1
00:00:02,780 --> 00:00:05,679

alright let's get started if you haven't  already there's two handouts on the left  and I should take two pieces of paper so  we'll be doing some actual origami  folding we'll be folding six eight four  nine today just like this it will only  take us eight hours or so this is the  Jenny and Eli folding you've seen on the  coaster pretty awesome now we'll be  folding butters more like this so thanks  everyone for giving so many cool  questions and comments and the feedback  is really helpful I was I didn't know  what I was gonna cover in this class I  had too many ideas and your questions  really help narrow it down into exciting  things so the structure is gonna be I'm  gonna go through questions so it's a  funny kind of interactivity where  there's a whole day in between where you  ask the questions and where I answer  them but feel free to ask more questions  follow up things but there's a lot  already here so it should be fun these  are not questions but sort of funny er  comments if you like double rainbow  jokes this was this this lecture  happened like a couple of months after  the double rainbow fiasco there's a lot  more as I recall it was sort of a  running theme throughout the whole  semester so look out for that I know  this is gonna this is entertaining I'm  getting used to listen to myself at  double speed should be fun this class  definitely is nice in the way that with  fairly simple technique you can prove a  very powerful theorem and that's sorry I  just remembered I need to push a  different button here that's that's  hopefully a theme throughout the class  but definitely especially nice here so  it's kind of cool though strip folding  if it was an open problem for a couple  years at least and no one thinking how  do we fold any shape that seems really  tough the point of this the strip  folding approach is that once you have  the right idea just start with a really  long rectangle somehow it becomes easy  there's still a lot of details and  getting that to work but it's kind of  neat how that that works out and now we  get to actual  all proposals so folding practice  planning on doing this but it's an  explicit comment to that effect so we're  going to fold some letters of the  alphabet you have in your packet  instructions for making the individual  digits six four eight and nine and you  also have which are all pretty easy you  also have a diagram for it's crazy  design called typeset which can desync  this same folding can make any letter of  the alphabet and any digit so just to  show you what they will look like these  are my foldings of six eight four and  nine according to the first set of  diagrams and then this is my folding of  think the letter the number six rather  out of jason coos design so if you  wanted to reconfigure it to i guess  eight is kind of hard let me do four  think okay so four four I've got a fold  this guy under fold this here this tab I  think goes back here so the advanced  origami folders can definitely do in  this classic and if they do the Jason  coos design but it it takes a little  while I think there I've got a four got  it  a little hard to hold in position but at  least in theory it will make any letter  all out of one one folding you just have  to move all the tabs around so it's kind  of neat but it takes at least half an  hour so to fold that unless you're  really fast so I would recommend pick  one of these working groups if you like  if you want you can form a group of 4  and make 6 8 4 & 9 follow the diagrams  this is an exercise in following  diagrams one way or the other only have  time to maybe make one digit each but  have fun with it and if you have  questions raise your hand I can tell you  the first step in 6 8 & 9 is to make an  8x8 grid there's a lot of ways to make  an 8x8 grid but an easy one is shown  here so you take your your sheet you  fold the bottom edge to the top edge I  think they want to do it white side up  again so they're all valleys when you  align those edges you'll get a nice  bisector then you repeat folding the  bottom edge to the middle and once you  do that to save a little bit of time you  can then fold that new bottom edge to  the middle again that will do eights on  one side it's a little bit inaccurate  because you're folding through two  layers but it's a little faster time is  of the essence then you do the same  thing on the bottom and you'll get  ates in one dimension but then you have  to fold it in eighths in the other  dimension  once you have eighths it's like three  steps it's really easy four six eight  and nine four uses a different approach  it's a little more free hand if you want  to be more creative try the four you've  got to kind of eyeball what looks and  feels good for number four  so out of curiosity how many people have  an 8x8 grid at this point you want to  take full so that's the top half of  these diagrams to 8x8 grid then it's  mostly folding up over individual edges  and some corner folds but they're all  they're all simple folds in this world  so you're folding through I think always  all the layers on this this is only  folding through one layer but these  would always fall into the some layer  simple folds category so for example  make a six this bottom edge up fold the  left three squares over this corner up  fold this corner down  fold this down this over  there's really big six and these numbers  are all pretty much proportioned  correctly the four you have to be it  helps to have a reference model one of  the other digits to make it the right  height but it'll end up roughly correct  anyway it's a little bit narrow and it's  designed it's slightly narrower than the  six and the nine but otherwise they're  nice compatible digits there's a whole  alphabet on the website that's linked  from this slide so you can go check it  out the origami Club so it's kind of fun  to think about font design and alphabet  design there's actually a lot of origami  alphabets out there this is one of the  simplest that has digits anyone have  questions how many people folded a  letter a digit okay so maybe I will  you can keep folding but maybe I'll  continue on it's gonna be a lot of fun  crackling anyone folded this one anyone  working on it uh-huh  a bunch of people cool well let me know  when you finish it's kind of fun it's  not that hard I wanted to point at the  origami website or gham it is the  origami Club at MIT and at the top you  see a different alphabet this is a four  fold alphabet designed by Jeannine  Mosely who's an MIT alum and came to  this class two years ago and so that's  the reference design in 2002 I'm not  sure if it has digits though at least  the the diagrams we found for the  letters do not also have digits so I  think an interesting challenge is to  design a for fit for fold digit set to  compliment her letter set if you're  interested that could be a cool project  to work on folding design minimal fold  alphabets you could try a three fold  alphabet to fold alphabet whatever also  on this website are the meeting schedule  it's always Sundays sometimes two  sometimes three o'clock check it out  and there's a convention coming up our  very own origami convention on October  27th and Jason ku wanted me to to remind  you about that coming up so if you even  if you've never done origami before  other than today you should check it out  it'll be fun lots of different sessions  on from simple models to complicated  models this is a cool design by Brian  Chen another MIT alum it's one square  paper folded into the men's at Montes  logo but here instead of the oil lamp  you've got a little origami crane cool  stuff cool so we proceeded on to other  questions this is a pretty simple one at  the top of the note says folding any  shape aka silhouette folding and gift  wrapping it has a couple of references  so where do those terms come from it's  the question and one answer is it's the  title of the paper folding flat  silhouettes and wrapping polyhedral  packages but that's that's not the real  answer so this is the two of us and Joe  Mitchell it's also that I think the  introduction of the term computational  origami but those those terms come from  earlier references so in that paper  there's a sentence classic open question  origami mathematics and we don't really  know where it came about but it was  first formally posed by Bern and Hayes  in this soda 96 paper which we'll be  talking about in the next lecture  lecture three and this is a quote from  their paper is every simple polygon when  scaled sufficiently small the silhouette  of a flat origami the point of saying  the word silhouette is that when you  fold something like this number six  there's a whole bunch of layers and  there's a lot of complexity to this  folding by saying silhouette we just  mean you know  collapse all the layers ignore the  coloring and just take the outline so  the silhouette of this thing is a  rectangle and in general that's the sort  of the transformation to throw away the  complexity of the folding and say I just  care about the shape can I get the  desired shape there's some other  interesting questions here though which  haven't been fully addressed how many  creases are necessary to fold well  actually  to that in a later question how thick  must the origami be this the strip  method can shows that if you start from  a rectangle of paper or the number of  layers can be very small I think 2 or 3  is enough for all the gadgets that we  use probably 3 if you start from a  square though we don't know the answer  to that question and in practice folding  through many layers is tough yeah so  that was the silhouette problem and then  the gift-wrapping problem and the  motivation is you have a weird-shaped  gift you want to wrap it with a piece of  paper and this is posed to us in a talk  by Janaki Amma at a Canadian geometry  conference jun akiyama is really cool  guy he he has a math or has had for many  many years on mathematics TV show in  Japan and he's known throughout Japan  because everyone in school watches his  videos and it covers really interesting  mathematics some of the results in this  class are actually in his videos as well  it's mostly in Japanese so it's a little  hard to for most of us to watch but  there's some subtitle versions and  they're really fun maybe we can have a  movie night and watch one of them I get  permission so there isn't a great  reference for that I mean written some  papers about different kinds of wrapping  problems but mostly it was this talk  that he gave in 1997 which is when I was  just starting out in computational  geometry that's where the terms come  from a lot of words have you ever  actually folded a model using this  method of zigzagging and folding with  this strip any real or sensible or  pretty origami models or is it purely  for the sake of universality my  knee-jerk reaction was no way is is  practical this is just for universality  and the point of this theorem has always  been in my mind to prove that everything  is possible but then the challenge is to  find good foldings for some notion of  good but actually there is a lot of  strip or there's there a bunch of  examples of strip folding not a lot of  different folds and not terribly many  gadgets but there's some cool things  especially with strip weaving these are  just a few examples of woven colored  strips you can make fun things like  space invaders you can weave together  baskets and do  wrap your packages and so on so this is  a little bit more bar gammak so there  were no real folds in there except at  the edges of the cube this one is a  modular origami it involves a bunch of  different folds to get the all the  pieces to lock together modular origami  means you have a bunch of identical  pieces so they kind of weave together  through folding and then you can make a  nice little crown this is a very classic  model you've probably seen it at some  point here that the paper is slit and  then it's woven so but there is there's  some folds down here so not a lot of  folding but strips are pretty neat you  can definitely use them all sorts of  different designs here's some more  sculptural design ii models from taking  strips of paper this has no glue in its  I think there's more strips at the end  lock locking this together and this guy  Zachary footer ur2 took a bunch of these  kinds of units and started weaving them  together and make really complicated  shapes so you can definitely do cool  things with strip folding and another  sort of common one around these days is  taking gum or candy wrappers and weaving  them to get folding them down into  little strips and then weaving them  together to make handbags and other  things become kind of a fashion trend  over the last few years so those are  things you can do with strips we have  used it in one paper that's where the  goal is actually to be efficient and use  a small piece of paper and not just  prove some universality result this is  in our paper folding a better  checkerboard which we will talk about I  think in two lectures if I recall  correctly in more detail but this is  sort of a baseline this is not the  better method just to develop in this  paper but it's sort of the starting  point which is you take a square so this  actually starts with a square you do  this pleating and you get this this is  with bicolor paper it's dark on one side  light on the other you get this strip of  squares in color pattern and then you  take that strip a huge number of layers  in the middle so it's not super  practical but it's actually pretty  efficient in terms of how big a square  you start with too  an n-by-n this is obviously not to scale  you need more squares here in order to  make this thing and then you just sort  of snake your path back and forth you  could use turn gadgets or here we're  just using 45-degree folds and this is  pretty close to what was believed to be  the best way to fold a checker board and  then this paper shows how to do a factor  of two better so we'll talk about that  later but there are some uses for strip  folding this is a little bit theoretical  but it's actually pretty competitive  against the best end by end checkerboard  foldings in the origami world like the  one I showed last class so that's  practicality of strip folding next  questions more about strip folding there  are a couple things that are in the  lecture notes the handwritten lecture  notes but we're not even mentioned in  the audio part of the lecture so a few  people asked about what what are these  things said pseudo-polynomial  upper bound pseudopolynomial is a fun  term let me tell you a little bit about  it that's from the algorithms world but  even a lot of alguns people don't know  it so let me tell you so maybe first I  should tell you about polynomial in  general what these terms are about is  measuring how fast an algorithm is so  the idea is you plot conceptually  n this is the problem size so if you  wanted to fold an arbitrary polyhedron  the one way to think of the problem size  is the number of vertices edges and  faces just the the total number of  things you're given as input and then  your output is whatever but n is  supposed to be the input problem size  and then on the y-axis you want to do  you want to plot running time of your  algorithm so this is how long it takes  to compute the way to fold your square  paper into your desired shape and  generally this is going to increase and  the question is does it increase in a  reasonable way or in a crazy way you  know that goes exponentially  so you want to know how does the running  time grow with n polynomial is a sense  of good growth and it just means you  grow like n to the C where C is some  constant so ideally you'd have n or  maybe you have N squared or n cubed or n  to the fourth all these are considered  good running times not quite as good as  polynomial is pseudo polynomial and I  would conjecture for this problem of  folding an arbitrary given polyhedron  you cannot achieve a polynomial number  of folds let's say so there are two  things we could measure here the running  time the algorithm we could measure  actually three things we can measure the  number of folds you make number of  operations you do sort of on the paper  and a third thing would be scale factor  how big a square do I have to start with  in order to make desired polygon and  pseudopolynomial means n times R to the  C what's R R is some geometric parameter  geometric ratio in the input and in  particular for this problem what makes  sense for our is basically the longest  length divided by the shortest length  this is typically what R refers to this  will come up in later lectures as well  so for example you take your entire  shape you measure the sort of the  diameter of the shape the two farthest  points that's your longest length  shortest length would be maybe you have  a triangle something like this in this  target poly huge and you want to make  this would be your shortest distance  this is actually called the minimum  feature size in computational geometry  or the minimum altitude of any of your  triangles okay so that's just some  number and you know you can have a  triangle just  super super narrow and so it's this  ratio are could be arbitrarily large  even though you only have three vertices  three edges one face so N&R are not  necessarily comparable so that's why in  pseudo polynomial we put them both  together and then we raise them to some  constant power that's a pseudo  polynomial running time so the the the  question is being posed here is can you  get a pseudo polynomial  upper bound and can you get a pseudo  polynomial lower bound and it doesn't  say for what but it's for all three  problems running time number folds scale  factor and not all of these are open so  in the original paper there's this  theorem that says lots of things you  know you can fold anything and it says  here the folding requires a number of  folds polynomial and N and the ratio R  so it already claims that there is a  pseudo polynomial bound on the number of  faults it doesn't say what that pseudo  polynomial bound is is it n times R is  it n plus R is it n times R squared  I would guess one of the first two and  times RN plus R so that's the upper  bound question maybe we can work on this  in a problem session a lower bound  question is do you prove that you need  some dependents both on N and R which I  would guess is pretty easy if you want  to take a square and fold it down to a  really really skinny triangle I think  you need at least our faults roughly and  similarly should need a at least n folds  so there should be a lower bound of like  n plus R but none of these have been  written down explicitly so that's what  those open questions are then there's  another slide 6 which is completely  uncovered it's not me if they're  questions I should maybe take a brief  moment to breathe so the next part of  the lecture notes s about seam placement  so seam placement it's the following  kind of issue when you fold like this  number it's a six this is a nine-fold  this number nine in addition to seeing  the color pattern if you look closely  there's also these kinds of seams this  white square is not just a white square  you can see  on the top layer this crease line and  here there's a seam here this the seam  these are like visible lines of course  you have to have seams at the color  transitions but there's other seams as  well maybe you want to minimize those  seams you want to get you want to place  the seams in a cool pattern when you  fold checker boards there's a such thing  as a seamless checker board where every  square is a whole square paper there's  no visible crease lines on the top layer  so this is an extension of the  universality result to also get sort of  universal seam placement and what what  the original paper proves is that you  can place the seams as long however you  want provided the seam regions the  regions between the seams are convex  polygons which is almost always the case  you look at a typical model you know  here this the seam regions are all  rectangles and triangles so this could  you could you could achieve exactly the  same pattern if you wanted it you could  also say oh well here's a nice rectangle  I'll make that a seam region here's a  nice rectangle make that a seam region  but you could not make the entire number  9 here a seam region because it's  non-convex least you can't do it with  this technique we don't know necessarily  whether this is possible by some other  folding I would guess no but it is  possible to make some non convex seam  regions for example I could take this  page and fold the corner over and now  I've got a non-convex seam region here  so some non convex seam regions are  possible open question is what seam  what if I give you a polygon we know  every polygon is possible now I give you  a polygon and I subdivide it into seam  regions which of those are possible not  everything is possible I'm pretty sure  though I'm not sure we have a proof of  that some things like this little heart  shape are possible characterized this is  another cool possible problem for a  problem session  questions about that okay  so ah I have a little bit about the  proof of how this is done if you wanted  to just do convex regions so the general  approach here for is you we want to  visit all of the regions in some order  this is called a tour it's pretty easy  to just I mean you're allowed to visit  regions more than once so you just keep  going keep trying to visit some  unvisited seem region when you visit a  seam region it's a convex polygon so  what we're gonna do is make our strip  fairly wide actually wide enough to  completely cover that seam region then  at this moment we'll basically need to  turn to do the next one and so there's  we know how to sort of change the  direction of this strip using a turn  gadget then we have to change the width  of the strip maybe it needs to be wider  maybe it needs to be thinner and then we  need to shift the strip one one way or  the other so you know if we just end  here we turn we might be misaligned so  we need to shift it over expand it then  do the next one then turn then shift it  over then set the right width keep going  like that okay that's pretty messy and  complicated but you can do it with these  two gadgets strip with gadget you take a  strip and you can make it anywhere  between 1/2 and 100% of its original  width so the idea is you start with a  really wide strip wide enough to cover  all the polygons then you do this gadget  and keep shrinking it by half until it's  roughly the right size and then when  you're almost correct you shrink it by a  little bit more here drinking to 1/3 and  then you get your shrunken strip and it  happens right at the the line that you  specify so you can basically on a dime  shrink your strip and then by doing the  reverse you can grow it back this is  maybe not with simple folds though then  the other gadget is a shift gadget where  you're just you're at this position you  want to shift up so that's pretty easy  you just do to turn gadget  it's okay so that's how at a high level  how that making a desired seam pattern  works go on to the next question a lot  of people asked about this and so this  is an open problem that I mentioned  orally it's not written in the notes  which is can you actually do the things  that we said we can do with simple  faults so can you get a universal  folding of a polygon to color pattern  polyhedron using simple faults and I  thought it would be fun to actually work  on this here live because I think this  is an easy problem and there's a bunch  of possible answers and there are even  two suggested ideas from the comment  skill so let me just remind you of the  issue what's happening and then I need  your input what's gonna work here so  general picture for strip method was we  do one triangle we end here then we do a  bunch of folds like this and then we and  here maybe and then we sig zag and the  trouble is we've already made this  triangle over here when we make this  triangle we have this excess stuff which  I haven't drawn very accurately but you  recall it looks something like that or  maybe even more like that if we use  right angle turn gadgets and then we  want to fold it underneath and we're  doing that the way I said with hi gadget  Mountain folds but the model of simple  Falls which I should make may be more  explicit you're not allowed to collide  during the motion  it's the idea with the simple fold is  that you should be folding along one  line segment and you should fold by the  model that we defined back in the day  I'll I'll talk more about where this  notion comes from you fold by plus or  minus 180 degrees  which means after you do the fold you'll  be flat again and no collision during  the motion so if we folded this triangle  and then we folded this one and this  stuff is on the top we can't mountain  fold that's not considered a simple fold  now one proposal is could we just make  the next triangle underneath the  previous one a different proposal is  Valley fold these are actually different  proposals because especially for two  color patterns it'll make a difference  if we weave Valley fold here there's  gonna be some junk on the front side  especially if you want to get a desired  seam pattern but maybe we'll leave seam  patterns for later if you want to get a  color pattern you might reveal some  wrong color when you do that Valley fold  so I haven't really thought about this  idea yet I think it might be good so the  idea there would be so you've already  made this triangle you've out you  Mountain fold everything now when you go  when you do these zigzags you want to be  underneath everything that you've done  yeah okay different idea as you  basically fold this out of the way do  this thing hide gadget and then fold it  back maybe I've wondered about that too  it's it's gonna get a little challenging  them in general there's a huge set of  triangles so unless you can like go far  away make your triangle and then plop it  down maybe it's possible ah I guess we  can pursue that idea but maybe first we  should exhaust the easier ideas so I  mean that is definitely plausible that's  possible doing that with simple folds  and not leaving any garbage is gonna be  a little challenging but might be doable  this to me is the simplest idea so we  should first see if it works anyone see  problems with this plan  so guess we should think about I have  some strips we could think about what it  means to be doing a turn gadget  underneath here so I don't know quite  about well let's suppose we're already  here  and now maybe I do some as a turn gadget  goat some mountain folds then a at 90  degrees and then a valley fold like that  and then I do a mountain fold I think it  might be okay because turn gadgets start  with a mountain fold which so if you're  underneath everything that's gonna avoid  collision and then the valley fold is  brings it back with again we're using  just the space below everything that we  see so then we make that strip we keep  turning around and then later on we're  going to mount and fold this behind  somehow to meet this edge no seams okay  yeah question  yeah going from one triangle to next  yeah so we could think about that too  it's it's very you're right I mean I'm  just looking at the turning-around part  for making a single triangle but there's  also the turn gadget going from here to  here it's actually slightly more  complicated that we've covered in class  but not really much it's just a slightly  generalized trend gadget so you're  coming here you basically want to turn  around so but let's just think about a  way to do it you could imagine first  doing a turn like this  it's not exactly a pure turn gadget and  then turning around to get to the next  place this is really hard to do on a  blackboard strips just tend not to stay  together very well  okay now we're going parallel to the  correct direction then we turn back and  forth but each of those is just using  turn gadgets so as long as the turn  gadget works fine a turn gadget is gonna  be a mountain fold we're just gonna go  behind everything and then a valley fold  to bring it to a desired direction those  are all using everything behind the  board so it seems like all those  operations are okay now we should also  check the color reversal gadget which  it's funny thing I remember everything I  did before 2000 or so so I still have  memorized the color reversal gauge at  least I think so  I should probably color this piece of  paper so you can see the colors change I  don't remember anything I've done since  2000 but anything up to 2000 I'm okay  this is 1998 I think I remember falling  Lots this is ticker tape they use for or  not ticker tape but just use this for  like adding machines so I think it's a  90 degree mountain fold then you fold up  like that and then you fold back down  with a mountain fold like that and you  get a color reversal on all of those  folds were working behind my plane here  so should avoid collision with  everything I think you can do color  reversal and turn gadgets behind and so  this suggestion works who made this  suggestion good idea so unless there any  objections I think that will work I had  a different plan which was to use this  second idea and set up the turn gadget  so there was no when you fold this with  a valley fold there's no ugly colors so  you could maybe modify the turn gadget  to be completely solidly colored on both  sides but I think this is much easier  it's probably why I didn't write it down  the notes but I'm not sure yeah  go far away make a triangle okay how do  you do that last part what's so I mean  you have to do it with simple folds  that's the main that's the challenge so  the idea is you you're way out here you  have a triangle out here which you want  to bring over here maybe I should do it  like this so you could do something like  this so now the triangles over here then  maybe you want to go most all the way  here and then fold it back and then fold  it forth and back until you get your  triangle exactly where you want it it  seems possible my only concern would be  when you do this thing there might be a  little little corner depends how you  fold this thing and then you've got to  hide that corner and if there's  triangles all around here there may not  be room I mean maybe there's triangles  all around here it's okay to have that  corner but maybe the triangles are  different colors so I I do believe that  should be possible it's but I think it  is a little bit more complicated because  you have to hide one last piece after  you get it in position anyway I think  there are at least three ways to solve  this problem  yeah are you allowed to unfold it's a  good question I don't remember whether  the original model says whether you're  allowed to unfold so there are two  versions simple folds and unfolds or  just simple folds I don't think we  actually said unfolding is allowed so  we're definitely thinking about at some  point it's probably not in the model as  defined there questions all right what's  next this is the paper that introduced  simple folds it's called when can you  fold a map because it originally was  motivated by a map folding and it had a  bunch of reasons for introducing simple  folds  among them here's this quote which if  you've watched l1 as in there I think  the easiest way to refold a road map is  differently coalesce to make it easier  to refold your road map correctly but  here's one quote from that paper as  motivation  so is origami motivation but we're also  wondering about applications like sheet  metal bending cardboard folding things  like that where you want to manufacture  things using a machine  and while origamist can do complicated  folds non simple folds to make artwork  practical manufacturing you want to have  the simplest possible machine so if you  can get away with just simple folds as  defined here that would be great now you  don't really need some of these are  maybe artificial you probably don't need  the 180 degree condition because most of  the things you want to fold aren't flat  we introduced that just to keep things  simple  mathematically but you'd like to fold  along just one segment at a time ideally  you definitely don't want collision so  you don't want the material to hit  things whereas in origami you can do  tuck so you can do things that you know  are not simple folds it's a lot harder  with a machine that doesn't have any  feedback so here's a very simple machine  this is a a break folder we actually  have a break folder in CC although this  one is electro break so this has an  electric assist so the idea is you slide  your sheet in and you hold here you pull  up and in this case is bending to a 90  degree angle but you can adjust it to  different angles and so on so that and  there are lots of automated machines a  little hard to get photos and videos of  them but they're based on this principle  maybe you you push in a V and you end up  with a crease in one spot and you'd like  to just make sort of conveyor belt with  lots of different pushes and pulls and  do a bunch of simple folds basically  except for this 180-degree constraint  and so we're just curious about what's  possible by simple folds and that led us  into the mat folding stuff where it's  fairly easy to characterize other things  where it's harder which we'll see in  lecture 3  I thought I'd show you some examples of  things people make with pretty much  simple folds other than this 180  condition out of things like this is  folding wood so you take sheet material  and you start bending these parts up and  you can make a little chair a little  table and you can fold it back when  you're not using your living room you  can hide everything so you could imagine  also having multiple sheets and  sometimes your room is a living room  other times it's a whatever furniture  you need you just unfold the appropriate  thing it's kind of a vision here's a  cute little folding card there's a huge  number of folding chairs but this one is  pretty much simple folds the one thing  I'm not sure where there falls under  simple fold is this fold so it's you do  fold along one line but it's in two  different pieces not sure we'd call that  a simple fold there's of course lots of  slits in the material here but of course  it has all the same advantages of simple  folds this is easy to execute one step  at a time here's some more complicated  designs some of these are computer  rendering some of these are real again  too taking furniture out of flat walls  here's some table designs these are of  sheet metal I like I like this one it's  very simple take a square of sheet metal  put in some slits do some very simple  folds boom you've got a table this one's  also pretty simple here we again here  we're folding along one line but it's in  two different pieces so is that a simple  fold is definitely harder to build such  a machine but it's doable here we have  something's definitely not a simple fold  but it's also fairly easy to execute  using our roller you can kind of curve  one segment so I mean when you go to  reality there's you can change the model  in all sorts of different ways and still  have something practical and no one rule  set is is gospel but mathematically we  have to hone in on at least one model at  a time and then we can see how changing  the rules changes what you can make okay  next question this is actually about the  definition of simple folds so it's  probably answered already is allowed to  bend the rest of paper to get it out of  the way  avoid collision the answer is no in  simple folds at least you're you're only  allowed to move that one segment we have  actually lately been thinking about a  different model where you do allow this  but simple folds  you can't move other parts you can just  move the single hinge that you're  folding and the end product has to be  flat yes in our model though it'd be  interesting to think without this  condition because you're doing 180  degree operations before you do the next  one you will be flat at all times  1d or 2d according to whether you  started with a 1d piece of paper or a 2d  piece of paper okay so in then the  second half of the lecture was basically  about proving characterizing flat  foldability of 1d segments and it showed  in particular that simple folds are  universal that you if you have some  Mountain Valley pattern and it's  foldable at all if it's flat foldable it  will be flat foldable via simple folds  and in particular using crimps and n  folds so let me and there and it was a  bit of a messy proof partly because I've  made a mistake in lecture as you saw I  kind of corrected for it on the fly but  it's maybe not the best written so I  wanted to go through a couple quick  examples to make clear all the issues  there  so here are the ones I prepared we can  certainly do more if it's still not  clear so here's a simple Mountain Valley  pattern and it's got some long segments  and let's just say equidistant segments  here 3 valleys than a mountain so first  question is is this mingling and then  this the ultimate question is is it flat  foldable  so is it mingling well maybe you can  answer for me just yes or no 50% chance  okay so maybe the definition of mingling  is not super clear let me review it so  you look at each I mean generally of a  sequence of mountains and valleys so you  look at a chunk of all valleys then you  look at a chunk of all mountains and  chunk of all valleys here there's only  two chunks three valleys and one  mountain and the definition is a little  awkward for a single crease but let's  start with the all valleys the point is  to check for the first segment between  two valleys versus the segments just  before it which is bigger and this is  the bad case this is the non mingling  situation because this thing is bigger  than this strictly bigger it's strictly  bigger the notation we used and the  lecture is an open square bracket so  square bracket meant that this is bigger  round bracket would mean this is less  than or equal to this that's just the  definition over on this side of the  valleys this length is equal to this  length so the last distance between two  valleys is equal to the one right after  it and so that's a good case so we write  a closed round bracket then we have a  sequence of mountains and here it's a  little confusing but it's the same idea  you look at so this is the very first  mountain you look at the length right  after it versus the length right before  it and this is smaller and so that's a  good case we write an open round bracket  for this mountain group and then same  thing now we're comparing the same two  distances but it's now bad because this  one is strictly longer than this one so  we write a closed square bracket so  that's the notation in this example any  questions about that so you just have to  check we're looking in general you have  a whole group of valleys because these  are all valleys or all mountains and you  want to compare this one versus this one  and it's square closed according to  which is bigger and you want to look at  the last one versus the after last one  okay so that's the notation and the  point of the proof was to argue that  either there's a if you're gonna be flat  foldable at all  actually if you're mingling mingling  meant that for each of these intervals  at least one of the sides was round that  was considered good  so this crease pattern is mingling  because there's two regions this one has  a round bracket this one has a round  bracket and what we argued is that if  you're mingling you which was necessary  if you're flat foldable you have to be  mingling it's a necessary of a not  sufficient condition for flat  foldability if you're mingling either  you have a pattern like this close round  bracket open round bracket that's good  because this is a crimp that you can do  and you can see it up here this is a  crimp you Valley fold mountain fold and  you don't hide anything when you make  that operation or there's an end fold  which correspond it to an open round  bracket at the beginning or a closed  round bracket at the end so here there's  a crimp let's do the crimp so when we do  the crimp let's keep this part of the  paper fixed so this we go over to here  roughly then we valley fold then we  mountain fold and we keep going from  there it's that segment is that signal  okay so we still have this this valley  this was the valley we just folded this  is the mountain we just folded now  conceptually we just sort of fuse this  back into the paper because those  creases are done we don't need to think  about them the point is in that region  there were no extra creases this these  round parentheses will guarantee there's  nothing here no creases here here or  here could be creases farther away but  you buy these inequalities that this  length is less or equal to this one and  this length is less or equal to this one  let's try greater than or equal to equal  to then you know this is okay okay crimp  so there's two valleys left so now we  have two valleys we have a long segment  in a long segment and this is something  that can't be made because if you I mean  there's no folded state of this thing  never mind simple folds because it's  gonna cross like that so this is not  flat foldable yeah it's also not  mingling because if you look at these  two valleys you look at the distance  over here that's bigger than this one so  that's bad so you have an open square  bracket and this one is also bigger than  this imagining these is fused and so  it's a closed square bracket and so this  group of valleys is not mingling so it's  not mingling so ultimately this pattern  we started with is not flat foldable  because one of the things we proved is  that doing a crimp never changes flat  foldability it's always a safe thing to  do so I mean there was I mean you might  wonder oh maybe there's some other fold  I could do that eventually works but we  proved crimps are always safe to do so  and we did it and we got stuck that  means this was not flat foldable even  though it was mingling and so the  mingling forever property just means if  it's mingling and when you do a crimp  and is still mingling and if you keep  doing crimson it stays mingling all the  way then you were flat foldable it's not  a very satisfying characterization but  it is a thing maybe I'll do one more  example just to where it works  we're super clear Oh out of time so I  won't do another example all right  ambitious yeah and I got to work on my  timing there are a couple other fun  questions here I would encourage you to  read the notes about them in particular  there's an algorithmic question how do  you actually compute this efficiently  you can do it very efficiently in linear  time so and where C is one just n time  instead of N squared or something using  a pretty simple idea basically just look  for the first crimp to do it and then  see if they're crimps nearby and keep  going forward and you can prove that  takes linear time there is this fun  question I enjoyed thinking about can  you make any Mountain Valley pattern  flat foldable by adding creases the  answer is yes I can think of it as a  puzzle there's one proposed way to do it  here I have another one in the notes you  can think about it and the last question  is what is it possibly mean to fold  something in four dimensions how do you  imagine it hard to imagine but you can  think about it you have a d-dimensional  piece of paper you fold it through D  plus one dimensions if you want to flat  fold it it ends up back in D dimensions  and your creases are D minus one  dimensional and the rest you just have  to visualize I have one example of  folding a solid cube in half in the  notes that's certainly possible it's not  very well studied and there's lots of  interesting open questions about higher  dimensional folding any questions before  we go all right  watch lecture three and please send your  feedback is really helpful  you

2
00:00:05,679 --> 00:00:07,990
alright let's get started if you haven't  already there's two handouts on the left  and I should take two pieces of paper so  we'll be doing some actual origami  folding we'll be folding six eight four  nine today just like this it will only  take us eight hours or so this is the  Jenny and Eli folding you've seen on the  coaster pretty awesome now we'll be  folding butters more like this so thanks  everyone for giving so many cool  questions and comments and the feedback  is really helpful I was I didn't know  what I was gonna cover in this class I  had too many ideas and your questions  really help narrow it down into exciting  things so the structure is gonna be I'm  gonna go through questions so it's a  funny kind of interactivity where  there's a whole day in between where you  ask the questions and where I answer  them but feel free to ask more questions  follow up things but there's a lot  already here so it should be fun these  are not questions but sort of funny er  comments if you like double rainbow  jokes this was this this lecture  happened like a couple of months after  the double rainbow fiasco there's a lot  more as I recall it was sort of a  running theme throughout the whole  semester so look out for that I know  this is gonna this is entertaining I'm  getting used to listen to myself at  double speed should be fun this class  definitely is nice in the way that with  fairly simple technique you can prove a  very powerful theorem and that's sorry I  just remembered I need to push a  different button here that's that's  hopefully a theme throughout the class  but definitely especially nice here so  it's kind of cool though strip folding  if it was an open problem for a couple  years at least and no one thinking how  do we fold any shape that seems really  tough the point of this the strip  folding approach is that once you have  the right idea just start with a really  long rectangle somehow it becomes easy  there's still a lot of details and  getting that to work but it's kind of  neat how that that works out and now we  get to actual  all proposals so folding practice  planning on doing this but it's an  explicit comment to that effect so we're  going to fold some letters of the  alphabet you have in your packet  instructions for making the individual  digits six four eight and nine and you  also have which are all pretty easy you  also have a diagram for it's crazy  design called typeset which can desync  this same folding can make any letter of  the alphabet and any digit so just to  show you what they will look like these  are my foldings of six eight four and  nine according to the first set of  diagrams and then this is my folding of  think the letter the number six rather  out of jason coos design so if you  wanted to reconfigure it to i guess  eight is kind of hard let me do four  think okay so four four I've got a fold  this guy under fold this here this tab I  think goes back here so the advanced  origami folders can definitely do in  this classic and if they do the Jason  coos design but it it takes a little  while I think there I've got a four got  it  a little hard to hold in position but at  least in theory it will make any letter  all out of one one folding you just have  to move all the tabs around so it's kind  of neat but it takes at least half an  hour so to fold that unless you're  really fast so I would recommend pick  one of these working groups if you like  if you want you can form a group of 4  and make 6 8 4 & 9 follow the diagrams  this is an exercise in following  diagrams one way or the other only have  time to maybe make one digit each but  have fun with it and if you have  questions raise your hand I can tell you  the first step in 6 8 & 9 is to make an  8x8 grid there's a lot of ways to make  an 8x8 grid but an easy one is shown  here so you take your your sheet you  fold the bottom edge to the top edge I  think they want to do it white side up  again so they're all valleys when you  align those edges you'll get a nice  bisector then you repeat folding the  bottom edge to the middle and once you  do that to save a little bit of time you  can then fold that new bottom edge to  the middle again that will do eights on  one side it's a little bit inaccurate  because you're folding through two  layers but it's a little faster time is  of the essence then you do the same  thing on the bottom and you'll get  ates in one dimension but then you have  to fold it in eighths in the other  dimension  once you have eighths it's like three  steps it's really easy four six eight  and nine four uses a different approach  it's a little more free hand if you want  to be more creative try the four you've  got to kind of eyeball what looks and  feels good for number four  so out of curiosity how many people have  an 8x8 grid at this point you want to  take full so that's the top half of  these diagrams to 8x8 grid then it's  mostly folding up over individual edges  and some corner folds but they're all  they're all simple folds in this world  so you're folding through I think always  all the layers on this this is only  folding through one layer but these  would always fall into the some layer  simple folds category so for example  make a six this bottom edge up fold the  left three squares over this corner up  fold this corner down  fold this down this over  there's really big six and these numbers  are all pretty much proportioned  correctly the four you have to be it  helps to have a reference model one of  the other digits to make it the right  height but it'll end up roughly correct  anyway it's a little bit narrow and it's  designed it's slightly narrower than the  six and the nine but otherwise they're  nice compatible digits there's a whole  alphabet on the website that's linked  from this slide so you can go check it  out the origami Club so it's kind of fun  to think about font design and alphabet  design there's actually a lot of origami  alphabets out there this is one of the  simplest that has digits anyone have  questions how many people folded a  letter a digit okay so maybe I will  you can keep folding but maybe I'll  continue on it's gonna be a lot of fun  crackling anyone folded this one anyone  working on it uh-huh  a bunch of people cool well let me know  when you finish it's kind of fun it's  not that hard I wanted to point at the  origami website or gham it is the  origami Club at MIT and at the top you  see a different alphabet this is a four  fold alphabet designed by Jeannine  Mosely who's an MIT alum and came to  this class two years ago and so that's  the reference design in 2002 I'm not  sure if it has digits though at least  the the diagrams we found for the  letters do not also have digits so I  think an interesting challenge is to  design a for fit for fold digit set to  compliment her letter set if you're  interested that could be a cool project  to work on folding design minimal fold  alphabets you could try a three fold  alphabet to fold alphabet whatever also  on this website are the meeting schedule  it's always Sundays sometimes two  sometimes three o'clock check it out  and there's a convention coming up our  very own origami convention on October  27th and Jason ku wanted me to to remind  you about that coming up so if you even  if you've never done origami before  other than today you should check it out  it'll be fun lots of different sessions  on from simple models to complicated  models this is a cool design by Brian  Chen another MIT alum it's one square  paper folded into the men's at Montes  logo but here instead of the oil lamp  you've got a little origami crane cool  stuff cool so we proceeded on to other  questions this is a pretty simple one at  the top of the note says folding any  shape aka silhouette folding and gift  wrapping it has a couple of references  so where do those terms come from it's  the question and one answer is it's the  title of the paper folding flat  silhouettes and wrapping polyhedral  packages but that's that's not the real  answer so this is the two of us and Joe  Mitchell it's also that I think the  introduction of the term computational  origami but those those terms come from  earlier references so in that paper  there's a sentence classic open question  origami mathematics and we don't really  know where it came about but it was  first formally posed by Bern and Hayes  in this soda 96 paper which we'll be  talking about in the next lecture  lecture three and this is a quote from  their paper is every simple polygon when  scaled sufficiently small the silhouette  of a flat origami the point of saying  the word silhouette is that when you  fold something like this number six  there's a whole bunch of layers and  there's a lot of complexity to this  folding by saying silhouette we just  mean you know  collapse all the layers ignore the  coloring and just take the outline so  the silhouette of this thing is a  rectangle and in general that's the sort  of the transformation to throw away the  complexity of the folding and say I just  care about the shape can I get the  desired shape there's some other  interesting questions here though which  haven't been fully addressed how many  creases are necessary to fold well  actually  to that in a later question how thick  must the origami be this the strip  method can shows that if you start from  a rectangle of paper or the number of  layers can be very small I think 2 or 3  is enough for all the gadgets that we  use probably 3 if you start from a  square though we don't know the answer  to that question and in practice folding  through many layers is tough yeah so  that was the silhouette problem and then  the gift-wrapping problem and the  motivation is you have a weird-shaped  gift you want to wrap it with a piece of  paper and this is posed to us in a talk  by Janaki Amma at a Canadian geometry  conference jun akiyama is really cool  guy he he has a math or has had for many  many years on mathematics TV show in  Japan and he's known throughout Japan  because everyone in school watches his  videos and it covers really interesting  mathematics some of the results in this  class are actually in his videos as well  it's mostly in Japanese so it's a little  hard to for most of us to watch but  there's some subtitle versions and  they're really fun maybe we can have a  movie night and watch one of them I get  permission so there isn't a great  reference for that I mean written some  papers about different kinds of wrapping  problems but mostly it was this talk  that he gave in 1997 which is when I was  just starting out in computational  geometry that's where the terms come  from a lot of words have you ever  actually folded a model using this  method of zigzagging and folding with  this strip any real or sensible or  pretty origami models or is it purely  for the sake of universality my  knee-jerk reaction was no way is is  practical this is just for universality  and the point of this theorem has always  been in my mind to prove that everything  is possible but then the challenge is to  find good foldings for some notion of  good but actually there is a lot of  strip or there's there a bunch of  examples of strip folding not a lot of  different folds and not terribly many  gadgets but there's some cool things  especially with strip weaving these are  just a few examples of woven colored  strips you can make fun things like  space invaders you can weave together  baskets and do  wrap your packages and so on so this is  a little bit more bar gammak so there  were no real folds in there except at  the edges of the cube this one is a  modular origami it involves a bunch of  different folds to get the all the  pieces to lock together modular origami  means you have a bunch of identical  pieces so they kind of weave together  through folding and then you can make a  nice little crown this is a very classic  model you've probably seen it at some  point here that the paper is slit and  then it's woven so but there is there's  some folds down here so not a lot of  folding but strips are pretty neat you  can definitely use them all sorts of  different designs here's some more  sculptural design ii models from taking  strips of paper this has no glue in its  I think there's more strips at the end  lock locking this together and this guy  Zachary footer ur2 took a bunch of these  kinds of units and started weaving them  together and make really complicated  shapes so you can definitely do cool  things with strip folding and another  sort of common one around these days is  taking gum or candy wrappers and weaving  them to get folding them down into  little strips and then weaving them  together to make handbags and other  things become kind of a fashion trend  over the last few years so those are  things you can do with strips we have  used it in one paper that's where the  goal is actually to be efficient and use  a small piece of paper and not just  prove some universality result this is  in our paper folding a better  checkerboard which we will talk about I  think in two lectures if I recall  correctly in more detail but this is  sort of a baseline this is not the  better method just to develop in this  paper but it's sort of the starting  point which is you take a square so this  actually starts with a square you do  this pleating and you get this this is  with bicolor paper it's dark on one side  light on the other you get this strip of  squares in color pattern and then you  take that strip a huge number of layers  in the middle so it's not super  practical but it's actually pretty  efficient in terms of how big a square  you start with too  an n-by-n this is obviously not to scale  you need more squares here in order to  make this thing and then you just sort  of snake your path back and forth you  could use turn gadgets or here we're  just using 45-degree folds and this is  pretty close to what was believed to be  the best way to fold a checker board and  then this paper shows how to do a factor  of two better so we'll talk about that  later but there are some uses for strip  folding this is a little bit theoretical  but it's actually pretty competitive  against the best end by end checkerboard  foldings in the origami world like the  one I showed last class so that's  practicality of strip folding next  questions more about strip folding there  are a couple things that are in the  lecture notes the handwritten lecture  notes but we're not even mentioned in  the audio part of the lecture so a few  people asked about what what are these  things said pseudo-polynomial  upper bound pseudopolynomial is a fun  term let me tell you a little bit about  it that's from the algorithms world but  even a lot of alguns people don't know  it so let me tell you so maybe first I  should tell you about polynomial in  general what these terms are about is  measuring how fast an algorithm is so  the idea is you plot conceptually  n this is the problem size so if you  wanted to fold an arbitrary polyhedron  the one way to think of the problem size  is the number of vertices edges and  faces just the the total number of  things you're given as input and then  your output is whatever but n is  supposed to be the input problem size  and then on the y-axis you want to do  you want to plot running time of your  algorithm so this is how long it takes  to compute the way to fold your square  paper into your desired shape and  generally this is going to increase and  the question is does it increase in a  reasonable way or in a crazy way you  know that goes exponentially  so you want to know how does the running  time grow with n polynomial is a sense  of good growth and it just means you  grow like n to the C where C is some  constant so ideally you'd have n or  maybe you have N squared or n cubed or n  to the fourth all these are considered  good running times not quite as good as  polynomial is pseudo polynomial and I  would conjecture for this problem of  folding an arbitrary given polyhedron  you cannot achieve a polynomial number  of folds let's say so there are two  things we could measure here the running  time the algorithm we could measure  actually three things we can measure the  number of folds you make number of  operations you do sort of on the paper  and a third thing would be scale factor  how big a square do I have to start with  in order to make desired polygon and  pseudopolynomial means n times R to the  C what's R R is some geometric parameter  geometric ratio in the input and in  particular for this problem what makes  sense for our is basically the longest  length divided by the shortest length  this is typically what R refers to this  will come up in later lectures as well  so for example you take your entire  shape you measure the sort of the  diameter of the shape the two farthest  points that's your longest length  shortest length would be maybe you have  a triangle something like this in this  target poly huge and you want to make  this would be your shortest distance  this is actually called the minimum  feature size in computational geometry  or the minimum altitude of any of your  triangles okay so that's just some  number and you know you can have a  triangle just  super super narrow and so it's this  ratio are could be arbitrarily large  even though you only have three vertices  three edges one face so N&R are not  necessarily comparable so that's why in  pseudo polynomial we put them both  together and then we raise them to some  constant power that's a pseudo  polynomial running time so the the the  question is being posed here is can you  get a pseudo polynomial  upper bound and can you get a pseudo  polynomial lower bound and it doesn't  say for what but it's for all three  problems running time number folds scale  factor and not all of these are open so  in the original paper there's this  theorem that says lots of things you  know you can fold anything and it says  here the folding requires a number of  folds polynomial and N and the ratio R  so it already claims that there is a  pseudo polynomial bound on the number of  faults it doesn't say what that pseudo  polynomial bound is is it n times R is  it n plus R is it n times R squared  I would guess one of the first two and  times RN plus R so that's the upper  bound question maybe we can work on this  in a problem session a lower bound  question is do you prove that you need  some dependents both on N and R which I  would guess is pretty easy if you want  to take a square and fold it down to a  really really skinny triangle I think  you need at least our faults roughly and  similarly should need a at least n folds  so there should be a lower bound of like  n plus R but none of these have been  written down explicitly so that's what  those open questions are then there's  another slide 6 which is completely  uncovered it's not me if they're  questions I should maybe take a brief  moment to breathe so the next part of  the lecture notes s about seam placement  so seam placement it's the following  kind of issue when you fold like this  number it's a six this is a nine-fold  this number nine in addition to seeing  the color pattern if you look closely  there's also these kinds of seams this  white square is not just a white square  you can see  on the top layer this crease line and  here there's a seam here this the seam  these are like visible lines of course  you have to have seams at the color  transitions but there's other seams as  well maybe you want to minimize those  seams you want to get you want to place  the seams in a cool pattern when you  fold checker boards there's a such thing  as a seamless checker board where every  square is a whole square paper there's  no visible crease lines on the top layer  so this is an extension of the  universality result to also get sort of  universal seam placement and what what  the original paper proves is that you  can place the seams as long however you  want provided the seam regions the  regions between the seams are convex  polygons which is almost always the case  you look at a typical model you know  here this the seam regions are all  rectangles and triangles so this could  you could you could achieve exactly the  same pattern if you wanted it you could  also say oh well here's a nice rectangle  I'll make that a seam region here's a  nice rectangle make that a seam region  but you could not make the entire number  9 here a seam region because it's  non-convex least you can't do it with  this technique we don't know necessarily  whether this is possible by some other  folding I would guess no but it is  possible to make some non convex seam  regions for example I could take this  page and fold the corner over and now  I've got a non-convex seam region here  so some non convex seam regions are  possible open question is what seam  what if I give you a polygon we know  every polygon is possible now I give you  a polygon and I subdivide it into seam  regions which of those are possible not  everything is possible I'm pretty sure  though I'm not sure we have a proof of  that some things like this little heart  shape are possible characterized this is  another cool possible problem for a  problem session  questions about that okay  so ah I have a little bit about the  proof of how this is done if you wanted  to just do convex regions so the general  approach here for is you we want to  visit all of the regions in some order  this is called a tour it's pretty easy  to just I mean you're allowed to visit  regions more than once so you just keep  going keep trying to visit some  unvisited seem region when you visit a  seam region it's a convex polygon so  what we're gonna do is make our strip  fairly wide actually wide enough to  completely cover that seam region then  at this moment we'll basically need to  turn to do the next one and so there's  we know how to sort of change the  direction of this strip using a turn  gadget then we have to change the width  of the strip maybe it needs to be wider  maybe it needs to be thinner and then we  need to shift the strip one one way or  the other so you know if we just end  here we turn we might be misaligned so  we need to shift it over expand it then  do the next one then turn then shift it  over then set the right width keep going  like that okay that's pretty messy and  complicated but you can do it with these  two gadgets strip with gadget you take a  strip and you can make it anywhere  between 1/2 and 100% of its original  width so the idea is you start with a  really wide strip wide enough to cover  all the polygons then you do this gadget  and keep shrinking it by half until it's  roughly the right size and then when  you're almost correct you shrink it by a  little bit more here drinking to 1/3 and  then you get your shrunken strip and it  happens right at the the line that you  specify so you can basically on a dime  shrink your strip and then by doing the  reverse you can grow it back this is  maybe not with simple folds though then  the other gadget is a shift gadget where  you're just you're at this position you  want to shift up so that's pretty easy  you just do to turn gadget  it's okay so that's how at a high level  how that making a desired seam pattern  works go on to the next question a lot  of people asked about this and so this  is an open problem that I mentioned  orally it's not written in the notes  which is can you actually do the things  that we said we can do with simple  faults so can you get a universal  folding of a polygon to color pattern  polyhedron using simple faults and I  thought it would be fun to actually work  on this here live because I think this  is an easy problem and there's a bunch  of possible answers and there are even  two suggested ideas from the comment  skill so let me just remind you of the  issue what's happening and then I need  your input what's gonna work here so  general picture for strip method was we  do one triangle we end here then we do a  bunch of folds like this and then we and  here maybe and then we sig zag and the  trouble is we've already made this  triangle over here when we make this  triangle we have this excess stuff which  I haven't drawn very accurately but you  recall it looks something like that or  maybe even more like that if we use  right angle turn gadgets and then we  want to fold it underneath and we're  doing that the way I said with hi gadget  Mountain folds but the model of simple  Falls which I should make may be more  explicit you're not allowed to collide  during the motion  it's the idea with the simple fold is  that you should be folding along one  line segment and you should fold by the  model that we defined back in the day  I'll I'll talk more about where this  notion comes from you fold by plus or  minus 180 degrees  which means after you do the fold you'll  be flat again and no collision during  the motion so if we folded this triangle  and then we folded this one and this  stuff is on the top we can't mountain  fold that's not considered a simple fold  now one proposal is could we just make  the next triangle underneath the  previous one a different proposal is  Valley fold these are actually different  proposals because especially for two  color patterns it'll make a difference  if we weave Valley fold here there's  gonna be some junk on the front side  especially if you want to get a desired  seam pattern but maybe we'll leave seam  patterns for later if you want to get a  color pattern you might reveal some  wrong color when you do that Valley fold  so I haven't really thought about this  idea yet I think it might be good so the  idea there would be so you've already  made this triangle you've out you  Mountain fold everything now when you go  when you do these zigzags you want to be  underneath everything that you've done  yeah okay different idea as you  basically fold this out of the way do  this thing hide gadget and then fold it  back maybe I've wondered about that too  it's it's gonna get a little challenging  them in general there's a huge set of  triangles so unless you can like go far  away make your triangle and then plop it  down maybe it's possible ah I guess we  can pursue that idea but maybe first we  should exhaust the easier ideas so I  mean that is definitely plausible that's  possible doing that with simple folds  and not leaving any garbage is gonna be  a little challenging but might be doable  this to me is the simplest idea so we  should first see if it works anyone see  problems with this plan  so guess we should think about I have  some strips we could think about what it  means to be doing a turn gadget  underneath here so I don't know quite  about well let's suppose we're already  here  and now maybe I do some as a turn gadget  goat some mountain folds then a at 90  degrees and then a valley fold like that  and then I do a mountain fold I think it  might be okay because turn gadgets start  with a mountain fold which so if you're  underneath everything that's gonna avoid  collision and then the valley fold is  brings it back with again we're using  just the space below everything that we  see so then we make that strip we keep  turning around and then later on we're  going to mount and fold this behind  somehow to meet this edge no seams okay  yeah question  yeah going from one triangle to next  yeah so we could think about that too  it's it's very you're right I mean I'm  just looking at the turning-around part  for making a single triangle but there's  also the turn gadget going from here to  here it's actually slightly more  complicated that we've covered in class  but not really much it's just a slightly  generalized trend gadget so you're  coming here you basically want to turn  around so but let's just think about a  way to do it you could imagine first  doing a turn like this  it's not exactly a pure turn gadget and  then turning around to get to the next  place this is really hard to do on a  blackboard strips just tend not to stay  together very well  okay now we're going parallel to the  correct direction then we turn back and  forth but each of those is just using  turn gadgets so as long as the turn  gadget works fine a turn gadget is gonna  be a mountain fold we're just gonna go  behind everything and then a valley fold  to bring it to a desired direction those  are all using everything behind the  board so it seems like all those  operations are okay now we should also  check the color reversal gadget which  it's funny thing I remember everything I  did before 2000 or so so I still have  memorized the color reversal gauge at  least I think so  I should probably color this piece of  paper so you can see the colors change I  don't remember anything I've done since  2000 but anything up to 2000 I'm okay  this is 1998 I think I remember falling  Lots this is ticker tape they use for or  not ticker tape but just use this for  like adding machines so I think it's a  90 degree mountain fold then you fold up  like that and then you fold back down  with a mountain fold like that and you  get a color reversal on all of those  folds were working behind my plane here  so should avoid collision with  everything I think you can do color  reversal and turn gadgets behind and so  this suggestion works who made this  suggestion good idea so unless there any  objections I think that will work I had  a different plan which was to use this  second idea and set up the turn gadget  so there was no when you fold this with  a valley fold there's no ugly colors so  you could maybe modify the turn gadget  to be completely solidly colored on both  sides but I think this is much easier  it's probably why I didn't write it down  the notes but I'm not sure yeah  go far away make a triangle okay how do  you do that last part what's so I mean  you have to do it with simple folds  that's the main that's the challenge so  the idea is you you're way out here you  have a triangle out here which you want  to bring over here maybe I should do it  like this so you could do something like  this so now the triangles over here then  maybe you want to go most all the way  here and then fold it back and then fold  it forth and back until you get your  triangle exactly where you want it it  seems possible my only concern would be  when you do this thing there might be a  little little corner depends how you  fold this thing and then you've got to  hide that corner and if there's  triangles all around here there may not  be room I mean maybe there's triangles  all around here it's okay to have that  corner but maybe the triangles are  different colors so I I do believe that  should be possible it's but I think it  is a little bit more complicated because  you have to hide one last piece after  you get it in position anyway I think  there are at least three ways to solve  this problem  yeah are you allowed to unfold it's a  good question I don't remember whether  the original model says whether you're  allowed to unfold so there are two  versions simple folds and unfolds or  just simple folds I don't think we  actually said unfolding is allowed so  we're definitely thinking about at some  point it's probably not in the model as  defined there questions all right what's  next this is the paper that introduced  simple folds it's called when can you  fold a map because it originally was  motivated by a map folding and it had a  bunch of reasons for introducing simple  folds  among them here's this quote which if  you've watched l1 as in there I think  the easiest way to refold a road map is  differently coalesce to make it easier  to refold your road map correctly but  here's one quote from that paper as  motivation  so is origami motivation but we're also  wondering about applications like sheet  metal bending cardboard folding things  like that where you want to manufacture  things using a machine  and while origamist can do complicated  folds non simple folds to make artwork  practical manufacturing you want to have  the simplest possible machine so if you  can get away with just simple folds as  defined here that would be great now you  don't really need some of these are  maybe artificial you probably don't need  the 180 degree condition because most of  the things you want to fold aren't flat  we introduced that just to keep things  simple  mathematically but you'd like to fold  along just one segment at a time ideally  you definitely don't want collision so  you don't want the material to hit  things whereas in origami you can do  tuck so you can do things that you know  are not simple folds it's a lot harder  with a machine that doesn't have any  feedback so here's a very simple machine  this is a a break folder we actually  have a break folder in CC although this  one is electro break so this has an  electric assist so the idea is you slide  your sheet in and you hold here you pull  up and in this case is bending to a 90  degree angle but you can adjust it to  different angles and so on so that and  there are lots of automated machines a  little hard to get photos and videos of  them but they're based on this principle  maybe you you push in a V and you end up  with a crease in one spot and you'd like  to just make sort of conveyor belt with  lots of different pushes and pulls and  do a bunch of simple folds basically  except for this 180-degree constraint  and so we're just curious about what's  possible by simple folds and that led us  into the mat folding stuff where it's  fairly easy to characterize other things  where it's harder which we'll see in  lecture 3  I thought I'd show you some examples of  things people make with pretty much  simple folds other than this 180  condition out of things like this is  folding wood so you take sheet material  and you start bending these parts up and  you can make a little chair a little  table and you can fold it back when  you're not using your living room you  can hide everything so you could imagine  also having multiple sheets and  sometimes your room is a living room  other times it's a whatever furniture  you need you just unfold the appropriate  thing it's kind of a vision here's a  cute little folding card there's a huge  number of folding chairs but this one is  pretty much simple folds the one thing  I'm not sure where there falls under  simple fold is this fold so it's you do  fold along one line but it's in two  different pieces not sure we'd call that  a simple fold there's of course lots of  slits in the material here but of course  it has all the same advantages of simple  folds this is easy to execute one step  at a time here's some more complicated  designs some of these are computer  rendering some of these are real again  too taking furniture out of flat walls  here's some table designs these are of  sheet metal I like I like this one it's  very simple take a square of sheet metal  put in some slits do some very simple  folds boom you've got a table this one's  also pretty simple here we again here  we're folding along one line but it's in  two different pieces so is that a simple  fold is definitely harder to build such  a machine but it's doable here we have  something's definitely not a simple fold  but it's also fairly easy to execute  using our roller you can kind of curve  one segment so I mean when you go to  reality there's you can change the model  in all sorts of different ways and still  have something practical and no one rule  set is is gospel but mathematically we  have to hone in on at least one model at  a time and then we can see how changing  the rules changes what you can make okay  next question this is actually about the  definition of simple folds so it's  probably answered already is allowed to  bend the rest of paper to get it out of  the way  avoid collision the answer is no in  simple folds at least you're you're only  allowed to move that one segment we have  actually lately been thinking about a  different model where you do allow this  but simple folds  you can't move other parts you can just  move the single hinge that you're  folding and the end product has to be  flat yes in our model though it'd be  interesting to think without this  condition because you're doing 180  degree operations before you do the next  one you will be flat at all times  1d or 2d according to whether you  started with a 1d piece of paper or a 2d  piece of paper okay so in then the  second half of the lecture was basically  about proving characterizing flat  foldability of 1d segments and it showed  in particular that simple folds are  universal that you if you have some  Mountain Valley pattern and it's  foldable at all if it's flat foldable it  will be flat foldable via simple folds  and in particular using crimps and n  folds so let me and there and it was a  bit of a messy proof partly because I've  made a mistake in lecture as you saw I  kind of corrected for it on the fly but  it's maybe not the best written so I  wanted to go through a couple quick  examples to make clear all the issues  there  so here are the ones I prepared we can  certainly do more if it's still not  clear so here's a simple Mountain Valley  pattern and it's got some long segments  and let's just say equidistant segments  here 3 valleys than a mountain so first  question is is this mingling and then  this the ultimate question is is it flat  foldable  so is it mingling well maybe you can  answer for me just yes or no 50% chance  okay so maybe the definition of mingling  is not super clear let me review it so  you look at each I mean generally of a  sequence of mountains and valleys so you  look at a chunk of all valleys then you  look at a chunk of all mountains and  chunk of all valleys here there's only  two chunks three valleys and one  mountain and the definition is a little  awkward for a single crease but let's  start with the all valleys the point is  to check for the first segment between  two valleys versus the segments just  before it which is bigger and this is  the bad case this is the non mingling  situation because this thing is bigger  than this strictly bigger it's strictly  bigger the notation we used and the  lecture is an open square bracket so  square bracket meant that this is bigger  round bracket would mean this is less  than or equal to this that's just the  definition over on this side of the  valleys this length is equal to this  length so the last distance between two  valleys is equal to the one right after  it and so that's a good case so we write  a closed round bracket then we have a  sequence of mountains and here it's a  little confusing but it's the same idea  you look at so this is the very first  mountain you look at the length right  after it versus the length right before  it and this is smaller and so that's a  good case we write an open round bracket  for this mountain group and then same  thing now we're comparing the same two  distances but it's now bad because this  one is strictly longer than this one so  we write a closed square bracket so  that's the notation in this example any  questions about that so you just have to  check we're looking in general you have  a whole group of valleys because these  are all valleys or all mountains and you  want to compare this one versus this one  and it's square closed according to  which is bigger and you want to look at  the last one versus the after last one  okay so that's the notation and the  point of the proof was to argue that  either there's a if you're gonna be flat  foldable at all  actually if you're mingling mingling  meant that for each of these intervals  at least one of the sides was round that  was considered good  so this crease pattern is mingling  because there's two regions this one has  a round bracket this one has a round  bracket and what we argued is that if  you're mingling you which was necessary  if you're flat foldable you have to be  mingling it's a necessary of a not  sufficient condition for flat  foldability if you're mingling either  you have a pattern like this close round  bracket open round bracket that's good  because this is a crimp that you can do  and you can see it up here this is a  crimp you Valley fold mountain fold and  you don't hide anything when you make  that operation or there's an end fold  which correspond it to an open round  bracket at the beginning or a closed  round bracket at the end so here there's  a crimp let's do the crimp so when we do  the crimp let's keep this part of the  paper fixed so this we go over to here  roughly then we valley fold then we  mountain fold and we keep going from  there it's that segment is that signal  okay so we still have this this valley  this was the valley we just folded this  is the mountain we just folded now  conceptually we just sort of fuse this  back into the paper because those  creases are done we don't need to think  about them the point is in that region  there were no extra creases this these  round parentheses will guarantee there's  nothing here no creases here here or  here could be creases farther away but  you buy these inequalities that this  length is less or equal to this one and  this length is less or equal to this one  let's try greater than or equal to equal  to then you know this is okay okay crimp  so there's two valleys left so now we  have two valleys we have a long segment  in a long segment and this is something  that can't be made because if you I mean  there's no folded state of this thing  never mind simple folds because it's  gonna cross like that so this is not  flat foldable yeah it's also not  mingling because if you look at these  two valleys you look at the distance  over here that's bigger than this one so  that's bad so you have an open square  bracket and this one is also bigger than  this imagining these is fused and so  it's a closed square bracket and so this  group of valleys is not mingling so it's  not mingling so ultimately this pattern  we started with is not flat foldable  because one of the things we proved is  that doing a crimp never changes flat  foldability it's always a safe thing to  do so I mean there was I mean you might  wonder oh maybe there's some other fold  I could do that eventually works but we  proved crimps are always safe to do so  and we did it and we got stuck that  means this was not flat foldable even  though it was mingling and so the  mingling forever property just means if  it's mingling and when you do a crimp  and is still mingling and if you keep  doing crimson it stays mingling all the  way then you were flat foldable it's not  a very satisfying characterization but  it is a thing maybe I'll do one more  example just to where it works  we're super clear Oh out of time so I  won't do another example all right  ambitious yeah and I got to work on my  timing there are a couple other fun  questions here I would encourage you to  read the notes about them in particular  there's an algorithmic question how do  you actually compute this efficiently  you can do it very efficiently in linear  time so and where C is one just n time  instead of N squared or something using  a pretty simple idea basically just look  for the first crimp to do it and then  see if they're crimps nearby and keep  going forward and you can prove that  takes linear time there is this fun  question I enjoyed thinking about can  you make any Mountain Valley pattern  flat foldable by adding creases the  answer is yes I can think of it as a  puzzle there's one proposed way to do it  here I have another one in the notes you  can think about it and the last question  is what is it possibly mean to fold  something in four dimensions how do you  imagine it hard to imagine but you can  think about it you have a d-dimensional  piece of paper you fold it through D  plus one dimensions if you want to flat  fold it it ends up back in D dimensions  and your creases are D minus one  dimensional and the rest you just have  to visualize I have one example of  folding a solid cube in half in the  notes that's certainly possible it's not  very well studied and there's lots of  interesting open questions about higher  dimensional folding any questions before  we go all right  watch lecture three and please send your  feedback is really helpful  you
 

3
00:00:07,990 --> 00:00:11,650

 

4
00:00:11,650 --> 00:00:12,879

 

5
00:00:12,879 --> 00:00:16,359

 

6
00:00:16,359 --> 00:00:18,370

 

7
00:00:18,370 --> 00:00:22,900

 

8
00:00:22,900 --> 00:00:24,550

 

9
00:00:24,550 --> 00:00:27,370

 

10
00:00:27,370 --> 00:00:32,439

 

11
00:00:32,439 --> 00:00:35,490

 

12
00:00:35,490 --> 00:00:37,540

 

13
00:00:37,540 --> 00:00:40,389

 

14
00:00:40,389 --> 00:00:41,799

 

15
00:00:41,799 --> 00:00:43,600

 

16
00:00:43,600 --> 00:00:45,880

 

17
00:00:45,880 --> 00:00:47,350

 

18
00:00:47,350 --> 00:00:50,139

 

19
00:00:50,139 --> 00:00:51,580

 

20
00:00:51,580 --> 00:00:53,110

 

21
00:00:53,110 --> 00:00:54,520

 

22
00:00:54,520 --> 00:00:56,260

 

23
00:00:56,260 --> 00:00:58,060

 

24
00:00:58,060 --> 00:01:01,900

 

25
00:01:01,900 --> 00:01:04,770

 

26
00:01:04,770 --> 00:01:07,660

 

27
00:01:07,660 --> 00:01:09,370

 

28
00:01:09,370 --> 00:01:11,050

 

29
00:01:11,050 --> 00:01:13,929

 

30
00:01:13,929 --> 00:01:16,270

 

31
00:01:16,270 --> 00:01:17,469

 

32
00:01:17,469 --> 00:01:21,070

 

33
00:01:21,070 --> 00:01:22,660

 

34
00:01:22,660 --> 00:01:24,070

 

35
00:01:24,070 --> 00:01:27,730

 

36
00:01:27,730 --> 00:01:31,060

 

37
00:01:31,060 --> 00:01:32,920

 

38
00:01:32,920 --> 00:01:37,060

 

39
00:01:37,060 --> 00:01:38,140

 

40
00:01:38,140 --> 00:01:41,590

 

41
00:01:41,590 --> 00:01:43,359

 

42
00:01:43,359 --> 00:01:52,800

 

43
00:01:52,800 --> 00:01:55,090

 

44
00:01:55,090 --> 00:01:57,010

 

45
00:01:57,010 --> 00:01:59,380

 

46
00:01:59,380 --> 00:02:00,850

 

47
00:02:00,850 --> 00:02:03,760

 

48
00:02:03,760 --> 00:02:05,170

 

49
00:02:05,170 --> 00:02:07,420

 

50
00:02:07,420 --> 00:02:09,130

 

51
00:02:09,130 --> 00:02:10,270

 

52
00:02:10,270 --> 00:02:11,710

 

53
00:02:11,710 --> 00:02:14,949

 

54
00:02:14,949 --> 00:02:16,090

 

55
00:02:16,090 --> 00:02:18,720

 

56
00:02:18,720 --> 00:02:21,250

 

57
00:02:21,250 --> 00:02:23,350

 

58
00:02:23,350 --> 00:02:25,330

 

59
00:02:25,330 --> 00:02:27,580

 

60
00:02:27,580 --> 00:02:29,920

 

61
00:02:29,920 --> 00:02:33,760

 

62
00:02:33,760 --> 00:02:38,110

 

63
00:02:38,110 --> 00:02:40,540

 

64
00:02:40,540 --> 00:02:43,030

 

65
00:02:43,030 --> 00:02:45,190

 

66
00:02:45,190 --> 00:02:49,360

 

67
00:02:49,360 --> 00:02:51,820

 

68
00:02:51,820 --> 00:02:56,020

 

69
00:02:56,020 --> 00:02:58,060

 

70
00:02:58,060 --> 00:03:01,470

 

71
00:03:01,470 --> 00:03:04,780

 

72
00:03:04,780 --> 00:03:13,780

 

73
00:03:13,780 --> 00:03:15,640

 

74
00:03:15,640 --> 00:03:20,110

 

75
00:03:20,110 --> 00:03:25,060

 

76
00:03:25,060 --> 00:03:30,880

 

77
00:03:30,880 --> 00:03:36,520

 

78
00:03:36,520 --> 00:03:38,650

 

79
00:03:38,650 --> 00:03:41,200

 

80
00:03:41,200 --> 00:03:43,090

 

81
00:03:43,090 --> 00:03:50,170

 

82
00:03:50,170 --> 00:03:50,180

 

83
00:03:50,180 --> 00:03:51,390


84
00:03:51,390 --> 00:03:54,929

 

85
00:03:54,929 --> 00:03:56,550

 

86
00:03:56,550 --> 00:03:58,679

 

87
00:03:58,679 --> 00:04:00,929

 

88
00:04:00,929 --> 00:04:04,890

 

89
00:04:04,890 --> 00:04:06,660

 

90
00:04:06,660 --> 00:04:09,059

 

91
00:04:09,059 --> 00:04:11,610

 

92
00:04:11,610 --> 00:04:13,830

 

93
00:04:13,830 --> 00:04:17,400

 

94
00:04:17,400 --> 00:04:18,779

 

95
00:04:18,779 --> 00:04:20,969

 

96
00:04:20,969 --> 00:04:24,529

 

97
00:04:24,529 --> 00:04:27,629

 

98
00:04:27,629 --> 00:04:30,500

 

99
00:04:30,500 --> 00:04:37,879

 

100
00:04:37,879 --> 00:04:41,070

 

101
00:04:41,070 --> 00:04:44,820

 

102
00:04:44,820 --> 00:04:48,089

 

103
00:04:48,089 --> 00:04:50,670

 

104
00:04:50,670 --> 00:04:52,260

 

105
00:04:52,260 --> 00:04:56,730

 

106
00:04:56,730 --> 00:04:58,850

 

107
00:04:58,850 --> 00:05:02,909

 

108
00:05:02,909 --> 00:05:07,800

 

109
00:05:07,800 --> 00:05:09,600

 

110
00:05:09,600 --> 00:05:11,490

 

111
00:05:11,490 --> 00:05:14,490

 

112
00:05:14,490 --> 00:05:17,520

 

113
00:05:17,520 --> 00:05:18,390

 

114
00:05:18,390 --> 00:05:22,230

 

115
00:05:22,230 --> 00:05:25,379

 

116
00:05:25,379 --> 00:05:31,450

 

117
00:05:31,450 --> 00:05:34,280

 

118
00:05:34,280 --> 00:05:36,050

 

119
00:05:36,050 --> 00:05:36,060

 

120
00:05:36,060 --> 00:05:42,250


121
00:05:42,250 --> 00:05:44,390

 

122
00:05:44,390 --> 00:05:47,630

 

123
00:05:47,630 --> 00:05:51,440

 

124
00:05:51,440 --> 00:05:53,090

 

125
00:05:53,090 --> 00:05:55,700

 

126
00:05:55,700 --> 00:05:57,620

 

127
00:05:57,620 --> 00:06:49,100

 

128
00:06:49,100 --> 00:06:50,930

 

129
00:06:50,930 --> 00:06:54,050

 

130
00:06:54,050 --> 00:07:02,059

 

131
00:07:02,059 --> 00:07:04,640

 

132
00:07:04,640 --> 00:07:06,920

 

133
00:07:06,920 --> 00:07:09,520

 

134
00:07:09,520 --> 00:07:13,700

 

135
00:07:13,700 --> 00:07:15,499

 

136
00:07:15,499 --> 00:07:18,200

 

137
00:07:18,200 --> 00:07:20,570

 

138
00:07:20,570 --> 00:07:22,399

 

139
00:07:22,399 --> 00:07:25,969

 

140
00:07:25,969 --> 00:07:35,360

 

141
00:07:35,360 --> 00:07:44,480

 

142
00:07:44,480 --> 00:08:03,240

 

143
00:08:03,240 --> 00:08:21,390

 

144
00:08:21,390 --> 00:08:25,180

 

145
00:08:25,180 --> 00:08:26,620

 

146
00:08:26,620 --> 00:08:30,460

 

147
00:08:30,460 --> 00:08:32,680

 

148
00:08:32,680 --> 00:08:33,880

 

149
00:08:33,880 --> 00:08:36,460

 

150
00:08:36,460 --> 00:08:39,940

 

151
00:08:39,940 --> 00:08:41,800

 

152
00:08:41,800 --> 00:08:43,600

 

153
00:08:43,600 --> 00:08:46,120

 

154
00:08:46,120 --> 00:08:48,370

 

155
00:08:48,370 --> 00:08:51,460

 

156
00:08:51,460 --> 00:08:57,370

 

157
00:08:57,370 --> 00:09:00,070

 

158
00:09:00,070 --> 00:09:01,690

 

159
00:09:01,690 --> 00:09:04,780

 

160
00:09:04,780 --> 00:09:11,560

 

161
00:09:11,560 --> 00:09:16,390

 

162
00:09:16,390 --> 00:09:21,880

 

163
00:09:21,880 --> 00:09:23,560

 

164
00:09:23,560 --> 00:09:25,540

 

165
00:09:25,540 --> 00:09:31,660

 

166
00:09:31,660 --> 00:09:33,430

 

167
00:09:33,430 --> 00:09:37,030

 

168
00:09:37,030 --> 00:09:40,420

 

169
00:09:40,420 --> 00:09:44,200

 

170
00:09:44,200 --> 00:09:46,240

 

171
00:09:46,240 --> 00:09:48,280

 

172
00:09:48,280 --> 00:09:50,710

 

173
00:09:50,710 --> 00:09:52,390

 

174
00:09:52,390 --> 00:09:55,840

 

175
00:09:55,840 --> 00:09:59,290

 

176
00:09:59,290 --> 00:10:01,450

 

177
00:10:01,450 --> 00:10:03,250

 

178
00:10:03,250 --> 00:10:05,650

 

179
00:10:05,650 --> 00:10:07,810

 

180
00:10:07,810 --> 00:10:09,340

 

181
00:10:09,340 --> 00:10:14,560

 

182
00:10:14,560 --> 00:10:16,180

 

183
00:10:16,180 --> 00:10:17,850

 

184
00:10:17,850 --> 00:10:21,760

 

185
00:10:21,760 --> 00:10:23,230

 

186
00:10:23,230 --> 00:10:26,620

 

187
00:10:26,620 --> 00:10:28,570

 

188
00:10:28,570 --> 00:10:30,160

 

189
00:10:30,160 --> 00:10:31,890

 

190
00:10:31,890 --> 00:10:33,990

 

191
00:10:33,990 --> 00:10:36,300

 

192
00:10:36,300 --> 00:10:40,740

 

193
00:10:40,740 --> 00:10:43,019

 

194
00:10:43,019 --> 00:10:44,190

 

195
00:10:44,190 --> 00:10:46,620

 

196
00:10:46,620 --> 00:10:48,870

 

197
00:10:48,870 --> 00:10:52,620

 

198
00:10:52,620 --> 00:10:55,470

 

199
00:10:55,470 --> 00:10:57,810

 

200
00:10:57,810 --> 00:10:59,790

 

201
00:10:59,790 --> 00:11:03,570

 

202
00:11:03,570 --> 00:11:06,660

 

203
00:11:06,660 --> 00:11:12,180

 

204
00:11:12,180 --> 00:11:13,920

 

205
00:11:13,920 --> 00:11:16,350

 

206
00:11:16,350 --> 00:11:19,620

 

207
00:11:19,620 --> 00:11:22,740

 

208
00:11:22,740 --> 00:11:24,360

 

209
00:11:24,360 --> 00:11:27,930

 

210
00:11:27,930 --> 00:11:29,790

 

211
00:11:29,790 --> 00:11:31,140

 

212
00:11:31,140 --> 00:11:33,510

 

213
00:11:33,510 --> 00:11:36,300

 

214
00:11:36,300 --> 00:11:39,210

 

215
00:11:39,210 --> 00:11:40,740

 

216
00:11:40,740 --> 00:11:44,610

 

217
00:11:44,610 --> 00:11:47,310

 

218
00:11:47,310 --> 00:11:49,980

 

219
00:11:49,980 --> 00:11:51,750

 

220
00:11:51,750 --> 00:11:53,850

 

221
00:11:53,850 --> 00:11:55,800

 

222
00:11:55,800 --> 00:11:59,040

 

223
00:11:59,040 --> 00:12:00,960

 

224
00:12:00,960 --> 00:12:04,350

 

225
00:12:04,350 --> 00:12:07,530

 

226
00:12:07,530 --> 00:12:09,090

 

227
00:12:09,090 --> 00:12:11,640

 

228
00:12:11,640 --> 00:12:13,530

 

229
00:12:13,530 --> 00:12:18,290

 

230
00:12:18,290 --> 00:12:20,400

 

231
00:12:20,400 --> 00:12:21,570

 

232
00:12:21,570 --> 00:12:23,220

 

233
00:12:23,220 --> 00:12:23,970

 

234
00:12:23,970 --> 00:12:25,530

 

235
00:12:25,530 --> 00:12:27,000

 

236
00:12:27,000 --> 00:12:28,769

 

237
00:12:28,769 --> 00:12:32,490

 

238
00:12:32,490 --> 00:12:33,750

 

239
00:12:33,750 --> 00:12:35,130

 

240
00:12:35,130 --> 00:12:37,500

 

241
00:12:37,500 --> 00:12:40,199

 

242
00:12:40,199 --> 00:12:42,090

 

243
00:12:42,090 --> 00:12:43,410

 

244
00:12:43,410 --> 00:12:45,120

 

245
00:12:45,120 --> 00:12:45,130

 

246
00:12:45,130 --> 00:12:45,560


247
00:12:45,560 --> 00:12:47,510

 

248
00:12:47,510 --> 00:12:50,480

 

249
00:12:50,480 --> 00:12:53,060

 

250
00:12:53,060 --> 00:12:54,590

 

251
00:12:54,590 --> 00:12:56,750

 

252
00:12:56,750 --> 00:12:59,360

 

253
00:12:59,360 --> 00:13:02,960

 

254
00:13:02,960 --> 00:13:04,430

 

255
00:13:04,430 --> 00:13:07,400

 

256
00:13:07,400 --> 00:13:10,130

 

257
00:13:10,130 --> 00:13:12,230

 

258
00:13:12,230 --> 00:13:14,810

 

259
00:13:14,810 --> 00:13:16,010

 

260
00:13:16,010 --> 00:13:17,960

 

261
00:13:17,960 --> 00:13:21,440

 

262
00:13:21,440 --> 00:13:24,800

 

263
00:13:24,800 --> 00:13:28,580

 

264
00:13:28,580 --> 00:13:31,370

 

265
00:13:31,370 --> 00:13:33,530

 

266
00:13:33,530 --> 00:13:36,080

 

267
00:13:36,080 --> 00:13:38,150

 

268
00:13:38,150 --> 00:13:40,580

 

269
00:13:40,580 --> 00:13:42,890

 

270
00:13:42,890 --> 00:13:44,720

 

271
00:13:44,720 --> 00:13:46,910

 

272
00:13:46,910 --> 00:13:49,220

 

273
00:13:49,220 --> 00:13:50,810

 

274
00:13:50,810 --> 00:13:52,580

 

275
00:13:52,580 --> 00:13:54,500

 

276
00:13:54,500 --> 00:13:57,710

 

277
00:13:57,710 --> 00:13:59,300

 

278
00:13:59,300 --> 00:14:00,650

 

279
00:14:00,650 --> 00:14:02,000

 

280
00:14:02,000 --> 00:14:04,100

 

281
00:14:04,100 --> 00:14:06,230

 

282
00:14:06,230 --> 00:14:07,970

 

283
00:14:07,970 --> 00:14:11,630

 

284
00:14:11,630 --> 00:14:13,250

 

285
00:14:13,250 --> 00:14:17,210

 

286
00:14:17,210 --> 00:14:19,460

 

287
00:14:19,460 --> 00:14:21,170

 

288
00:14:21,170 --> 00:14:22,880

 

289
00:14:22,880 --> 00:14:25,400

 

290
00:14:25,400 --> 00:14:27,710

 

291
00:14:27,710 --> 00:14:29,630

 

292
00:14:29,630 --> 00:14:31,940

 

293
00:14:31,940 --> 00:14:33,890

 

294
00:14:33,890 --> 00:14:35,660

 

295
00:14:35,660 --> 00:14:37,760

 

296
00:14:37,760 --> 00:14:39,680

 

297
00:14:39,680 --> 00:14:42,470

 

298
00:14:42,470 --> 00:14:44,000

 

299
00:14:44,000 --> 00:14:46,280

 

300
00:14:46,280 --> 00:14:48,680

 

301
00:14:48,680 --> 00:14:51,260

 

302
00:14:51,260 --> 00:14:52,940

 

303
00:14:52,940 --> 00:14:55,130

 

304
00:14:55,130 --> 00:14:57,370

 

305
00:14:57,370 --> 00:15:02,379

 

306
00:15:02,379 --> 00:15:04,720

 

307
00:15:04,720 --> 00:15:06,189

 

308
00:15:06,189 --> 00:15:08,259

 

309
00:15:08,259 --> 00:15:10,090

 

310
00:15:10,090 --> 00:15:11,860

 

311
00:15:11,860 --> 00:15:13,720

 

312
00:15:13,720 --> 00:15:15,189

 

313
00:15:15,189 --> 00:15:17,470

 

314
00:15:17,470 --> 00:15:19,329

 

315
00:15:19,329 --> 00:15:21,370

 

316
00:15:21,370 --> 00:15:23,319

 

317
00:15:23,319 --> 00:15:24,970

 

318
00:15:24,970 --> 00:15:27,910

 

319
00:15:27,910 --> 00:15:29,590

 

320
00:15:29,590 --> 00:15:31,990

 

321
00:15:31,990 --> 00:15:33,910

 

322
00:15:33,910 --> 00:15:36,430

 

323
00:15:36,430 --> 00:15:43,509

 

324
00:15:43,509 --> 00:15:45,400

 

325
00:15:45,400 --> 00:15:47,910

 

326
00:15:47,910 --> 00:15:51,689

 

327
00:15:51,689 --> 00:15:55,540

 

328
00:15:55,540 --> 00:15:57,129

 

329
00:15:57,129 --> 00:15:58,900

 

330
00:15:58,900 --> 00:16:00,790

 

331
00:16:00,790 --> 00:16:04,720

 

332
00:16:04,720 --> 00:16:07,600

 

333
00:16:07,600 --> 00:16:10,990

 

334
00:16:10,990 --> 00:16:12,280

 

335
00:16:12,280 --> 00:16:13,509

 

336
00:16:13,509 --> 00:16:16,329

 

337
00:16:16,329 --> 00:16:18,160

 

338
00:16:18,160 --> 00:16:21,540

 

339
00:16:21,540 --> 00:16:24,460

 

340
00:16:24,460 --> 00:16:26,769

 

341
00:16:26,769 --> 00:16:28,660

 

342
00:16:28,660 --> 00:16:32,350

 

343
00:16:32,350 --> 00:16:34,930

 

344
00:16:34,930 --> 00:16:36,100

 

345
00:16:36,100 --> 00:16:37,889

 

346
00:16:37,889 --> 00:16:40,420

 

347
00:16:40,420 --> 00:16:43,360

 

348
00:16:43,360 --> 00:16:44,980

 

349
00:16:44,980 --> 00:16:46,929

 

350
00:16:46,929 --> 00:16:48,189

 

351
00:16:48,189 --> 00:16:50,949

 

352
00:16:50,949 --> 00:16:52,629

 

353
00:16:52,629 --> 00:16:55,389

 

354
00:16:55,389 --> 00:16:57,189

 

355
00:16:57,189 --> 00:16:59,050

 

356
00:16:59,050 --> 00:17:01,629

 

357
00:17:01,629 --> 00:17:03,610

 

358
00:17:03,610 --> 00:17:04,870

 

359
00:17:04,870 --> 00:17:06,610

 

360
00:17:06,610 --> 00:17:08,260

 

361
00:17:08,260 --> 00:17:09,600

 

362
00:17:09,600 --> 00:17:12,510

 

363
00:17:12,510 --> 00:17:14,640

 

364
00:17:14,640 --> 00:17:16,829

 

365
00:17:16,829 --> 00:17:18,689

 

366
00:17:18,689 --> 00:17:20,010

 

367
00:17:20,010 --> 00:17:22,410

 

368
00:17:22,410 --> 00:17:25,140

 

369
00:17:25,140 --> 00:17:26,760

 

370
00:17:26,760 --> 00:17:28,559

 

371
00:17:28,559 --> 00:17:30,450

 

372
00:17:30,450 --> 00:17:32,880

 

373
00:17:32,880 --> 00:17:34,650

 

374
00:17:34,650 --> 00:17:36,060

 

375
00:17:36,060 --> 00:17:38,640

 

376
00:17:38,640 --> 00:17:40,950

 

377
00:17:40,950 --> 00:17:44,270

 

378
00:17:44,270 --> 00:17:48,390

 

379
00:17:48,390 --> 00:17:50,520

 

380
00:17:50,520 --> 00:17:51,780

 

381
00:17:51,780 --> 00:17:53,460

 

382
00:17:53,460 --> 00:17:57,320

 

383
00:17:57,320 --> 00:18:00,720

 

384
00:18:00,720 --> 00:18:03,030

 

385
00:18:03,030 --> 00:18:04,919

 

386
00:18:04,919 --> 00:18:08,940

 

387
00:18:08,940 --> 00:18:10,799

 

388
00:18:10,799 --> 00:18:13,590

 

389
00:18:13,590 --> 00:18:15,900

 

390
00:18:15,900 --> 00:18:21,900

 

391
00:18:21,900 --> 00:18:24,780

 

392
00:18:24,780 --> 00:18:27,570

 

393
00:18:27,570 --> 00:18:31,020

 

394
00:18:31,020 --> 00:18:35,010

 

395
00:18:35,010 --> 00:18:44,700

 

396
00:18:44,700 --> 00:18:47,659

 

397
00:18:47,659 --> 00:18:49,799

 

398
00:18:49,799 --> 00:18:51,210

 

399
00:18:51,210 --> 00:18:53,220

 

400
00:18:53,220 --> 00:18:55,289

 

401
00:18:55,289 --> 00:18:56,880

 

402
00:18:56,880 --> 00:18:58,470

 

403
00:18:58,470 --> 00:19:01,169

 

404
00:19:01,169 --> 00:19:04,770

 

405
00:19:04,770 --> 00:19:06,990

 

406
00:19:06,990 --> 00:19:10,320

 

407
00:19:10,320 --> 00:19:11,970

 

408
00:19:11,970 --> 00:19:14,070

 

409
00:19:14,070 --> 00:19:17,130

 

410
00:19:17,130 --> 00:19:19,560

 

411
00:19:19,560 --> 00:19:21,850

 

412
00:19:21,850 --> 00:19:23,740

 

413
00:19:23,740 --> 00:19:26,529

 

414
00:19:26,529 --> 00:19:28,810

 

415
00:19:28,810 --> 00:19:32,169

 

416
00:19:32,169 --> 00:19:39,549

 

417
00:19:39,549 --> 00:19:43,389

 

418
00:19:43,389 --> 00:19:44,860

 

419
00:19:44,860 --> 00:19:48,039

 

420
00:19:48,039 --> 00:19:58,120

 

421
00:19:58,120 --> 00:20:00,639

 

422
00:20:00,639 --> 00:20:02,799

 

423
00:20:02,799 --> 00:20:07,240

 

424
00:20:07,240 --> 00:20:08,740

 

425
00:20:08,740 --> 00:20:10,000

 

426
00:20:10,000 --> 00:20:11,230

 

427
00:20:11,230 --> 00:20:13,600

 

428
00:20:13,600 --> 00:20:15,970

 

429
00:20:15,970 --> 00:20:17,740

 

430
00:20:17,740 --> 00:20:20,379

 

431
00:20:20,379 --> 00:20:22,450

 

432
00:20:22,450 --> 00:20:31,560

 

433
00:20:31,560 --> 00:20:35,320

 

434
00:20:35,320 --> 00:20:41,259

 

435
00:20:41,259 --> 00:20:46,509

 

436
00:20:46,509 --> 00:20:48,190

 

437
00:20:48,190 --> 00:20:51,299

 

438
00:20:51,299 --> 00:20:58,539

 

439
00:20:58,539 --> 00:21:01,269

 

440
00:21:01,269 --> 00:21:05,370

 

441
00:21:05,370 --> 00:21:07,960

 

442
00:21:07,960 --> 00:21:09,310

 

443
00:21:09,310 --> 00:21:10,960

 

444
00:21:10,960 --> 00:21:12,669

 

445
00:21:12,669 --> 00:21:15,549

 

446
00:21:15,549 --> 00:21:18,549

 

447
00:21:18,549 --> 00:21:20,110

 

448
00:21:20,110 --> 00:21:23,139

 

449
00:21:23,139 --> 00:21:24,460

 

450
00:21:24,460 --> 00:21:25,990

 

451
00:21:25,990 --> 00:21:28,840

 

452
00:21:28,840 --> 00:21:31,690

 

453
00:21:31,690 --> 00:21:34,060

 

454
00:21:34,060 --> 00:21:35,440

 

455
00:21:35,440 --> 00:21:38,769

 

456
00:21:38,769 --> 00:21:40,659

 

457
00:21:40,659 --> 00:21:42,129

 

458
00:21:42,129 --> 00:21:45,460

 

459
00:21:45,460 --> 00:21:47,440

 

460
00:21:47,440 --> 00:21:48,789

 

461
00:21:48,789 --> 00:21:51,009

 

462
00:21:51,009 --> 00:21:52,870

 

463
00:21:52,870 --> 00:21:56,620

 

464
00:21:56,620 --> 00:22:00,190

 

465
00:22:00,190 --> 00:22:01,299

 

466
00:22:01,299 --> 00:22:02,860

 

467
00:22:02,860 --> 00:22:04,629

 

468
00:22:04,629 --> 00:22:06,279

 

469
00:22:06,279 --> 00:22:09,700

 

470
00:22:09,700 --> 00:22:13,570

 

471
00:22:13,570 --> 00:22:16,600

 

472
00:22:16,600 --> 00:22:18,310

 

473
00:22:18,310 --> 00:22:20,139

 

474
00:22:20,139 --> 00:22:21,730

 

475
00:22:21,730 --> 00:22:26,639

 

476
00:22:26,639 --> 00:22:29,710

 

477
00:22:29,710 --> 00:22:31,539

 

478
00:22:31,539 --> 00:22:33,759

 

479
00:22:33,759 --> 00:22:37,029

 

480
00:22:37,029 --> 00:22:40,419

 

481
00:22:40,419 --> 00:22:43,360

 

482
00:22:43,360 --> 00:22:46,269

 

483
00:22:46,269 --> 00:22:48,009

 

484
00:22:48,009 --> 00:22:49,990

 

485
00:22:49,990 --> 00:22:52,899

 

486
00:22:52,899 --> 00:22:55,120

 

487
00:22:55,120 --> 00:22:57,220

 

488
00:22:57,220 --> 00:22:59,019

 

489
00:22:59,019 --> 00:23:00,909

 

490
00:23:00,909 --> 00:23:04,600

 

491
00:23:04,600 --> 00:23:06,700

 

492
00:23:06,700 --> 00:23:08,169

 

493
00:23:08,169 --> 00:23:11,139

 

494
00:23:11,139 --> 00:23:12,610

 

495
00:23:12,610 --> 00:23:15,009

 

496
00:23:15,009 --> 00:23:16,930

 

497
00:23:16,930 --> 00:23:19,269

 

498
00:23:19,269 --> 00:23:21,250

 

499
00:23:21,250 --> 00:23:26,139

 

500
00:23:26,139 --> 00:23:28,799

 

501
00:23:28,799 --> 00:23:31,810

 

502
00:23:31,810 --> 00:23:34,210

 

503
00:23:34,210 --> 00:23:39,190

 

504
00:23:39,190 --> 00:23:41,259

 

505
00:23:41,259 --> 00:23:43,240

 

506
00:23:43,240 --> 00:23:45,269

 

507
00:23:45,269 --> 00:23:48,369

 

508
00:23:48,369 --> 00:23:49,180

 

509
00:23:49,180 --> 00:23:53,050

 

510
00:23:53,050 --> 00:23:55,120

 

511
00:23:55,120 --> 00:23:58,720

 

512
00:23:58,720 --> 00:24:00,280

 

513
00:24:00,280 --> 00:24:03,280

 

514
00:24:03,280 --> 00:24:05,140

 

515
00:24:05,140 --> 00:24:07,000

 

516
00:24:07,000 --> 00:24:11,470

 

517
00:24:11,470 --> 00:24:13,000

 

518
00:24:13,000 --> 00:24:14,950

 

519
00:24:14,950 --> 00:24:16,960

 

520
00:24:16,960 --> 00:24:19,680

 

521
00:24:19,680 --> 00:24:22,750

 

522
00:24:22,750 --> 00:24:24,760

 

523
00:24:24,760 --> 00:24:28,540

 

524
00:24:28,540 --> 00:24:30,580

 

525
00:24:30,580 --> 00:24:33,430

 

526
00:24:33,430 --> 00:24:35,680

 

527
00:24:35,680 --> 00:24:37,690

 

528
00:24:37,690 --> 00:24:40,870

 

529
00:24:40,870 --> 00:24:43,990

 

530
00:24:43,990 --> 00:24:45,730

 

531
00:24:45,730 --> 00:24:48,970

 

532
00:24:48,970 --> 00:24:51,160

 

533
00:24:51,160 --> 00:24:52,990

 

534
00:24:52,990 --> 00:24:54,850

 

535
00:24:54,850 --> 00:24:56,770

 

536
00:24:56,770 --> 00:24:58,780

 

537
00:24:58,780 --> 00:25:01,060

 

538
00:25:01,060 --> 00:25:05,140

 

539
00:25:05,140 --> 00:25:06,760

 

540
00:25:06,760 --> 00:25:09,520

 

541
00:25:09,520 --> 00:25:11,770

 

542
00:25:11,770 --> 00:25:15,220

 

543
00:25:15,220 --> 00:25:19,390

 

544
00:25:19,390 --> 00:25:21,190

 

545
00:25:21,190 --> 00:25:26,590

 

546
00:25:26,590 --> 00:25:29,160

 

547
00:25:29,160 --> 00:25:31,600

 

548
00:25:31,600 --> 00:25:35,350

 

549
00:25:35,350 --> 00:25:38,560

 

550
00:25:38,560 --> 00:25:40,390

 

551
00:25:40,390 --> 00:25:42,130

 

552
00:25:42,130 --> 00:25:44,320

 

553
00:25:44,320 --> 00:25:46,120

 

554
00:25:46,120 --> 00:25:48,040

 

555
00:25:48,040 --> 00:25:50,890

 

556
00:25:50,890 --> 00:25:54,130

 

557
00:25:54,130 --> 00:25:56,260

 

558
00:25:56,260 --> 00:25:58,669

 

559
00:25:58,669 --> 00:26:02,910

 

560
00:26:02,910 --> 00:26:06,330

 

561
00:26:06,330 --> 00:26:08,460

 

562
00:26:08,460 --> 00:26:11,940

 

563
00:26:11,940 --> 00:26:15,780

 

564
00:26:15,780 --> 00:26:17,910

 

565
00:26:17,910 --> 00:26:21,240

 

566
00:26:21,240 --> 00:26:22,650

 

567
00:26:22,650 --> 00:26:24,330

 

568
00:26:24,330 --> 00:26:26,610

 

569
00:26:26,610 --> 00:26:29,520

 

570
00:26:29,520 --> 00:26:31,440

 

571
00:26:31,440 --> 00:26:33,270

 

572
00:26:33,270 --> 00:26:35,400

 

573
00:26:35,400 --> 00:26:39,600

 

574
00:26:39,600 --> 00:26:41,400

 

575
00:26:41,400 --> 00:26:43,980

 

576
00:26:43,980 --> 00:26:45,870

 

577
00:26:45,870 --> 00:26:48,780

 

578
00:26:48,780 --> 00:26:52,200

 

579
00:26:52,200 --> 00:26:53,790

 

580
00:26:53,790 --> 00:26:56,760

 

581
00:26:56,760 --> 00:27:00,450

 

582
00:27:00,450 --> 00:27:02,880

 

583
00:27:02,880 --> 00:27:05,880

 

584
00:27:05,880 --> 00:27:08,190

 

585
00:27:08,190 --> 00:27:11,040

 

586
00:27:11,040 --> 00:27:14,190

 

587
00:27:14,190 --> 00:27:18,600

 

588
00:27:18,600 --> 00:27:20,040

 

589
00:27:20,040 --> 00:27:23,280

 

590
00:27:23,280 --> 00:27:25,169

 

591
00:27:25,169 --> 00:27:29,520

 

592
00:27:29,520 --> 00:27:31,740

 

593
00:27:31,740 --> 00:27:33,510

 

594
00:27:33,510 --> 00:27:36,000

 

595
00:27:36,000 --> 00:27:38,250

 

596
00:27:38,250 --> 00:27:40,470

 

597
00:27:40,470 --> 00:27:42,360

 

598
00:27:42,360 --> 00:27:46,049

 

599
00:27:46,049 --> 00:27:48,720

 

600
00:27:48,720 --> 00:27:50,940

 

601
00:27:50,940 --> 00:27:53,100

 

602
00:27:53,100 --> 00:27:56,190

 

603
00:27:56,190 --> 00:27:59,840

 

604
00:27:59,840 --> 00:28:03,450

 

605
00:28:03,450 --> 00:28:05,400

 

606
00:28:05,400 --> 00:28:07,230

 

607
00:28:07,230 --> 00:28:09,750

 

608
00:28:09,750 --> 00:28:11,139

 

609
00:28:11,139 --> 00:28:15,190

 

610
00:28:15,190 --> 00:28:18,399

 

611
00:28:18,399 --> 00:28:23,739

 

612
00:28:23,739 --> 00:28:27,159

 

613
00:28:27,159 --> 00:28:28,930

 

614
00:28:28,930 --> 00:28:30,719

 

615
00:28:30,719 --> 00:28:34,239

 

616
00:28:34,239 --> 00:28:35,829

 

617
00:28:35,829 --> 00:28:39,729

 

618
00:28:39,729 --> 00:28:41,680

 

619
00:28:41,680 --> 00:28:46,419

 

620
00:28:46,419 --> 00:28:47,979

 

621
00:28:47,979 --> 00:28:52,629

 

622
00:28:52,629 --> 00:28:54,249

 

623
00:28:54,249 --> 00:28:55,509

 

624
00:28:55,509 --> 00:28:57,849

 

625
00:28:57,849 --> 00:29:00,820

 

626
00:29:00,820 --> 00:29:03,909

 

627
00:29:03,909 --> 00:29:07,229

 

628
00:29:07,229 --> 00:29:10,539

 

629
00:29:10,539 --> 00:29:14,229

 

630
00:29:14,229 --> 00:29:17,709

 

631
00:29:17,709 --> 00:29:21,099

 

632
00:29:21,099 --> 00:29:22,839

 

633
00:29:22,839 --> 00:29:25,119

 

634
00:29:25,119 --> 00:29:28,269

 

635
00:29:28,269 --> 00:29:29,799

 

636
00:29:29,799 --> 00:29:32,999

 

637
00:29:32,999 --> 00:29:35,259

 

638
00:29:35,259 --> 00:29:38,109

 

639
00:29:38,109 --> 00:29:40,239

 

640
00:29:40,239 --> 00:29:42,609

 

641
00:29:42,609 --> 00:29:46,719

 

642
00:29:46,719 --> 00:29:48,369

 

643
00:29:48,369 --> 00:29:55,450

 

644
00:29:55,450 --> 00:29:56,739

 

645
00:29:56,739 --> 00:29:58,989

 

646
00:29:58,989 --> 00:30:02,909

 

647
00:30:02,909 --> 00:30:11,739

 

648
00:30:11,739 --> 00:30:14,259

 

649
00:30:14,259 --> 00:30:16,180

 

650
00:30:16,180 --> 00:30:19,719

 

651
00:30:19,719 --> 00:30:22,340

 

652
00:30:22,340 --> 00:30:24,200

 

653
00:30:24,200 --> 00:30:29,210

 

654
00:30:29,210 --> 00:30:37,999

 

655
00:30:37,999 --> 00:30:39,289

 

656
00:30:39,289 --> 00:30:41,600

 

657
00:30:41,600 --> 00:30:44,019

 

658
00:30:44,019 --> 00:30:48,529

 

659
00:30:48,529 --> 00:30:51,259

 

660
00:30:51,259 --> 00:30:54,529

 

661
00:30:54,529 --> 00:30:56,720

 

662
00:30:56,720 --> 00:30:59,180

 

663
00:30:59,180 --> 00:31:00,879

 

664
00:31:00,879 --> 00:31:04,879

 

665
00:31:04,879 --> 00:31:06,980

 

666
00:31:06,980 --> 00:31:08,990

 

667
00:31:08,990 --> 00:31:10,369

 

668
00:31:10,369 --> 00:31:12,350

 

669
00:31:12,350 --> 00:31:13,730

 

670
00:31:13,730 --> 00:31:16,779

 

671
00:31:16,779 --> 00:31:18,980

 

672
00:31:18,980 --> 00:31:26,090

 

673
00:31:26,090 --> 00:31:30,830

 

674
00:31:30,830 --> 00:31:34,190

 

675
00:31:34,190 --> 00:31:36,499

 

676
00:31:36,499 --> 00:31:39,289

 

677
00:31:39,289 --> 00:31:40,840

 

678
00:31:40,840 --> 00:31:52,549

 

679
00:31:52,549 --> 00:31:55,009

 

680
00:31:55,009 --> 00:31:57,320

 

681
00:31:57,320 --> 00:32:00,919

 

682
00:32:00,919 --> 00:32:02,629

 

683
00:32:02,629 --> 00:32:04,430

 

684
00:32:04,430 --> 00:32:06,889

 

685
00:32:06,889 --> 00:32:09,499

 

686
00:32:09,499 --> 00:32:14,960

 

687
00:32:14,960 --> 00:32:16,789

 

688
00:32:16,789 --> 00:32:19,100

 

689
00:32:19,100 --> 00:32:21,230

 

690
00:32:21,230 --> 00:32:22,700

 

691
00:32:22,700 --> 00:32:25,759

 

692
00:32:25,759 --> 00:32:28,570

 

693
00:32:28,570 --> 00:32:31,190

 

694
00:32:31,190 --> 00:32:32,960

 

695
00:32:32,960 --> 00:32:35,350

 

696
00:32:35,350 --> 00:32:37,539

 

697
00:32:37,539 --> 00:32:40,960

 

698
00:32:40,960 --> 00:32:45,010

 

699
00:32:45,010 --> 00:32:48,430

 

700
00:32:48,430 --> 00:32:51,220

 

701
00:32:51,220 --> 00:32:51,230

 

702
00:32:51,230 --> 00:32:51,700


703
00:32:51,700 --> 00:32:56,140

 

704
00:32:56,140 --> 00:33:01,570

 

705
00:33:01,570 --> 00:33:04,799

 

706
00:33:04,799 --> 00:33:08,470

 

707
00:33:08,470 --> 00:33:11,049

 

708
00:33:11,049 --> 00:33:14,140

 

709
00:33:14,140 --> 00:33:16,960

 

710
00:33:16,960 --> 00:33:19,150

 

711
00:33:19,150 --> 00:33:22,090

 

712
00:33:22,090 --> 00:33:24,310

 

713
00:33:24,310 --> 00:33:27,640

 

714
00:33:27,640 --> 00:33:29,620

 

715
00:33:29,620 --> 00:33:32,169

 

716
00:33:32,169 --> 00:33:38,789

 

717
00:33:38,789 --> 00:33:43,049

 

718
00:33:43,049 --> 00:34:00,510

 

719
00:34:00,510 --> 00:34:03,210

 

720
00:34:03,210 --> 00:34:05,830

 

721
00:34:05,830 --> 00:34:07,470

 

722
00:34:07,470 --> 00:34:09,790

 

723
00:34:09,790 --> 00:34:11,320

 

724
00:34:11,320 --> 00:34:13,510

 

725
00:34:13,510 --> 00:34:14,710

 

726
00:34:14,710 --> 00:34:16,659

 

727
00:34:16,659 --> 00:34:18,669

 

728
00:34:18,669 --> 00:34:20,889

 

729
00:34:20,889 --> 00:34:25,810

 

730
00:34:25,810 --> 00:34:28,810

 

731
00:34:28,810 --> 00:34:36,379

 

732
00:34:36,379 --> 00:34:39,270

 

733
00:34:39,270 --> 00:34:44,010

 

734
00:34:44,010 --> 00:34:47,220

 

735
00:34:47,220 --> 00:34:51,450

 

736
00:34:51,450 --> 00:34:52,470

 

737
00:34:52,470 --> 00:34:54,300

 

738
00:34:54,300 --> 00:34:56,190

 

739
00:34:56,190 --> 00:34:58,920

 

740
00:34:58,920 --> 00:35:00,390

 

741
00:35:00,390 --> 00:35:03,240

 

742
00:35:03,240 --> 00:35:05,100

 

743
00:35:05,100 --> 00:35:07,770

 

744
00:35:07,770 --> 00:35:11,850

 

745
00:35:11,850 --> 00:35:14,250

 

746
00:35:14,250 --> 00:35:15,750

 

747
00:35:15,750 --> 00:35:17,940

 

748
00:35:17,940 --> 00:35:20,030

 

749
00:35:20,030 --> 00:35:22,380

 

750
00:35:22,380 --> 00:35:26,700

 

751
00:35:26,700 --> 00:35:29,040

 

752
00:35:29,040 --> 00:35:29,820

 

753
00:35:29,820 --> 00:35:32,310

 

754
00:35:32,310 --> 00:35:35,370

 

755
00:35:35,370 --> 00:35:36,900

 

756
00:35:36,900 --> 00:35:40,770

 

757
00:35:40,770 --> 00:35:46,140

 

758
00:35:46,140 --> 00:35:48,720

 

759
00:35:48,720 --> 00:35:50,250

 

760
00:35:50,250 --> 00:35:52,740

 

761
00:35:52,740 --> 00:35:55,160

 

762
00:35:55,160 --> 00:35:58,860

 

763
00:35:58,860 --> 00:36:00,780

 

764
00:36:00,780 --> 00:36:02,190

 

765
00:36:02,190 --> 00:36:07,230

 

766
00:36:07,230 --> 00:36:09,120

 

767
00:36:09,120 --> 00:36:10,470

 

768
00:36:10,470 --> 00:36:13,560

 

769
00:36:13,560 --> 00:36:15,540

 

770
00:36:15,540 --> 00:36:21,780

 

771
00:36:21,780 --> 00:36:23,640

 

772
00:36:23,640 --> 00:36:26,150

 

773
00:36:26,150 --> 00:36:30,330

 

774
00:36:30,330 --> 00:36:32,460

 

775
00:36:32,460 --> 00:36:35,640

 

776
00:36:35,640 --> 00:36:37,230

 

777
00:36:37,230 --> 00:36:39,000

 

778
00:36:39,000 --> 00:36:41,540

 

779
00:36:41,540 --> 00:36:43,620

 

780
00:36:43,620 --> 00:36:52,830

 

781
00:36:52,830 --> 00:36:59,320

 

782
00:36:59,320 --> 00:37:08,800

 

783
00:37:08,800 --> 00:37:09,700

 

784
00:37:09,700 --> 00:37:12,250

 

785
00:37:12,250 --> 00:37:15,099

 

786
00:37:15,099 --> 00:37:16,359

 

787
00:37:16,359 --> 00:37:19,420

 

788
00:37:19,420 --> 00:37:23,410

 

789
00:37:23,410 --> 00:37:27,570

 

790
00:37:27,570 --> 00:37:30,790

 

791
00:37:30,790 --> 00:37:34,630

 

792
00:37:34,630 --> 00:37:36,460

 

793
00:37:36,460 --> 00:37:38,800

 

794
00:37:38,800 --> 00:37:42,160

 

795
00:37:42,160 --> 00:37:44,380

 

796
00:37:44,380 --> 00:37:47,200

 

797
00:37:47,200 --> 00:37:48,910

 

798
00:37:48,910 --> 00:37:50,530

 

799
00:37:50,530 --> 00:37:53,230

 

800
00:37:53,230 --> 00:37:55,630

 

801
00:37:55,630 --> 00:37:56,890

 

802
00:37:56,890 --> 00:37:59,170

 

803
00:37:59,170 --> 00:38:02,170

 

804
00:38:02,170 --> 00:38:04,390

 

805
00:38:04,390 --> 00:38:05,560

 

806
00:38:05,560 --> 00:38:07,870

 

807
00:38:07,870 --> 00:38:11,380

 

808
00:38:11,380 --> 00:38:12,700

 

809
00:38:12,700 --> 00:38:13,390

 

810
00:38:13,390 --> 00:38:17,380

 

811
00:38:17,380 --> 00:38:19,300

 

812
00:38:19,300 --> 00:38:22,480

 

813
00:38:22,480 --> 00:38:24,430

 

814
00:38:24,430 --> 00:38:27,160

 

815
00:38:27,160 --> 00:38:30,460

 

816
00:38:30,460 --> 00:38:33,400

 

817
00:38:33,400 --> 00:38:35,290

 

818
00:38:35,290 --> 00:38:36,820

 

819
00:38:36,820 --> 00:38:42,550

 

820
00:38:42,550 --> 00:38:45,760

 

821
00:38:45,760 --> 00:38:47,230

 

822
00:38:47,230 --> 00:38:49,089

 

823
00:38:49,089 --> 00:38:52,660

 

824
00:38:52,660 --> 00:38:56,589

 

825
00:38:56,589 --> 00:38:56,599

 

826
00:38:56,599 --> 00:38:57,680


827
00:38:57,680 --> 00:38:59,690

 

828
00:38:59,690 --> 00:39:02,120

 

829
00:39:02,120 --> 00:39:03,950

 

830
00:39:03,950 --> 00:39:07,850

 

831
00:39:07,850 --> 00:39:11,180

 

832
00:39:11,180 --> 00:39:12,740

 

833
00:39:12,740 --> 00:39:12,750

 

834
00:39:12,750 --> 00:39:13,820


835
00:39:13,820 --> 00:39:16,250

 

836
00:39:16,250 --> 00:39:18,290

 

837
00:39:18,290 --> 00:39:20,660

 

838
00:39:20,660 --> 00:39:23,390

 

839
00:39:23,390 --> 00:39:24,650

 

840
00:39:24,650 --> 00:39:27,020

 

841
00:39:27,020 --> 00:39:31,090

 

842
00:39:31,090 --> 00:39:33,380

 

843
00:39:33,380 --> 00:39:35,390

 

844
00:39:35,390 --> 00:39:36,950

 

845
00:39:36,950 --> 00:39:39,500

 

846
00:39:39,500 --> 00:39:41,360

 

847
00:39:41,360 --> 00:39:44,080

 

848
00:39:44,080 --> 00:39:47,330

 

849
00:39:47,330 --> 00:39:48,560

 

850
00:39:48,560 --> 00:39:50,450

 

851
00:39:50,450 --> 00:39:50,460

 

852
00:39:50,460 --> 00:39:51,520


853
00:39:51,520 --> 00:39:53,390

 

854
00:39:53,390 --> 00:39:55,310

 

855
00:39:55,310 --> 00:39:57,710

 

856
00:39:57,710 --> 00:39:58,880

 

857
00:39:58,880 --> 00:40:00,590

 

858
00:40:00,590 --> 00:40:03,560

 

859
00:40:03,560 --> 00:40:07,070

 

860
00:40:07,070 --> 00:40:09,230

 

861
00:40:09,230 --> 00:40:12,880

 

862
00:40:12,880 --> 00:40:19,370

 

863
00:40:19,370 --> 00:40:21,290

 

864
00:40:21,290 --> 00:40:24,020

 

865
00:40:24,020 --> 00:40:26,510

 

866
00:40:26,510 --> 00:40:30,770

 

867
00:40:30,770 --> 00:40:33,350

 

868
00:40:33,350 --> 00:40:35,050

 

869
00:40:35,050 --> 00:40:38,300

 

870
00:40:38,300 --> 00:40:40,490

 

871
00:40:40,490 --> 00:40:42,140

 

872
00:40:42,140 --> 00:40:45,080

 

873
00:40:45,080 --> 00:40:48,290

 

874
00:40:48,290 --> 00:40:50,030

 

875
00:40:50,030 --> 00:40:51,950

 

876
00:40:51,950 --> 00:40:54,260

 

877
00:40:54,260 --> 00:40:56,290

 

878
00:40:56,290 --> 00:40:59,030

 

879
00:40:59,030 --> 00:41:00,650

 

880
00:41:00,650 --> 00:41:02,450

 

881
00:41:02,450 --> 00:41:04,490

 

882
00:41:04,490 --> 00:41:06,740

 

883
00:41:06,740 --> 00:41:07,850

 

884
00:41:07,850 --> 00:41:09,479

 

885
00:41:09,479 --> 00:41:12,249

 

886
00:41:12,249 --> 00:41:15,009

 

887
00:41:15,009 --> 00:41:16,499

 

888
00:41:16,499 --> 00:41:19,569

 

889
00:41:19,569 --> 00:41:22,870

 

890
00:41:22,870 --> 00:41:24,579

 

891
00:41:24,579 --> 00:41:26,049

 

892
00:41:26,049 --> 00:41:27,700

 

893
00:41:27,700 --> 00:41:29,680

 

894
00:41:29,680 --> 00:41:32,200

 

895
00:41:32,200 --> 00:41:34,029

 

896
00:41:34,029 --> 00:41:35,349

 

897
00:41:35,349 --> 00:41:37,719

 

898
00:41:37,719 --> 00:41:39,309

 

899
00:41:39,309 --> 00:41:42,759

 

900
00:41:42,759 --> 00:41:44,890

 

901
00:41:44,890 --> 00:41:47,049

 

902
00:41:47,049 --> 00:41:50,829

 

903
00:41:50,829 --> 00:41:52,450

 

904
00:41:52,450 --> 00:41:55,329

 

905
00:41:55,329 --> 00:41:57,789

 

906
00:41:57,789 --> 00:42:00,130

 

907
00:42:00,130 --> 00:42:01,539

 

908
00:42:01,539 --> 00:42:04,390

 

909
00:42:04,390 --> 00:42:05,979

 

910
00:42:05,979 --> 00:42:09,370

 

911
00:42:09,370 --> 00:42:10,930

 

912
00:42:10,930 --> 00:42:12,130

 

913
00:42:12,130 --> 00:42:14,319

 

914
00:42:14,319 --> 00:42:19,259

 

915
00:42:19,259 --> 00:42:21,640

 

916
00:42:21,640 --> 00:42:24,880

 

917
00:42:24,880 --> 00:42:27,009

 

918
00:42:27,009 --> 00:42:29,319

 

919
00:42:29,319 --> 00:42:32,559

 

920
00:42:32,559 --> 00:42:37,209

 

921
00:42:37,209 --> 00:42:39,430

 

922
00:42:39,430 --> 00:42:41,319

 

923
00:42:41,319 --> 00:42:42,609

 

924
00:42:42,609 --> 00:42:45,219

 

925
00:42:45,219 --> 00:42:47,229

 

926
00:42:47,229 --> 00:42:49,120

 

927
00:42:49,120 --> 00:42:52,599

 

928
00:42:52,599 --> 00:42:54,969

 

929
00:42:54,969 --> 00:42:56,829

 

930
00:42:56,829 --> 00:42:58,120

 

931
00:42:58,120 --> 00:43:00,339

 

932
00:43:00,339 --> 00:43:05,380

 

933
00:43:05,380 --> 00:43:07,329

 

934
00:43:07,329 --> 00:43:09,130

 

935
00:43:09,130 --> 00:43:12,849

 

936
00:43:12,849 --> 00:43:15,069

 

937
00:43:15,069 --> 00:43:17,440

 

938
00:43:17,440 --> 00:43:21,009

 

939
00:43:21,009 --> 00:43:22,450

 

940
00:43:22,450 --> 00:43:22,990

 

941
00:43:22,990 --> 00:43:24,850

 

942
00:43:24,850 --> 00:43:27,670

 

943
00:43:27,670 --> 00:43:30,340

 

944
00:43:30,340 --> 00:43:31,630

 

945
00:43:31,630 --> 00:43:33,220

 

946
00:43:33,220 --> 00:43:35,530

 

947
00:43:35,530 --> 00:43:37,600

 

948
00:43:37,600 --> 00:43:38,830

 

949
00:43:38,830 --> 00:43:42,070

 

950
00:43:42,070 --> 00:43:45,790

 

951
00:43:45,790 --> 00:43:46,930

 

952
00:43:46,930 --> 00:43:48,400

 

953
00:43:48,400 --> 00:43:50,050

 

954
00:43:50,050 --> 00:43:52,000

 

955
00:43:52,000 --> 00:43:53,770

 

956
00:43:53,770 --> 00:43:55,660

 

957
00:43:55,660 --> 00:44:01,900

 

958
00:44:01,900 --> 00:44:04,300

 

959
00:44:04,300 --> 00:44:07,600

 

960
00:44:07,600 --> 00:44:11,680

 

961
00:44:11,680 --> 00:44:13,240

 

962
00:44:13,240 --> 00:44:15,850

 

963
00:44:15,850 --> 00:44:21,880

 

964
00:44:21,880 --> 00:44:24,280

 

965
00:44:24,280 --> 00:44:28,150

 

966
00:44:28,150 --> 00:44:29,830

 

967
00:44:29,830 --> 00:44:36,940

 

968
00:44:36,940 --> 00:44:38,470

 

969
00:44:38,470 --> 00:44:41,320

 

970
00:44:41,320 --> 00:44:43,150

 

971
00:44:43,150 --> 00:44:45,070

 

972
00:44:45,070 --> 00:44:47,440

 

973
00:44:47,440 --> 00:44:50,410

 

974
00:44:50,410 --> 00:44:50,420

 

975
00:44:50,420 --> 00:44:52,440


976
00:44:52,440 --> 00:44:55,720

 

977
00:44:55,720 --> 00:44:58,000

 

978
00:44:58,000 --> 00:45:04,030

 

979
00:45:04,030 --> 00:45:08,470

 

980
00:45:08,470 --> 00:45:10,630

 

981
00:45:10,630 --> 00:45:13,750

 

982
00:45:13,750 --> 00:45:18,820

 

983
00:45:18,820 --> 00:45:20,830

 

984
00:45:20,830 --> 00:45:20,840

 

985
00:45:20,840 --> 00:45:27,239


986
00:45:27,239 --> 00:45:30,670

 

987
00:45:30,670 --> 00:45:36,839

 

988
00:45:36,839 --> 00:45:39,099

 

989
00:45:39,099 --> 00:45:42,999

 

990
00:45:42,999 --> 00:45:45,880

 

991
00:45:45,880 --> 00:45:47,499

 

992
00:45:47,499 --> 00:45:49,660

 

993
00:45:49,660 --> 00:45:51,160

 

994
00:45:51,160 --> 00:45:52,479

 

995
00:45:52,479 --> 00:45:54,309

 

996
00:45:54,309 --> 00:45:56,140

 

997
00:45:56,140 --> 00:45:58,059

 

998
00:45:58,059 --> 00:46:00,519

 

999
00:46:00,519 --> 00:46:04,479

 

1000
00:46:04,479 --> 00:46:06,579

 

1001
00:46:06,579 --> 00:46:10,509

 

1002
00:46:10,509 --> 00:46:12,959

 

1003
00:46:12,959 --> 00:46:15,459

 

1004
00:46:15,459 --> 00:46:17,829

 

1005
00:46:17,829 --> 00:46:20,739

 

1006
00:46:20,739 --> 00:46:23,769

 

1007
00:46:23,769 --> 00:46:25,680

 

1008
00:46:25,680 --> 00:46:27,880

 

1009
00:46:27,880 --> 00:46:30,069

 

1010
00:46:30,069 --> 00:46:31,870

 

1011
00:46:31,870 --> 00:46:35,469

 

1012
00:46:35,469 --> 00:46:37,479

 

1013
00:46:37,479 --> 00:46:39,609

 

1014
00:46:39,609 --> 00:46:41,979

 

1015
00:46:41,979 --> 00:46:46,180

 

1016
00:46:46,180 --> 00:46:48,009

 

1017
00:46:48,009 --> 00:46:49,569

 

1018
00:46:49,569 --> 00:46:52,329

 

1019
00:46:52,329 --> 00:46:54,249

 

1020
00:46:54,249 --> 00:46:55,959

 

1021
00:46:55,959 --> 00:46:58,569

 

1022
00:46:58,569 --> 00:47:00,849

 

1023
00:47:00,849 --> 00:47:03,969

 

1024
00:47:03,969 --> 00:47:05,890

 

1025
00:47:05,890 --> 00:47:07,779

 

1026
00:47:07,779 --> 00:47:09,160

 

1027
00:47:09,160 --> 00:47:11,769

 

1028
00:47:11,769 --> 00:47:13,959

 

1029
00:47:13,959 --> 00:47:17,799

 

1030
00:47:17,799 --> 00:47:19,239

 

1031
00:47:19,239 --> 00:47:23,199

 

1032
00:47:23,199 --> 00:47:26,469

 

1033
00:47:26,469 --> 00:47:28,930

 

1034
00:47:28,930 --> 00:47:31,509

 

1035
00:47:31,509 --> 00:47:33,489

 

1036
00:47:33,489 --> 00:47:37,940

 

1037
00:47:37,940 --> 00:47:40,309

 

1038
00:47:40,309 --> 00:47:41,780

 

1039
00:47:41,780 --> 00:47:45,589

 

1040
00:47:45,589 --> 00:47:46,760

 

1041
00:47:46,760 --> 00:47:49,430

 

1042
00:47:49,430 --> 00:47:51,440

 

1043
00:47:51,440 --> 00:47:54,890

 

1044
00:47:54,890 --> 00:47:55,880

 

1045
00:47:55,880 --> 00:47:59,770

 

1046
00:47:59,770 --> 00:48:02,780

 

1047
00:48:02,780 --> 00:48:04,280

 

1048
00:48:04,280 --> 00:48:06,349

 

1049
00:48:06,349 --> 00:48:09,950

 

1050
00:48:09,950 --> 00:48:11,690

 

1051
00:48:11,690 --> 00:48:13,880

 

1052
00:48:13,880 --> 00:48:15,049

 

1053
00:48:15,049 --> 00:48:18,109

 

1054
00:48:18,109 --> 00:48:20,480

 

1055
00:48:20,480 --> 00:48:22,549

 

1056
00:48:22,549 --> 00:48:26,180

 

1057
00:48:26,180 --> 00:48:27,799

 

1058
00:48:27,799 --> 00:48:30,470

 

1059
00:48:30,470 --> 00:48:31,970

 

1060
00:48:31,970 --> 00:48:34,280

 

1061
00:48:34,280 --> 00:48:35,720

 

1062
00:48:35,720 --> 00:48:37,099

 

1063
00:48:37,099 --> 00:48:38,870

 

1064
00:48:38,870 --> 00:48:42,950

 

1065
00:48:42,950 --> 00:48:45,650

 

1066
00:48:45,650 --> 00:48:50,349

 

1067
00:48:50,349 --> 00:48:54,620

 

1068
00:48:54,620 --> 00:48:58,549

 

1069
00:48:58,549 --> 00:49:01,569

 

1070
00:49:01,569 --> 00:49:07,000

 

1071
00:49:07,000 --> 00:49:10,010

 

1072
00:49:10,010 --> 00:49:12,109

 

1073
00:49:12,109 --> 00:49:13,789

 

1074
00:49:13,789 --> 00:49:15,680

 

1075
00:49:15,680 --> 00:49:17,059

 

1076
00:49:17,059 --> 00:49:18,829

 

1077
00:49:18,829 --> 00:49:22,309

 

1078
00:49:22,309 --> 00:49:24,079

 

1079
00:49:24,079 --> 00:49:26,599

 

1080
00:49:26,599 --> 00:49:28,640

 

1081
00:49:28,640 --> 00:49:31,370

 

1082
00:49:31,370 --> 00:49:33,520

 

1083
00:49:33,520 --> 00:49:37,460

 

1084
00:49:37,460 --> 00:49:40,910

 

1085
00:49:40,910 --> 00:49:46,620

 

1086
00:49:46,620 --> 00:49:50,350

 

1087
00:49:50,350 --> 00:49:52,840

 

1088
00:49:52,840 --> 00:49:54,070

 

1089
00:49:54,070 --> 00:49:57,010

 

1090
00:49:57,010 --> 00:49:58,390

 

1091
00:49:58,390 --> 00:50:00,760

 

1092
00:50:00,760 --> 00:50:02,950

 

1093
00:50:02,950 --> 00:50:07,240

 

1094
00:50:07,240 --> 00:50:10,060

 

1095
00:50:10,060 --> 00:50:12,580

 

1096
00:50:12,580 --> 00:50:14,350

 

1097
00:50:14,350 --> 00:50:16,000

 

1098
00:50:16,000 --> 00:50:18,640

 

1099
00:50:18,640 --> 00:50:21,430

 

1100
00:50:21,430 --> 00:50:23,080

 

1101
00:50:23,080 --> 00:50:25,510

 

1102
00:50:25,510 --> 00:50:28,600

 

1103
00:50:28,600 --> 00:50:30,540

 

1104
00:50:30,540 --> 00:50:32,680

 

1105
00:50:32,680 --> 00:50:35,380

 

1106
00:50:35,380 --> 00:50:37,270

 

1107
00:50:37,270 --> 00:50:41,020

 

1108
00:50:41,020 --> 00:50:42,340

 

1109
00:50:42,340 --> 00:50:44,620

 

1110
00:50:44,620 --> 00:50:46,770

 

1111
00:50:46,770 --> 00:50:49,390

 

1112
00:50:49,390 --> 00:50:51,400

 

1113
00:50:51,400 --> 00:50:53,140

 

1114
00:50:53,140 --> 00:50:55,570

 

1115
00:50:55,570 --> 00:50:57,370

 

1116
00:50:57,370 --> 00:50:58,750

 

1117
00:50:58,750 --> 00:51:00,220

 

1118
00:51:00,220 --> 00:51:02,110

 

1119
00:51:02,110 --> 00:51:03,820

 

1120
00:51:03,820 --> 00:51:08,290

 

1121
00:51:08,290 --> 00:51:14,589

 

1122
00:51:14,589 --> 00:51:20,299

 

1123
00:51:20,299 --> 00:51:22,880

 

1124
00:51:22,880 --> 00:51:30,380

 

1125
00:51:30,380 --> 00:51:33,019

 

1126
00:51:33,019 --> 00:51:34,609

 

1127
00:51:34,609 --> 00:51:36,319

 

1128
00:51:36,319 --> 00:51:38,210

 

1129
00:51:38,210 --> 00:51:39,650

 

1130
00:51:39,650 --> 00:51:42,529

 

1131
00:51:42,529 --> 00:51:46,039

 

1132
00:51:46,039 --> 00:51:48,559

 

1133
00:51:48,559 --> 00:51:51,140

 

1134
00:51:51,140 --> 00:51:52,729

 

1135
00:51:52,729 --> 00:51:54,650

 

1136
00:51:54,650 --> 00:51:56,059

 

1137
00:51:56,059 --> 00:52:00,769

 

1138
00:52:00,769 --> 00:52:02,809

 

1139
00:52:02,809 --> 00:52:03,890

 

1140
00:52:03,890 --> 00:52:06,319

 

1141
00:52:06,319 --> 00:52:08,059

 

1142
00:52:08,059 --> 00:52:09,920

 

1143
00:52:09,920 --> 00:52:11,539

 

1144
00:52:11,539 --> 00:52:14,660

 

1145
00:52:14,660 --> 00:52:16,099

 

1146
00:52:16,099 --> 00:52:17,450

 

1147
00:52:17,450 --> 00:52:19,640

 

1148
00:52:19,640 --> 00:52:22,039

 

1149
00:52:22,039 --> 00:52:24,829

 

1150
00:52:24,829 --> 00:52:26,420

 

1151
00:52:26,420 --> 00:52:29,209

 

1152
00:52:29,209 --> 00:52:31,789

 

1153
00:52:31,789 --> 00:52:33,890

 

1154
00:52:33,890 --> 00:52:35,539

 

1155
00:52:35,539 --> 00:52:37,219

 

1156
00:52:37,219 --> 00:52:39,979

 

1157
00:52:39,979 --> 00:52:41,359

 

1158
00:52:41,359 --> 00:52:43,069

 

1159
00:52:43,069 --> 00:52:45,259

 

1160
00:52:45,259 --> 00:52:48,349

 

1161
00:52:48,349 --> 00:52:51,319

 

1162
00:52:51,319 --> 00:53:00,370

 

1163
00:53:00,370 --> 00:53:00,380

 

1164
00:53:00,380 --> 00:53:02,440