
00:31:38,070 --> 00:31:42,470

 

828
00:31:42,470 --> 00:31:44,070

 

829
00:31:44,070 --> 00:31:46,470

 

830
00:31:46,470 --> 00:31:48,950

 

831
00:31:48,950 --> 00:31:51,509

 

832
00:31:51,509 --> 00:31:53,350

 

833
00:31:53,350 --> 00:31:55,430

 

834
00:31:55,430 --> 00:31:59,269

 

835
00:31:59,269 --> 00:32:03,269

 

836
00:32:03,269 --> 00:32:06,389

 

837
00:32:06,389 --> 00:32:11,110

 

838
00:32:11,110 --> 00:32:15,430

 

839
00:32:15,430 --> 00:32:18,950

 

840
00:32:18,950 --> 00:32:18,960

 

841
00:32:18,960 --> 00:32:19,590


842
00:32:19,590 --> 00:32:22,549

 

843
00:32:22,549 --> 00:32:23,990

 

844
00:32:23,990 --> 00:32:25,830

 

845
00:32:25,830 --> 00:32:27,269

 

846
00:32:27,269 --> 00:32:29,750

 

847
00:32:29,750 --> 00:32:31,509

 

848
00:32:31,509 --> 00:32:32,630

 

849
00:32:32,630 --> 00:32:33,590

 

850
00:32:33,590 --> 00:32:36,389

 

851
00:32:36,389 --> 00:32:38,149

 

852
00:32:38,149 --> 00:32:39,430

 

853
00:32:39,430 --> 00:32:43,669

 

854
00:32:43,669 --> 00:32:46,710

 

855
00:32:46,710 --> 00:32:46,720

 

856
00:32:46,720 --> 00:32:47,830


857
00:32:47,830 --> 00:32:49,269

 

858
00:32:49,269 --> 00:32:51,269

 

859
00:32:51,269 --> 00:32:53,029

 

860
00:32:53,029 --> 00:32:54,310

 

861
00:32:54,310 --> 00:32:58,389

 

862
00:32:58,389 --> 00:33:00,710

 

863
00:33:00,710 --> 00:33:02,389

 

864
00:33:02,389 --> 00:33:04,389

 

865
00:33:04,389 --> 00:33:05,750

 

866
00:33:05,750 --> 00:33:05,760

 

867
00:33:05,760 --> 00:33:06,789


868
00:33:06,789 --> 00:33:09,110

 

869
00:33:09,110 --> 00:33:10,630

 

870
00:33:10,630 --> 00:33:12,310

 

871
00:33:12,310 --> 00:33:13,590

 

872
00:33:13,590 --> 00:33:15,590

 

873
00:33:15,590 --> 00:33:18,070

 

874
00:33:18,070 --> 00:33:19,909

 

875
00:33:19,909 --> 00:33:21,430

 

876
00:33:21,430 --> 00:33:24,310

 

877
00:33:24,310 --> 00:33:25,430

 

878
00:33:25,430 --> 00:33:27,430

 

879
00:33:27,430 --> 00:33:28,789

 

880
00:33:28,789 --> 00:33:30,070

 

881
00:33:30,070 --> 00:33:30,080

 

882
00:33:30,080 --> 00:33:31,269


883
00:33:31,269 --> 00:33:33,590

 

884
00:33:33,590 --> 00:33:35,350

 

885
00:33:35,350 --> 00:33:37,190

 

886
00:33:37,190 --> 00:33:39,029

 

887
00:33:39,029 --> 00:33:40,230

 

888
00:33:40,230 --> 00:33:40,240

 

889
00:33:40,240 --> 00:33:40,870


890
00:33:40,870 --> 00:33:43,430

 

891
00:33:43,430 --> 00:33:45,269

 

892
00:33:45,269 --> 00:33:45,909

 

893
00:33:45,909 --> 00:33:47,350

 

894
00:33:47,350 --> 00:33:48,950

 

895
00:33:48,950 --> 00:33:50,470

 

896
00:33:50,470 --> 00:33:52,710

 

897
00:33:52,710 --> 00:33:53,509

 

898
00:33:53,509 --> 00:33:56,710

 

899
00:33:56,710 --> 00:34:00,789

 

900
00:34:00,789 --> 00:34:02,950

 

901
00:34:02,950 --> 00:34:04,830

 

902
00:34:04,830 --> 00:34:04,840

 

903
00:34:04,840 --> 00:34:06,230


904
00:34:06,230 --> 00:34:08,950

 

905
00:34:08,950 --> 00:34:10,629

 

906
00:34:10,629 --> 00:34:10,639

 

907
00:34:10,639 --> 00:34:11,750


908
00:34:11,750 --> 00:34:14,629

 

909
00:34:14,629 --> 00:34:16,869

 

910
00:34:16,869 --> 00:34:18,149

 

911
00:34:18,149 --> 00:34:21,109

 

912
00:34:21,109 --> 00:34:24,149

 

913
00:34:24,149 --> 00:34:26,310

 

914
00:34:26,310 --> 00:34:27,990

 

915
00:34:27,990 --> 00:34:31,510

 

916
00:34:31,510 --> 00:34:34,230

 

917
00:34:34,230 --> 00:34:36,389

 

918
00:34:36,389 --> 00:34:39,669

 

919
00:34:39,669 --> 00:34:43,510

 

920
00:34:43,510 --> 00:34:46,829

 

921
00:34:46,829 --> 00:34:46,839

 

922
00:34:46,839 --> 00:34:47,990


923
00:34:47,990 --> 00:34:50,790

 

924
00:34:50,790 --> 00:34:52,310

 

925
00:34:52,310 --> 00:34:52,320

 

926
00:34:52,320 --> 00:34:53,030


927
00:34:53,030 --> 00:34:55,270

 

928
00:34:55,270 --> 00:34:56,310

 

929
00:34:56,310 --> 00:34:57,430

 

930
00:34:57,430 --> 00:34:58,950

 

931
00:34:58,950 --> 00:34:58,960

 

932
00:34:58,960 --> 00:34:59,589


933
00:34:59,589 --> 00:35:01,190

 

934
00:35:01,190 --> 00:35:03,670

 

935
00:35:03,670 --> 00:35:07,109

 

936
00:35:07,109 --> 00:35:09,030

 

937
00:35:09,030 --> 00:35:10,230

 

938
00:35:10,230 --> 00:35:12,790

 

939
00:35:12,790 --> 00:35:13,670

 

940
00:35:13,670 --> 00:35:15,270

 

941
00:35:15,270 --> 00:35:17,750

 

942
00:35:17,750 --> 00:35:17,760

 

943
00:35:17,760 --> 00:35:18,310


944
00:35:18,310 --> 00:35:20,390

 

945
00:35:20,390 --> 00:35:22,230

 

946
00:35:22,230 --> 00:35:23,430

 

947
00:35:23,430 --> 00:35:25,510

 

948
00:35:25,510 --> 00:35:26,870

 

949
00:35:26,870 --> 00:35:29,589

 

950
00:35:29,589 --> 00:35:30,950

 

951
00:35:30,950 --> 00:35:34,550

 

952
00:35:34,550 --> 00:35:36,310

 

953
00:35:36,310 --> 00:35:37,990

 

954
00:35:37,990 --> 00:35:40,390

 

955
00:35:40,390 --> 00:35:44,710

 

956
00:35:44,710 --> 00:35:46,950

 

957
00:35:46,950 --> 00:35:48,710

 

958
00:35:48,710 --> 00:35:48,720

 

959
00:35:48,720 --> 00:35:49,750


960
00:35:49,750 --> 00:35:52,150

 

961
00:35:52,150 --> 00:35:52,160

 

962
00:35:52,160 --> 00:35:53,430


963
00:35:53,430 --> 00:35:55,910

 

964
00:35:55,910 --> 00:35:57,109

 

965
00:35:57,109 --> 00:35:59,349

 

966
00:35:59,349 --> 00:36:04,150

 

967
00:36:04,150 --> 00:36:05,829

 

968
00:36:05,829 --> 00:36:07,270

 

969
00:36:07,270 --> 00:36:10,470

 

970
00:36:10,470 --> 00:36:12,470

 

971
00:36:12,470 --> 00:36:15,030

 

972
00:36:15,030 --> 00:36:15,040

 

973
00:36:15,040 --> 00:36:15,990


974
00:36:15,990 --> 00:36:17,349

 

975
00:36:17,349 --> 00:36:19,910

 

976
00:36:19,910 --> 00:36:20,470

 

977
00:36:20,470 --> 00:36:23,829

 

978
00:36:23,829 --> 00:36:26,230

 

979
00:36:26,230 --> 00:36:28,230

 

980
00:36:28,230 --> 00:36:31,109

 

981
00:36:31,109 --> 00:36:31,119

 

982
00:36:31,119 --> 00:36:31,670


983
00:36:31,670 --> 00:36:32,710

 

984
00:36:32,710 --> 00:36:35,589

 

985
00:36:35,589 --> 00:36:37,510

 

986
00:36:37,510 --> 00:36:39,030

 

987
00:36:39,030 --> 00:36:43,270

 

988
00:36:43,270 --> 00:36:47,109

 

989
00:36:47,109 --> 00:36:48,790

 

990
00:36:48,790 --> 00:36:51,030

 

991
00:36:51,030 --> 00:36:52,950

 

992
00:36:52,950 --> 00:36:54,390

 

993
00:36:54,390 --> 00:36:56,230

 

994
00:36:56,230 --> 00:36:59,349

 

995
00:36:59,349 --> 00:37:02,470

 

996
00:37:02,470 --> 00:37:03,030

 

997
00:37:03,030 --> 00:37:06,230

 

998
00:37:06,230 --> 00:37:08,390

 

999
00:37:08,390 --> 00:37:08,400

 

1000
00:37:08,400 --> 00:37:09,270


1001
00:37:09,270 --> 00:37:11,510

 

1002
00:37:11,510 --> 00:37:13,589

 

1003
00:37:13,589 --> 00:37:15,910

 

1004
00:37:15,910 --> 00:37:16,870

 

1005
00:37:16,870 --> 00:37:19,430

 

1006
00:37:19,430 --> 00:37:20,230

 

1007
00:37:20,230 --> 00:37:24,310

 

1008
00:37:24,310 --> 00:37:27,990

 

1009
00:37:27,990 --> 00:37:28,000

 

1010
00:37:28,000 --> 00:37:28,710


1011
00:37:28,710 --> 00:37:30,069

 

1012
00:37:30,069 --> 00:37:37,270

 

1013
00:37:37,270 --> 00:37:41,910

 

1014
00:37:41,910 --> 00:37:44,069

 

1015
00:37:44,069 --> 00:37:45,349

 

1016
00:37:45,349 --> 00:37:48,230

 

1017
00:37:48,230 --> 00:37:49,589

 

1018
00:37:49,589 --> 00:37:51,270

 

1019
00:37:51,270 --> 00:37:52,310

 

1020
00:37:52,310 --> 00:37:55,430

 

1021
00:37:55,430 --> 00:37:56,710

 

1022
00:37:56,710 --> 00:37:59,190

 

1023
00:37:59,190 --> 00:38:01,910

 

1024
00:38:01,910 --> 00:38:05,109

 

1025
00:38:05,109 --> 00:38:08,870

 

1026
00:38:08,870 --> 00:38:11,109

 

1027
00:38:11,109 --> 00:38:11,119

 

1028
00:38:11,119 --> 00:38:12,230


1029
00:38:12,230 --> 00:38:15,430

 

1030
00:38:15,430 --> 00:38:17,430

 

1031
00:38:17,430 --> 00:38:20,390

 

1032
00:38:20,390 --> 00:38:22,470

 

1033
00:38:22,470 --> 00:38:24,069

 

1034
00:38:24,069 --> 00:38:25,190

 

1035
00:38:25,190 --> 00:38:27,109

 

1036
00:38:27,109 --> 00:38:30,630

 

1037
00:38:30,630 --> 00:38:35,750

 

1038
00:38:35,750 --> 00:38:36,950

 

1039
00:38:36,950 --> 00:38:38,470

 

1040
00:38:38,470 --> 00:38:40,069

 

1041
00:38:40,069 --> 00:38:43,109

 

1042
00:38:43,109 --> 00:38:45,030

 

1043
00:38:45,030 --> 00:38:45,040

 

1044
00:38:45,040 --> 00:38:45,349


1045
00:38:45,349 --> 00:38:46,470

 

1046
00:38:46,470 --> 00:38:48,230

 

1047
00:38:48,230 --> 00:38:48,240

 

1048
00:38:48,240 --> 00:38:49,030


1049
00:38:49,030 --> 00:38:51,670

 

1050
00:38:51,670 --> 00:38:52,390

 

1051
00:38:52,390 --> 00:38:54,470

 

1052
00:38:54,470 --> 00:38:56,790

 

1053
00:38:56,790 --> 00:38:57,829

 

1054
00:38:57,829 --> 00:39:01,829

 

1055
00:39:01,829 --> 00:39:05,270

 

1056
00:39:05,270 --> 00:39:06,630

 

1057
00:39:06,630 --> 00:39:12,870

 

1058
00:39:12,870 --> 00:39:14,390

 

1059
00:39:14,390 --> 00:39:15,750

 

1060
00:39:15,750 --> 00:39:17,589

 

1061
00:39:17,589 --> 00:39:18,950

 

1062
00:39:18,950 --> 00:39:18,960

 

1063
00:39:18,960 --> 00:39:19,750


1064
00:39:19,750 --> 00:39:21,829

 

1065
00:39:21,829 --> 00:39:21,839

 

1066
00:39:21,839 --> 00:39:22,790


1067
00:39:22,790 --> 00:39:25,829

 

1068
00:39:25,829 --> 00:39:26,630

 

1069
00:39:26,630 --> 00:39:28,310

 

1070
00:39:28,310 --> 00:39:29,910

 

1071
00:39:29,910 --> 00:39:32,310

 

1072
00:39:32,310 --> 00:39:33,430

 

1073
00:39:33,430 --> 00:39:34,550

 

1074
00:39:34,550 --> 00:39:36,870

 

1075
00:39:36,870 --> 00:39:42,870

 

1076
00:39:42,870 --> 00:39:44,150

 

1077
00:39:44,150 --> 00:39:45,510

 

1078
00:39:45,510 --> 00:39:45,520

 

1079
00:39:45,520 --> 00:39:46,390


1080
00:39:46,390 --> 00:39:50,310

 

1081
00:39:50,310 --> 00:39:51,829

 

1082
00:39:51,829 --> 00:39:52,950

 

1083
00:39:52,950 --> 00:39:54,790

 

1084
00:39:54,790 --> 00:39:55,670

 

1085
00:39:55,670 --> 00:39:58,230

 

1086
00:39:58,230 --> 00:39:58,240

 

1087
00:39:58,240 --> 00:40:00,310


1088
00:40:00,310 --> 00:40:04,150

 

1089
00:40:04,150 --> 00:40:05,910

 

1090
00:40:05,910 --> 00:40:08,870

 

1091
00:40:08,870 --> 00:40:10,390

 

1092
00:40:10,390 --> 00:40:12,150

 

1093
00:40:12,150 --> 00:40:13,589

 

1094
00:40:13,589 --> 00:40:13,599

 

1095
00:40:13,599 --> 00:40:14,069


1096
00:40:14,069 --> 00:40:15,589

 

1097
00:40:15,589 --> 00:40:17,589

 

1098
00:40:17,589 --> 00:40:21,349

 

1099
00:40:21,349 --> 00:40:23,030

 

1100
00:40:23,030 --> 00:40:25,430

 

1101
00:40:25,430 --> 00:40:28,470

 

1102
00:40:28,470 --> 00:40:29,510

 

1103
00:40:29,510 --> 00:40:32,710

 

1104
00:40:32,710 --> 00:40:35,270

 

1105
00:40:35,270 --> 00:40:36,950

 

1106
00:40:36,950 --> 00:40:40,390

 

1107
00:40:40,390 --> 00:40:41,990

 

1108
00:40:41,990 --> 00:40:42,000

 

1109
00:40:42,000 --> 00:40:44,150


1110
00:40:44,150 --> 00:40:46,630

 

1111
00:40:46,630 --> 00:40:49,030

 

1112
00:40:49,030 --> 00:40:53,109

 

1113
00:40:53,109 --> 00:40:55,430

 

1114
00:40:55,430 --> 00:40:56,550

 

1115
00:40:56,550 --> 00:40:59,270

 

1116
00:40:59,270 --> 00:41:00,950

 

1117
00:41:00,950 --> 00:41:02,390

 

1118
00:41:02,390 --> 00:41:04,550

 

1119
00:41:04,550 --> 00:41:06,309

 

1120
00:41:06,309 --> 00:41:07,829

 

1121
00:41:07,829 --> 00:41:09,510

 

1122
00:41:09,510 --> 00:41:12,550

 

1123
00:41:12,550 --> 00:41:16,069

 

1124
00:41:16,069 --> 00:41:18,069

 

1125
00:41:18,069 --> 00:41:18,079

 

1126
00:41:18,079 --> 00:41:19,270


1127
00:41:19,270 --> 00:41:22,069

 

1128
00:41:22,069 --> 00:41:24,390

 

1129
00:41:24,390 --> 00:41:26,390

 

1130
00:41:26,390 --> 00:41:28,150

 

1131
00:41:28,150 --> 00:41:29,430

 

1132
00:41:29,430 --> 00:41:31,430

 

1133
00:41:31,430 --> 00:41:33,670

 

1134
00:41:33,670 --> 00:41:33,680

 

1135
00:41:33,680 --> 00:41:34,470


1136
00:41:34,470 --> 00:41:35,910

 

1137
00:41:35,910 --> 00:41:37,510

 

1138
00:41:37,510 --> 00:41:39,510

 

1139
00:41:39,510 --> 00:41:42,390

 

1140
00:41:42,390 --> 00:41:45,109

 

1141
00:41:45,109 --> 00:41:45,119

 

1142
00:41:45,119 --> 00:41:45,589


1143
00:41:45,589 --> 00:41:47,510

 

1144
00:41:47,510 --> 00:41:49,190

 

1145
00:41:49,190 --> 00:41:50,630

 

1146
00:41:50,630 --> 00:41:52,950

 

1147
00:41:52,950 --> 00:41:54,550

 

1148
00:41:54,550 --> 00:41:56,790

 

1149
00:41:56,790 --> 00:41:57,990

 

1150
00:41:57,990 --> 00:41:58,630

 

1151
00:41:58,630 --> 00:42:01,349

 

1152
00:42:01,349 --> 00:42:02,550

 

1153
00:42:02,550 --> 00:42:03,829

 

1154
00:42:03,829 --> 00:42:06,470

 

1155
00:42:06,470 --> 00:42:06,480

 

1156
00:42:06,480 --> 00:42:09,030


1157
00:42:09,030 --> 00:42:11,829

 

1158
00:42:11,829 --> 00:42:11,839

 

1159
00:42:11,839 --> 00:42:13,030


1160
00:42:13,030 --> 00:42:14,630

 

1161
00:42:14,630 --> 00:42:16,630

 

1162
00:42:16,630 --> 00:42:19,270

 

1163
00:42:19,270 --> 00:42:20,870

 

1164
00:42:20,870 --> 00:42:24,550

 

1165
00:42:24,550 --> 00:42:27,589

 

1166
00:42:27,589 --> 00:42:29,589

 

1167
00:42:29,589 --> 00:42:31,589

 

1168
00:42:31,589 --> 00:42:32,950

 

1169
00:42:32,950 --> 00:42:35,510

 

1170
00:42:35,510 --> 00:42:38,790

 

1171
00:42:38,790 --> 00:42:40,550

 

1172
00:42:40,550 --> 00:42:43,030

 

1173
00:42:43,030 --> 00:42:45,349

 

1174
00:42:45,349 --> 00:42:46,790

 

1175
00:42:46,790 --> 00:42:48,069

 

1176
00:42:48,069 --> 00:42:51,109

 

1177
00:42:51,109 --> 00:42:54,390

 

1178
00:42:54,390 --> 00:42:57,030

 

1179
00:42:57,030 --> 00:42:58,790

 

1180
00:42:58,790 --> 00:43:01,109

 

1181
00:43:01,109 --> 00:43:03,109

 

1182
00:43:03,109 --> 00:43:04,870

 

1183
00:43:04,870 --> 00:43:08,069

 

1184
00:43:08,069 --> 00:43:09,349

 

1185
00:43:09,349 --> 00:43:10,630

 

1186
00:43:10,630 --> 00:43:12,150

 

1187
00:43:12,150 --> 00:43:15,190

 

1188
00:43:15,190 --> 00:43:18,470

 

1189
00:43:18,470 --> 00:43:21,829

 

1190
00:43:21,829 --> 00:43:23,349

 

1191
00:43:23,349 --> 00:43:24,470

 

1192
00:43:24,470 --> 00:43:25,990

 

1193
00:43:25,990 --> 00:43:29,510

 

1194
00:43:29,510 --> 00:43:31,030

 

1195
00:43:31,030 --> 00:43:33,190

 

1196
00:43:33,190 --> 00:43:35,910

 

1197
00:43:35,910 --> 00:43:36,790

 

1198
00:43:36,790 --> 00:43:39,190

 

1199
00:43:39,190 --> 00:43:42,069

 

1200
00:43:42,069 --> 00:43:43,910

 

1201
00:43:43,910 --> 00:43:46,150

 

1202
00:43:46,150 --> 00:43:48,309

 

1203
00:43:48,309 --> 00:43:49,750

 

1204
00:43:49,750 --> 00:43:53,670

 

1205
00:43:53,670 --> 00:43:56,150

 

1206
00:43:56,150 --> 00:43:57,190

 

1207
00:43:57,190 --> 00:44:00,069

 

1208
00:44:00,069 --> 00:44:01,910

 

1209
00:44:01,910 --> 00:44:01,920

 

1210
00:44:01,920 --> 00:44:02,630


1211
00:44:02,630 --> 00:44:05,910

 

1212
00:44:05,910 --> 00:44:08,790

 

1213
00:44:08,790 --> 00:44:12,230

 

1214
00:44:12,230 --> 00:44:15,109

 

1215
00:44:15,109 --> 00:44:16,870

 

1216
00:44:16,870 --> 00:44:17,270

 

1217
00:44:17,270 --> 00:44:20,470

 

1218
00:44:20,470 --> 00:44:21,829

 

1219
00:44:21,829 --> 00:44:23,270

 

1220
00:44:23,270 --> 00:44:23,280

 

1221
00:44:23,280 --> 00:44:23,990


1222
00:44:23,990 --> 00:44:25,349

 

1223
00:44:25,349 --> 00:44:26,630

 

1224
00:44:26,630 --> 00:44:26,640

 

1225
00:44:26,640 --> 00:44:27,670


1226
00:44:27,670 --> 00:44:29,750

 

1227
00:44:29,750 --> 00:44:30,950

 

1228
00:44:30,950 --> 00:44:30,960

 

1229
00:44:30,960 --> 00:44:31,829


1230
00:44:31,829 --> 00:44:33,109

 

1231
00:44:33,109 --> 00:44:34,790

 

1232
00:44:34,790 --> 00:44:36,550

 

1233
00:44:36,550 --> 00:44:38,390

 

1234
00:44:38,390 --> 00:44:40,150

 

1235
00:44:40,150 --> 00:44:41,190

 

1236
00:44:41,190 --> 00:44:43,109

 

1237
00:44:43,109 --> 00:44:44,870

 

1238
00:44:44,870 --> 00:44:47,109

 

1239
00:44:47,109 --> 00:44:48,950

 

1240
00:44:48,950 --> 00:44:51,829

 

1241
00:44:51,829 --> 00:44:52,790

 

1242
00:44:52,790 --> 00:44:54,550

 

1243
00:44:54,550 --> 00:44:56,790

 

1244
00:44:56,790 --> 00:44:58,470

 

1245
00:44:58,470 --> 00:44:59,829

 

1246
00:44:59,829 --> 00:45:01,349

 

1247
00:45:01,349 --> 00:45:03,190

 

1248
00:45:03,190 --> 00:45:04,550

 

1249
00:45:04,550 --> 00:45:07,190

 

1250
00:45:07,190 --> 00:45:08,950

 

1251
00:45:08,950 --> 00:45:11,589

 

1252
00:45:11,589 --> 00:45:13,430

 

1253
00:45:13,430 --> 00:45:15,589

 

1254
00:45:15,589 --> 00:45:15,599

 

1255
00:45:15,599 --> 00:45:16,230


1256
00:45:16,230 --> 00:45:19,510

 

1257
00:45:19,510 --> 00:45:21,349

 

1258
00:45:21,349 --> 00:45:22,710

 

1259
00:45:22,710 --> 00:45:25,270

 

1260
00:45:25,270 --> 00:45:26,710

 

1261
00:45:26,710 --> 00:45:26,720

 

1262
00:45:26,720 --> 00:45:27,109


1263
00:45:27,109 --> 00:45:28,790

 

1264
00:45:28,790 --> 00:45:31,030

 

1265
00:45:31,030 --> 00:45:33,750

 

1266
00:45:33,750 --> 00:45:35,030

 

1267
00:45:35,030 --> 00:45:35,040

 

1268
00:45:35,040 --> 00:45:36,069


1269
00:45:36,069 --> 00:45:38,069

 

1270
00:45:38,069 --> 00:45:39,990

 

1271
00:45:39,990 --> 00:45:43,270

 

1272
00:45:43,270 --> 00:45:44,150

 

1273
00:45:44,150 --> 00:45:45,910

 

1274
00:45:45,910 --> 00:45:47,910

 

1275
00:45:47,910 --> 00:45:49,670

 

1276
00:45:49,670 --> 00:45:51,510

 

1277
00:45:51,510 --> 00:45:52,870

 

1278
00:45:52,870 --> 00:45:54,390

 

1279
00:45:54,390 --> 00:45:56,230

 

1280
00:45:56,230 --> 00:45:56,240

 

1281
00:45:56,240 --> 00:45:57,109


1282
00:45:57,109 --> 00:45:58,550

 

1283
00:45:58,550 --> 00:46:00,710

 

1284
00:46:00,710 --> 00:46:01,670

 

1285
00:46:01,670 --> 00:46:03,589

 

1286
00:46:03,589 --> 00:46:05,349

 

1287
00:46:05,349 --> 00:46:07,589

 

1288
00:46:07,589 --> 00:46:08,630

 

1289
00:46:08,630 --> 00:46:13,349

 

1290
00:46:13,349 --> 00:46:16,630

 

1291
00:46:16,630 --> 00:46:18,150

 

1292
00:46:18,150 --> 00:46:19,270

 

1293
00:46:19,270 --> 00:46:20,790

 

1294
00:46:20,790 --> 00:46:24,230

 

1295
00:46:24,230 --> 00:46:25,510

 

1296
00:46:25,510 --> 00:46:27,670

 

1297
00:46:27,670 --> 00:46:29,910

 

1298
00:46:29,910 --> 00:46:29,920

 

1299
00:46:29,920 --> 00:46:31,190


1300
00:46:31,190 --> 00:46:32,870

 

1301
00:46:32,870 --> 00:46:35,190

 

1302
00:46:35,190 --> 00:46:37,510

 

1303
00:46:37,510 --> 00:46:38,630

 

1304
00:46:38,630 --> 00:47:07,510

 

1305
00:47:07,510 --> 00:47:11,589

 

1306
00:47:11,589 --> 00:47:12,950

 

1307
00:47:12,950 --> 00:47:15,030

 

1308
00:47:15,030 --> 00:47:15,040

 

1309
00:47:15,040 --> 00:47:18,550


1310
00:47:18,550 --> 00:47:22,150

 

1311
00:47:22,150 --> 00:47:23,990

 

1312
00:47:23,990 --> 00:47:25,430

 

1313
00:47:25,430 --> 00:47:26,630

 

1314
00:47:26,630 --> 00:47:28,470

 

1315
00:47:28,470 --> 00:47:29,750

 

1316
00:47:29,750 --> 00:47:32,470

 

1317
00:47:32,470 --> 00:47:36,829

 

1318
00:47:36,829 --> 00:47:36,839

 

1319
00:47:36,839 --> 00:47:40,069


1320
00:47:40,069 --> 00:47:44,470

 

1321
00:47:44,470 --> 00:47:48,470

 

1322
00:47:48,470 --> 00:47:50,790

 

1323
00:47:50,790 --> 00:47:50,800

 

1324
00:47:50,800 --> 00:47:51,670


1325
00:47:51,670 --> 00:47:53,589

 

1326
00:47:53,589 --> 00:47:54,870

 

1327
00:47:54,870 --> 00:47:56,230

 

1328
00:47:56,230 --> 00:47:57,829

 

1329
00:47:57,829 --> 00:47:59,750

 

1330
00:47:59,750 --> 00:48:01,190

 

1331
00:48:01,190 --> 00:48:03,349

 

1332
00:48:03,349 --> 00:48:04,630

 

1333
00:48:04,630 --> 00:48:08,549

 

1334
00:48:08,549 --> 00:48:08,559

 

1335
00:48:08,559 --> 00:48:12,069


1336
00:48:12,069 --> 00:48:17,750

 

1337
00:48:17,750 --> 00:48:24,230

 

1338
00:48:24,230 --> 00:48:25,510

 

1339
00:48:25,510 --> 00:48:26,790

 

1340
00:48:26,790 --> 00:48:27,910

 

1341
00:48:27,910 --> 00:48:30,230

 

1342
00:48:30,230 --> 00:48:33,349

 

1343
00:48:33,349 --> 00:48:36,470

 

1344
00:48:36,470 --> 00:48:38,390

 

1345
00:48:38,390 --> 00:48:40,309

 

1346
00:48:40,309 --> 00:48:43,030

 

1347
00:48:43,030 --> 00:48:44,470

 

1348
00:48:44,470 --> 00:48:49,430

 

1349
00:48:49,430 --> 00:48:51,349

 

1350
00:48:51,349 --> 00:48:53,190

 

1351
00:48:53,190 --> 00:48:57,670

 

1352
00:48:57,670 --> 00:49:05,430

 

1353
00:49:05,430 --> 00:49:06,950

 

1354
00:49:06,950 --> 00:49:09,510

 

1355
00:49:09,510 --> 00:49:13,430

 

1356
00:49:13,430 --> 00:49:16,870

 

1357
00:49:16,870 --> 00:49:16,880

 

1358
00:49:16,880 --> 00:49:17,990


1359
00:49:17,990 --> 00:49:20,150

 

1360
00:49:20,150 --> 00:49:21,829

 

1361
00:49:21,829 --> 00:49:24,390

 

1362
00:49:24,390 --> 00:49:27,589

 

1363
00:49:27,589 --> 00:49:30,710

 

1364
00:49:30,710 --> 00:49:30,720

 

1365
00:49:30,720 --> 00:49:32,309


1366
00:49:32,309 --> 00:49:37,030

 

1367
00:49:37,030 --> 00:49:38,790

 

1368
00:49:38,790 --> 00:49:40,390

 

1369
00:49:40,390 --> 00:49:41,910

 

1370
00:49:41,910 --> 00:49:43,990

 

1371
00:49:43,990 --> 00:49:45,349

 

1372
00:49:45,349 --> 00:49:47,829

 

1373
00:49:47,829 --> 00:49:49,190

 

1374
00:49:49,190 --> 00:49:49,990

 

1375
00:49:49,990 --> 00:49:51,910

 

1376
00:49:51,910 --> 00:49:53,750

 

1377
00:49:53,750 --> 00:49:55,510

 

1378
00:49:55,510 --> 00:49:57,910

 

1379
00:49:57,910 --> 00:49:59,270

 

1380
00:49:59,270 --> 00:50:00,790

 

1381
00:50:00,790 --> 00:50:02,710

 

1382
00:50:02,710 --> 00:50:04,150

 

1383
00:50:04,150 --> 00:50:07,510

 

1384
00:50:07,510 --> 00:50:10,549

 

1385
00:50:10,549 --> 00:50:12,790

 

1386
00:50:12,790 --> 00:50:14,470

 

1387
00:50:14,470 --> 00:50:16,390

 

1388
00:50:16,390 --> 00:50:17,750

 

1389
00:50:17,750 --> 00:50:19,910

 

1390
00:50:19,910 --> 00:50:20,790

 

1391
00:50:20,790 --> 00:50:22,950

 

1392
00:50:22,950 --> 00:50:23,829

 

1393
00:50:23,829 --> 00:50:27,349

 

1394
00:50:27,349 --> 00:50:30,630

 

1395
00:50:30,630 --> 00:50:32,549

 

1396
00:50:32,549 --> 00:50:36,549

 

1397
00:50:36,549 --> 00:50:40,069

 

1398
00:50:40,069 --> 00:50:43,670

 

1399
00:50:43,670 --> 00:50:46,390

 

1400
00:50:46,390 --> 00:50:48,470

 

1401
00:50:48,470 --> 00:50:52,150

 

1402
00:50:52,150 --> 00:50:52,160

 

1403
00:50:52,160 --> 00:50:53,030


1404
00:50:53,030 --> 00:50:56,870

 

1405
00:50:56,870 --> 00:50:58,549

 

1406
00:50:58,549 --> 00:51:00,150

 

1407
00:51:00,150 --> 00:51:00,160

 

1408
00:51:00,160 --> 00:51:00,790


1409
00:51:00,790 --> 00:51:03,190

 

1410
00:51:03,190 --> 00:51:05,109

 

1411
00:51:05,109 --> 00:51:07,270

 

1412
00:51:07,270 --> 00:51:08,710

 

1413
00:51:08,710 --> 00:51:10,950

 

1414
00:51:10,950 --> 00:51:12,150

 

1415
00:51:12,150 --> 00:51:13,589

 

1416
00:51:13,589 --> 00:51:15,510

 

1417
00:51:15,510 --> 00:51:17,030

 

1418
00:51:17,030 --> 00:51:18,230

 

1419
00:51:18,230 --> 00:51:19,589

 

1420
00:51:19,589 --> 00:51:20,870

 

1421
00:51:20,870 --> 00:51:21,990

 

1422
00:51:21,990 --> 00:51:22,000

 

1423
00:51:22,000 --> 00:51:23,109


1424
00:51:23,109 --> 00:51:24,950

 

1425
00:51:24,950 --> 00:51:26,470

 

1426
00:51:26,470 --> 00:51:28,390

 

1427
00:51:28,390 --> 00:51:30,710

 

1428
00:51:30,710 --> 00:51:31,510

 

1429
00:51:31,510 --> 00:51:33,190

 

1430
00:51:33,190 --> 00:51:35,190

 

1431
00:51:35,190 --> 00:51:37,030

 

1432
00:51:37,030 --> 00:51:39,750

 

1433
00:51:39,750 --> 00:51:41,349

 

1434
00:51:41,349 --> 00:51:42,870

 

1435
00:51:42,870 --> 00:51:43,829

 

1436
00:51:43,829 --> 00:51:45,109

 

1437
00:51:45,109 --> 00:51:46,950

 

1438
00:51:46,950 --> 00:51:49,910

 

1439
00:51:49,910 --> 00:51:49,920

 

1440
00:51:49,920 --> 00:51:51,030


1441
00:51:51,030 --> 00:51:52,630

 

1442
00:51:52,630 --> 00:51:54,390

 

1443
00:51:54,390 --> 00:51:56,069

 

1444
00:51:56,069 --> 00:51:56,079

 

1445
00:51:56,079 --> 00:51:57,430


1446
00:51:57,430 --> 00:51:58,630

 

1447
00:51:58,630 --> 00:52:00,710

 

1448
00:52:00,710 --> 00:52:03,430

 

1449
00:52:03,430 --> 00:52:04,390

 

1450
00:52:04,390 --> 00:52:05,750

 

1451
00:52:05,750 --> 00:52:07,750

 

1452
00:52:07,750 --> 00:52:13,109

 

1453
00:52:13,109 --> 00:52:16,950

 

1454
00:52:16,950 --> 00:52:19,750

 

1455
00:52:19,750 --> 00:52:21,349

 

1456
00:52:21,349 --> 00:52:23,109

 

1457
00:52:23,109 --> 00:52:24,309

 

1458
00:52:24,309 --> 00:52:24,319

 

1459
00:52:24,319 --> 00:52:27,430


1460
00:52:27,430 --> 00:52:29,910

 

1461
00:52:29,910 --> 00:52:29,920

 

1462
00:52:29,920 --> 00:52:30,549


1463
00:52:30,549 --> 00:52:32,950

 

1464
00:52:32,950 --> 00:52:33,829

 

1465
00:52:33,829 --> 00:52:35,510

 

1466
00:52:35,510 --> 00:52:36,790

 

1467
00:52:36,790 --> 00:52:38,390

 

1468
00:52:38,390 --> 00:52:39,510

 

1469
00:52:39,510 --> 00:52:42,309

 

1470
00:52:42,309 --> 00:52:43,430

 

1471
00:52:43,430 --> 00:52:47,349

 

1472
00:52:47,349 --> 00:52:51,109

 

1473
00:52:51,109 --> 00:52:52,309

 

1474
00:52:52,309 --> 00:52:54,069

 

1475
00:52:54,069 --> 00:52:54,079

 

1476
00:52:54,079 --> 00:52:54,870


1477
00:52:54,870 --> 00:52:58,230

 

1478
00:52:58,230 --> 00:53:03,349

 

1479
00:53:03,349 --> 00:53:06,069

 

1480
00:53:06,069 --> 00:53:08,150

 

1481
00:53:08,150 --> 00:53:13,349

 

1482
00:53:13,349 --> 00:53:18,950

 

1483
00:53:18,950 --> 00:53:21,109

 

1484
00:53:21,109 --> 00:53:21,119

 

1485
00:53:21,119 --> 00:53:23,589


1486
00:53:23,589 --> 00:53:25,349

 

1487
00:53:25,349 --> 00:53:27,910

 

1488
00:53:27,910 --> 00:53:27,920

 

1489
00:53:27,920 --> 00:53:28,309


1490
00:53:28,309 --> 00:53:29,990

 

1491
00:53:29,990 --> 00:53:30,000

 

1492
00:53:30,000 --> 00:53:31,910


1493
00:53:31,910 --> 00:53:33,670

 

1494
00:53:33,670 --> 00:53:50,390

 

1495
00:53:50,390 --> 00:53:51,910

 

1496
00:53:51,910 --> 00:53:54,950

 

1497
00:53:54,950 --> 00:53:56,950

 

1498
00:53:56,950 --> 00:53:59,030

 

1499
00:53:59,030 --> 00:54:01,190

 

1500
00:54:01,190 --> 00:54:02,870

 

1501
00:54:02,870 --> 00:54:06,630

 

1502
00:54:06,630 --> 00:54:07,829

 

1503
00:54:07,829 --> 00:54:11,270

 

1504
00:54:11,270 --> 00:54:13,589

 

1505
00:54:13,589 --> 00:54:15,349

 

1506
00:54:15,349 --> 00:54:16,630

 

1507
00:54:16,630 --> 00:54:19,430

 

1508
00:54:19,430 --> 00:54:20,710

 

1509
00:54:20,710 --> 00:54:22,150

 

1510
00:54:22,150 --> 00:54:24,309

 

1511
00:54:24,309 --> 00:54:25,910

 

1512
00:54:25,910 --> 00:54:28,790

 

1513
00:54:28,790 --> 00:54:30,549

 

1514
00:54:30,549 --> 00:54:31,670

 

1515
00:54:31,670 --> 00:54:33,270

 

1516
00:54:33,270 --> 00:54:33,280

 

1517
00:54:33,280 --> 00:54:33,990


1518
00:54:33,990 --> 00:54:38,789

 

1519
00:54:38,789 --> 00:54:42,870

 

1520
00:54:42,870 --> 00:54:45,589

 

1521
00:54:45,589 --> 00:54:45,599

 

1522
00:54:45,599 --> 00:54:47,030


1523
00:54:47,030 --> 00:54:49,750

 

1524
00:54:49,750 --> 00:54:49,760

 

1525
00:54:49,760 --> 00:54:51,270


1526
00:54:51,270 --> 00:54:54,630

 

1527
00:54:54,630 --> 00:54:57,270

 

1528
00:54:57,270 --> 00:54:58,470

 

1529
00:54:58,470 --> 00:55:00,069

 

1530
00:55:00,069 --> 00:55:01,750

 

1531
00:55:01,750 --> 00:55:03,349

 

1532
00:55:03,349 --> 00:55:04,470

 

1533
00:55:04,470 --> 00:55:04,480

 

1534
00:55:04,480 --> 00:55:05,109


1535
00:55:05,109 --> 00:55:06,309

 

1536
00:55:06,309 --> 00:55:08,549

 

1537
00:55:08,549 --> 00:55:11,109

 

1538
00:55:11,109 --> 00:55:11,119

 

1539
00:55:11,119 --> 00:55:12,069


1540
00:55:12,069 --> 00:55:13,990

 

1541
00:55:13,990 --> 00:55:15,750

 

1542
00:55:15,750 --> 00:55:17,589

 

1543
00:55:17,589 --> 00:55:19,829

 

1544
00:55:19,829 --> 00:55:21,109

 

1545
00:55:21,109 --> 00:55:23,510

 

1546
00:55:23,510 --> 00:55:26,309

 

1547
00:55:26,309 --> 00:55:26,789

 

1548
00:55:26,789 --> 00:55:29,829

 

1549
00:55:29,829 --> 00:55:31,829

 

1550
00:55:31,829 --> 00:55:33,510

 

1551
00:55:33,510 --> 00:55:35,510

 

1552
00:55:35,510 --> 00:55:36,630

 

1553
00:55:36,630 --> 00:55:39,270

 

1554
00:55:39,270 --> 00:55:40,549

 

1555
00:55:40,549 --> 00:55:40,559

 

1556
00:55:40,559 --> 00:55:41,670


1557
00:55:41,670 --> 00:55:43,670

 

1558
00:55:43,670 --> 00:55:45,910

 

1559
00:55:45,910 --> 00:55:50,870

 

1560
00:55:50,870 --> 00:55:54,069

 

1561
00:55:54,069 --> 00:55:56,230

 

1562
00:55:56,230 --> 00:55:57,589

 

1563
00:55:57,589 --> 00:56:00,829

 

1564
00:56:00,829 --> 00:56:00,839

 

1565
00:56:00,839 --> 00:56:03,270


1566
00:56:03,270 --> 00:56:05,670

 

1567
00:56:05,670 --> 00:56:09,589

 

1568
00:56:09,589 --> 00:56:11,270

 

1569
00:56:11,270 --> 00:56:12,789

 

1570
00:56:12,789 --> 00:56:15,349

 

1571
00:56:15,349 --> 00:56:17,829

 

1572
00:56:17,829 --> 00:56:18,470

 

1573
00:56:18,470 --> 00:56:20,710

 

1574
00:56:20,710 --> 00:56:22,789

 

1575
00:56:22,789 --> 00:56:24,630

 

1576
00:56:24,630 --> 00:56:26,230

 

1577
00:56:26,230 --> 00:56:27,829

 

1578
00:56:27,829 --> 00:56:29,589

 

1579
00:56:29,589 --> 00:56:30,789

 

1580
00:56:30,789 --> 00:56:32,309

 

1581
00:56:32,309 --> 00:56:33,829

 

1582
00:56:33,829 --> 00:56:35,510

 

1583
00:56:35,510 --> 00:56:37,829

 

1584
00:56:37,829 --> 00:56:39,430

 

1585
00:56:39,430 --> 00:56:41,030

 

1586
00:56:41,030 --> 00:56:43,190

 

1587
00:56:43,190 --> 00:56:44,470

 

1588
00:56:44,470 --> 00:56:45,510

 

1589
00:56:45,510 --> 00:56:49,190

 

1590
00:56:49,190 --> 00:56:49,200

 

1591
00:56:49,200 --> 00:56:50,150


1592
00:56:50,150 --> 00:56:51,750

 

1593
00:56:51,750 --> 00:56:54,069

 

1594
00:56:54,069 --> 00:56:58,789

 

1595
00:56:58,789 --> 00:57:01,829

 

1596
00:57:01,829 --> 00:57:03,030

 

1597
00:57:03,030 --> 00:57:06,230

 

1598
00:57:06,230 --> 00:57:10,390

 

1599
00:57:10,390 --> 00:57:12,150

 

1600
00:57:12,150 --> 00:57:13,670

 

1601
00:57:13,670 --> 00:57:15,829

 

1602
00:57:15,829 --> 00:57:17,030

 

1603
00:57:17,030 --> 00:57:19,750

 

1604
00:57:19,750 --> 00:57:19,760

 

1605
00:57:19,760 --> 00:57:20,549


1606
00:57:20,549 --> 00:57:22,870

 

1607
00:57:22,870 --> 00:57:26,630

 

1608
00:57:26,630 --> 00:57:29,349

 

1609
00:57:29,349 --> 00:57:31,109

 

1610
00:57:31,109 --> 00:57:32,789

 

1611
00:57:32,789 --> 00:57:34,710

 

1612
00:57:34,710 --> 00:57:36,390

 

1613
00:57:36,390 --> 00:57:38,069

 

1614
00:57:38,069 --> 00:57:40,069

 

1615
00:57:40,069 --> 00:57:40,079

 

1616
00:57:40,079 --> 00:57:41,349


1617
00:57:41,349 --> 00:57:43,430

 

1618
00:57:43,430 --> 00:57:45,670

 

1619
00:57:45,670 --> 00:57:47,109

 

1620
00:57:47,109 --> 00:57:49,349

 

1621
00:57:49,349 --> 00:57:50,150

 

1622
00:57:50,150 --> 00:57:53,190

 

1623
00:57:53,190 --> 00:57:54,870

 

1624
00:57:54,870 --> 00:57:56,230

 

1625
00:57:56,230 --> 00:57:58,710

 

1626
00:57:58,710 --> 00:58:00,390

 

1627
00:58:00,390 --> 00:58:01,510

 

1628
00:58:01,510 --> 00:58:02,710

 

1629
00:58:02,710 --> 00:58:04,309

 

1630
00:58:04,309 --> 00:58:04,319

 

1631
00:58:04,319 --> 00:58:05,190


1632
00:58:05,190 --> 00:58:07,990

 

1633
00:58:07,990 --> 00:58:09,030

 

1634
00:58:09,030 --> 00:58:10,630

 

1635
00:58:10,630 --> 00:58:12,390

 

1636
00:58:12,390 --> 00:58:13,910

 

1637
00:58:13,910 --> 00:58:16,150

 

1638
00:58:16,150 --> 00:58:16,160

 

1639
00:58:16,160 --> 00:58:17,910


1640
00:58:17,910 --> 00:58:20,710

 

1641
00:58:20,710 --> 00:58:20,720

 

1642
00:58:20,720 --> 00:58:21,589


1643
00:58:21,589 --> 00:58:23,510

 

1644
00:58:23,510 --> 00:58:23,520

 

1645
00:58:23,520 --> 00:58:25,430


1646
00:58:25,430 --> 00:58:27,829

 

1647
00:58:27,829 --> 00:58:28,549

 

1648
00:58:28,549 --> 00:58:30,470

 

1649
00:58:30,470 --> 00:58:32,470

 

1650
00:58:32,470 --> 00:58:34,309

 

1651
00:58:34,309 --> 00:58:36,710

 

1652
00:58:36,710 --> 00:58:38,150

 

1653
00:58:38,150 --> 00:58:40,150

 

1654
00:58:40,150 --> 00:58:42,309

 

1655
00:58:42,309 --> 00:58:43,670

 

1656
00:58:43,670 --> 00:58:45,430

 

1657
00:58:45,430 --> 00:58:47,430

 

1658
00:58:47,430 --> 00:58:48,549

 

1659
00:58:48,549 --> 00:58:50,150

 

1660
00:58:50,150 --> 00:58:50,870

 

1661
00:58:50,870 --> 00:58:52,230

 

1662
00:58:52,230 --> 00:58:54,150

 

1663
00:58:54,150 --> 00:58:55,349

 

1664
00:58:55,349 --> 00:58:57,670

 

1665
00:58:57,670 --> 00:58:58,950

 

1666
00:58:58,950 --> 00:59:00,630

 

1667
00:59:00,630 --> 00:59:02,150

 

1668
00:59:02,150 --> 00:59:03,829

 

1669
00:59:03,829 --> 00:59:03,839

 

1670
00:59:03,839 --> 00:59:04,710


1671
00:59:04,710 --> 00:59:06,309

 

1672
00:59:06,309 --> 00:59:07,829

 

1673
00:59:07,829 --> 00:59:07,839

 

1674
00:59:07,839 --> 00:59:09,349


1675
00:59:09,349 --> 00:59:10,470

 

1676
00:59:10,470 --> 00:59:13,109

 

1677
00:59:13,109 --> 00:59:16,150

 

1678
00:59:16,150 --> 00:59:16,160

 

1679
00:59:16,160 --> 00:59:17,670


1680
00:59:17,670 --> 00:59:19,910

 

1681
00:59:19,910 --> 00:59:21,430

 

1682
00:59:21,430 --> 00:59:23,270

 

1683
00:59:23,270 --> 00:59:24,789

 

1684
00:59:24,789 --> 00:59:25,589

 

1685
00:59:25,589 --> 00:59:28,069

 

1686
00:59:28,069 --> 00:59:30,309

 

1687
00:59:30,309 --> 00:59:34,069

 

1688
00:59:34,069 --> 00:59:35,750

 

1689
00:59:35,750 --> 00:59:37,349

 

1690
00:59:37,349 --> 00:59:39,589

 

1691
00:59:39,589 --> 00:59:41,990

 

1692
00:59:41,990 --> 00:59:44,069

 

1693
00:59:44,069 --> 00:59:47,030

 

1694
00:59:47,030 --> 00:59:47,589

 

1695
00:59:47,589 --> 00:59:50,870

 

1696
00:59:50,870 --> 00:59:52,789

 

1697
00:59:52,789 --> 00:59:54,069

 

1698
00:59:54,069 --> 00:59:57,190

 

1699
00:59:57,190 --> 00:59:59,589

 

1700
00:59:59,589 --> 01:00:00,950

 

1701
01:00:00,950 --> 01:00:00,960

 

1702
01:00:00,960 --> 01:00:01,670


1703
01:00:01,670 --> 01:00:05,750

 

1704
01:00:05,750 --> 01:00:07,430

 

1705
01:00:07,430 --> 01:00:09,670

 

1706
01:00:09,670 --> 01:00:11,190

 

1707
01:00:11,190 --> 01:00:13,190

 

1708
01:00:13,190 --> 01:00:14,710

 

1709
01:00:14,710 --> 01:00:18,069

 

1710
01:00:18,069 --> 01:00:20,470

 

1711
01:00:20,470 --> 01:00:21,190

 

1712
01:00:21,190 --> 01:00:23,829

 

1713
01:00:23,829 --> 01:00:25,190

 

1714
01:00:25,190 --> 01:00:26,150

 

1715
01:00:26,150 --> 01:00:27,589

 

1716
01:00:27,589 --> 01:00:30,710

 

1717
01:00:30,710 --> 01:00:32,069

 

1718
01:00:32,069 --> 01:00:34,789

 

1719
01:00:34,789 --> 01:00:37,910

 

1720
01:00:37,910 --> 01:00:38,549

 

1721
01:00:38,549 --> 01:00:40,470

 

1722
01:00:40,470 --> 01:00:42,069

 

1723
01:00:42,069 --> 01:00:45,589

 

1724
01:00:45,589 --> 01:00:47,430

 

1725
01:00:47,430 --> 01:00:49,670

 

1726
01:00:49,670 --> 01:00:50,789

 

1727
01:00:50,789 --> 01:00:52,710

 

1728
01:00:52,710 --> 01:00:52,720

 

1729
01:00:52,720 --> 01:00:53,270


1730
01:00:53,270 --> 01:00:54,710

 

1731
01:00:54,710 --> 01:00:56,630

 

1732
01:00:56,630 --> 01:00:59,829

 

1733
01:00:59,829 --> 01:00:59,839

 

1734
01:00:59,839 --> 01:01:00,390


1735
01:01:00,390 --> 01:01:03,349

 

1736
01:01:03,349 --> 01:01:04,630

 

1737
01:01:04,630 --> 01:01:06,309

 

1738
01:01:06,309 --> 01:01:08,549

 

1739
01:01:08,549 --> 01:01:11,030

 

1740
01:01:11,030 --> 01:01:11,670

 

1741
01:01:11,670 --> 01:01:17,750

 

1742
01:01:17,750 --> 01:01:19,109

 

1743
01:01:19,109 --> 01:01:22,470

 

1744
01:01:22,470 --> 01:01:24,710

 

1745
01:01:24,710 --> 01:01:26,549

 

1746
01:01:26,549 --> 01:01:28,230

 

1747
01:01:28,230 --> 01:01:29,109

 

1748
01:01:29,109 --> 01:01:30,789

 

1749
01:01:30,789 --> 01:01:32,150

 

1750
01:01:32,150 --> 01:01:34,549

 

1751
01:01:34,549 --> 01:01:35,750

 

1752
01:01:35,750 --> 01:01:37,510

 

1753
01:01:37,510 --> 01:01:39,670

 

1754
01:01:39,670 --> 01:01:41,109

 

1755
01:01:41,109 --> 01:01:43,510

 

1756
01:01:43,510 --> 01:01:44,789

 

1757
01:01:44,789 --> 01:01:46,470

 

1758
01:01:46,470 --> 01:01:47,910

 

1759
01:01:47,910 --> 01:01:50,870

 

1760
01:01:50,870 --> 01:01:53,190

 

1761
01:01:53,190 --> 01:01:55,670

 

1762
01:01:55,670 --> 01:01:57,750

 

1763
01:01:57,750 --> 01:02:01,270

 

1764
01:02:01,270 --> 01:02:03,510

 

1765
01:02:03,510 --> 01:02:05,190

 

1766
01:02:05,190 --> 01:02:06,630

 

1767
01:02:06,630 --> 01:02:09,510

 

1768
01:02:09,510 --> 01:02:11,270

 

1769
01:02:11,270 --> 01:02:14,390

 

1770
01:02:14,390 --> 01:02:16,150

 

1771
01:02:16,150 --> 01:02:18,549

 

1772
01:02:18,549 --> 01:02:20,230

 

1773
01:02:20,230 --> 01:02:22,710

 

1774
01:02:22,710 --> 01:02:24,390

 

1775
01:02:24,390 --> 01:02:25,589

 

1776
01:02:25,589 --> 01:02:27,190

 

1777
01:02:27,190 --> 01:02:33,510

 

1778
01:02:33,510 --> 01:02:36,309

 

1779
01:02:36,309 --> 01:02:36,319

 

1780
01:02:36,319 --> 01:02:37,349


1781
01:02:37,349 --> 01:02:39,910

 

1782
01:02:39,910 --> 01:02:41,270

 

1783
01:02:41,270 --> 01:02:42,870

 

1784
01:02:42,870 --> 01:02:44,309

 

1785
01:02:44,309 --> 01:02:45,670

 

1786
01:02:45,670 --> 01:02:45,680

 

1787
01:02:45,680 --> 01:02:47,190


1788
01:02:47,190 --> 01:02:48,789

 

1789
01:02:48,789 --> 01:02:53,270

 

1790
01:02:53,270 --> 01:02:56,870

 

1791
01:02:56,870 --> 01:02:59,990

 

1792
01:02:59,990 --> 01:03:01,829

 

1793
01:03:01,829 --> 01:03:04,390

 

1794
01:03:04,390 --> 01:03:05,990

 

1795
01:03:05,990 --> 01:03:07,990

 

1796
01:03:07,990 --> 01:03:10,309

 

1797
01:03:10,309 --> 01:03:11,510

 

1798
01:03:11,510 --> 01:03:14,069

 

1799
01:03:14,069 --> 01:03:15,910

 

1800
01:03:15,910 --> 01:03:17,430

 

1801
01:03:17,430 --> 01:03:18,230

 

1802
01:03:18,230 --> 01:03:20,710

 

1803
01:03:20,710 --> 01:03:20,720

 

1804
01:03:20,720 --> 01:03:21,430


1805
01:03:21,430 --> 01:03:24,870

 

1806
01:03:24,870 --> 01:03:27,029

 

1807
01:03:27,029 --> 01:03:27,990

 

1808
01:03:27,990 --> 01:03:30,069

 

1809
01:03:30,069 --> 01:03:31,430

 

1810
01:03:31,430 --> 01:03:32,309

 

1811
01:03:32,309 --> 01:03:34,950

 

1812
01:03:34,950 --> 01:03:36,309

 

1813
01:03:36,309 --> 01:03:38,309

 

1814
01:03:38,309 --> 01:03:41,270

 

1815
01:03:41,270 --> 01:03:44,390

 

1816
01:03:44,390 --> 01:03:45,910

 

1817
01:03:45,910 --> 01:03:47,990

 

1818
01:03:47,990 --> 01:03:51,029

 

1819
01:03:51,029 --> 01:03:52,549

 

1820
01:03:52,549 --> 01:03:55,910

 

1821
01:03:55,910 --> 01:03:58,549

 

1822
01:03:58,549 --> 01:04:00,150

 

1823
01:04:00,150 --> 01:04:00,160

 

1824
01:04:00,160 --> 01:04:00,789


1825
01:04:00,789 --> 01:04:02,950

 

1826
01:04:02,950 --> 01:04:04,230

 

1827
01:04:04,230 --> 01:04:06,309

 

1828
01:04:06,309 --> 01:04:07,750

 

1829
01:04:07,750 --> 01:04:09,670

 

1830
01:04:09,670 --> 01:04:11,190

 

1831
01:04:11,190 --> 01:04:12,789

 

1832
01:04:12,789 --> 01:04:14,230

 

1833
01:04:14,230 --> 01:04:14,240

 

1834
01:04:14,240 --> 01:04:15,190


1835
01:04:15,190 --> 01:04:17,109

 

1836
01:04:17,109 --> 01:04:17,119

 

1837
01:04:17,119 --> 01:04:18,390


1838
01:04:18,390 --> 01:04:21,029

 

1839
01:04:21,029 --> 01:04:22,710

 

1840
01:04:22,710 --> 01:04:26,150

 

1841
01:04:26,150 --> 01:04:27,190

 

1842
01:04:27,190 --> 01:04:29,910

 

1843
01:04:29,910 --> 01:04:32,950

 

1844
01:04:32,950 --> 01:04:34,710

 

1845
01:04:34,710 --> 01:04:35,990

 

1846
01:04:35,990 --> 01:04:38,230

 

1847
01:04:38,230 --> 01:04:40,789

 

1848
01:04:40,789 --> 01:04:42,390

 

1849
01:04:42,390 --> 01:04:43,510

 

1850
01:04:43,510 --> 01:04:48,309

 

1851
01:04:48,309 --> 01:04:50,950

 

1852
01:04:50,950 --> 01:04:52,470

 

1853
01:04:52,470 --> 01:04:54,309

 

1854
01:04:54,309 --> 01:04:57,270

 

1855
01:04:57,270 --> 01:04:59,670

 

1856
01:04:59,670 --> 01:05:00,950

 

1857
01:05:00,950 --> 01:05:02,069

 

1858
01:05:02,069 --> 01:05:03,430

 

1859
01:05:03,430 --> 01:05:07,190

 

1860
01:05:07,190 --> 01:05:10,870

 

1861
01:05:10,870 --> 01:05:14,309

 

1862
01:05:14,309 --> 01:05:16,390

 

1863
01:05:16,390 --> 01:05:18,230

 

1864
01:05:18,230 --> 01:05:19,589

 

1865
01:05:19,589 --> 01:05:23,109

 

1866
01:05:23,109 --> 01:05:23,119

 

1867
01:05:23,119 --> 01:05:25,109


1868
01:05:25,109 --> 01:05:28,470

 

1869
01:05:28,470 --> 01:05:32,150

 

1870
01:05:32,150 --> 01:05:35,589

 

1871
01:05:35,589 --> 01:05:35,599

 

1872
01:05:35,599 --> 01:05:35,990


1873
01:05:35,990 --> 01:05:39,270

 

1874
01:05:39,270 --> 01:05:41,510

 

1875
01:05:41,510 --> 01:05:43,829

 

1876
01:05:43,829 --> 01:05:45,750

 

1877
01:05:45,750 --> 01:05:47,109

 

1878
01:05:47,109 --> 01:05:48,870

 

1879
01:05:48,870 --> 01:05:50,549

 

1880
01:05:50,549 --> 01:05:52,390

 

1881
01:05:52,390 --> 01:05:54,150

 

1882
01:05:54,150 --> 01:05:56,630

 

1883
01:05:56,630 --> 01:05:58,150

 

1884
01:05:58,150 --> 01:05:59,270

 

1885
01:05:59,270 --> 01:06:01,990

 

1886
01:06:01,990 --> 01:06:03,510

 

1887
01:06:03,510 --> 01:06:05,589

 

1888
01:06:05,589 --> 01:06:06,710

 

1889
01:06:06,710 --> 01:06:07,670

 

1890
01:06:07,670 --> 01:06:10,069

 

1891
01:06:10,069 --> 01:06:12,309

 

1892
01:06:12,309 --> 01:06:15,029

 

1893
01:06:15,029 --> 01:06:15,039

 

1894
01:06:15,039 --> 01:06:16,710


1895
01:06:16,710 --> 01:06:17,910

 

1896
01:06:17,910 --> 01:06:20,309

 

1897
01:06:20,309 --> 01:06:23,029

 

1898
01:06:23,029 --> 01:06:24,549

 

1899
01:06:24,549 --> 01:06:26,470

 

1900
01:06:26,470 --> 01:06:28,230

 

1901
01:06:28,230 --> 01:06:30,630

 

1902
01:06:30,630 --> 01:06:32,390

 

1903
01:06:32,390 --> 01:06:35,510

 

1904
01:06:35,510 --> 01:06:37,990

 

1905
01:06:37,990 --> 01:06:39,270

 

1906
01:06:39,270 --> 01:06:41,109

 

1907
01:06:41,109 --> 01:06:42,710

 

1908
01:06:42,710 --> 01:06:44,150

 

1909
01:06:44,150 --> 01:06:46,069

 

1910
01:06:46,069 --> 01:06:46,549

 

1911
01:06:46,549 --> 01:06:49,910

 

1912
01:06:49,910 --> 01:06:53,589

 

1913
01:06:53,589 --> 01:06:54,150

 

1914
01:06:54,150 --> 01:06:58,789

 

1915
01:06:58,789 --> 01:07:01,990

 

1916
01:07:01,990 --> 01:07:03,670

 

1917
01:07:03,670 --> 01:07:06,069

 

1918
01:07:06,069 --> 01:07:07,430

 

1919
01:07:07,430 --> 01:07:09,270

 

1920
01:07:09,270 --> 01:07:12,829

 

1921
01:07:12,829 --> 01:07:12,839

 

1922
01:07:12,839 --> 01:07:14,789


1923
01:07:14,789 --> 01:07:22,710

 

1924
01:07:22,710 --> 01:07:24,390

 

1925
01:07:24,390 --> 01:07:26,829

 

1926
01:07:26,829 --> 01:07:26,839

 

1927
01:07:26,839 --> 01:07:28,309


1928
01:07:28,309 --> 01:07:33,349

 

1929
01:07:33,349 --> 01:07:39,589

 

1930
01:07:39,589 --> 01:07:41,910

 

1931
01:07:41,910 --> 01:07:43,589

 

1932
01:07:43,589 --> 01:07:45,430

 

1933
01:07:45,430 --> 01:07:46,950

 

1934
01:07:46,950 --> 01:07:48,470

 

1935
01:07:48,470 --> 01:07:50,230

 

1936
01:07:50,230 --> 01:07:51,589

 

1937
01:07:51,589 --> 01:07:53,349

 

1938
01:07:53,349 --> 01:07:54,870

 

1939
01:07:54,870 --> 01:07:58,630

 

1940
01:07:58,630 --> 01:08:01,270

 

1941
01:08:01,270 --> 01:08:01,280

 

1942
01:08:01,280 --> 01:08:02,150


1943
01:08:02,150 --> 01:08:05,829

 

1944
01:08:05,829 --> 01:08:08,950

 

1945
01:08:08,950 --> 01:08:10,549

 

1946
01:08:10,549 --> 01:08:12,309

 

1947
01:08:12,309 --> 01:08:12,319

 

1948
01:08:12,319 --> 01:08:14,069


1949
01:08:14,069 --> 01:08:16,070

 

1950
01:08:16,070 --> 01:08:17,910

 

1951
01:08:17,910 --> 01:08:19,430

 

1952
01:08:19,430 --> 01:08:21,110

 

1953
01:08:21,110 --> 01:08:22,829

 

1954
01:08:22,829 --> 01:08:22,839

 

1955
01:08:22,839 --> 01:08:24,870


1956
01:08:24,870 --> 01:08:27,829

 

1957
01:08:27,829 --> 01:08:27,839

 

1958
01:08:27,839 --> 01:08:28,229


1959
01:08:28,229 --> 01:08:29,829

 

1960
01:08:29,829 --> 01:08:31,910

 

1961
01:08:31,910 --> 01:08:34,229

 

1962
01:08:34,229 --> 01:08:35,189

 

1963
01:08:35,189 --> 01:08:38,309

 

1964
01:08:38,309 --> 01:08:41,749

 

1965
01:08:41,749 --> 01:08:46,149

 

1966
01:08:46,149 --> 01:08:46,159

 

1967
01:08:46,159 --> 01:08:46,789


1968
01:08:46,789 --> 01:08:50,950

 

1969
01:08:50,950 --> 01:08:53,990

 

1970
01:08:53,990 --> 01:08:57,189

 

1971
01:08:57,189 --> 01:08:58,630

 

1972
01:08:58,630 --> 01:09:01,030

 

1973
01:09:01,030 --> 01:09:02,309

 

1974
01:09:02,309 --> 01:09:04,470

 

1975
01:09:04,470 --> 01:09:06,070

 

1976
01:09:06,070 --> 01:09:07,829

 

1977
01:09:07,829 --> 01:09:12,070

 

1978
01:09:12,070 --> 01:09:16,470

 

1979
01:09:16,470 --> 01:09:18,470

 

1980
01:09:18,470 --> 01:09:22,550

 

1981
01:09:22,550 --> 01:09:24,870

 

1982
01:09:24,870 --> 01:09:24,880

 

1983
01:09:24,880 --> 01:09:25,590


1984
01:09:25,590 --> 01:09:28,870

 

1985
01:09:28,870 --> 01:09:30,630

 

1986
01:09:30,630 --> 01:09:33,349

 

1987
01:09:33,349 --> 01:09:34,789

 

1988
01:09:34,789 --> 01:09:39,430

 

1989
01:09:39,430 --> 01:09:44,390

 

1990
01:09:44,390 --> 01:09:47,189

 

1991
01:09:47,189 --> 01:09:47,199

 

1992
01:09:47,199 --> 01:09:48,149


1993
01:09:48,149 --> 01:09:50,309

 

1994
01:09:50,309 --> 01:09:52,070

 

1995
01:09:52,070 --> 01:09:54,310

 

1996
01:09:54,310 --> 01:09:56,070

 

1997
01:09:56,070 --> 01:09:57,110

 

1998
01:09:57,110 --> 01:09:58,630

 

1999
01:09:58,630 --> 01:10:00,229

 

2000
01:10:00,229 --> 01:10:00,239

 

2001
01:10:00,239 --> 01:10:00,630


2002
01:10:00,630 --> 01:10:01,910

 

2003
01:10:01,910 --> 01:10:03,990

 

2004
01:10:03,990 --> 01:10:04,000

 

2005
01:10:04,000 --> 01:10:08,149


2006
01:10:08,149 --> 01:10:11,590

 

2007
01:10:11,590 --> 01:10:13,669

 

2008
01:10:13,669 --> 01:10:15,189

 

2009
01:10:15,189 --> 01:10:16,149

 

2010
01:10:16,149 --> 01:10:20,950

 

2011
01:10:20,950 --> 01:10:25,350

 

2012
01:10:25,350 --> 01:10:28,229

 

2013
01:10:28,229 --> 01:10:30,149

 

2014
01:10:30,149 --> 01:10:31,990

 

2015
01:10:31,990 --> 01:10:34,310

 

2016
01:10:34,310 --> 01:10:37,350

 

2017
01:10:37,350 --> 01:10:38,470

 

2018
01:10:38,470 --> 01:10:40,229

 

2019
01:10:40,229 --> 01:10:41,750

 

2020
01:10:41,750 --> 01:10:42,950

 

2021
01:10:42,950 --> 01:10:44,229

 

2022
01:10:44,229 --> 01:10:46,149

 

2023
01:10:46,149 --> 01:10:46,159

 

2024
01:10:46,159 --> 01:10:47,189


2025
01:10:47,189 --> 01:10:49,189

 

2026
01:10:49,189 --> 01:10:51,030

 

2027
01:10:51,030 --> 01:10:52,550

 

2028
01:10:52,550 --> 01:10:53,990

 

2029
01:10:53,990 --> 01:10:54,790

 

2030
01:10:54,790 --> 01:10:56,630

 

2031
01:10:56,630 --> 01:10:58,070

 

2032
01:10:58,070 --> 01:10:58,870

 

2033
01:10:58,870 --> 01:11:01,430

 

2034
01:11:01,430 --> 01:11:02,709

 

2035
01:11:02,709 --> 01:11:03,910

 

2036
01:11:03,910 --> 01:11:07,430

 

2037
01:11:07,430 --> 01:11:08,630

 

2038
01:11:08,630 --> 01:11:09,510

 

2039
01:11:09,510 --> 01:11:10,870

 

2040
01:11:10,870 --> 01:11:12,310

 

2041
01:11:12,310 --> 01:11:13,830

 

2042
01:11:13,830 --> 01:11:15,110

 

2043
01:11:15,110 --> 01:11:18,070

 

2044
01:11:18,070 --> 01:11:22,390

 

2045
01:11:22,390 --> 01:11:23,510

 

2046
01:11:23,510 --> 01:11:23,520

 

2047
01:11:23,520 --> 01:11:25,669


2048
01:11:25,669 --> 01:11:28,709

 

2049
01:11:28,709 --> 01:11:32,149

 

2050
01:11:32,149 --> 01:11:35,669

 

2051
01:11:35,669 --> 01:11:36,310

 

2052
01:11:36,310 --> 01:11:38,070

 

2053
01:11:38,070 --> 01:11:39,510

 

2054
01:11:39,510 --> 01:11:39,910

 

2055
01:11:39,910 --> 01:11:42,070

 

2056
01:11:42,070 --> 01:11:44,070

 

2057
01:11:44,070 --> 01:11:46,229

 

2058
01:11:46,229 --> 01:11:47,669

 

2059
01:11:47,669 --> 01:11:49,350

 

2060
01:11:49,350 --> 01:11:51,590

 

2061
01:11:51,590 --> 01:11:53,910

 

2062
01:11:53,910 --> 01:11:55,350

 

2063
01:11:55,350 --> 01:11:58,310

 

2064
01:11:58,310 --> 01:11:58,320

 

2065
01:11:58,320 --> 01:11:58,790


2066
01:11:58,790 --> 01:12:00,149

 

2067
01:12:00,149 --> 01:12:01,590

 

2068
01:12:01,590 --> 01:12:02,870

 

2069
01:12:02,870 --> 01:12:04,790

 

2070
01:12:04,790 --> 01:12:09,350

 

2071
01:12:09,350 --> 01:12:13,110

 

2072
01:12:13,110 --> 01:12:16,149

 

2073
01:12:16,149 --> 01:12:16,159

 

2074
01:12:16,159 --> 01:12:19,590


2075
01:12:19,590 --> 01:12:23,510

 

2076
01:12:23,510 --> 01:12:24,790

 

2077
01:12:24,790 --> 01:12:25,430

 

2078
01:12:25,430 --> 01:12:27,669

 

2079
01:12:27,669 --> 01:12:28,870

 

2080
01:12:28,870 --> 01:12:30,550

 

2081
01:12:30,550 --> 01:12:31,910

 

2082
01:12:31,910 --> 01:12:34,070

 

2083
01:12:34,070 --> 01:12:35,830

 

2084
01:12:35,830 --> 01:12:36,470

 

2085
01:12:36,470 --> 01:12:38,149

 

2086
01:12:38,149 --> 01:12:40,470

 

2087
01:12:40,470 --> 01:12:42,950

 

2088
01:12:42,950 --> 01:12:45,750

 

2089
01:12:45,750 --> 01:12:50,149

 

2090
01:12:50,149 --> 01:12:58,870

 

2091
01:12:58,870 --> 01:13:01,270

 

2092
01:13:01,270 --> 01:13:02,950

 

2093
01:13:02,950 --> 01:13:05,430

 

2094
01:13:05,430 --> 01:13:05,440

 

2095
01:13:05,440 --> 01:13:06,550


2096
01:13:06,550 --> 01:13:08,390

 

2097
01:13:08,390 --> 01:13:10,550

 

2098
01:13:10,550 --> 01:13:11,990

 

2099
01:13:11,990 --> 01:13:12,709

 

2100
01:13:12,709 --> 01:13:15,030

 

2101
01:13:15,030 --> 01:13:16,310

 

2102
01:13:16,310 --> 01:13:17,750

 

2103
01:13:17,750 --> 01:13:19,990

 

2104
01:13:19,990 --> 01:13:23,030

 

2105
01:13:23,030 --> 01:13:24,550

 

2106
01:13:24,550 --> 01:13:26,470

 

2107
01:13:26,470 --> 01:13:26,480

 

2108
01:13:26,480 --> 01:13:26,950


2109
01:13:26,950 --> 01:13:30,630

 

2110
01:13:30,630 --> 01:13:36,070

 

2111
01:13:36,070 --> 01:13:39,030

 

2112
01:13:39,030 --> 01:13:39,040

 

2113
01:13:39,040 --> 01:13:39,830


2114
01:13:39,830 --> 01:13:42,070

 

2115
01:13:42,070 --> 01:13:43,830

 

2116
01:13:43,830 --> 01:13:46,790

 

2117
01:13:46,790 --> 01:13:46,800

 

2118
01:13:46,800 --> 01:13:47,350


2119
01:13:47,350 --> 01:13:49,270

 

2120
01:13:49,270 --> 01:13:50,470

 

2121
01:13:50,470 --> 01:13:51,350

 

2122
01:13:51,350 --> 01:13:53,510

 

2123
01:13:53,510 --> 01:13:54,790

 

2124
01:13:54,790 --> 01:13:56,550

 

2125
01:13:56,550 --> 01:13:59,350

 

2126
01:13:59,350 --> 01:14:01,030

 

2127
01:14:01,030 --> 01:14:02,709

 

2128
01:14:02,709 --> 01:14:05,110

 

2129
01:14:05,110 --> 01:14:06,550

 

2130
01:14:06,550 --> 01:14:08,630

 

2131
01:14:08,630 --> 01:14:10,229

 

2132
01:14:10,229 --> 01:14:12,390

 

2133
01:14:12,390 --> 01:14:17,990

 

2134
01:14:17,990 --> 01:14:19,990

 

2135
01:14:19,990 --> 01:14:21,030

 

2136
01:14:21,030 --> 01:14:23,030

 

2137
01:14:23,030 --> 01:14:23,040

 

2138
01:14:23,040 --> 01:14:24,390


2139
01:14:24,390 --> 01:14:26,830

 

2140
01:14:26,830 --> 01:14:29,669

 

2141
01:14:29,669 --> 01:14:31,910

 

2142
01:14:31,910 --> 01:14:33,669

 

2143
01:14:33,669 --> 01:14:35,830

 

2144
01:14:35,830 --> 01:14:36,790

 

2145
01:14:36,790 --> 01:14:38,950

 

2146
01:14:38,950 --> 01:14:40,070

 

2147
01:14:40,070 --> 01:14:42,550

 

2148
01:14:42,550 --> 01:14:45,510

 

2149
01:14:45,510 --> 01:14:48,390

 

2150
01:14:48,390 --> 01:14:49,510

 

2151
01:14:49,510 --> 01:14:51,189

 

2152
01:14:51,189 --> 01:14:53,189

 

2153
01:14:53,189 --> 01:14:54,950

 

2154
01:14:54,950 --> 01:14:56,550

 

2155
01:14:56,550 --> 01:14:59,750

 

2156
01:14:59,750 --> 01:15:02,870

 

2157
01:15:02,870 --> 01:15:02,880

 

2158
01:15:02,880 --> 01:15:03,750


2159
01:15:03,750 --> 01:15:07,189

 

2160
01:15:07,189 --> 01:15:09,270

 

2161
01:15:09,270 --> 01:15:10,390

 

2162
01:15:10,390 --> 01:15:12,790

 

2163
01:15:12,790 --> 01:15:14,390

 

2164
01:15:14,390 --> 01:15:15,750

 

2165
01:15:15,750 --> 01:15:17,270

 

2166
01:15:17,270 --> 01:15:19,030

 

2167
01:15:19,030 --> 01:15:20,550

 

2168
01:15:20,550 --> 01:15:21,910

 

2169
01:15:21,910 --> 01:15:25,189

 

2170
01:15:25,189 --> 01:15:28,390

 

2171
01:15:28,390 --> 01:15:29,830

 

2172
01:15:29,830 --> 01:15:29,840

 

2173
01:15:29,840 --> 01:15:33,030


2174
01:15:33,030 --> 01:15:35,750

 

2175
01:15:35,750 --> 01:15:36,709

 

2176
01:15:36,709 --> 01:15:38,390

 

2177
01:15:38,390 --> 01:15:40,149

 

2178
01:15:40,149 --> 01:15:43,110

 

2179
01:15:43,110 --> 01:15:44,870

 

2180
01:15:44,870 --> 01:15:47,990

 

2181
01:15:47,990 --> 01:15:49,750

 

2182
01:15:49,750 --> 01:15:51,510

 

2183
01:15:51,510 --> 01:15:53,270

 

2184
01:15:53,270 --> 01:15:54,550

 

2185
01:15:54,550 --> 01:15:56,550

 

2186
01:15:56,550 --> 01:15:58,470

 

2187
01:15:58,470 --> 01:16:01,510

 

2188
01:16:01,510 --> 01:16:02,630

 

2189
01:16:02,630 --> 01:16:04,630

 

2190
01:16:04,630 --> 01:16:07,110

 

2191
01:16:07,110 --> 01:16:10,229

 

2192
01:16:10,229 --> 01:16:12,149

 

2193
01:16:12,149 --> 01:16:15,669

 

2194
01:16:15,669 --> 01:16:18,310

 

2195
01:16:18,310 --> 01:16:19,430

 

2196
01:16:19,430 --> 01:16:22,229

 

2197
01:16:22,229 --> 01:16:23,590

 

2198
01:16:23,590 --> 01:16:25,270

 

2199
01:16:25,270 --> 01:16:27,030

 

2200
01:16:27,030 --> 01:16:28,550

 

2201
01:16:28,550 --> 01:16:31,110

 

2202
01:16:31,110 --> 01:16:32,790

 

2203
01:16:32,790 --> 01:16:34,070

 

2204
01:16:34,070 --> 01:16:34,870

 

2205
01:16:34,870 --> 01:16:38,070

 

2206
01:16:38,070 --> 01:16:40,229

 

2207
01:16:40,229 --> 01:16:42,229

 

2208
01:16:42,229 --> 01:16:44,310

 

2209
01:16:44,310 --> 01:16:48,070

 

2210
01:16:48,070 --> 01:16:52,310

 

2211
01:16:52,310 --> 01:16:53,990

 

2212
01:16:53,990 --> 01:16:56,550

 

2213
01:16:56,550 --> 01:16:59,830

 

2214
01:16:59,830 --> 01:17:01,430

 

2215
01:17:01,430 --> 01:17:04,950

 

2216
01:17:04,950 --> 01:17:09,110

 

2217
01:17:09,110 --> 01:17:10,470

 

2218
01:17:10,470 --> 01:17:12,070

 

2219
01:17:12,070 --> 01:17:15,590

 

2220
01:17:15,590 --> 01:17:17,430

 

2221
01:17:17,430 --> 01:17:19,030

 

2222
01:17:19,030 --> 01:17:21,750

 

2223
01:17:21,750 --> 01:17:22,870

 

2224
01:17:22,870 --> 01:17:24,310

 

2225
01:17:24,310 --> 01:17:26,070

 

2226
01:17:26,070 --> 01:17:26,080

 

2227
01:17:26,080 --> 01:17:26,950


2228
01:17:26,950 --> 01:17:29,189

 

2229
01:17:29,189 --> 01:17:30,870

 

2230
01:17:30,870 --> 01:17:33,590

 

2231
01:17:33,590 --> 01:17:34,950

 

2232
01:17:34,950 --> 01:17:36,950

 

2233
01:17:36,950 --> 01:17:39,110

 

2234
01:17:39,110 --> 01:17:39,120

 

2235
01:17:39,120 --> 01:17:40,310


2236
01:17:40,310 --> 01:17:42,070

 

2237
01:17:42,070 --> 01:17:42,080

 

2238
01:17:42,080 --> 01:17:43,430


2239
01:17:43,430 --> 01:17:45,430

 

2240
01:17:45,430 --> 01:17:45,440

 

2241
01:17:45,440 --> 01:17:46,709


2242
01:17:46,709 --> 01:17:51,430

 

2243
01:17:51,430 --> 01:17:53,990

 

2244
01:17:53,990 --> 01:17:54,000

 

2245
01:17:54,000 --> 01:17:54,950


2246
01:17:54,950 --> 01:17:56,830

 

2247
01:17:56,830 --> 01:18:00,149

 

2248
01:18:00,149 --> 01:18:02,790

 

2249
01:18:02,790 --> 01:18:04,550

 

2250
01:18:04,550 --> 01:18:06,470

 

2251
01:18:06,470 --> 01:18:09,350

 

2252
01:18:09,350 --> 01:18:11,430

 

2253
01:18:11,430 --> 01:18:13,270

 

2254
01:18:13,270 --> 01:18:13,280

 

2255
01:18:13,280 --> 01:18:14,149


2256
01:18:14,149 --> 01:18:15,669

 

2257
01:18:15,669 --> 01:18:17,189

 

2258
01:18:17,189 --> 01:18:18,390

 

2259
01:18:18,390 --> 01:18:20,070

 

2260
01:18:20,070 --> 01:18:20,080

 

2261
01:18:20,080 --> 01:18:21,030


2262
01:18:21,030 --> 01:18:22,870

 

2263
01:18:22,870 --> 01:18:24,550

 

2264
01:18:24,550 --> 01:18:27,189

 

2265
01:18:27,189 --> 01:18:28,070

 

2266
01:18:28,070 --> 01:18:31,110

 

2267
01:18:31,110 --> 01:18:32,630

 

2268
01:18:32,630 --> 01:18:34,870

 

2269
01:18:34,870 --> 01:18:37,270

 

2270
01:18:37,270 --> 01:18:38,550

 

2271
01:18:38,550 --> 01:18:40,310

 

2272
01:18:40,310 --> 01:18:42,229

 

2273
01:18:42,229 --> 01:18:43,350

 

2274
01:18:43,350 --> 01:18:44,310

 

2275
01:18:44,310 --> 01:18:45,669

 

2276
01:18:45,669 --> 01:18:47,189

 

2277
01:18:47,189 --> 01:18:47,199

 

2278
01:18:47,199 --> 01:18:48,310


2279
01:18:48,310 --> 01:18:51,830

 

2280
01:18:51,830 --> 01:18:54,310

 

2281
01:18:54,310 --> 01:18:56,310

 

2282
01:18:56,310 --> 01:18:59,110

 

2283
01:18:59,110 --> 01:19:00,310

 

2284
01:19:00,310 --> 01:19:03,189

 

2285
01:19:03,189 --> 01:19:06,070

 

2286
01:19:06,070 --> 01:19:08,950

 

2287
01:19:08,950 --> 01:19:11,110

 

2288
01:19:11,110 --> 01:19:14,550

 

2289
01:19:14,550 --> 01:19:16,229

 

2290
01:19:16,229 --> 01:19:19,270

 

2291
01:19:19,270 --> 01:19:20,630

 

2292
01:19:20,630 --> 01:19:23,510

 

2293
01:19:23,510 --> 01:19:23,520

 

2294
01:19:23,520 --> 01:19:24,709


2295
01:19:24,709 --> 01:19:27,750

 

2296
01:19:27,750 --> 01:19:29,830

 

2297
01:19:29,830 --> 01:19:31,830

 

2298
01:19:31,830 --> 01:19:33,669

 

2299
01:19:33,669 --> 01:19:35,910

 

2300
01:19:35,910 --> 01:19:36,390

 

2301
01:19:36,390 --> 01:19:37,990

 

2302
01:19:37,990 --> 01:19:39,430

 

2303
01:19:39,430 --> 01:19:40,790

 

2304
01:19:40,790 --> 01:19:42,310

 

2305
01:19:42,310 --> 01:19:43,910

 

2306
01:19:43,910 --> 01:19:46,470

 

2307
01:19:46,470 --> 01:19:49,030

 

2308
01:19:49,030 --> 01:19:50,550

 

2309
01:19:50,550 --> 01:19:51,990

 

2310
01:19:51,990 --> 01:19:53,350

 

2311
01:19:53,350 --> 01:19:55,030

 

2312
01:19:55,030 --> 01:19:58,229

 

2313
01:19:58,229 --> 01:20:00,709

 

2314
01:20:00,709 --> 01:20:00,719

 

2315
01:20:00,719 --> 01:20:01,270


2316
01:20:01,270 --> 01:20:05,590

 

2317
01:20:05,590 --> 01:20:08,470

 

2318
01:20:08,470 --> 01:20:09,669

 

2319
01:20:09,669 --> 01:20:12,390

 

2320
01:20:12,390 --> 01:20:13,510

 

2321
01:20:13,510 --> 01:20:15,350

 

2322
01:20:15,350 --> 01:20:17,270

 

2323
01:20:17,270 --> 01:20:19,270

 

2324
01:20:19,270 --> 01:20:20,870

 

2325
01:20:20,870 --> 01:20:23,270

 

2326
01:20:23,270 --> 01:20:24,149

 

2327
01:20:24,149 --> 01:20:25,990

 

2328
01:20:25,990 --> 01:20:27,189

 

2329
01:20:27,189 --> 01:20:29,430

 

2330
01:20:29,430 --> 01:20:31,030

 

2331
01:20:31,030 --> 01:20:32,950

 

2332
01:20:32,950 --> 01:20:35,189

 

2333
01:20:35,189 --> 01:20:37,669

 

2334
01:20:37,669 --> 01:20:39,030

 

2335
01:20:39,030 --> 01:20:39,990

 

2336
01:20:39,990 --> 01:20:46,149

 

2337
01:20:46,149 --> 01:20:47,590

 

2338
01:20:47,590 --> 01:20:58,480