1
00:00:06,240 --> 00:00:09,190

hey everybody welcome to 6849  uh geometric folding algorithms i am  eric domain you can call me  eric and we have as ta jason lynch who's  right there  um and this class is a bit unusual at  least for me  uh because we're trying or i'm trying  for the first time  uh a new experiment which is inverted  lecturing  and i wrote this on the poster for the  class and everyone started asking me  what's inverted lectures  well it's uh it's not a new idea but  i've never tried it before  uh the the concept is to make these in  class times where we're all here  together  more interactive by taking the lecture  component of the class which is covering  all the material  into videos that you watch online so  this class is basically going to  alternate between  real in person things as you are here of  course we're also being video recorded  so  slightly contradiction in terms but uh  that's for the people in the interwebs  to be able to watch this  and so alternate between the real in  class part  and video lectures from the last time i  taught this class which was fall 2010  uh so this is actually what the course  webpage looks like  as of uh an hour ago so we've got  these l l1 l2  l3 those are lectures that's like  content  uh lots of material packed in there  which is how i'm used to teaching a  class put as much material as i can per  lecture  um those are already available you can  start watching  those videos lecture one is optional i'm  going to basically cover  an uh a shorter version of that today so  you don't  says right there optional then the cs  which don't exist yet because we're  doing  it right now the c's are the in-class  sessions the contact hours  some of those may be recorded like this  one we're going to record but some will  not be  and you need to attend both so you've  got to watch all the videos  and you've got to attend all the classes  that's the  that's the setup so this is a bit  unusual so  i'll spend a few more minutes on how  this class is formatted  so uh  requirements  we've got  watch video lectures  i'll talk a little bit about timing of  that and then we've got  attend classes  and then the the sort of actual material  the regular part of the class that  you'll be graded on  uh consists of two parts problem sets  and a project and the project also has a  presentation  this is a project focused class so the  bulk of your grade  is based on the project and presentation  project can take on many different forms  i have this is a theoretical computer  science class  so there's the typical kinds of projects  like read a bunch of papers and  summarize them survey them  although we ask that you keep them  disjoint  more or less or avoid stuff that is well  covered in this book geometric folding  algorithms  which is uh or in the class itself  but otherwise you can survey material  in addition to standard survey paper or  instead of a standard survey paper  you can write a bunch of wikipedia  articles about  folding stuff so whereas survey stuff  should be new material that you haven't  seen  wikipedia stuff could be material that  you've seen in this class  but is not well covered in wikipedia and  that will help  take over the world our usual goal you  could if you see a cool algorithm in  this class you can implement it if  you're a coder  and see how it actually works make it  demorable it should be fun or you can  work on an open problem so we'll  if there's interest we'll run an  optional open problem solving session  where we'll all get together and try to  solve open problems i might also do them  in these class times because now we have  a lot of freedom to be more interactive  to  actually fold paper to have fun in class  instead of me trying to pack as much  material as i can  because that's already been done in the  videos  you could pose an open problem  especially if you come from another  field and you have some interesting  folding related problems those could be  really cool  or you could build something a physical  sculpture a physical structure of some  kind furniture  architecture whatever or you could do it  in the virtual world and just design  something interesting so there you  probably might want to do more than one  make it harder for yourself  all of these are possible projects  there's a lot of different options for  different types of people  different backgrounds whatever then  we'll also have problem sets which are  weekly  roughly and uh they shouldn't be too  long  and they will also have an option of not  doing all the problems  so uh every problem set will probably  have  uh the rule that you can drop you we  will drop the lowest grade on one of the  problems that's on the problem set  so that means if you just don't want to  do a problem you can skip it  and your grade will be determined by the  others if you want you can try to do all  of them we'll just drop the lowest  lowest grade so that'll also  if there's some problems that are easier  for you harder for others you can kind  of mix and match  cool there's a lot more details about  this kind of stuff on the website if you  click on  project you'll get lots of stuff  about lots of details about these  different styles of project  implementation survey  open problems whatever if because what i  said was pretty  pretty brief and  so look for that look at that for more  details project you don't have to worry  about right away but  one of the luxuries of having video  lectures you can skip ahead a little bit  and see what looks interesting to you  and work on a project related to that i  wanted to briefly show you a little bit  about this website  and how it works so you get a flavor for  it so let's say  the very next thing you should do right  after you leave here no  sometime between now and noon on monday  you should watch lecture two so you can  just click on lecture two here on the  left  and this is the the way it's set up  zoom to fit everything on your screen on  the upper left you've got your video  and it should start playing welcome back  hey it's me i'm even wearing the right  t-shirt it's part of it  and coincidence  the fun thing about these videos uh  there's a few nice features one is as  you jump around in the video  you see the slide down here in the lower  left updates to whatever i'm covering  and also if i jump farther the uh  page of handwritten notes that i'm  covering on the right changes  so as i scrobble around i think is the  term  all right yeah too many buttons  who designed this website all right i  forgot this  uh let's reload um  i'll turn down the volume uh you can  also  speed up the video this can be pretty  entertaining if you go really fast but  you can go 1.1 1.2 that's pretty  comfortable  1.5 yesterday i listened to a video at  2x  it's uh it's interesting  thankfully i do not sound like a  chipmunk but  it is if you haven't seen the lecture  before it's a little harder but i'll be  watching the lectures with you  experiencing the same total number of  hours you have to experience so  we're in this together let me know if  you have any comments on the website  it's all been done by hand so if there's  anything you want changed it's easy to  change  you can also do fun things like when  it's paused you can jump around in pages  on the right and say oh i didn't really  understand that page and then you click  on this time  and it will start playing from that time  at full speed you can slow it down too  which can be pretty entertaining  anyway you get the idea let me know how  it goes i want to make it as easy as  possible in fact if you don't like my  software you can just skip it all  download this video like 720p version or  the 360p version  play it in vlc with whatever fancy speed  up speed downs you want to use  or put on your ipad or other tablet  whatever  this page should work on an ipad but i  haven't tested it lately if it doesn't  let me know  cool so that's what it looks like and  then at the top of this page you see  here completion form  a link to completion form when you  finish watching the video  you click on this form it's pretty  minimal you enter  basically your name your username or  your email address you say  yes i watch the video and then that's  all that's required then you click  submit  but we highly encourage you to say  what you think at this point if you have  any questions about the lecture  that wasn't clear or anything you didn't  understand  or i briefly mentioned something that  sounded cool to you and you want to know  more about it  this is your chance to influence what i  cover in class  and this is why you have to submit this  form by noon the previous day  so watching video lectures and filling  out that form  is due by  noon on  mondays and wednesdays the day before  class  that will give me 22 hours to prepare  class  which should be enough i'll find out and  so i can adapt  the in-class time to be whatever people  care about most  so it's kind of like a poll if lots of  people say i didn't understand  x i will cover x if lots of people are  curious about why i will cover why  so whatever you want to know about  typically related things but if you have  unrelated things you want to  bring up as well feel free to put it in  there and we'll try to schedule it in  when it's time or when when it fits  yeah that's the plan i've never done  this before so if you don't like the  form  tell me anything you don't like or if  things should change just  send email or put in the form i guess  but uh email's good too  cool question i notice you have a no  option is that for like oh shoot 11 45  on monday and i haven't watched this  should i still submit the form and say  no  interesting uh why would you  just you're so honest why would you fill  out no on this forum  i i guess so you can fill out the form  multiple times  so you could i guess you could say no  and give your excuse here  if you want it i'll watch it tomorrow  maybe and then later out fill it in with  yes  so in particular if you also have  questions like you watch the video early  on and then later on you have a question  about it  if it's still before class time feel  free to fill this out again it just  makes another entry in our table this is  a google form  that's another question all right  i guess that's the reason for now but  maybe also we don't we couldn't put a  single option  now we want to it's it's to force you to  claim  honestly this is all honor system right  so this is like forcing you to make a  decision to be dishonest i guess  or to be honest rather  cool to make this work you've got to ask  questions you've got to request topics  otherwise  i'll i'll just fill your class times  with more material  that would be my natural temptation so  gotta hold me back and ask lots of  questions  one particular request i wanted to  highlight is to um  send me cool things to cover in  particular  you all surf the web and if you see  um cool folding things or any kind of  folding related things  on the web send it to me  by email and the plan is maybe to do one  a day  or maybe two a day for in class time  there's lots of fun things out there and  it's hard to see it all so when you  discover things  tell me and i will schedule it in  throughout the semesters you can think  of this as a standing homework problem  you should do it sometime before  the end of semester  cool another experiment i think it could  be fun  that is the end of the class  organizationally i think unless there  any questions about  requirements or style or format  all right then in the rest of this  class i'm going to cover as i said a  short version of  lecture one which is just an overview of  the whole class so i want to give you a  flavor of what the course is about  what kind of theorems we prove what kind  of algorithms we develop  so you can decide whether you are  interested in this class whether to take  it  there is also a  sort of survey that where you put your  name email address will add you to the  mailing list and i  announce every lecture so if you want to  if you're listening to the class you  just want to  listen to things you're interested in  come come to class for interesting  things  just listen to that email and see when  cool things come up and also there's a  little survey on that piece of paper  does anyone not have the piece of paper  a couple people so we will get them to  you  a little survey just to get your  background uh there's sort of no  required background for this class  because lots of people come from  different areas  hopefully you know at least one of the  background areas but  anything can kind of be filled in and  one of the points of this  questionnaire is to uh  if there's something you don't  understand like oh gosh i should know  xyz algorithm and you assumed it in  lecture but i don't know it could you  cover it  and if enough people request that i will  so that's the format out of curiosity  how many people here  are course six let's say  and how many people are of course 18 and  how many people  are other okay  cool so reasonably balanced  a lot of six though cool  so we are talking about geometric  folding algorithms  so there's geometry there's folding  and there's algorithms algorithms are  you know how we compute things at least  theoretically  and in general what this field is about  is both mathematics and computer science  uh  say the mathematics and algorithms  that underlie how things fold  and the technical term for things is  geometric objects  and we're interested not only in how  they fold but also how they unfold  in general you have some geometric  structure that can reconfigure like my  arm here can reconfigure in interesting  ways  we want to know what are all the ways it  can reconfigure what are all the ways it  can fold  this piece of paper is another geometric  object  and we want to know what are all the  ways that it can fold  in general so to before i talk about  sort of more formally what this means i  want to show you that i mean folding  comes up  everywhere pretty much almost any  discipline you can name there's some  that involves physical things at least  has geometric objects  geometric objects tend to comply unless  you're even if you're working with  completely rigid objects you don't want  them to fold  that's that's a folding problem  so i have here a list of a bunch of  different  application areas and i'm going to show  you some pictures and videos  related to them so the first one is  robotics  so if you have things like a robotic arm  you want to know  and it starts in one configuration you  want to continuously move it to reach  some other configuration so it can you  know pick up somebody  drop it off over here i want to know how  should i plan the motion of my robotic  arm  a different kind of robotic application  is to make  sheets of material that are themselves  robots that fold themselves into origami  this is a self-folding sheet developed  by  collaboration with at mit and harvard  and you just send a little bit of  electrical current  these little metal pieces heat up which  causes them to pull the creases shut  and boom you get your origami boat  with no origamis required the same sheet  can fold into many different shapes and  the under underlying algorithm here  or the underlying mathematics is that  this pattern of creases square  square grid with alternating diagonals  can fold into essentially any shape  made out of uh that is made out of cubes  here we're making a paper airplane out  of the same sheet  sending it different signals it does not  fly  but both probably floats  so that's the idea and you can imagine  a sheet like this could just be  arbitrarily reprogrammed  to fold into this gadget so i don't have  to bring around so many gadgets maybe my  laptop could fold smaller and later  could unfold into something to my  desktop  and i don't know my phone could  reconfigure into something else  that's that's the vision this is what we  call programmable matter where you can  change shape  in the same way that we programmed  software we don't want to be able to  program  shape obviously we're just  getting there now so that's robotics  next up is graphics one example is you  want to animate  uh a character from one position to  another that's a  kind of by a sort of a keyframe  animation  that's a folding problem i have here an  example of the two-dimensional analog  where you have one polygon and another  polygon and you want to morph  continuously from one to the other  and these these animations are all found  by  algorithms that are motivated by folding  stuff that we will cover  in this class and it's not so easy but  that's lots of fun things uh the  all of these motions avoid collision and  they approximately preserve the edge  lengths  also if it's possible if they match in  the left and the right they will  preserve those edge links  uh cool so that's graphics sort of  morphing  in mechanics mechanics motivated a lot  of mechanical linkage  problems in particular is this great  book from 1877 called how to draw a  straight line  you may think you know how to draw a  straight line but it's  the point of this book is to figure out  by turning a circular crank can i draw a  straight line  and this is motivated by steam engines  where you have a steam piston that's  moving up and down in a straight line  and you want to turn a wheel on your on  your train  so how do you do that well these are two  old ways to do it  and here's what they look like in  in computational land i guess these are  the original drawings so we have on the  top a design by  watt the unit in 1784  and it's approximately straight it's not  perfect but this highlighted point moves  along a circle and the green point moves  along the figure eight  and the figure eight is fairly straight  here in the middle so if you just use  that part  it works pretty well you lose a little  bit from the wiggle  but it's it was used in a lot of  locomotive engines  down here you have the first correct  solution by posse a in 1864  almost 100 years later and if you move  the highlighted point along a circle  this point moves along the red line  and pretty cool and there's a whole  bunch of mathematics related to this  generalizing this result  and we will cover it in this class  next up we have manufacturing uh this is  a fun example  of bending a piece of wire  this is a machine called di wire do it  yourself wire bending so you can build  one of these machines here it's making a  planar shape you can also make  three-dimensional shapes  lots of things are possible this is a  recent machine there's actually a lot of  wire bending machines  that are out there this is a nice and  simple one  but can you make everything no because  this  piece of wire can't collide with itself  and it can't collide with the machine  that's a constraint and you can't you  want to figure out  what shapes can you make can i make my  pair of glasses  or am i limited somehow that's a folding  problem which we will talk about  next up medical so one example of a  medical application is uh this  is to build a stent this is called an  origami stent developed at oxford  a bunch of years ago and the idea is you  want to do non-intrusive heart surgery  so you want to take  this big thing fold it down really small  like this and  and stick it through some small blood  vessels in your body until you get up  near your heart where you've got nice  big  vessels you want to declog and so you  want to expand it back  out to its larger size so it's  essentially a transportation problem  you want to make something small till it  gets where you need it and then you make  it big  that's an one example of a medical  application  in related to that kind of  transportation problem you also have  motivation of  aerospace so you want to send a large  object into space  your space shuttle isn't big enough to  do it you'd like to fold it down to  become  smaller this is an example by robert  lang who's pictured here  leading origami designer and  this uh this is a prototype it's only  five meters in size  uh the goal is to make a hundred meter  uh telescope lens which definitely does  not fit in your  space station so you do lots of foldings  like this until it fits in  in your space shuttle you send it out  when you're in space you've got lots of  room and you can unfold  out and you get your giant lens much  larger than hubble's lens for example  uh so this is in general this area is  called deployable structures when you  want to make something small for  transportation  then we have biology protein folding is  is a big problem a lot of people work on  it and a protein folds kind of like a  robotic  arm geometrically it's a little  different and it's not  fully understood how it works but  there's a lot of cool  folding problems geometry problems that  we've studied motivated by  by how proteins fold and figuring that  out the application's thing like things  like drug design you want to kill a  virus without killing the host  design a protein that folds into the  right shape so that it  kills one thing and not the other that's  the sort of  general goal and we have some cool  things which we will cover in this class  related to that next is  sculpture and origami design is an  obvious motivation for why you might be  here  this is why we initially got interested  origami has reached incredible heights  both of these are folded from  one square no cuts so just one square  paper you can make three-headed dog  you can make uh i guess i shouldn't call  him a man  but you can make a nas ghoul on his  horse or is it one  entity i don't know but it's one square  paper that much i know  uh this is designed by jason kuh who's  the president of the origami club at mit  origami which you should all  check out they meet on sundays at three  three good uh i know there's a bunch of  origami people here  so uh how are these possible these are  possible through mathematics and the  kind of algorithms that we are  uh that we will cover in this class in  fact there's a guest lecture by jason  kuh  uh which you'll be watching i think is  lecture six uh  and he's not actually in town but still  he can give the video lecture because he  already gave it  that's like time travel um  i have uh just a few examples of some of  our sculpture this is with uh martin  domain who's your cameraman  here and uh these are based on curved  creases which we may talk about to some  extent in this class that we didn't talk  about it too much  two years ago but a little bit curved  creases are not very well understood  mathematically  almost everything we will cover is based  on straight crease design  like most origami curve creases are  pretty amazing though and they do some  really cool shapes  and we're still trying to figure them  out how to design them algorithmically  and this these pieces are on display in  dc right now if you want to  check them out for the next several  months  yeah oh these pieces of paper are  circles with a circular hole and there's  more than one this has two pieces  this has three and this has i think five  okay so that was a brief uh sculpture  origami design lots of different other  possible sculptures kinetic sculpture  you might want to try building  um also architecture reconfigurable  architecture  i think this is an underexplored area  how many people here are from  architecture  a few okay you might want to make some  reconfigurable buildings  hoberman is one example of someone  embracing this a lot  he started in folding toy design but now  he does mostly folding architecture  this is one example from his company  hoberman associates  from the winter olympics at salt lake  city 2002  it's just it's very dramatic you have  this  folding structure that can almost  completely close up except for a little  circle here and when it opens up you've  got the huge  stage you can see the scale of of people  he did uh he co-designed this this giant  u2 stage if you've seen  u2 anytime in the last few years like i  did this incredible folding structure  based on similar similar principles and  and chuck hoberman is actually teaching  a class  at the graduate school of design at  harvard on monday afternoons  and i think anyone's welcome if you want  to check it out you should  should go see it and talk about his  techniques maybe we'll get him to  do a guest lecture we'll see so that was  a  brief overview of a bunch of different  applications  there's tons more in fact if you have  more please send them to me  i know a bunch more and we we will get  to them  but this is kind of a little survey of  different fields that folding touches on  now i'm going to switch over to more  mathematical stuff which is the bulk of  the class  that's our questions cool  so we move to  what kind of things we're interested in  folding they are essentially three types  linkages which are  we usually think of as kind of one  dimensional structure so we have one  dimensional  usually straight segments connected  together at hinges  and here i've drawn it in two dimensions  it could also live in three dimensions  like my arm  is made up of one-dimensional bones and  then there's  sockets for them to join essentially  universal joints  how can those things fold what shapes  can they fold into general constraints  here are that the edges should stay the  same length i can't  stretch my arm longer and they you have  to stay connected at the joints  i can't detach my elbow and radar later  reattach it at least not ideally  so the mathematical version is you're  not allowed to  uh one dimension up we have sheet  folding which we usually refer to as  paper folding  so you have but it could be you know  sheet metal  anything you have your sheet of material  and you want to fold it into things like  origami  like space stations like telescope  lenses whatever  general rules here like this piece of  paper you cannot stretch the paper  can't get any longer it can't collide  with itself  and you can't tear the paper you're not  allowed to cut it  uh because that would make tends to make  things too easy  uh so the sort of mathematically pure  version and also the  the modern origami pure version is that  you're not allowed to cut your piece of  paper  so just folding no stretching  no crossing if you want to go another  dimension up  we typically call this polyhedron  folding  so polyhedron it's uh made up of a bunch  of polygons  like cube and uh typically we're  interested  in unfolding a polyhedron  so think of this as just a surface it's  hollow inside  if you wanted to build that surface  you'd like to find  a shape like this cross  that folds into that cube  because if you have sheet material you  want to start from some  original shape that's flat uh they can  fold into your  shape you can also consider the reverse  problem which is folding  suppose i give you this cross what  shapes can it fold into and we've looked  at both  and i'll talk about them in a second  okay so in general for each of these  categories and you'll see these  categories over and over because they're  they're on the book here it says  linkages origami and polyhedra  so and there's two three parts to the  book there's linkages for copy and  polyhedra  part 142.3 just like this our class will  not follow this order though  uh after today we're going to i think  we'll talk about paper for a while then  go to linkages for a while then go to  polyhedra for a while  and then we'll repeat paper for a while  and and so on so we kind of  different different parts will be of  interest to different people so  do lots of different coverage but i  think paper is the most exciting so  we'll we'll start there  in general for each of these types of  things you want to fold  there are two kinds of problems we're  generally interested in  um based around the idea that well  there's there's some folding  structure that you're interested in and  then there's the way it can fold  and you can start from either one  so if you start from a folding structure  and you want to know what it can fold  into  this we typically call a foldability  problem  or you could call it an analysis problem  i have something  like i don't know  like this and i want to know what it can  fold into that's an example  what can the cross fold into  or i have some linkage and i want to  know i have a robotic arm it's already  fixed i've already designed it built it  what shapes can it fold into  the reverse problem is a design problem  or a synthesis problem  so there i start with what foldings i  would like to have  so for example i want to fold a  butterfly maybe so i design  a butterfly and now i'd say well i have  a rectangular paper can i fold it into a  butterfly  so that's a design problem here you want  to design a folding structure that  achieves your goals like shape  so that's kind of generically what we  like to do  and then there's three types of results  we typically get  so i'm giving you a super high level  first  and then we'll get to some actual fun  examples  so these are typical results  so three types are universality  decision and hardness  and these relate to sort of what is  possible  in particular is everything possible and  this makes sense both from a foldability  standpoint and from a design standpoint  if i give you a robotic arm i'd like to  know can it fold into  all the possible configurations or  there's some that it can't reach  that if it can reach all of them we have  a universality result and typically when  you can prove  that you can make anything you actually  get an  algorithm to do it so you can say oh  well how do i fold into x well here's  how you do it there's an algorithm  to compute that for you also in design  i'd like to know  okay i can make a butterfly and i can  make a nozzle  can i make everything if yes you get a  universality result so that's a design  universality if universality is not true  if you can't make everything  the next best thing you could hope for  is a decision algorithm  an algorithm that tells you quickly is  this possible is this impossible  so this is like a characterization of  what's possible and what's impossible  using algorithms sometimes though that's  not possible  there is no good algorithm to tell  whether something's foldable and then we  aim for hardness result to prove  that there is no good algorithm to  solve your problem so that gets into  complexity theory and we will  be not everyone's going to know that we  will be reviewing it in the video  lectures and in here  as needed so those those are the sort of  typical  outcomes for any of these kinds of  problems  let's see some examples unless they're  generic questions at this point  only generic questions about generic  material  next we'll get to specific material and  you can ask specific questions  uh cool i think  it's all good  so first up is linkages  and of course there's lots of results in  each of these fields i'm just going to  show you a couple in each  and the first question you are typically  interested in  about a linkage is is it rigid does it  move at all  let me give you some examples of this  yeah  do a little pop quiz uh  it's okay to get it wrong just  makes for some fun interaction uh  that one so  yes or no is this rigid or rigid or  flexible i should say  rigid everyone agrees good  correct richard are flexible  flexible uh depends on the dimension  very good  it's rigid or flexible uh it depends  it's rigid  in 2d because these are triangles  but it's flexible in 3d because  you can rotate one triangle around this  hinge  okay and this is rigid or flexible  flexible  everyone agrees and is correct  cool uh in general given a structure  like this a  linkage you want to know is it rigid or  is it flexible this has lots of  applications like you want to build a  bridge you don't want it to move you're  building a building you don't want it to  move  so here's the mathematical state of  affairs distinguishing  this boundary which is rigid in 2d  versus flexible in 2d  we understand super wells great  algorithms to do it we will cover them  this boundary between rigid and 3d and  flexible in 3d  we don't really understand and there  aren't great algorithms to solve it  there's a little disconcerting given the  number of buildings and bridges we live  in  but that's sort of the state of the  world  and we will talk about all that  so that is rigidity next up we have  universality so this is the robotic arm  question i mentioned i have a robotic  arm does it fold  into everything  into every possible configuration  and the answer again depends what  dimension you live in  and in two dimensions there are cool  ways to reconfigure your robotic arm so  this is actually a polygon a closed arm  if you will a closed chain  and this is one way to unfold it and  then in principle you can refold that  into any other shape you want  and we will cover this is one algorithm  to do it there's actually three  algorithms to do this  this is the most recent one this is  originally my  my phd thesis to figure out whether this  was possible  it's called the carpenter's rule problem  this one this algorithm preserves the  fivefold rotational symmetry of the  polygon which is pretty cool but in  general you take any  2d polygon you can unfold it while  preserving all the edge links keeping  all the connections and avoiding  collision  show you some other fun examples this is  a kind of spider 500 vertices  we're kind of zooming out as we expand  here you can see the links get really  tiny but in reality they're  all the lengths are staying the same  throughout time  and we fold so this is the state of  affairs in 2d we have good algorithms  it's always possible universality  so uh what about 3d 3d  is harder in particular because of  examples like this  we call this knitting needles you've got  two blue segments which you can think of  as long needles  and then a short purple thread  connecting them  and there's no way to undo this so in  particular if you started with a  straight  robotic arm you could not fold it into  this shape or vice versa  call this a locked configuration and in  general lock configurations exist  obviously and it's we don't know how to  distinguish  good or bad so here actually we don't  know whether it's there's a good  decision algorithm or  hardness but it's definitely not  universal  and 3d is very interesting because it  relates to robotic arms in real life and  protein folding and all these things so  i'll be talking about that in 4d if  you're lucky enough to live in four  dimensions  you get universality everything's great  and we'll  talk about that too uh so  next up we have paper folding  so that was a brief overview of linkages  so paper folding i'm going to mention  it one problem in the foldability world  and some problems in the design world  this is probably my favorite area  paper origami design but foldability is  interesting too the obvious f  and the sort of the first question you  might wonder about  in origami foldability is if i what  crease patterns  fold flat so if you take a classic  flapping bird  you unfold it these are the creases you  get but in general if i gave you some  pattern of creases drawn on your piece  of paper does it fold into anything  and here to make it interesting we say  does it fold into anything flat  flat origami you have to fold along all  the creases  sadly deciding this is a problem  called that's np hard so there's  basically no good algorithm to solve  this and we will prove that  uh in a few classes a few lectures  um so that was the  foldability problem now onto design if i  give you a square paper  what shapes can i make that's the  obvious origami design problem  and it turns out you can make any  polygons so you can make any flat shape  you want  if you have a piece of paper that's  white on one side and patterned or color  black on the other you can make any two  color pattern you want if it's polygonal  it's made out of straight sides if you  want to make a 3d thing you can make any  3d polyhedron also with two color  patterns whatever you want  we proved this way back in 1999 i think  is actually the first paper to use the  term computational origami  and uh it's not hard to prove we will  prove it uh in a couple of lectures i  think  but it this is uh we don't have a  practical way to do this  there is an algorithm here but it  doesn't give you a good way to fold  anything and so the quest continues for  a good way to fold  everything and one such approach is  called orgamizer  and it you give it an arbitrary 3d  surface  it designs a crease pattern this you  fold from a square piece of paper  solids or mountains dashed or valleys  and you get  this bunny this is a classic model in  computer graphics called the stanford  bunny  and this takes about 10 hours to fold if  you're tomohirotachi  who designed it and there's free  software available called organizer by  tomohira  and he was just here a couple weeks ago  and we've been proving that this  algorithm always works  and it also seems very practical most of  the material here gets used in the  surface  so it doesn't you don't need a very  large square paper to make your  bunny i have one  a little example here sort of classic in  the origami world which is to make a  checkerboard  and this is a 4x4 checkerboard obviously  not a full 8x8  but you can fold an 8x8 and we'll talk  about different ways to do it  this one is folded from one square paper  white on one side  green on the other and actually has a  pretty good scale factor it just  collapses by  a factor of two here so i can get it  back into correct state so  there you go you can make your own  origami checker boards  if you ran out of your regular wooden  ones  cool next example is what we call maze  folding this is the  poster uh if you saw it this is  a folding of this crease pattern and the  design was  uh in general you take any orthogonal  graph meaning horizontal and vertical  lines on a grid and you want to extrude  that graph  out of the plane then there's an  algorithm it's online you can play with  it i think ericdomain.org  maze will get you there you just draw  that pattern it will give you this  crease pattern you print it out  and uh eight hours maybe  you can fold it into reality so these  this is by  jenny and eli who i think are students  in this class  they got started a little early and i  didn't bring it here but at some point i  will bring  the real 3d one so that's mace folding  and the next one i have is called  folding cut this is actually the first  problem we worked on  so kind of a personal favorite you take  a rectangle of paper  and then you fold it flat  and you take your scissors and make one  complete straight cut this is a magic  trick performed by  harry houdini and other magicians  and you get in this case two pieces and  the question is what shapes can you make  by folding  and one straight cut here i've made a  little swan  okay you're not impressed so to a harder  one  this one is actually quite challenging  to cut i was folding it  as you were arriving some of you  probably saw  that's about about right  it's a little hard to open up but this  should be  the mit logo  all right good  so that's the idea you can impress all  your friends print these out and  uh fold them it's a good good challenge  to fold along the crease pattern and as  you might guess there's an algorithm  here  given any polygon or collection of  polygons you can  fold a piece of paper to line up all the  edges of that polygon cut along it  you're done okay that is paper  next i want to briefly mention polyhedra  and again actually i have polyeter right  over here  so again there's an unfolding problem  and a folding problem  and i'll start by showing you a little  bit about the  folding problem well this shows a little  bit of both but this is a video we made  back in 99  i was a grad student i had lots of hours  to make videos  animations so this is one example of an  unfolding of a cube  but the unfolding we want to consider  here is this cross unfolding that i keep  talking about  it's kind of a classic so fun to analyze  so if you take that cross shape you want  to know what other shapes what other  convex polyhedra can it fold into  and here the rules are a little  different from origami you have to get  exactly the surface you want no overlap  here we happen to get a doubly covered  quadrilateral  which is a kind of a convex surface  a little bit cheating but if we change  the creases in this way we get a  five-sided polyhedron with a plane of  symmetry  down this axis  or  we can change the creases in this way  and get a tetrahedron and it's exact  coverage so this little tab fits into  exactly that pocket  uh get exactly what you want as you  might guess there's an algorithm that  will enumerate  all the different convex polyhedra you  can make from a given polygon of paper  and it's based on a theorem in russian  by alexandrov which is in the background  here if you're curious what the cyrillic  background is  this theorem is the key to the algorithm  so that's what's possible from a  by folding a given polygon  and then you can ask about unfolding let  me give you a brief overview unfolding  so uh when you're unfolding you might be  interested in only cutting along the  edges of the surface which is what  happened  in this cross unfolding i cut along  these edges  or you might allow cutting anywhere on  the surface and the answers depend  on those two versions you could try to  do convex polyhedra or you could try to  do arbitrary  polyhedra so here's the story  this one we don't know this one we don't  know  this one the answer is yes we'll prove  it this one is the answer is no we'll  prove it  in fact i can show you some examples uh  this is an edge unfolding of a convex  polyhedron the oldest  set that we know is from this 1525 book  by albert durer  and this is a snub cube he didn't have  wikipedia at the time so he couldn't  just  look at the image and draw it he had to  build a physical model and so he'd cut  out  pieces of paper to build physical models  so he could paint it and various of his  prints  have convex polyhedra in them and this  is an example of this open problem  which we don't know the answer to uh can  you edge unfold convex polyhedra  uh if you want to take non-convex  polyhedra  you cannot always do it this is an  example of a kind of spiky tetrahedron  that cannot be edge unfolded we'll prove  that  but it can be generally unfolded if you  can cut anywhere  you can find a one piece unfolding and  so big open problem  is can you do any polyhedron  with general cuts i think the answer is  yes very cool  problem uh and that's polyhedra  and the last thing i wanted to show you  is hinge dissection this is  one more thing after one two three and  four after one two and three we have  four hinge dissection this is a chain of  blocks which are kind of two-dimensional  uh so it's but it's not really paper  folding so it's kind of like a thick  linkage kind of  between one and two if you will and this  chain folds into an equilateral triangle  or square and it's over 100 years old  can you do anything turns out there's a  universality result this is just a  picture  part of the proof which we will cover in  a later lecture  that you can fold you can take any set  of polygons  of the same area there's one chain of  blocks  which will fold into each of those  polygons so  you can you get a universality result  you can do equilateral triangle to a  regular hexagon to  a swan to whatever as long as they're  all the same area  you can do it and this also generalizes  to 3d  we use that to make a little sculpture  here little sculpture  about a thousand blocks you pick up with  your gloves these blocks rearrange them  and the theorem is  you can make any shape you want out of  out of blocks  by this kind of hinge dissection  and that was a brief overview of a few  of the results that we'll be talking  about in this class there's a lot more  and you can check out the 2010 webpage  if you really want to know everything  that's covered  but we'll be posting lectures up there  and let me know if you have any  questions  you

2
00:00:09,190 --> 00:00:11,830
hey everybody welcome to 6849  uh geometric folding algorithms i am  eric domain you can call me  eric and we have as ta jason lynch who's  right there  um and this class is a bit unusual at  least for me  uh because we're trying or i'm trying  for the first time  uh a new experiment which is inverted  lecturing  and i wrote this on the poster for the  class and everyone started asking me  what's inverted lectures  well it's uh it's not a new idea but  i've never tried it before  uh the the concept is to make these in  class times where we're all here  together  more interactive by taking the lecture  component of the class which is covering  all the material  into videos that you watch online so  this class is basically going to  alternate between  real in person things as you are here of  course we're also being video recorded  so  slightly contradiction in terms but uh  that's for the people in the interwebs  to be able to watch this  and so alternate between the real in  class part  and video lectures from the last time i  taught this class which was fall 2010  uh so this is actually what the course  webpage looks like  as of uh an hour ago so we've got  these l l1 l2  l3 those are lectures that's like  content  uh lots of material packed in there  which is how i'm used to teaching a  class put as much material as i can per  lecture  um those are already available you can  start watching  those videos lecture one is optional i'm  going to basically cover  an uh a shorter version of that today so  you don't  says right there optional then the cs  which don't exist yet because we're  doing  it right now the c's are the in-class  sessions the contact hours  some of those may be recorded like this  one we're going to record but some will  not be  and you need to attend both so you've  got to watch all the videos  and you've got to attend all the classes  that's the  that's the setup so this is a bit  unusual so  i'll spend a few more minutes on how  this class is formatted  so uh  requirements  we've got  watch video lectures  i'll talk a little bit about timing of  that and then we've got  attend classes  and then the the sort of actual material  the regular part of the class that  you'll be graded on  uh consists of two parts problem sets  and a project and the project also has a  presentation  this is a project focused class so the  bulk of your grade  is based on the project and presentation  project can take on many different forms  i have this is a theoretical computer  science class  so there's the typical kinds of projects  like read a bunch of papers and  summarize them survey them  although we ask that you keep them  disjoint  more or less or avoid stuff that is well  covered in this book geometric folding  algorithms  which is uh or in the class itself  but otherwise you can survey material  in addition to standard survey paper or  instead of a standard survey paper  you can write a bunch of wikipedia  articles about  folding stuff so whereas survey stuff  should be new material that you haven't  seen  wikipedia stuff could be material that  you've seen in this class  but is not well covered in wikipedia and  that will help  take over the world our usual goal you  could if you see a cool algorithm in  this class you can implement it if  you're a coder  and see how it actually works make it  demorable it should be fun or you can  work on an open problem so we'll  if there's interest we'll run an  optional open problem solving session  where we'll all get together and try to  solve open problems i might also do them  in these class times because now we have  a lot of freedom to be more interactive  to  actually fold paper to have fun in class  instead of me trying to pack as much  material as i can  because that's already been done in the  videos  you could pose an open problem  especially if you come from another  field and you have some interesting  folding related problems those could be  really cool  or you could build something a physical  sculpture a physical structure of some  kind furniture  architecture whatever or you could do it  in the virtual world and just design  something interesting so there you  probably might want to do more than one  make it harder for yourself  all of these are possible projects  there's a lot of different options for  different types of people  different backgrounds whatever then  we'll also have problem sets which are  weekly  roughly and uh they shouldn't be too  long  and they will also have an option of not  doing all the problems  so uh every problem set will probably  have  uh the rule that you can drop you we  will drop the lowest grade on one of the  problems that's on the problem set  so that means if you just don't want to  do a problem you can skip it  and your grade will be determined by the  others if you want you can try to do all  of them we'll just drop the lowest  lowest grade so that'll also  if there's some problems that are easier  for you harder for others you can kind  of mix and match  cool there's a lot more details about  this kind of stuff on the website if you  click on  project you'll get lots of stuff  about lots of details about these  different styles of project  implementation survey  open problems whatever if because what i  said was pretty  pretty brief and  so look for that look at that for more  details project you don't have to worry  about right away but  one of the luxuries of having video  lectures you can skip ahead a little bit  and see what looks interesting to you  and work on a project related to that i  wanted to briefly show you a little bit  about this website  and how it works so you get a flavor for  it so let's say  the very next thing you should do right  after you leave here no  sometime between now and noon on monday  you should watch lecture two so you can  just click on lecture two here on the  left  and this is the the way it's set up  zoom to fit everything on your screen on  the upper left you've got your video  and it should start playing welcome back  hey it's me i'm even wearing the right  t-shirt it's part of it  and coincidence  the fun thing about these videos uh  there's a few nice features one is as  you jump around in the video  you see the slide down here in the lower  left updates to whatever i'm covering  and also if i jump farther the uh  page of handwritten notes that i'm  covering on the right changes  so as i scrobble around i think is the  term  all right yeah too many buttons  who designed this website all right i  forgot this  uh let's reload um  i'll turn down the volume uh you can  also  speed up the video this can be pretty  entertaining if you go really fast but  you can go 1.1 1.2 that's pretty  comfortable  1.5 yesterday i listened to a video at  2x  it's uh it's interesting  thankfully i do not sound like a  chipmunk but  it is if you haven't seen the lecture  before it's a little harder but i'll be  watching the lectures with you  experiencing the same total number of  hours you have to experience so  we're in this together let me know if  you have any comments on the website  it's all been done by hand so if there's  anything you want changed it's easy to  change  you can also do fun things like when  it's paused you can jump around in pages  on the right and say oh i didn't really  understand that page and then you click  on this time  and it will start playing from that time  at full speed you can slow it down too  which can be pretty entertaining  anyway you get the idea let me know how  it goes i want to make it as easy as  possible in fact if you don't like my  software you can just skip it all  download this video like 720p version or  the 360p version  play it in vlc with whatever fancy speed  up speed downs you want to use  or put on your ipad or other tablet  whatever  this page should work on an ipad but i  haven't tested it lately if it doesn't  let me know  cool so that's what it looks like and  then at the top of this page you see  here completion form  a link to completion form when you  finish watching the video  you click on this form it's pretty  minimal you enter  basically your name your username or  your email address you say  yes i watch the video and then that's  all that's required then you click  submit  but we highly encourage you to say  what you think at this point if you have  any questions about the lecture  that wasn't clear or anything you didn't  understand  or i briefly mentioned something that  sounded cool to you and you want to know  more about it  this is your chance to influence what i  cover in class  and this is why you have to submit this  form by noon the previous day  so watching video lectures and filling  out that form  is due by  noon on  mondays and wednesdays the day before  class  that will give me 22 hours to prepare  class  which should be enough i'll find out and  so i can adapt  the in-class time to be whatever people  care about most  so it's kind of like a poll if lots of  people say i didn't understand  x i will cover x if lots of people are  curious about why i will cover why  so whatever you want to know about  typically related things but if you have  unrelated things you want to  bring up as well feel free to put it in  there and we'll try to schedule it in  when it's time or when when it fits  yeah that's the plan i've never done  this before so if you don't like the  form  tell me anything you don't like or if  things should change just  send email or put in the form i guess  but uh email's good too  cool question i notice you have a no  option is that for like oh shoot 11 45  on monday and i haven't watched this  should i still submit the form and say  no  interesting uh why would you  just you're so honest why would you fill  out no on this forum  i i guess so you can fill out the form  multiple times  so you could i guess you could say no  and give your excuse here  if you want it i'll watch it tomorrow  maybe and then later out fill it in with  yes  so in particular if you also have  questions like you watch the video early  on and then later on you have a question  about it  if it's still before class time feel  free to fill this out again it just  makes another entry in our table this is  a google form  that's another question all right  i guess that's the reason for now but  maybe also we don't we couldn't put a  single option  now we want to it's it's to force you to  claim  honestly this is all honor system right  so this is like forcing you to make a  decision to be dishonest i guess  or to be honest rather  cool to make this work you've got to ask  questions you've got to request topics  otherwise  i'll i'll just fill your class times  with more material  that would be my natural temptation so  gotta hold me back and ask lots of  questions  one particular request i wanted to  highlight is to um  send me cool things to cover in  particular  you all surf the web and if you see  um cool folding things or any kind of  folding related things  on the web send it to me  by email and the plan is maybe to do one  a day  or maybe two a day for in class time  there's lots of fun things out there and  it's hard to see it all so when you  discover things  tell me and i will schedule it in  throughout the semesters you can think  of this as a standing homework problem  you should do it sometime before  the end of semester  cool another experiment i think it could  be fun  that is the end of the class  organizationally i think unless there  any questions about  requirements or style or format  all right then in the rest of this  class i'm going to cover as i said a  short version of  lecture one which is just an overview of  the whole class so i want to give you a  flavor of what the course is about  what kind of theorems we prove what kind  of algorithms we develop  so you can decide whether you are  interested in this class whether to take  it  there is also a  sort of survey that where you put your  name email address will add you to the  mailing list and i  announce every lecture so if you want to  if you're listening to the class you  just want to  listen to things you're interested in  come come to class for interesting  things  just listen to that email and see when  cool things come up and also there's a  little survey on that piece of paper  does anyone not have the piece of paper  a couple people so we will get them to  you  a little survey just to get your  background uh there's sort of no  required background for this class  because lots of people come from  different areas  hopefully you know at least one of the  background areas but  anything can kind of be filled in and  one of the points of this  questionnaire is to uh  if there's something you don't  understand like oh gosh i should know  xyz algorithm and you assumed it in  lecture but i don't know it could you  cover it  and if enough people request that i will  so that's the format out of curiosity  how many people here  are course six let's say  and how many people are of course 18 and  how many people  are other okay  cool so reasonably balanced  a lot of six though cool  so we are talking about geometric  folding algorithms  so there's geometry there's folding  and there's algorithms algorithms are  you know how we compute things at least  theoretically  and in general what this field is about  is both mathematics and computer science  uh  say the mathematics and algorithms  that underlie how things fold  and the technical term for things is  geometric objects  and we're interested not only in how  they fold but also how they unfold  in general you have some geometric  structure that can reconfigure like my  arm here can reconfigure in interesting  ways  we want to know what are all the ways it  can reconfigure what are all the ways it  can fold  this piece of paper is another geometric  object  and we want to know what are all the  ways that it can fold  in general so to before i talk about  sort of more formally what this means i  want to show you that i mean folding  comes up  everywhere pretty much almost any  discipline you can name there's some  that involves physical things at least  has geometric objects  geometric objects tend to comply unless  you're even if you're working with  completely rigid objects you don't want  them to fold  that's that's a folding problem  so i have here a list of a bunch of  different  application areas and i'm going to show  you some pictures and videos  related to them so the first one is  robotics  so if you have things like a robotic arm  you want to know  and it starts in one configuration you  want to continuously move it to reach  some other configuration so it can you  know pick up somebody  drop it off over here i want to know how  should i plan the motion of my robotic  arm  a different kind of robotic application  is to make  sheets of material that are themselves  robots that fold themselves into origami  this is a self-folding sheet developed  by  collaboration with at mit and harvard  and you just send a little bit of  electrical current  these little metal pieces heat up which  causes them to pull the creases shut  and boom you get your origami boat  with no origamis required the same sheet  can fold into many different shapes and  the under underlying algorithm here  or the underlying mathematics is that  this pattern of creases square  square grid with alternating diagonals  can fold into essentially any shape  made out of uh that is made out of cubes  here we're making a paper airplane out  of the same sheet  sending it different signals it does not  fly  but both probably floats  so that's the idea and you can imagine  a sheet like this could just be  arbitrarily reprogrammed  to fold into this gadget so i don't have  to bring around so many gadgets maybe my  laptop could fold smaller and later  could unfold into something to my  desktop  and i don't know my phone could  reconfigure into something else  that's that's the vision this is what we  call programmable matter where you can  change shape  in the same way that we programmed  software we don't want to be able to  program  shape obviously we're just  getting there now so that's robotics  next up is graphics one example is you  want to animate  uh a character from one position to  another that's a  kind of by a sort of a keyframe  animation  that's a folding problem i have here an  example of the two-dimensional analog  where you have one polygon and another  polygon and you want to morph  continuously from one to the other  and these these animations are all found  by  algorithms that are motivated by folding  stuff that we will cover  in this class and it's not so easy but  that's lots of fun things uh the  all of these motions avoid collision and  they approximately preserve the edge  lengths  also if it's possible if they match in  the left and the right they will  preserve those edge links  uh cool so that's graphics sort of  morphing  in mechanics mechanics motivated a lot  of mechanical linkage  problems in particular is this great  book from 1877 called how to draw a  straight line  you may think you know how to draw a  straight line but it's  the point of this book is to figure out  by turning a circular crank can i draw a  straight line  and this is motivated by steam engines  where you have a steam piston that's  moving up and down in a straight line  and you want to turn a wheel on your on  your train  so how do you do that well these are two  old ways to do it  and here's what they look like in  in computational land i guess these are  the original drawings so we have on the  top a design by  watt the unit in 1784  and it's approximately straight it's not  perfect but this highlighted point moves  along a circle and the green point moves  along the figure eight  and the figure eight is fairly straight  here in the middle so if you just use  that part  it works pretty well you lose a little  bit from the wiggle  but it's it was used in a lot of  locomotive engines  down here you have the first correct  solution by posse a in 1864  almost 100 years later and if you move  the highlighted point along a circle  this point moves along the red line  and pretty cool and there's a whole  bunch of mathematics related to this  generalizing this result  and we will cover it in this class  next up we have manufacturing uh this is  a fun example  of bending a piece of wire  this is a machine called di wire do it  yourself wire bending so you can build  one of these machines here it's making a  planar shape you can also make  three-dimensional shapes  lots of things are possible this is a  recent machine there's actually a lot of  wire bending machines  that are out there this is a nice and  simple one  but can you make everything no because  this  piece of wire can't collide with itself  and it can't collide with the machine  that's a constraint and you can't you  want to figure out  what shapes can you make can i make my  pair of glasses  or am i limited somehow that's a folding  problem which we will talk about  next up medical so one example of a  medical application is uh this  is to build a stent this is called an  origami stent developed at oxford  a bunch of years ago and the idea is you  want to do non-intrusive heart surgery  so you want to take  this big thing fold it down really small  like this and  and stick it through some small blood  vessels in your body until you get up  near your heart where you've got nice  big  vessels you want to declog and so you  want to expand it back  out to its larger size so it's  essentially a transportation problem  you want to make something small till it  gets where you need it and then you make  it big  that's an one example of a medical  application  in related to that kind of  transportation problem you also have  motivation of  aerospace so you want to send a large  object into space  your space shuttle isn't big enough to  do it you'd like to fold it down to  become  smaller this is an example by robert  lang who's pictured here  leading origami designer and  this uh this is a prototype it's only  five meters in size  uh the goal is to make a hundred meter  uh telescope lens which definitely does  not fit in your  space station so you do lots of foldings  like this until it fits in  in your space shuttle you send it out  when you're in space you've got lots of  room and you can unfold  out and you get your giant lens much  larger than hubble's lens for example  uh so this is in general this area is  called deployable structures when you  want to make something small for  transportation  then we have biology protein folding is  is a big problem a lot of people work on  it and a protein folds kind of like a  robotic  arm geometrically it's a little  different and it's not  fully understood how it works but  there's a lot of cool  folding problems geometry problems that  we've studied motivated by  by how proteins fold and figuring that  out the application's thing like things  like drug design you want to kill a  virus without killing the host  design a protein that folds into the  right shape so that it  kills one thing and not the other that's  the sort of  general goal and we have some cool  things which we will cover in this class  related to that next is  sculpture and origami design is an  obvious motivation for why you might be  here  this is why we initially got interested  origami has reached incredible heights  both of these are folded from  one square no cuts so just one square  paper you can make three-headed dog  you can make uh i guess i shouldn't call  him a man  but you can make a nas ghoul on his  horse or is it one  entity i don't know but it's one square  paper that much i know  uh this is designed by jason kuh who's  the president of the origami club at mit  origami which you should all  check out they meet on sundays at three  three good uh i know there's a bunch of  origami people here  so uh how are these possible these are  possible through mathematics and the  kind of algorithms that we are  uh that we will cover in this class in  fact there's a guest lecture by jason  kuh  uh which you'll be watching i think is  lecture six uh  and he's not actually in town but still  he can give the video lecture because he  already gave it  that's like time travel um  i have uh just a few examples of some of  our sculpture this is with uh martin  domain who's your cameraman  here and uh these are based on curved  creases which we may talk about to some  extent in this class that we didn't talk  about it too much  two years ago but a little bit curved  creases are not very well understood  mathematically  almost everything we will cover is based  on straight crease design  like most origami curve creases are  pretty amazing though and they do some  really cool shapes  and we're still trying to figure them  out how to design them algorithmically  and this these pieces are on display in  dc right now if you want to  check them out for the next several  months  yeah oh these pieces of paper are  circles with a circular hole and there's  more than one this has two pieces  this has three and this has i think five  okay so that was a brief uh sculpture  origami design lots of different other  possible sculptures kinetic sculpture  you might want to try building  um also architecture reconfigurable  architecture  i think this is an underexplored area  how many people here are from  architecture  a few okay you might want to make some  reconfigurable buildings  hoberman is one example of someone  embracing this a lot  he started in folding toy design but now  he does mostly folding architecture  this is one example from his company  hoberman associates  from the winter olympics at salt lake  city 2002  it's just it's very dramatic you have  this  folding structure that can almost  completely close up except for a little  circle here and when it opens up you've  got the huge  stage you can see the scale of of people  he did uh he co-designed this this giant  u2 stage if you've seen  u2 anytime in the last few years like i  did this incredible folding structure  based on similar similar principles and  and chuck hoberman is actually teaching  a class  at the graduate school of design at  harvard on monday afternoons  and i think anyone's welcome if you want  to check it out you should  should go see it and talk about his  techniques maybe we'll get him to  do a guest lecture we'll see so that was  a  brief overview of a bunch of different  applications  there's tons more in fact if you have  more please send them to me  i know a bunch more and we we will get  to them  but this is kind of a little survey of  different fields that folding touches on  now i'm going to switch over to more  mathematical stuff which is the bulk of  the class  that's our questions cool  so we move to  what kind of things we're interested in  folding they are essentially three types  linkages which are  we usually think of as kind of one  dimensional structure so we have one  dimensional  usually straight segments connected  together at hinges  and here i've drawn it in two dimensions  it could also live in three dimensions  like my arm  is made up of one-dimensional bones and  then there's  sockets for them to join essentially  universal joints  how can those things fold what shapes  can they fold into general constraints  here are that the edges should stay the  same length i can't  stretch my arm longer and they you have  to stay connected at the joints  i can't detach my elbow and radar later  reattach it at least not ideally  so the mathematical version is you're  not allowed to  uh one dimension up we have sheet  folding which we usually refer to as  paper folding  so you have but it could be you know  sheet metal  anything you have your sheet of material  and you want to fold it into things like  origami  like space stations like telescope  lenses whatever  general rules here like this piece of  paper you cannot stretch the paper  can't get any longer it can't collide  with itself  and you can't tear the paper you're not  allowed to cut it  uh because that would make tends to make  things too easy  uh so the sort of mathematically pure  version and also the  the modern origami pure version is that  you're not allowed to cut your piece of  paper  so just folding no stretching  no crossing if you want to go another  dimension up  we typically call this polyhedron  folding  so polyhedron it's uh made up of a bunch  of polygons  like cube and uh typically we're  interested  in unfolding a polyhedron  so think of this as just a surface it's  hollow inside  if you wanted to build that surface  you'd like to find  a shape like this cross  that folds into that cube  because if you have sheet material you  want to start from some  original shape that's flat uh they can  fold into your  shape you can also consider the reverse  problem which is folding  suppose i give you this cross what  shapes can it fold into and we've looked  at both  and i'll talk about them in a second  okay so in general for each of these  categories and you'll see these  categories over and over because they're  they're on the book here it says  linkages origami and polyhedra  so and there's two three parts to the  book there's linkages for copy and  polyhedra  part 142.3 just like this our class will  not follow this order though  uh after today we're going to i think  we'll talk about paper for a while then  go to linkages for a while then go to  polyhedra for a while  and then we'll repeat paper for a while  and and so on so we kind of  different different parts will be of  interest to different people so  do lots of different coverage but i  think paper is the most exciting so  we'll we'll start there  in general for each of these types of  things you want to fold  there are two kinds of problems we're  generally interested in  um based around the idea that well  there's there's some folding  structure that you're interested in and  then there's the way it can fold  and you can start from either one  so if you start from a folding structure  and you want to know what it can fold  into  this we typically call a foldability  problem  or you could call it an analysis problem  i have something  like i don't know  like this and i want to know what it can  fold into that's an example  what can the cross fold into  or i have some linkage and i want to  know i have a robotic arm it's already  fixed i've already designed it built it  what shapes can it fold into  the reverse problem is a design problem  or a synthesis problem  so there i start with what foldings i  would like to have  so for example i want to fold a  butterfly maybe so i design  a butterfly and now i'd say well i have  a rectangular paper can i fold it into a  butterfly  so that's a design problem here you want  to design a folding structure that  achieves your goals like shape  so that's kind of generically what we  like to do  and then there's three types of results  we typically get  so i'm giving you a super high level  first  and then we'll get to some actual fun  examples  so these are typical results  so three types are universality  decision and hardness  and these relate to sort of what is  possible  in particular is everything possible and  this makes sense both from a foldability  standpoint and from a design standpoint  if i give you a robotic arm i'd like to  know can it fold into  all the possible configurations or  there's some that it can't reach  that if it can reach all of them we have  a universality result and typically when  you can prove  that you can make anything you actually  get an  algorithm to do it so you can say oh  well how do i fold into x well here's  how you do it there's an algorithm  to compute that for you also in design  i'd like to know  okay i can make a butterfly and i can  make a nozzle  can i make everything if yes you get a  universality result so that's a design  universality if universality is not true  if you can't make everything  the next best thing you could hope for  is a decision algorithm  an algorithm that tells you quickly is  this possible is this impossible  so this is like a characterization of  what's possible and what's impossible  using algorithms sometimes though that's  not possible  there is no good algorithm to tell  whether something's foldable and then we  aim for hardness result to prove  that there is no good algorithm to  solve your problem so that gets into  complexity theory and we will  be not everyone's going to know that we  will be reviewing it in the video  lectures and in here  as needed so those those are the sort of  typical  outcomes for any of these kinds of  problems  let's see some examples unless they're  generic questions at this point  only generic questions about generic  material  next we'll get to specific material and  you can ask specific questions  uh cool i think  it's all good  so first up is linkages  and of course there's lots of results in  each of these fields i'm just going to  show you a couple in each  and the first question you are typically  interested in  about a linkage is is it rigid does it  move at all  let me give you some examples of this  yeah  do a little pop quiz uh  it's okay to get it wrong just  makes for some fun interaction uh  that one so  yes or no is this rigid or rigid or  flexible i should say  rigid everyone agrees good  correct richard are flexible  flexible uh depends on the dimension  very good  it's rigid or flexible uh it depends  it's rigid  in 2d because these are triangles  but it's flexible in 3d because  you can rotate one triangle around this  hinge  okay and this is rigid or flexible  flexible  everyone agrees and is correct  cool uh in general given a structure  like this a  linkage you want to know is it rigid or  is it flexible this has lots of  applications like you want to build a  bridge you don't want it to move you're  building a building you don't want it to  move  so here's the mathematical state of  affairs distinguishing  this boundary which is rigid in 2d  versus flexible in 2d  we understand super wells great  algorithms to do it we will cover them  this boundary between rigid and 3d and  flexible in 3d  we don't really understand and there  aren't great algorithms to solve it  there's a little disconcerting given the  number of buildings and bridges we live  in  but that's sort of the state of the  world  and we will talk about all that  so that is rigidity next up we have  universality so this is the robotic arm  question i mentioned i have a robotic  arm does it fold  into everything  into every possible configuration  and the answer again depends what  dimension you live in  and in two dimensions there are cool  ways to reconfigure your robotic arm so  this is actually a polygon a closed arm  if you will a closed chain  and this is one way to unfold it and  then in principle you can refold that  into any other shape you want  and we will cover this is one algorithm  to do it there's actually three  algorithms to do this  this is the most recent one this is  originally my  my phd thesis to figure out whether this  was possible  it's called the carpenter's rule problem  this one this algorithm preserves the  fivefold rotational symmetry of the  polygon which is pretty cool but in  general you take any  2d polygon you can unfold it while  preserving all the edge links keeping  all the connections and avoiding  collision  show you some other fun examples this is  a kind of spider 500 vertices  we're kind of zooming out as we expand  here you can see the links get really  tiny but in reality they're  all the lengths are staying the same  throughout time  and we fold so this is the state of  affairs in 2d we have good algorithms  it's always possible universality  so uh what about 3d 3d  is harder in particular because of  examples like this  we call this knitting needles you've got  two blue segments which you can think of  as long needles  and then a short purple thread  connecting them  and there's no way to undo this so in  particular if you started with a  straight  robotic arm you could not fold it into  this shape or vice versa  call this a locked configuration and in  general lock configurations exist  obviously and it's we don't know how to  distinguish  good or bad so here actually we don't  know whether it's there's a good  decision algorithm or  hardness but it's definitely not  universal  and 3d is very interesting because it  relates to robotic arms in real life and  protein folding and all these things so  i'll be talking about that in 4d if  you're lucky enough to live in four  dimensions  you get universality everything's great  and we'll  talk about that too uh so  next up we have paper folding  so that was a brief overview of linkages  so paper folding i'm going to mention  it one problem in the foldability world  and some problems in the design world  this is probably my favorite area  paper origami design but foldability is  interesting too the obvious f  and the sort of the first question you  might wonder about  in origami foldability is if i what  crease patterns  fold flat so if you take a classic  flapping bird  you unfold it these are the creases you  get but in general if i gave you some  pattern of creases drawn on your piece  of paper does it fold into anything  and here to make it interesting we say  does it fold into anything flat  flat origami you have to fold along all  the creases  sadly deciding this is a problem  called that's np hard so there's  basically no good algorithm to solve  this and we will prove that  uh in a few classes a few lectures  um so that was the  foldability problem now onto design if i  give you a square paper  what shapes can i make that's the  obvious origami design problem  and it turns out you can make any  polygons so you can make any flat shape  you want  if you have a piece of paper that's  white on one side and patterned or color  black on the other you can make any two  color pattern you want if it's polygonal  it's made out of straight sides if you  want to make a 3d thing you can make any  3d polyhedron also with two color  patterns whatever you want  we proved this way back in 1999 i think  is actually the first paper to use the  term computational origami  and uh it's not hard to prove we will  prove it uh in a couple of lectures i  think  but it this is uh we don't have a  practical way to do this  there is an algorithm here but it  doesn't give you a good way to fold  anything and so the quest continues for  a good way to fold  everything and one such approach is  called orgamizer  and it you give it an arbitrary 3d  surface  it designs a crease pattern this you  fold from a square piece of paper  solids or mountains dashed or valleys  and you get  this bunny this is a classic model in  computer graphics called the stanford  bunny  and this takes about 10 hours to fold if  you're tomohirotachi  who designed it and there's free  software available called organizer by  tomohira  and he was just here a couple weeks ago  and we've been proving that this  algorithm always works  and it also seems very practical most of  the material here gets used in the  surface  so it doesn't you don't need a very  large square paper to make your  bunny i have one  a little example here sort of classic in  the origami world which is to make a  checkerboard  and this is a 4x4 checkerboard obviously  not a full 8x8  but you can fold an 8x8 and we'll talk  about different ways to do it  this one is folded from one square paper  white on one side  green on the other and actually has a  pretty good scale factor it just  collapses by  a factor of two here so i can get it  back into correct state so  there you go you can make your own  origami checker boards  if you ran out of your regular wooden  ones  cool next example is what we call maze  folding this is the  poster uh if you saw it this is  a folding of this crease pattern and the  design was  uh in general you take any orthogonal  graph meaning horizontal and vertical  lines on a grid and you want to extrude  that graph  out of the plane then there's an  algorithm it's online you can play with  it i think ericdomain.org  maze will get you there you just draw  that pattern it will give you this  crease pattern you print it out  and uh eight hours maybe  you can fold it into reality so these  this is by  jenny and eli who i think are students  in this class  they got started a little early and i  didn't bring it here but at some point i  will bring  the real 3d one so that's mace folding  and the next one i have is called  folding cut this is actually the first  problem we worked on  so kind of a personal favorite you take  a rectangle of paper  and then you fold it flat  and you take your scissors and make one  complete straight cut this is a magic  trick performed by  harry houdini and other magicians  and you get in this case two pieces and  the question is what shapes can you make  by folding  and one straight cut here i've made a  little swan  okay you're not impressed so to a harder  one  this one is actually quite challenging  to cut i was folding it  as you were arriving some of you  probably saw  that's about about right  it's a little hard to open up but this  should be  the mit logo  all right good  so that's the idea you can impress all  your friends print these out and  uh fold them it's a good good challenge  to fold along the crease pattern and as  you might guess there's an algorithm  here  given any polygon or collection of  polygons you can  fold a piece of paper to line up all the  edges of that polygon cut along it  you're done okay that is paper  next i want to briefly mention polyhedra  and again actually i have polyeter right  over here  so again there's an unfolding problem  and a folding problem  and i'll start by showing you a little  bit about the  folding problem well this shows a little  bit of both but this is a video we made  back in 99  i was a grad student i had lots of hours  to make videos  animations so this is one example of an  unfolding of a cube  but the unfolding we want to consider  here is this cross unfolding that i keep  talking about  it's kind of a classic so fun to analyze  so if you take that cross shape you want  to know what other shapes what other  convex polyhedra can it fold into  and here the rules are a little  different from origami you have to get  exactly the surface you want no overlap  here we happen to get a doubly covered  quadrilateral  which is a kind of a convex surface  a little bit cheating but if we change  the creases in this way we get a  five-sided polyhedron with a plane of  symmetry  down this axis  or  we can change the creases in this way  and get a tetrahedron and it's exact  coverage so this little tab fits into  exactly that pocket  uh get exactly what you want as you  might guess there's an algorithm that  will enumerate  all the different convex polyhedra you  can make from a given polygon of paper  and it's based on a theorem in russian  by alexandrov which is in the background  here if you're curious what the cyrillic  background is  this theorem is the key to the algorithm  so that's what's possible from a  by folding a given polygon  and then you can ask about unfolding let  me give you a brief overview unfolding  so uh when you're unfolding you might be  interested in only cutting along the  edges of the surface which is what  happened  in this cross unfolding i cut along  these edges  or you might allow cutting anywhere on  the surface and the answers depend  on those two versions you could try to  do convex polyhedra or you could try to  do arbitrary  polyhedra so here's the story  this one we don't know this one we don't  know  this one the answer is yes we'll prove  it this one is the answer is no we'll  prove it  in fact i can show you some examples uh  this is an edge unfolding of a convex  polyhedron the oldest  set that we know is from this 1525 book  by albert durer  and this is a snub cube he didn't have  wikipedia at the time so he couldn't  just  look at the image and draw it he had to  build a physical model and so he'd cut  out  pieces of paper to build physical models  so he could paint it and various of his  prints  have convex polyhedra in them and this  is an example of this open problem  which we don't know the answer to uh can  you edge unfold convex polyhedra  uh if you want to take non-convex  polyhedra  you cannot always do it this is an  example of a kind of spiky tetrahedron  that cannot be edge unfolded we'll prove  that  but it can be generally unfolded if you  can cut anywhere  you can find a one piece unfolding and  so big open problem  is can you do any polyhedron  with general cuts i think the answer is  yes very cool  problem uh and that's polyhedra  and the last thing i wanted to show you  is hinge dissection this is  one more thing after one two three and  four after one two and three we have  four hinge dissection this is a chain of  blocks which are kind of two-dimensional  uh so it's but it's not really paper  folding so it's kind of like a thick  linkage kind of  between one and two if you will and this  chain folds into an equilateral triangle  or square and it's over 100 years old  can you do anything turns out there's a  universality result this is just a  picture  part of the proof which we will cover in  a later lecture  that you can fold you can take any set  of polygons  of the same area there's one chain of  blocks  which will fold into each of those  polygons so  you can you get a universality result  you can do equilateral triangle to a  regular hexagon to  a swan to whatever as long as they're  all the same area  you can do it and this also generalizes  to 3d  we use that to make a little sculpture  here little sculpture  about a thousand blocks you pick up with  your gloves these blocks rearrange them  and the theorem is  you can make any shape you want out of  out of blocks  by this kind of hinge dissection  and that was a brief overview of a few  of the results that we'll be talking  about in this class there's a lot more  and you can check out the 2010 webpage  if you really want to know everything  that's covered  but we'll be posting lectures up there  and let me know if you have any  questions  you
 

3
00:00:11,830 --> 00:00:13,350

 

4
00:00:13,350 --> 00:00:16,470

 

5
00:00:16,470 --> 00:00:17,670

 

6
00:00:17,670 --> 00:00:21,189

 

7
00:00:21,189 --> 00:00:22,230

 

8
00:00:22,230 --> 00:00:24,550

 

9
00:00:24,550 --> 00:00:25,750

 

10
00:00:25,750 --> 00:00:29,189

 

11
00:00:29,189 --> 00:00:29,199

 

12
00:00:29,199 --> 00:00:31,669


13
00:00:31,669 --> 00:00:33,030

 

14
00:00:33,030 --> 00:00:34,310

 

15
00:00:34,310 --> 00:00:36,069

 

16
00:00:36,069 --> 00:00:38,150

 

17
00:00:38,150 --> 00:00:39,750

 

18
00:00:39,750 --> 00:00:44,150

 

19
00:00:44,150 --> 00:00:45,830

 

20
00:00:45,830 --> 00:00:45,840

 

21
00:00:45,840 --> 00:00:46,549


22
00:00:46,549 --> 00:00:49,190

 

23
00:00:49,190 --> 00:00:51,110

 

24
00:00:51,110 --> 00:00:52,310

 

25
00:00:52,310 --> 00:00:55,110

 

26
00:00:55,110 --> 00:00:56,389

 

27
00:00:56,389 --> 00:00:57,590

 

28
00:00:57,590 --> 00:00:59,750

 

29
00:00:59,750 --> 00:01:01,349

 

30
00:01:01,349 --> 00:01:01,359

 

31
00:01:01,359 --> 00:01:02,069


32
00:01:02,069 --> 00:01:04,789

 

33
00:01:04,789 --> 00:01:06,789

 

34
00:01:06,789 --> 00:01:09,190

 

35
00:01:09,190 --> 00:01:12,230

 

36
00:01:12,230 --> 00:01:13,109

 

37
00:01:13,109 --> 00:01:15,830

 

38
00:01:15,830 --> 00:01:18,789

 

39
00:01:18,789 --> 00:01:20,789

 

40
00:01:20,789 --> 00:01:22,230

 

41
00:01:22,230 --> 00:01:25,590

 

42
00:01:25,590 --> 00:01:28,710

 

43
00:01:28,710 --> 00:01:31,510

 

44
00:01:31,510 --> 00:01:31,520

 

45
00:01:31,520 --> 00:01:32,469


46
00:01:32,469 --> 00:01:34,630

 

47
00:01:34,630 --> 00:01:35,910

 

48
00:01:35,910 --> 00:01:37,910

 

49
00:01:37,910 --> 00:01:37,920

 

50
00:01:37,920 --> 00:01:38,870


51
00:01:38,870 --> 00:01:41,429

 

52
00:01:41,429 --> 00:01:42,310

 

53
00:01:42,310 --> 00:01:44,789

 

54
00:01:44,789 --> 00:01:46,310

 

55
00:01:46,310 --> 00:01:49,190

 

56
00:01:49,190 --> 00:01:49,670

 

57
00:01:49,670 --> 00:01:51,990

 

58
00:01:51,990 --> 00:01:53,670

 

59
00:01:53,670 --> 00:01:53,680

 

60
00:01:53,680 --> 00:01:53,990


61
00:01:53,990 --> 00:01:56,789

 

62
00:01:56,789 --> 00:01:58,550

 

63
00:01:58,550 --> 00:02:00,069

 

64
00:02:00,069 --> 00:02:01,910

 

65
00:02:01,910 --> 00:02:02,950

 

66
00:02:02,950 --> 00:02:05,510

 

67
00:02:05,510 --> 00:02:06,789

 

68
00:02:06,789 --> 00:02:09,350

 

69
00:02:09,350 --> 00:02:10,949

 

70
00:02:10,949 --> 00:02:13,670

 

71
00:02:13,670 --> 00:02:14,630

 

72
00:02:14,630 --> 00:02:17,750

 

73
00:02:17,750 --> 00:02:20,830

 

74
00:02:20,830 --> 00:02:23,910

 

75
00:02:23,910 --> 00:02:23,920

 

76
00:02:23,920 --> 00:02:31,110


77
00:02:31,110 --> 00:02:34,869

 

78
00:02:34,869 --> 00:02:41,750

 

79
00:02:41,750 --> 00:02:43,350

 

80
00:02:43,350 --> 00:02:45,030

 

81
00:02:45,030 --> 00:02:49,750

 

82
00:02:49,750 --> 00:02:51,990

 

83
00:02:51,990 --> 00:02:53,350

 

84
00:02:53,350 --> 00:02:54,790

 

85
00:02:54,790 --> 00:02:59,910

 

86
00:02:59,910 --> 00:03:03,270

 

87
00:03:03,270 --> 00:03:03,280

 

88
00:03:03,280 --> 00:03:11,110


89
00:03:11,110 --> 00:03:13,830

 

90
00:03:13,830 --> 00:03:14,790

 

91
00:03:14,790 --> 00:03:18,550

 

92
00:03:18,550 --> 00:03:21,910

 

93
00:03:21,910 --> 00:03:24,710

 

94
00:03:24,710 --> 00:03:25,509

 

95
00:03:25,509 --> 00:03:27,910

 

96
00:03:27,910 --> 00:03:29,670

 

97
00:03:29,670 --> 00:03:31,270

 

98
00:03:31,270 --> 00:03:33,430

 

99
00:03:33,430 --> 00:03:33,440

 

100
00:03:33,440 --> 00:03:34,390


101
00:03:34,390 --> 00:03:36,789

 

102
00:03:36,789 --> 00:03:38,470

 

103
00:03:38,470 --> 00:03:38,480

 

104
00:03:38,480 --> 00:03:39,430


105
00:03:39,430 --> 00:03:43,830

 

106
00:03:43,830 --> 00:03:47,430

 

107
00:03:47,430 --> 00:03:49,430

 

108
00:03:49,430 --> 00:03:51,350

 

109
00:03:51,350 --> 00:03:52,949

 

110
00:03:52,949 --> 00:03:54,470

 

111
00:03:54,470 --> 00:03:56,630

 

112
00:03:56,630 --> 00:03:58,550

 

113
00:03:58,550 --> 00:03:58,560

 

114
00:03:58,560 --> 00:03:59,350


115
00:03:59,350 --> 00:04:01,030

 

116
00:04:01,030 --> 00:04:02,390

 

117
00:04:02,390 --> 00:04:04,149

 

118
00:04:04,149 --> 00:04:05,910

 

119
00:04:05,910 --> 00:04:09,030

 

120
00:04:09,030 --> 00:04:10,710

 

121
00:04:10,710 --> 00:04:11,990

 

122
00:04:11,990 --> 00:04:12,710

 

123
00:04:12,710 --> 00:04:15,830

 

124
00:04:15,830 --> 00:04:19,509

 

125
00:04:19,509 --> 00:04:22,629

 

126
00:04:22,629 --> 00:04:24,310

 

127
00:04:24,310 --> 00:04:26,469

 

128
00:04:26,469 --> 00:04:27,909

 

129
00:04:27,909 --> 00:04:29,590

 

130
00:04:29,590 --> 00:04:31,909

 

131
00:04:31,909 --> 00:04:34,469

 

132
00:04:34,469 --> 00:04:34,479

 

133
00:04:34,479 --> 00:04:35,189


134
00:04:35,189 --> 00:04:39,510

 

135
00:04:39,510 --> 00:04:41,909

 

136
00:04:41,909 --> 00:04:43,030

 

137
00:04:43,030 --> 00:04:44,710

 

138
00:04:44,710 --> 00:04:44,720

 

139
00:04:44,720 --> 00:04:46,629


140
00:04:46,629 --> 00:04:48,310

 

141
00:04:48,310 --> 00:04:49,510

 

142
00:04:49,510 --> 00:04:50,790

 

143
00:04:50,790 --> 00:04:52,310

 

144
00:04:52,310 --> 00:04:53,270

 

145
00:04:53,270 --> 00:04:56,790

 

146
00:04:56,790 --> 00:04:59,110

 

147
00:04:59,110 --> 00:05:01,990

 

148
00:05:01,990 --> 00:05:04,390

 

149
00:05:04,390 --> 00:05:06,150

 

150
00:05:06,150 --> 00:05:07,909

 

151
00:05:07,909 --> 00:05:09,430

 

152
00:05:09,430 --> 00:05:11,990

 

153
00:05:11,990 --> 00:05:13,590

 

154
00:05:13,590 --> 00:05:14,870

 

155
00:05:14,870 --> 00:05:16,230

 

156
00:05:16,230 --> 00:05:18,950

 

157
00:05:18,950 --> 00:05:20,710

 

158
00:05:20,710 --> 00:05:20,720

 

159
00:05:20,720 --> 00:05:22,950


160
00:05:22,950 --> 00:05:25,350

 

161
00:05:25,350 --> 00:05:25,360

 

162
00:05:25,360 --> 00:05:26,070


163
00:05:26,070 --> 00:05:28,390

 

164
00:05:28,390 --> 00:05:29,590

 

165
00:05:29,590 --> 00:05:32,310

 

166
00:05:32,310 --> 00:05:32,320

 

167
00:05:32,320 --> 00:05:32,790


168
00:05:32,790 --> 00:05:35,830

 

169
00:05:35,830 --> 00:05:38,230

 

170
00:05:38,230 --> 00:05:40,070

 

171
00:05:40,070 --> 00:05:41,990

 

172
00:05:41,990 --> 00:05:43,430

 

173
00:05:43,430 --> 00:05:44,790

 

174
00:05:44,790 --> 00:05:46,390

 

175
00:05:46,390 --> 00:05:48,070

 

176
00:05:48,070 --> 00:05:51,110

 

177
00:05:51,110 --> 00:05:52,310

 

178
00:05:52,310 --> 00:05:54,390

 

179
00:05:54,390 --> 00:05:56,710

 

180
00:05:56,710 --> 00:05:59,909

 

181
00:05:59,909 --> 00:06:01,830

 

182
00:06:01,830 --> 00:06:03,270

 

183
00:06:03,270 --> 00:06:06,950

 

184
00:06:06,950 --> 00:06:08,629

 

185
00:06:08,629 --> 00:06:09,830

 

186
00:06:09,830 --> 00:06:11,189

 

187
00:06:11,189 --> 00:06:14,070

 

188
00:06:14,070 --> 00:06:15,350

 

189
00:06:15,350 --> 00:06:18,550

 

190
00:06:18,550 --> 00:06:20,150

 

191
00:06:20,150 --> 00:06:22,150

 

192
00:06:22,150 --> 00:06:23,270

 

193
00:06:23,270 --> 00:06:24,710

 

194
00:06:24,710 --> 00:06:26,309

 

195
00:06:26,309 --> 00:06:28,230

 

196
00:06:28,230 --> 00:06:30,550

 

197
00:06:30,550 --> 00:06:32,150

 

198
00:06:32,150 --> 00:06:34,309

 

199
00:06:34,309 --> 00:06:36,870

 

200
00:06:36,870 --> 00:06:38,390

 

201
00:06:38,390 --> 00:06:40,790

 

202
00:06:40,790 --> 00:06:42,150

 

203
00:06:42,150 --> 00:06:46,230

 

204
00:06:46,230 --> 00:06:49,350

 

205
00:06:49,350 --> 00:06:50,950

 

206
00:06:50,950 --> 00:06:50,960

 

207
00:06:50,960 --> 00:06:52,309


208
00:06:52,309 --> 00:06:55,510

 

209
00:06:55,510 --> 00:06:57,749

 

210
00:06:57,749 --> 00:06:59,990

 

211
00:06:59,990 --> 00:07:04,550

 

212
00:07:04,550 --> 00:07:06,309

 

213
00:07:06,309 --> 00:07:09,110

 

214
00:07:09,110 --> 00:07:12,150

 

215
00:07:12,150 --> 00:07:13,909

 

216
00:07:13,909 --> 00:07:15,990

 

217
00:07:15,990 --> 00:07:17,589

 

218
00:07:17,589 --> 00:07:19,430

 

219
00:07:19,430 --> 00:07:21,670

 

220
00:07:21,670 --> 00:07:24,790

 

221
00:07:24,790 --> 00:07:26,390

 

222
00:07:26,390 --> 00:07:28,230

 

223
00:07:28,230 --> 00:07:31,029

 

224
00:07:31,029 --> 00:07:31,039

 

225
00:07:31,039 --> 00:07:31,990


226
00:07:31,990 --> 00:07:35,589

 

227
00:07:35,589 --> 00:07:38,629

 

228
00:07:38,629 --> 00:07:39,430

 

229
00:07:39,430 --> 00:07:47,189

 

230
00:07:47,189 --> 00:07:50,390

 

231
00:07:50,390 --> 00:07:50,400

 

232
00:07:50,400 --> 00:07:51,029


233
00:07:51,029 --> 00:07:53,270

 

234
00:07:53,270 --> 00:07:55,110

 

235
00:07:55,110 --> 00:07:57,189

 

236
00:07:57,189 --> 00:07:57,199

 

237
00:07:57,199 --> 00:07:58,309


238
00:07:58,309 --> 00:08:01,350

 

239
00:08:01,350 --> 00:08:01,360

 

240
00:08:01,360 --> 00:08:01,990


241
00:08:01,990 --> 00:08:09,350

 

242
00:08:09,350 --> 00:08:10,550

 

243
00:08:10,550 --> 00:08:12,469

 

244
00:08:12,469 --> 00:08:13,909

 

245
00:08:13,909 --> 00:08:15,589

 

246
00:08:15,589 --> 00:08:17,189

 

247
00:08:17,189 --> 00:08:19,430

 

248
00:08:19,430 --> 00:08:20,790

 

249
00:08:20,790 --> 00:08:23,670

 

250
00:08:23,670 --> 00:08:25,029

 

251
00:08:25,029 --> 00:08:26,710

 

252
00:08:26,710 --> 00:08:28,309

 

253
00:08:28,309 --> 00:08:28,319

 

254
00:08:28,319 --> 00:08:29,350


255
00:08:29,350 --> 00:08:32,230

 

256
00:08:32,230 --> 00:08:34,630

 

257
00:08:34,630 --> 00:08:36,949

 

258
00:08:36,949 --> 00:08:38,310

 

259
00:08:38,310 --> 00:08:39,670

 

260
00:08:39,670 --> 00:08:45,509

 

261
00:08:45,509 --> 00:08:47,430

 

262
00:08:47,430 --> 00:08:52,790

 

263
00:08:52,790 --> 00:08:54,949

 

264
00:08:54,949 --> 00:08:56,790

 

265
00:08:56,790 --> 00:08:58,310

 

266
00:08:58,310 --> 00:09:00,070

 

267
00:09:00,070 --> 00:09:02,550

 

268
00:09:02,550 --> 00:09:04,070

 

269
00:09:04,070 --> 00:09:07,030

 

270
00:09:07,030 --> 00:09:08,949

 

271
00:09:08,949 --> 00:09:11,590

 

272
00:09:11,590 --> 00:09:11,600

 

273
00:09:11,600 --> 00:09:12,389


274
00:09:12,389 --> 00:09:14,070

 

275
00:09:14,070 --> 00:09:15,670

 

276
00:09:15,670 --> 00:09:17,509

 

277
00:09:17,509 --> 00:09:19,110

 

278
00:09:19,110 --> 00:09:21,430

 

279
00:09:21,430 --> 00:09:23,829

 

280
00:09:23,829 --> 00:09:25,750

 

281
00:09:25,750 --> 00:09:27,190

 

282
00:09:27,190 --> 00:09:28,870

 

283
00:09:28,870 --> 00:09:30,310

 

284
00:09:30,310 --> 00:09:31,990

 

285
00:09:31,990 --> 00:09:33,910

 

286
00:09:33,910 --> 00:09:36,949

 

287
00:09:36,949 --> 00:09:37,990

 

288
00:09:37,990 --> 00:09:38,000

 

289
00:09:38,000 --> 00:09:38,710


290
00:09:38,710 --> 00:09:41,990

 

291
00:09:41,990 --> 00:09:43,509

 

292
00:09:43,509 --> 00:09:45,750

 

293
00:09:45,750 --> 00:09:47,990

 

294
00:09:47,990 --> 00:09:48,000

 

295
00:09:48,000 --> 00:09:49,030


296
00:09:49,030 --> 00:09:51,350

 

297
00:09:51,350 --> 00:09:52,630

 

298
00:09:52,630 --> 00:09:53,670

 

299
00:09:53,670 --> 00:09:55,750

 

300
00:09:55,750 --> 00:09:58,070

 

301
00:09:58,070 --> 00:09:59,910

 

302
00:09:59,910 --> 00:10:02,389

 

303
00:10:02,389 --> 00:10:05,030

 

304
00:10:05,030 --> 00:10:06,310

 

305
00:10:06,310 --> 00:10:09,430

 

306
00:10:09,430 --> 00:10:12,870

 

307
00:10:12,870 --> 00:10:15,509

 

308
00:10:15,509 --> 00:10:15,519

 

309
00:10:15,519 --> 00:10:16,389


310
00:10:16,389 --> 00:10:19,350

 

311
00:10:19,350 --> 00:10:19,360

 

312
00:10:19,360 --> 00:10:20,790


313
00:10:20,790 --> 00:10:23,350

 

314
00:10:23,350 --> 00:10:24,550

 

315
00:10:24,550 --> 00:10:26,710

 

316
00:10:26,710 --> 00:10:27,670

 

317
00:10:27,670 --> 00:10:29,829

 

318
00:10:29,829 --> 00:10:31,030

 

319
00:10:31,030 --> 00:10:33,590

 

320
00:10:33,590 --> 00:10:35,670

 

321
00:10:35,670 --> 00:10:37,990

 

322
00:10:37,990 --> 00:10:39,590

 

323
00:10:39,590 --> 00:10:41,269

 

324
00:10:41,269 --> 00:10:42,949

 

325
00:10:42,949 --> 00:10:45,110

 

326
00:10:45,110 --> 00:10:49,110

 

327
00:10:49,110 --> 00:10:50,949

 

328
00:10:50,949 --> 00:10:52,470

 

329
00:10:52,470 --> 00:10:52,480

 

330
00:10:52,480 --> 00:10:52,870


331
00:10:52,870 --> 00:10:54,550

 

332
00:10:54,550 --> 00:10:56,150

 

333
00:10:56,150 --> 00:10:58,389

 

334
00:10:58,389 --> 00:11:01,590

 

335
00:11:01,590 --> 00:11:05,110

 

336
00:11:05,110 --> 00:11:08,630

 

337
00:11:08,630 --> 00:11:10,150

 

338
00:11:10,150 --> 00:11:11,670

 

339
00:11:11,670 --> 00:11:11,680

 

340
00:11:11,680 --> 00:11:12,069


341
00:11:12,069 --> 00:11:15,269

 

342
00:11:15,269 --> 00:11:17,030

 

343
00:11:17,030 --> 00:11:20,150

 

344
00:11:20,150 --> 00:11:23,190

 

345
00:11:23,190 --> 00:11:24,630

 

346
00:11:24,630 --> 00:11:26,389

 

347
00:11:26,389 --> 00:11:27,670

 

348
00:11:27,670 --> 00:11:29,430

 

349
00:11:29,430 --> 00:11:31,110

 

350
00:11:31,110 --> 00:11:31,120

 

351
00:11:31,120 --> 00:11:31,590


352
00:11:31,590 --> 00:11:32,949

 

353
00:11:32,949 --> 00:11:35,030

 

354
00:11:35,030 --> 00:11:36,550

 

355
00:11:36,550 --> 00:11:37,269

 

356
00:11:37,269 --> 00:11:38,630

 

357
00:11:38,630 --> 00:11:40,470

 

358
00:11:40,470 --> 00:11:42,150

 

359
00:11:42,150 --> 00:11:43,110

 

360
00:11:43,110 --> 00:11:46,150

 

361
00:11:46,150 --> 00:11:47,750

 

362
00:11:47,750 --> 00:11:49,190

 

363
00:11:49,190 --> 00:11:51,590

 

364
00:11:51,590 --> 00:11:54,150

 

365
00:11:54,150 --> 00:11:54,160

 

366
00:11:54,160 --> 00:11:54,870


367
00:11:54,870 --> 00:11:56,629

 

368
00:11:56,629 --> 00:11:58,550

 

369
00:11:58,550 --> 00:12:00,870

 

370
00:12:00,870 --> 00:12:04,310

 

371
00:12:04,310 --> 00:12:08,230

 

372
00:12:08,230 --> 00:12:09,829

 

373
00:12:09,829 --> 00:12:09,839

 

374
00:12:09,839 --> 00:12:10,389


375
00:12:10,389 --> 00:12:12,550

 

376
00:12:12,550 --> 00:12:14,629

 

377
00:12:14,629 --> 00:12:16,470

 

378
00:12:16,470 --> 00:12:18,150

 

379
00:12:18,150 --> 00:12:18,160

 

380
00:12:18,160 --> 00:12:19,350


381
00:12:19,350 --> 00:12:21,269

 

382
00:12:21,269 --> 00:12:24,230

 

383
00:12:24,230 --> 00:12:26,389

 

384
00:12:26,389 --> 00:12:26,399

 

385
00:12:26,399 --> 00:12:27,910


386
00:12:27,910 --> 00:12:31,269

 

387
00:12:31,269 --> 00:12:34,550

 

388
00:12:34,550 --> 00:12:42,949

 

389
00:12:42,949 --> 00:12:49,750

 

390
00:12:49,750 --> 00:12:53,030

 

391
00:12:53,030 --> 00:12:53,670

 

392
00:12:53,670 --> 00:12:56,150

 

393
00:12:56,150 --> 00:12:57,910

 

394
00:12:57,910 --> 00:13:00,150

 

395
00:13:00,150 --> 00:13:01,269

 

396
00:13:01,269 --> 00:13:03,590

 

397
00:13:03,590 --> 00:13:04,949

 

398
00:13:04,949 --> 00:13:06,710

 

399
00:13:06,710 --> 00:13:08,470

 

400
00:13:08,470 --> 00:13:11,750

 

401
00:13:11,750 --> 00:13:14,389

 

402
00:13:14,389 --> 00:13:15,750

 

403
00:13:15,750 --> 00:13:18,829

 

404
00:13:18,829 --> 00:13:20,470

 

405
00:13:20,470 --> 00:13:21,750

 

406
00:13:21,750 --> 00:13:25,670

 

407
00:13:25,670 --> 00:13:29,110

 

408
00:13:29,110 --> 00:13:31,910

 

409
00:13:31,910 --> 00:13:32,870

 

410
00:13:32,870 --> 00:13:35,190

 

411
00:13:35,190 --> 00:13:36,629

 

412
00:13:36,629 --> 00:13:38,629

 

413
00:13:38,629 --> 00:13:40,629

 

414
00:13:40,629 --> 00:13:42,389

 

415
00:13:42,389 --> 00:13:43,829

 

416
00:13:43,829 --> 00:13:45,269

 

417
00:13:45,269 --> 00:13:45,279

 

418
00:13:45,279 --> 00:13:45,670


419
00:13:45,670 --> 00:13:48,870

 

420
00:13:48,870 --> 00:13:50,389

 

421
00:13:50,389 --> 00:13:51,910

 

422
00:13:51,910 --> 00:13:53,030

 

423
00:13:53,030 --> 00:13:54,949

 

424
00:13:54,949 --> 00:13:56,150

 

425
00:13:56,150 --> 00:13:57,030

 

426
00:13:57,030 --> 00:13:58,710

 

427
00:13:58,710 --> 00:14:00,310

 

428
00:14:00,310 --> 00:14:00,320

 

429
00:14:00,320 --> 00:14:01,430


430
00:14:01,430 --> 00:14:04,470

 

431
00:14:04,470 --> 00:14:07,590

 

432
00:14:07,590 --> 00:14:09,189

 

433
00:14:09,189 --> 00:14:12,150

 

434
00:14:12,150 --> 00:14:14,949

 

435
00:14:14,949 --> 00:14:14,959

 

436
00:14:14,959 --> 00:14:15,430


437
00:14:15,430 --> 00:14:16,790

 

438
00:14:16,790 --> 00:14:18,629

 

439
00:14:18,629 --> 00:14:20,230

 

440
00:14:20,230 --> 00:14:21,350

 

441
00:14:21,350 --> 00:14:22,310

 

442
00:14:22,310 --> 00:14:23,430

 

443
00:14:23,430 --> 00:14:26,069

 

444
00:14:26,069 --> 00:14:27,990

 

445
00:14:27,990 --> 00:14:29,750

 

446
00:14:29,750 --> 00:14:33,110

 

447
00:14:33,110 --> 00:14:34,150

 

448
00:14:34,150 --> 00:14:36,870

 

449
00:14:36,870 --> 00:14:38,870

 

450
00:14:38,870 --> 00:14:40,389

 

451
00:14:40,389 --> 00:14:41,110

 

452
00:14:41,110 --> 00:14:44,230

 

453
00:14:44,230 --> 00:14:46,470

 

454
00:14:46,470 --> 00:14:47,509

 

455
00:14:47,509 --> 00:14:51,030

 

456
00:14:51,030 --> 00:14:54,230

 

457
00:14:54,230 --> 00:14:55,430

 

458
00:14:55,430 --> 00:14:58,790

 

459
00:14:58,790 --> 00:15:02,069

 

460
00:15:02,069 --> 00:15:05,430

 

461
00:15:05,430 --> 00:15:07,829

 

462
00:15:07,829 --> 00:15:09,430

 

463
00:15:09,430 --> 00:15:13,110

 

464
00:15:13,110 --> 00:15:16,230

 

465
00:15:16,230 --> 00:15:19,110

 

466
00:15:19,110 --> 00:15:19,120

 

467
00:15:19,120 --> 00:15:21,269


468
00:15:21,269 --> 00:15:26,550

 

469
00:15:26,550 --> 00:15:29,509

 

470
00:15:29,509 --> 00:15:29,519

 

471
00:15:29,519 --> 00:15:29,990


472
00:15:29,990 --> 00:15:33,430

 

473
00:15:33,430 --> 00:15:48,550

 

474
00:15:48,550 --> 00:15:50,710

 

475
00:15:50,710 --> 00:15:56,629

 

476
00:15:56,629 --> 00:15:58,069

 

477
00:15:58,069 --> 00:16:00,150

 

478
00:16:00,150 --> 00:16:02,150

 

479
00:16:02,150 --> 00:16:03,829

 

480
00:16:03,829 --> 00:16:05,670

 

481
00:16:05,670 --> 00:16:05,680

 

482
00:16:05,680 --> 00:16:06,150


483
00:16:06,150 --> 00:16:07,670

 

484
00:16:07,670 --> 00:16:09,430

 

485
00:16:09,430 --> 00:16:10,710

 

486
00:16:10,710 --> 00:16:13,269

 

487
00:16:13,269 --> 00:16:13,279

 

488
00:16:13,279 --> 00:16:13,910


489
00:16:13,910 --> 00:16:15,110

 

490
00:16:15,110 --> 00:16:18,150

 

491
00:16:18,150 --> 00:16:22,069

 

492
00:16:22,069 --> 00:16:23,590

 

493
00:16:23,590 --> 00:16:25,110

 

494
00:16:25,110 --> 00:16:25,749

 

495
00:16:25,749 --> 00:16:27,269

 

496
00:16:27,269 --> 00:16:29,590

 

497
00:16:29,590 --> 00:16:31,749

 

498
00:16:31,749 --> 00:16:33,030

 

499
00:16:33,030 --> 00:16:34,870

 

500
00:16:34,870 --> 00:16:36,629

 

501
00:16:36,629 --> 00:16:38,389

 

502
00:16:38,389 --> 00:16:39,910

 

503
00:16:39,910 --> 00:16:43,590

 

504
00:16:43,590 --> 00:16:46,389

 

505
00:16:46,389 --> 00:16:46,399

 

506
00:16:46,399 --> 00:16:46,949


507
00:16:46,949 --> 00:16:48,389

 

508
00:16:48,389 --> 00:16:50,150

 

509
00:16:50,150 --> 00:16:52,550

 

510
00:16:52,550 --> 00:16:52,560

 

511
00:16:52,560 --> 00:16:53,430


512
00:16:53,430 --> 00:16:55,269

 

513
00:16:55,269 --> 00:16:56,550

 

514
00:16:56,550 --> 00:16:58,150

 

515
00:16:58,150 --> 00:17:00,389

 

516
00:17:00,389 --> 00:17:01,829

 

517
00:17:01,829 --> 00:17:02,870

 

518
00:17:02,870 --> 00:17:05,829

 

519
00:17:05,829 --> 00:17:07,669

 

520
00:17:07,669 --> 00:17:07,679

 

521
00:17:07,679 --> 00:17:09,270


522
00:17:09,270 --> 00:17:11,270

 

523
00:17:11,270 --> 00:17:12,789

 

524
00:17:12,789 --> 00:17:14,549

 

525
00:17:14,549 --> 00:17:17,669

 

526
00:17:17,669 --> 00:17:20,630

 

527
00:17:20,630 --> 00:17:20,640

 

528
00:17:20,640 --> 00:17:21,270


529
00:17:21,270 --> 00:17:25,510

 

530
00:17:25,510 --> 00:17:27,510

 

531
00:17:27,510 --> 00:17:28,630

 

532
00:17:28,630 --> 00:17:31,510

 

533
00:17:31,510 --> 00:17:33,430

 

534
00:17:33,430 --> 00:17:36,470

 

535
00:17:36,470 --> 00:17:39,990

 

536
00:17:39,990 --> 00:17:42,230

 

537
00:17:42,230 --> 00:17:44,230

 

538
00:17:44,230 --> 00:17:45,750

 

539
00:17:45,750 --> 00:17:47,750

 

540
00:17:47,750 --> 00:17:49,669

 

541
00:17:49,669 --> 00:17:51,909

 

542
00:17:51,909 --> 00:17:54,870

 

543
00:17:54,870 --> 00:17:56,549

 

544
00:17:56,549 --> 00:17:57,590

 

545
00:17:57,590 --> 00:17:59,909

 

546
00:17:59,909 --> 00:17:59,919

 

547
00:17:59,919 --> 00:18:01,029


548
00:18:01,029 --> 00:18:04,710

 

549
00:18:04,710 --> 00:18:08,070

 

550
00:18:08,070 --> 00:18:10,230

 

551
00:18:10,230 --> 00:18:12,070

 

552
00:18:12,070 --> 00:18:14,230

 

553
00:18:14,230 --> 00:18:16,310

 

554
00:18:16,310 --> 00:18:18,230

 

555
00:18:18,230 --> 00:18:19,669

 

556
00:18:19,669 --> 00:18:19,679

 

557
00:18:19,679 --> 00:18:20,390


558
00:18:20,390 --> 00:18:22,230

 

559
00:18:22,230 --> 00:18:24,070

 

560
00:18:24,070 --> 00:18:25,750

 

561
00:18:25,750 --> 00:18:27,350

 

562
00:18:27,350 --> 00:18:28,390

 

563
00:18:28,390 --> 00:18:29,510

 

564
00:18:29,510 --> 00:18:31,029

 

565
00:18:31,029 --> 00:18:31,039

 

566
00:18:31,039 --> 00:18:31,510


567
00:18:31,510 --> 00:18:34,710

 

568
00:18:34,710 --> 00:18:38,070

 

569
00:18:38,070 --> 00:18:41,110

 

570
00:18:41,110 --> 00:18:42,150

 

571
00:18:42,150 --> 00:18:44,390

 

572
00:18:44,390 --> 00:18:45,350

 

573
00:18:45,350 --> 00:18:47,510

 

574
00:18:47,510 --> 00:18:47,520

 

575
00:18:47,520 --> 00:18:48,390


576
00:18:48,390 --> 00:18:50,950

 

577
00:18:50,950 --> 00:18:53,590

 

578
00:18:53,590 --> 00:18:56,630

 

579
00:18:56,630 --> 00:18:58,390

 

580
00:18:58,390 --> 00:19:00,310

 

581
00:19:00,310 --> 00:19:03,270

 

582
00:19:03,270 --> 00:19:03,280

 

583
00:19:03,280 --> 00:19:03,750


584
00:19:03,750 --> 00:19:05,990

 

585
00:19:05,990 --> 00:19:07,190

 

586
00:19:07,190 --> 00:19:10,070

 

587
00:19:10,070 --> 00:19:11,830

 

588
00:19:11,830 --> 00:19:14,310

 

589
00:19:14,310 --> 00:19:16,230

 

590
00:19:16,230 --> 00:19:16,240

 

591
00:19:16,240 --> 00:19:16,710


592
00:19:16,710 --> 00:19:18,470

 

593
00:19:18,470 --> 00:19:20,310

 

594
00:19:20,310 --> 00:19:23,510

 

595
00:19:23,510 --> 00:19:25,830

 

596
00:19:25,830 --> 00:19:25,840

 

597
00:19:25,840 --> 00:19:27,590


598
00:19:27,590 --> 00:19:30,070

 

599
00:19:30,070 --> 00:19:31,750

 

600
00:19:31,750 --> 00:19:34,070

 

601
00:19:34,070 --> 00:19:36,549

 

602
00:19:36,549 --> 00:19:37,430

 

603
00:19:37,430 --> 00:19:38,870

 

604
00:19:38,870 --> 00:19:40,950

 

605
00:19:40,950 --> 00:19:44,230

 

606
00:19:44,230 --> 00:19:47,510

 

607
00:19:47,510 --> 00:19:49,350

 

608
00:19:49,350 --> 00:19:51,750

 

609
00:19:51,750 --> 00:19:53,029

 

610
00:19:53,029 --> 00:19:54,630

 

611
00:19:54,630 --> 00:19:57,029

 

612
00:19:57,029 --> 00:19:58,789

 

613
00:19:58,789 --> 00:20:00,710

 

614
00:20:00,710 --> 00:20:02,149

 

615
00:20:02,149 --> 00:20:06,149

 

616
00:20:06,149 --> 00:20:08,470

 

617
00:20:08,470 --> 00:20:10,310

 

618
00:20:10,310 --> 00:20:11,430

 

619
00:20:11,430 --> 00:20:15,350

 

620
00:20:15,350 --> 00:20:17,029

 

621
00:20:17,029 --> 00:20:19,110

 

622
00:20:19,110 --> 00:20:20,789

 

623
00:20:20,789 --> 00:20:21,909

 

624
00:20:21,909 --> 00:20:23,430

 

625
00:20:23,430 --> 00:20:24,870

 

626
00:20:24,870 --> 00:20:25,510

 

627
00:20:25,510 --> 00:20:27,110

 

628
00:20:27,110 --> 00:20:28,789

 

629
00:20:28,789 --> 00:20:31,590

 

630
00:20:31,590 --> 00:20:33,350

 

631
00:20:33,350 --> 00:20:35,190

 

632
00:20:35,190 --> 00:20:37,990

 

633
00:20:37,990 --> 00:20:40,630

 

634
00:20:40,630 --> 00:20:42,390

 

635
00:20:42,390 --> 00:20:45,430

 

636
00:20:45,430 --> 00:20:47,350

 

637
00:20:47,350 --> 00:20:48,870

 

638
00:20:48,870 --> 00:20:50,310

 

639
00:20:50,310 --> 00:20:53,350

 

640
00:20:53,350 --> 00:20:56,310

 

641
00:20:56,310 --> 00:20:57,350

 

642
00:20:57,350 --> 00:21:01,270

 

643
00:21:01,270 --> 00:21:03,909

 

644
00:21:03,909 --> 00:21:05,669

 

645
00:21:05,669 --> 00:21:07,669

 

646
00:21:07,669 --> 00:21:09,990

 

647
00:21:09,990 --> 00:21:12,070

 

648
00:21:12,070 --> 00:21:14,149

 

649
00:21:14,149 --> 00:21:15,430

 

650
00:21:15,430 --> 00:21:16,710

 

651
00:21:16,710 --> 00:21:19,110

 

652
00:21:19,110 --> 00:21:19,750

 

653
00:21:19,750 --> 00:21:22,470

 

654
00:21:22,470 --> 00:21:22,480

 

655
00:21:22,480 --> 00:21:22,789


656
00:21:22,789 --> 00:21:24,470

 

657
00:21:24,470 --> 00:21:26,149

 

658
00:21:26,149 --> 00:21:29,110

 

659
00:21:29,110 --> 00:21:30,149

 

660
00:21:30,149 --> 00:21:32,070

 

661
00:21:32,070 --> 00:21:33,270

 

662
00:21:33,270 --> 00:21:35,909

 

663
00:21:35,909 --> 00:21:38,789

 

664
00:21:38,789 --> 00:21:42,470

 

665
00:21:42,470 --> 00:21:44,470

 

666
00:21:44,470 --> 00:21:46,149

 

667
00:21:46,149 --> 00:21:48,789

 

668
00:21:48,789 --> 00:21:51,430

 

669
00:21:51,430 --> 00:21:53,190

 

670
00:21:53,190 --> 00:21:54,310

 

671
00:21:54,310 --> 00:21:56,870

 

672
00:21:56,870 --> 00:21:58,390

 

673
00:21:58,390 --> 00:22:00,310

 

674
00:22:00,310 --> 00:22:01,990

 

675
00:22:01,990 --> 00:22:03,190

 

676
00:22:03,190 --> 00:22:03,200

 

677
00:22:03,200 --> 00:22:03,830


678
00:22:03,830 --> 00:22:06,549

 

679
00:22:06,549 --> 00:22:07,590

 

680
00:22:07,590 --> 00:22:09,669

 

681
00:22:09,669 --> 00:22:11,350

 

682
00:22:11,350 --> 00:22:13,110

 

683
00:22:13,110 --> 00:22:14,950

 

684
00:22:14,950 --> 00:22:17,350

 

685
00:22:17,350 --> 00:22:19,669

 

686
00:22:19,669 --> 00:22:19,679

 

687
00:22:19,679 --> 00:22:21,669


688
00:22:21,669 --> 00:22:23,190

 

689
00:22:23,190 --> 00:22:25,110

 

690
00:22:25,110 --> 00:22:26,549

 

691
00:22:26,549 --> 00:22:29,510

 

692
00:22:29,510 --> 00:22:30,789

 

693
00:22:30,789 --> 00:22:32,710

 

694
00:22:32,710 --> 00:22:34,710

 

695
00:22:34,710 --> 00:22:34,720

 

696
00:22:34,720 --> 00:22:35,430


697
00:22:35,430 --> 00:22:37,430

 

698
00:22:37,430 --> 00:22:38,789

 

699
00:22:38,789 --> 00:22:42,230

 

700
00:22:42,230 --> 00:22:45,190

 

701
00:22:45,190 --> 00:22:46,710

 

702
00:22:46,710 --> 00:22:49,190

 

703
00:22:49,190 --> 00:22:50,870

 

704
00:22:50,870 --> 00:22:52,230

 

705
00:22:52,230 --> 00:22:54,149

 

706
00:22:54,149 --> 00:22:55,909

 

707
00:22:55,909 --> 00:22:57,350

 

708
00:22:57,350 --> 00:22:58,630

 

709
00:22:58,630 --> 00:23:00,149

 

710
00:23:00,149 --> 00:23:03,510

 

711
00:23:03,510 --> 00:23:07,029

 

712
00:23:07,029 --> 00:23:08,950

 

713
00:23:08,950 --> 00:23:10,310

 

714
00:23:10,310 --> 00:23:11,430

 

715
00:23:11,430 --> 00:23:11,440

 

716
00:23:11,440 --> 00:23:13,430


717
00:23:13,430 --> 00:23:17,110

 

718
00:23:17,110 --> 00:23:19,110

 

719
00:23:19,110 --> 00:23:21,510

 

720
00:23:21,510 --> 00:23:21,520

 

721
00:23:21,520 --> 00:23:21,990


722
00:23:21,990 --> 00:23:23,909

 

723
00:23:23,909 --> 00:23:25,270

 

724
00:23:25,270 --> 00:23:27,190

 

725
00:23:27,190 --> 00:23:28,310

 

726
00:23:28,310 --> 00:23:30,070

 

727
00:23:30,070 --> 00:23:33,430

 

728
00:23:33,430 --> 00:23:35,190

 

729
00:23:35,190 --> 00:23:37,190

 

730
00:23:37,190 --> 00:23:38,950

 

731
00:23:38,950 --> 00:23:41,669

 

732
00:23:41,669 --> 00:23:43,669

 

733
00:23:43,669 --> 00:23:44,870

 

734
00:23:44,870 --> 00:23:47,510

 

735
00:23:47,510 --> 00:23:48,149

 

736
00:23:48,149 --> 00:23:50,390

 

737
00:23:50,390 --> 00:23:52,710

 

738
00:23:52,710 --> 00:23:56,950

 

739
00:23:56,950 --> 00:23:58,950

 

740
00:23:58,950 --> 00:24:00,870

 

741
00:24:00,870 --> 00:24:00,880

 

742
00:24:00,880 --> 00:24:01,750


743
00:24:01,750 --> 00:24:03,269

 

744
00:24:03,269 --> 00:24:05,350

 

745
00:24:05,350 --> 00:24:07,029

 

746
00:24:07,029 --> 00:24:10,710

 

747
00:24:10,710 --> 00:24:12,710

 

748
00:24:12,710 --> 00:24:15,430

 

749
00:24:15,430 --> 00:24:16,149

 

750
00:24:16,149 --> 00:24:18,950

 

751
00:24:18,950 --> 00:24:20,310

 

752
00:24:20,310 --> 00:24:22,149

 

753
00:24:22,149 --> 00:24:24,230

 

754
00:24:24,230 --> 00:24:26,630

 

755
00:24:26,630 --> 00:24:28,710

 

756
00:24:28,710 --> 00:24:30,390

 

757
00:24:30,390 --> 00:24:34,070

 

758
00:24:34,070 --> 00:24:37,110

 

759
00:24:37,110 --> 00:24:38,390

 

760
00:24:38,390 --> 00:24:40,870

 

761
00:24:40,870 --> 00:24:42,310

 

762
00:24:42,310 --> 00:24:44,070

 

763
00:24:44,070 --> 00:24:45,590

 

764
00:24:45,590 --> 00:24:47,269

 

765
00:24:47,269 --> 00:24:47,279

 

766
00:24:47,279 --> 00:24:47,909


767
00:24:47,909 --> 00:24:49,269

 

768
00:24:49,269 --> 00:24:51,830

 

769
00:24:51,830 --> 00:24:53,669

 

770
00:24:53,669 --> 00:24:55,590

 

771
00:24:55,590 --> 00:24:56,870

 

772
00:24:56,870 --> 00:25:00,070

 

773
00:25:00,070 --> 00:25:03,190

 

774
00:25:03,190 --> 00:25:05,029

 

775
00:25:05,029 --> 00:25:06,789

 

776
00:25:06,789 --> 00:25:09,750

 

777
00:25:09,750 --> 00:25:11,750

 

778
00:25:11,750 --> 00:25:13,110

 

779
00:25:13,110 --> 00:25:14,070

 

780
00:25:14,070 --> 00:25:16,149

 

781
00:25:16,149 --> 00:25:17,510

 

782
00:25:17,510 --> 00:25:17,520

 

783
00:25:17,520 --> 00:25:18,390


784
00:25:18,390 --> 00:25:20,070

 

785
00:25:20,070 --> 00:25:21,669

 

786
00:25:21,669 --> 00:25:23,909

 

787
00:25:23,909 --> 00:25:25,269

 

788
00:25:25,269 --> 00:25:26,950

 

789
00:25:26,950 --> 00:25:28,310

 

790
00:25:28,310 --> 00:25:32,549

 

791
00:25:32,549 --> 00:25:34,870

 

792
00:25:34,870 --> 00:25:36,390

 

793
00:25:36,390 --> 00:25:38,549

 

794
00:25:38,549 --> 00:25:38,559

 

795
00:25:38,559 --> 00:25:40,830


796
00:25:40,830 --> 00:25:44,549

 

797
00:25:44,549 --> 00:25:47,990

 

798
00:25:47,990 --> 00:25:50,310

 

799
00:25:50,310 --> 00:25:55,350

 

800
00:25:55,350 --> 00:25:57,190

 

801
00:25:57,190 --> 00:25:58,870

 

802
00:25:58,870 --> 00:26:00,789

 

803
00:26:00,789 --> 00:26:02,470

 

804
00:26:02,470 --> 00:26:05,190

 

805
00:26:05,190 --> 00:26:05,200

 

806
00:26:05,200 --> 00:26:06,070


807
00:26:06,070 --> 00:26:07,990

 

808
00:26:07,990 --> 00:26:09,190

 

809
00:26:09,190 --> 00:26:09,200

 

810
00:26:09,200 --> 00:26:10,390


811
00:26:10,390 --> 00:26:13,510

 

812
00:26:13,510 --> 00:26:14,950

 

813
00:26:14,950 --> 00:26:17,669

 

814
00:26:17,669 --> 00:26:19,029

 

815
00:26:19,029 --> 00:26:21,269

 

816
00:26:21,269 --> 00:26:23,590

 

817
00:26:23,590 --> 00:26:25,669

 

818
00:26:25,669 --> 00:26:27,430

 

819
00:26:27,430 --> 00:26:29,669

 

820
00:26:29,669 --> 00:26:31,110

 

821
00:26:31,110 --> 00:26:33,029

 

822
00:26:33,029 --> 00:26:33,039

 

823
00:26:33,039 --> 00:26:34,390


824
00:26:34,390 --> 00:26:36,390

 

825
00:26:36,390 --> 00:26:38,870

 

826
00:26:38,870 --> 00:26:40,789

 

827
00:26:40,789 --> 00:26:41,909

 

828
00:26:41,909 --> 00:26:45,190

 

829
00:26:45,190 --> 00:26:48,310

 

830
00:26:48,310 --> 00:26:49,990

 

831
00:26:49,990 --> 00:26:51,990

 

832
00:26:51,990 --> 00:26:55,430

 

833
00:26:55,430 --> 00:26:57,909

 

834
00:26:57,909 --> 00:26:59,510

 

835
00:26:59,510 --> 00:27:00,149

 

836
00:27:00,149 --> 00:27:02,230

 

837
00:27:02,230 --> 00:27:04,710

 

838
00:27:04,710 --> 00:27:06,230

 

839
00:27:06,230 --> 00:27:08,470

 

840
00:27:08,470 --> 00:27:10,549

 

841
00:27:10,549 --> 00:27:12,390

 

842
00:27:12,390 --> 00:27:15,110

 

843
00:27:15,110 --> 00:27:15,120

 

844
00:27:15,120 --> 00:27:15,510


845
00:27:15,510 --> 00:27:17,510

 

846
00:27:17,510 --> 00:27:17,520

 

847
00:27:17,520 --> 00:27:18,630


848
00:27:18,630 --> 00:27:20,389

 

849
00:27:20,389 --> 00:27:22,389

 

850
00:27:22,389 --> 00:27:25,110

 

851
00:27:25,110 --> 00:27:25,830

 

852
00:27:25,830 --> 00:27:27,590

 

853
00:27:27,590 --> 00:27:29,909

 

854
00:27:29,909 --> 00:27:31,830

 

855
00:27:31,830 --> 00:27:33,510

 

856
00:27:33,510 --> 00:27:34,310

 

857
00:27:34,310 --> 00:27:38,950

 

858
00:27:38,950 --> 00:27:43,590

 

859
00:27:43,590 --> 00:27:45,190

 

860
00:27:45,190 --> 00:27:51,350

 

861
00:27:51,350 --> 00:27:57,029

 

862
00:27:57,029 --> 00:27:58,870

 

863
00:27:58,870 --> 00:28:00,389

 

864
00:28:00,389 --> 00:28:00,399

 

865
00:28:00,399 --> 00:28:01,190


866
00:28:01,190 --> 00:28:03,190

 

867
00:28:03,190 --> 00:28:05,350

 

868
00:28:05,350 --> 00:28:07,029

 

869
00:28:07,029 --> 00:28:08,950

 

870
00:28:08,950 --> 00:28:09,909

 

871
00:28:09,909 --> 00:28:12,870

 

872
00:28:12,870 --> 00:28:13,750

 

873
00:28:13,750 --> 00:28:15,830

 

874
00:28:15,830 --> 00:28:18,710

 

875
00:28:18,710 --> 00:28:20,389

 

876
00:28:20,389 --> 00:28:22,389

 

877
00:28:22,389 --> 00:28:24,230

 

878
00:28:24,230 --> 00:28:25,269

 

879
00:28:25,269 --> 00:28:28,389

 

880
00:28:28,389 --> 00:28:30,149

 

881
00:28:30,149 --> 00:28:32,310

 

882
00:28:32,310 --> 00:28:34,470

 

883
00:28:34,470 --> 00:28:36,630

 

884
00:28:36,630 --> 00:28:39,110

 

885
00:28:39,110 --> 00:28:42,070

 

886
00:28:42,070 --> 00:28:43,590

 

887
00:28:43,590 --> 00:28:44,870

 

888
00:28:44,870 --> 00:28:47,510

 

889
00:28:47,510 --> 00:28:48,470

 

890
00:28:48,470 --> 00:28:52,710

 

891
00:28:52,710 --> 00:28:54,789

 

892
00:28:54,789 --> 00:28:54,799

 

893
00:28:54,799 --> 00:28:55,830


894
00:28:55,830 --> 00:28:58,789

 

895
00:28:58,789 --> 00:29:00,870

 

896
00:29:00,870 --> 00:29:02,549

 

897
00:29:02,549 --> 00:29:05,269

 

898
00:29:05,269 --> 00:29:08,070

 

899
00:29:08,070 --> 00:29:09,590

 

900
00:29:09,590 --> 00:29:12,310

 

901
00:29:12,310 --> 00:29:13,510

 

902
00:29:13,510 --> 00:29:15,750

 

903
00:29:15,750 --> 00:29:16,950

 

904
00:29:16,950 --> 00:29:19,269

 

905
00:29:19,269 --> 00:29:20,310

 

906
00:29:20,310 --> 00:29:22,870

 

907
00:29:22,870 --> 00:29:24,230

 

908
00:29:24,230 --> 00:29:24,240

 

909
00:29:24,240 --> 00:29:24,710


910
00:29:24,710 --> 00:29:27,990

 

911
00:29:27,990 --> 00:29:31,750

 

912
00:29:31,750 --> 00:29:33,510

 

913
00:29:33,510 --> 00:29:38,830

 

914
00:29:38,830 --> 00:29:38,840

 

915
00:29:38,840 --> 00:29:40,789


916
00:29:40,789 --> 00:29:43,590

 

917
00:29:43,590 --> 00:29:44,710

 

918
00:29:44,710 --> 00:29:47,669

 

919
00:29:47,669 --> 00:29:47,679

 

920
00:29:47,679 --> 00:29:48,310


921
00:29:48,310 --> 00:29:51,590

 

922
00:29:51,590 --> 00:29:53,350

 

923
00:29:53,350 --> 00:29:55,269

 

924
00:29:55,269 --> 00:29:57,350

 

925
00:29:57,350 --> 00:29:58,310

 

926
00:29:58,310 --> 00:30:02,710

 

927
00:30:02,710 --> 00:30:06,149

 

928
00:30:06,149 --> 00:30:07,750

 

929
00:30:07,750 --> 00:30:09,430

 

930
00:30:09,430 --> 00:30:11,990

 

931
00:30:11,990 --> 00:30:13,110

 

932
00:30:13,110 --> 00:30:15,269

 

933
00:30:15,269 --> 00:30:16,549

 

934
00:30:16,549 --> 00:30:18,310

 

935
00:30:18,310 --> 00:30:19,990

 

936
00:30:19,990 --> 00:30:21,269

 

937
00:30:21,269 --> 00:30:24,710

 

938
00:30:24,710 --> 00:30:27,590

 

939
00:30:27,590 --> 00:30:28,870

 

940
00:30:28,870 --> 00:30:30,710

 

941
00:30:30,710 --> 00:30:32,230

 

942
00:30:32,230 --> 00:30:34,549

 

943
00:30:34,549 --> 00:30:36,310

 

944
00:30:36,310 --> 00:30:38,230

 

945
00:30:38,230 --> 00:30:38,240

 

946
00:30:38,240 --> 00:30:39,190


947
00:30:39,190 --> 00:30:42,310

 

948
00:30:42,310 --> 00:30:43,830

 

949
00:30:43,830 --> 00:30:46,789

 

950
00:30:46,789 --> 00:30:48,470

 

951
00:30:48,470 --> 00:30:49,909

 

952
00:30:49,909 --> 00:30:51,190

 

953
00:30:51,190 --> 00:30:53,190

 

954
00:30:53,190 --> 00:30:55,110

 

955
00:30:55,110 --> 00:30:56,470

 

956
00:30:56,470 --> 00:30:58,149

 

957
00:30:58,149 --> 00:30:59,350

 

958
00:30:59,350 --> 00:31:00,870

 

959
00:31:00,870 --> 00:31:03,350

 

960
00:31:03,350 --> 00:31:06,070

 

961
00:31:06,070 --> 00:31:07,509

 

962
00:31:07,509 --> 00:31:10,310

 

963
00:31:10,310 --> 00:31:11,990

 

964
00:31:11,990 --> 00:31:14,470

 

965
00:31:14,470 --> 00:31:15,830

 

966
00:31:15,830 --> 00:31:18,070

 

967
00:31:18,070 --> 00:31:23,430

 

968
00:31:23,430 --> 00:31:26,710

 

969
00:31:26,710 --> 00:31:28,789

 

970
00:31:28,789 --> 00:31:30,389

 

971
00:31:30,389 --> 00:31:30,399

 

972
00:31:30,399 --> 00:31:31,110


973
00:31:31,110 --> 00:31:33,350

 

974
00:31:33,350 --> 00:31:33,360

 

975
00:31:33,360 --> 00:31:34,310


976
00:31:34,310 --> 00:31:37,110

 

977
00:31:37,110 --> 00:31:39,029

 

978
00:31:39,029 --> 00:31:45,029

 

979
00:31:45,029 --> 00:31:46,950

 

980
00:31:46,950 --> 00:31:48,789

 

981
00:31:48,789 --> 00:31:52,149

 

982
00:31:52,149 --> 00:31:53,669

 

983
00:31:53,669 --> 00:31:55,590

 

984
00:31:55,590 --> 00:31:57,029

 

985
00:31:57,029 --> 00:31:59,350

 

986
00:31:59,350 --> 00:32:01,669

 

987
00:32:01,669 --> 00:32:03,750

 

988
00:32:03,750 --> 00:32:06,149

 

989
00:32:06,149 --> 00:32:07,590

 

990
00:32:07,590 --> 00:32:09,350

 

991
00:32:09,350 --> 00:32:11,830

 

992
00:32:11,830 --> 00:32:13,990

 

993
00:32:13,990 --> 00:32:15,990

 

994
00:32:15,990 --> 00:32:16,000

 

995
00:32:16,000 --> 00:32:16,789


996
00:32:16,789 --> 00:32:18,230

 

997
00:32:18,230 --> 00:32:20,070

 

998
00:32:20,070 --> 00:32:22,870

 

999
00:32:22,870 --> 00:32:24,630

 

1000
00:32:24,630 --> 00:32:26,149

 

1001
00:32:26,149 --> 00:32:28,950

 

1002
00:32:28,950 --> 00:32:30,630

 

1003
00:32:30,630 --> 00:32:32,789

 

1004
00:32:32,789 --> 00:32:32,799

 

1005
00:32:32,799 --> 00:32:33,909


1006
00:32:33,909 --> 00:32:35,830

 

1007
00:32:35,830 --> 00:32:35,840

 

1008
00:32:35,840 --> 00:32:39,909


1009
00:32:39,909 --> 00:32:45,269

 

1010
00:32:45,269 --> 00:32:49,029

 

1011
00:32:49,029 --> 00:32:52,230

 

1012
00:32:52,230 --> 00:32:54,830

 

1013
00:32:54,830 --> 00:32:54,840

 

1014
00:32:54,840 --> 00:32:57,110


1015
00:32:57,110 --> 00:32:59,990

 

1016
00:32:59,990 --> 00:33:01,669

 

1017
00:33:01,669 --> 00:33:03,430

 

1018
00:33:03,430 --> 00:33:05,110

 

1019
00:33:05,110 --> 00:33:06,710

 

1020
00:33:06,710 --> 00:33:08,789

 

1021
00:33:08,789 --> 00:33:10,389

 

1022
00:33:10,389 --> 00:33:12,870

 

1023
00:33:12,870 --> 00:33:14,950

 

1024
00:33:14,950 --> 00:33:15,590

 

1025
00:33:15,590 --> 00:33:18,789

 

1026
00:33:18,789 --> 00:33:19,190

 

1027
00:33:19,190 --> 00:33:21,190

 

1028
00:33:21,190 --> 00:33:22,870

 

1029
00:33:22,870 --> 00:33:24,070

 

1030
00:33:24,070 --> 00:33:26,230

 

1031
00:33:26,230 --> 00:33:27,430

 

1032
00:33:27,430 --> 00:33:29,430

 

1033
00:33:29,430 --> 00:33:31,190

 

1034
00:33:31,190 --> 00:33:33,830

 

1035
00:33:33,830 --> 00:33:35,909

 

1036
00:33:35,909 --> 00:33:39,669

 

1037
00:33:39,669 --> 00:33:41,269

 

1038
00:33:41,269 --> 00:33:42,870

 

1039
00:33:42,870 --> 00:33:44,870

 

1040
00:33:44,870 --> 00:33:47,909

 

1041
00:33:47,909 --> 00:33:50,549

 

1042
00:33:50,549 --> 00:33:52,070

 

1043
00:33:52,070 --> 00:33:53,750

 

1044
00:33:53,750 --> 00:33:56,310

 

1045
00:33:56,310 --> 00:33:57,269

 

1046
00:33:57,269 --> 00:33:58,710

 

1047
00:33:58,710 --> 00:34:00,470

 

1048
00:34:00,470 --> 00:34:02,470

 

1049
00:34:02,470 --> 00:34:05,590

 

1050
00:34:05,590 --> 00:34:08,470

 

1051
00:34:08,470 --> 00:34:10,310

 

1052
00:34:10,310 --> 00:34:11,990

 

1053
00:34:11,990 --> 00:34:13,829

 

1054
00:34:13,829 --> 00:34:15,109

 

1055
00:34:15,109 --> 00:34:18,069

 

1056
00:34:18,069 --> 00:34:18,079

 

1057
00:34:18,079 --> 00:34:18,629


1058
00:34:18,629 --> 00:34:20,230

 

1059
00:34:20,230 --> 00:34:20,240

 

1060
00:34:20,240 --> 00:34:22,310


1061
00:34:22,310 --> 00:34:24,310

 

1062
00:34:24,310 --> 00:34:25,829

 

1063
00:34:25,829 --> 00:34:27,909

 

1064
00:34:27,909 --> 00:34:27,919

 

1065
00:34:27,919 --> 00:34:29,030


1066
00:34:29,030 --> 00:34:30,470

 

1067
00:34:30,470 --> 00:34:32,629

 

1068
00:34:32,629 --> 00:34:36,310

 

1069
00:34:36,310 --> 00:34:47,589

 

1070
00:34:47,589 --> 00:34:52,869

 

1071
00:34:52,869 --> 00:34:55,909

 

1072
00:34:55,909 --> 00:34:57,030

 

1073
00:34:57,030 --> 00:34:59,349

 

1074
00:34:59,349 --> 00:35:01,750

 

1075
00:35:01,750 --> 00:35:02,710

 

1076
00:35:02,710 --> 00:35:05,510

 

1077
00:35:05,510 --> 00:35:06,790

 

1078
00:35:06,790 --> 00:35:09,430

 

1079
00:35:09,430 --> 00:35:09,440

 

1080
00:35:09,440 --> 00:35:15,430


1081
00:35:15,430 --> 00:35:18,470

 

1082
00:35:18,470 --> 00:35:21,670

 

1083
00:35:21,670 --> 00:35:26,310

 

1084
00:35:26,310 --> 00:35:29,589

 

1085
00:35:29,589 --> 00:35:32,390

 

1086
00:35:32,390 --> 00:35:35,030

 

1087
00:35:35,030 --> 00:35:38,870

 

1088
00:35:38,870 --> 00:35:42,829

 

1089
00:35:42,829 --> 00:35:45,190

 

1090
00:35:45,190 --> 00:35:45,829

 

1091
00:35:45,829 --> 00:35:48,470

 

1092
00:35:48,470 --> 00:35:49,750

 

1093
00:35:49,750 --> 00:35:53,270

 

1094
00:35:53,270 --> 00:35:56,630

 

1095
00:35:56,630 --> 00:36:00,069

 

1096
00:36:00,069 --> 00:36:00,079

 

1097
00:36:00,079 --> 00:36:02,790


1098
00:36:02,790 --> 00:36:05,510

 

1099
00:36:05,510 --> 00:36:05,520

 

1100
00:36:05,520 --> 00:36:06,310


1101
00:36:06,310 --> 00:36:10,829

 

1102
00:36:10,829 --> 00:36:14,150

 

1103
00:36:14,150 --> 00:36:14,790

 

1104
00:36:14,790 --> 00:36:16,230

 

1105
00:36:16,230 --> 00:36:17,829

 

1106
00:36:17,829 --> 00:36:19,030

 

1107
00:36:19,030 --> 00:36:20,630

 

1108
00:36:20,630 --> 00:36:21,990

 

1109
00:36:21,990 --> 00:36:22,000

 

1110
00:36:22,000 --> 00:36:23,990


1111
00:36:23,990 --> 00:36:25,750

 

1112
00:36:25,750 --> 00:36:27,829

 

1113
00:36:27,829 --> 00:36:30,150

 

1114
00:36:30,150 --> 00:36:31,589

 

1115
00:36:31,589 --> 00:36:33,109

 

1116
00:36:33,109 --> 00:36:35,430

 

1117
00:36:35,430 --> 00:36:37,829

 

1118
00:36:37,829 --> 00:36:38,950

 

1119
00:36:38,950 --> 00:36:41,030

 

1120
00:36:41,030 --> 00:36:42,390

 

1121
00:36:42,390 --> 00:36:43,829

 

1122
00:36:43,829 --> 00:36:45,349

 

1123
00:36:45,349 --> 00:36:45,359

 

1124
00:36:45,359 --> 00:36:45,670


1125
00:36:45,670 --> 00:36:48,390

 

1126
00:36:48,390 --> 00:36:48,400

 

1127
00:36:48,400 --> 00:36:48,950


1128
00:36:48,950 --> 00:36:53,510

 

1129
00:36:53,510 --> 00:36:57,190

 

1130
00:36:57,190 --> 00:37:00,870

 

1131
00:37:00,870 --> 00:37:02,390

 

1132
00:37:02,390 --> 00:37:03,670

 

1133
00:37:03,670 --> 00:37:09,190

 

1134
00:37:09,190 --> 00:37:13,030

 

1135
00:37:13,030 --> 00:37:15,270

 

1136
00:37:15,270 --> 00:37:16,710

 

1137
00:37:16,710 --> 00:37:19,829

 

1138
00:37:19,829 --> 00:37:22,069

 

1139
00:37:22,069 --> 00:37:24,230

 

1140
00:37:24,230 --> 00:37:26,069

 

1141
00:37:26,069 --> 00:37:28,310

 

1142
00:37:28,310 --> 00:37:29,990

 

1143
00:37:29,990 --> 00:37:32,390

 

1144
00:37:32,390 --> 00:37:34,310

 

1145
00:37:34,310 --> 00:37:35,270

 

1146
00:37:35,270 --> 00:37:36,470

 

1147
00:37:36,470 --> 00:37:39,670

 

1148
00:37:39,670 --> 00:37:40,470

 

1149
00:37:40,470 --> 00:37:42,230

 

1150
00:37:42,230 --> 00:37:43,829

 

1151
00:37:43,829 --> 00:37:45,589

 

1152
00:37:45,589 --> 00:37:47,430

 

1153
00:37:47,430 --> 00:37:49,670

 

1154
00:37:49,670 --> 00:37:51,670

 

1155
00:37:51,670 --> 00:37:52,950

 

1156
00:37:52,950 --> 00:37:55,109

 

1157
00:37:55,109 --> 00:37:56,550

 

1158
00:37:56,550 --> 00:37:57,910

 

1159
00:37:57,910 --> 00:37:57,920

 

1160
00:37:57,920 --> 00:38:00,630


1161
00:38:00,630 --> 00:38:04,950

 

1162
00:38:04,950 --> 00:38:08,310

 

1163
00:38:08,310 --> 00:38:10,470

 

1164
00:38:10,470 --> 00:38:11,750

 

1165
00:38:11,750 --> 00:38:13,510

 

1166
00:38:13,510 --> 00:38:15,670

 

1167
00:38:15,670 --> 00:38:18,630

 

1168
00:38:18,630 --> 00:38:22,150

 

1169
00:38:22,150 --> 00:38:24,470

 

1170
00:38:24,470 --> 00:38:30,870

 

1171
00:38:30,870 --> 00:38:33,990

 

1172
00:38:33,990 --> 00:38:36,630

 

1173
00:38:36,630 --> 00:38:38,230

 

1174
00:38:38,230 --> 00:38:40,069

 

1175
00:38:40,069 --> 00:38:41,910

 

1176
00:38:41,910 --> 00:38:43,190

 

1177
00:38:43,190 --> 00:38:44,950

 

1178
00:38:44,950 --> 00:38:46,870

 

1179
00:38:46,870 --> 00:38:49,430

 

1180
00:38:49,430 --> 00:38:50,630

 

1181
00:38:50,630 --> 00:38:50,640

 

1182
00:38:50,640 --> 00:38:51,190


1183
00:38:51,190 --> 00:38:53,270

 

1184
00:38:53,270 --> 00:38:54,950

 

1185
00:38:54,950 --> 00:38:57,589

 

1186
00:38:57,589 --> 00:38:59,270

 

1187
00:38:59,270 --> 00:39:02,069

 

1188
00:39:02,069 --> 00:39:02,079

 

1189
00:39:02,079 --> 00:39:03,030


1190
00:39:03,030 --> 00:39:04,550

 

1191
00:39:04,550 --> 00:39:05,750

 

1192
00:39:05,750 --> 00:39:07,270

 

1193
00:39:07,270 --> 00:39:09,510

 

1194
00:39:09,510 --> 00:39:09,520

 

1195
00:39:09,520 --> 00:39:10,310


1196
00:39:10,310 --> 00:39:11,750

 

1197
00:39:11,750 --> 00:39:13,990

 

1198
00:39:13,990 --> 00:39:16,230

 

1199
00:39:16,230 --> 00:39:17,910

 

1200
00:39:17,910 --> 00:39:19,109

 

1201
00:39:19,109 --> 00:39:19,119

 

1202
00:39:19,119 --> 00:39:19,750


1203
00:39:19,750 --> 00:39:21,910

 

1204
00:39:21,910 --> 00:39:22,790

 

1205
00:39:22,790 --> 00:39:25,829

 

1206
00:39:25,829 --> 00:39:29,589

 

1207
00:39:29,589 --> 00:39:36,069

 

1208
00:39:36,069 --> 00:39:39,430

 

1209
00:39:39,430 --> 00:39:43,910

 

1210
00:39:43,910 --> 00:39:46,630

 

1211
00:39:46,630 --> 00:39:48,390

 

1212
00:39:48,390 --> 00:39:51,349

 

1213
00:39:51,349 --> 00:39:52,950

 

1214
00:39:52,950 --> 00:39:55,829

 

1215
00:39:55,829 --> 00:39:56,790

 

1216
00:39:56,790 --> 00:39:59,829

 

1217
00:39:59,829 --> 00:40:01,109

 

1218
00:40:01,109 --> 00:40:04,230

 

1219
00:40:04,230 --> 00:40:05,190

 

1220
00:40:05,190 --> 00:40:06,950

 

1221
00:40:06,950 --> 00:40:08,630

 

1222
00:40:08,630 --> 00:40:10,150

 

1223
00:40:10,150 --> 00:40:12,150

 

1224
00:40:12,150 --> 00:40:13,670

 

1225
00:40:13,670 --> 00:40:15,190

 

1226
00:40:15,190 --> 00:40:17,750

 

1227
00:40:17,750 --> 00:40:18,550

 

1228
00:40:18,550 --> 00:40:21,589

 

1229
00:40:21,589 --> 00:40:24,069

 

1230
00:40:24,069 --> 00:40:25,589

 

1231
00:40:25,589 --> 00:40:27,190

 

1232
00:40:27,190 --> 00:40:31,670

 

1233
00:40:31,670 --> 00:40:34,710

 

1234
00:40:34,710 --> 00:40:37,430

 

1235
00:40:37,430 --> 00:40:38,870

 

1236
00:40:38,870 --> 00:40:40,470

 

1237
00:40:40,470 --> 00:40:42,390

 

1238
00:40:42,390 --> 00:40:44,309

 

1239
00:40:44,309 --> 00:40:46,230

 

1240
00:40:46,230 --> 00:40:47,109

 

1241
00:40:47,109 --> 00:40:48,230

 

1242
00:40:48,230 --> 00:40:50,630

 

1243
00:40:50,630 --> 00:40:52,550

 

1244
00:40:52,550 --> 00:40:55,030

 

1245
00:40:55,030 --> 00:40:57,670

 

1246
00:40:57,670 --> 00:40:59,190

 

1247
00:40:59,190 --> 00:41:01,030

 

1248
00:41:01,030 --> 00:41:02,230

 

1249
00:41:02,230 --> 00:41:04,710

 

1250
00:41:04,710 --> 00:41:06,150

 

1251
00:41:06,150 --> 00:41:08,309

 

1252
00:41:08,309 --> 00:41:10,309

 

1253
00:41:10,309 --> 00:41:12,069

 

1254
00:41:12,069 --> 00:41:12,079

 

1255
00:41:12,079 --> 00:41:13,109


1256
00:41:13,109 --> 00:41:16,470

 

1257
00:41:16,470 --> 00:41:17,910

 

1258
00:41:17,910 --> 00:41:19,109

 

1259
00:41:19,109 --> 00:41:20,950

 

1260
00:41:20,950 --> 00:41:22,950

 

1261
00:41:22,950 --> 00:41:24,230

 

1262
00:41:24,230 --> 00:41:26,950

 

1263
00:41:26,950 --> 00:41:29,109

 

1264
00:41:29,109 --> 00:41:32,230

 

1265
00:41:32,230 --> 00:41:32,240

 

1266
00:41:32,240 --> 00:41:33,030


1267
00:41:33,030 --> 00:41:34,870

 

1268
00:41:34,870 --> 00:41:37,349

 

1269
00:41:37,349 --> 00:41:39,589

 

1270
00:41:39,589 --> 00:41:40,390

 

1271
00:41:40,390 --> 00:41:42,630

 

1272
00:41:42,630 --> 00:41:43,990

 

1273
00:41:43,990 --> 00:41:44,000

 

1274
00:41:44,000 --> 00:41:44,630


1275
00:41:44,630 --> 00:41:47,349

 

1276
00:41:47,349 --> 00:41:48,710

 

1277
00:41:48,710 --> 00:41:50,069

 

1278
00:41:50,069 --> 00:41:51,750

 

1279
00:41:51,750 --> 00:41:51,760

 

1280
00:41:51,760 --> 00:41:52,630


1281
00:41:52,630 --> 00:41:54,790

 

1282
00:41:54,790 --> 00:41:56,230

 

1283
00:41:56,230 --> 00:41:57,990

 

1284
00:41:57,990 --> 00:41:59,829

 

1285
00:41:59,829 --> 00:42:01,430

 

1286
00:42:01,430 --> 00:42:01,440

 

1287
00:42:01,440 --> 00:42:02,150


1288
00:42:02,150 --> 00:42:03,510

 

1289
00:42:03,510 --> 00:42:05,829

 

1290
00:42:05,829 --> 00:42:08,870

 

1291
00:42:08,870 --> 00:42:11,510

 

1292
00:42:11,510 --> 00:42:12,950

 

1293
00:42:12,950 --> 00:42:12,960

 

1294
00:42:12,960 --> 00:42:14,230


1295
00:42:14,230 --> 00:42:16,390

 

1296
00:42:16,390 --> 00:42:17,750

 

1297
00:42:17,750 --> 00:42:20,069

 

1298
00:42:20,069 --> 00:42:21,589

 

1299
00:42:21,589 --> 00:42:23,670

 

1300
00:42:23,670 --> 00:42:24,710

 

1301
00:42:24,710 --> 00:42:26,069

 

1302
00:42:26,069 --> 00:42:27,349

 

1303
00:42:27,349 --> 00:42:28,550

 

1304
00:42:28,550 --> 00:42:31,750

 

1305
00:42:31,750 --> 00:42:34,870

 

1306
00:42:34,870 --> 00:42:36,710

 

1307
00:42:36,710 --> 00:42:37,990

 

1308
00:42:37,990 --> 00:42:39,990

 

1309
00:42:39,990 --> 00:42:40,000

 

1310
00:42:40,000 --> 00:42:42,390


1311
00:42:42,390 --> 00:42:45,910

 

1312
00:42:45,910 --> 00:42:47,510

 

1313
00:42:47,510 --> 00:42:50,630

 

1314
00:42:50,630 --> 00:42:53,030

 

1315
00:42:53,030 --> 00:42:54,309

 

1316
00:42:54,309 --> 00:42:56,630

 

1317
00:42:56,630 --> 00:42:57,990

 

1318
00:42:57,990 --> 00:43:01,589

 

1319
00:43:01,589 --> 00:43:02,309

 

1320
00:43:02,309 --> 00:43:05,190

 

1321
00:43:05,190 --> 00:43:07,109

 

1322
00:43:07,109 --> 00:43:08,790

 

1323
00:43:08,790 --> 00:43:11,270

 

1324
00:43:11,270 --> 00:43:12,470

 

1325
00:43:12,470 --> 00:43:14,069

 

1326
00:43:14,069 --> 00:43:17,430

 

1327
00:43:17,430 --> 00:43:20,309

 

1328
00:43:20,309 --> 00:43:21,190

 

1329
00:43:21,190 --> 00:43:23,510

 

1330
00:43:23,510 --> 00:43:24,710

 

1331
00:43:24,710 --> 00:43:28,150

 

1332
00:43:28,150 --> 00:43:30,630

 

1333
00:43:30,630 --> 00:43:31,510

 

1334
00:43:31,510 --> 00:43:36,550

 

1335
00:43:36,550 --> 00:43:39,349

 

1336
00:43:39,349 --> 00:43:41,030

 

1337
00:43:41,030 --> 00:43:42,150

 

1338
00:43:42,150 --> 00:43:45,030

 

1339
00:43:45,030 --> 00:43:47,270

 

1340
00:43:47,270 --> 00:43:51,270

 

1341
00:43:51,270 --> 00:43:53,589

 

1342
00:43:53,589 --> 00:43:55,190

 

1343
00:43:55,190 --> 00:43:56,630

 

1344
00:43:56,630 --> 00:43:59,910

 

1345
00:43:59,910 --> 00:44:02,150

 

1346
00:44:02,150 --> 00:44:03,829

 

1347
00:44:03,829 --> 00:44:04,790

 

1348
00:44:04,790 --> 00:44:07,430

 

1349
00:44:07,430 --> 00:44:09,589

 

1350
00:44:09,589 --> 00:44:12,710

 

1351
00:44:12,710 --> 00:44:12,720

 

1352
00:44:12,720 --> 00:44:14,390


1353
00:44:14,390 --> 00:44:16,230

 

1354
00:44:16,230 --> 00:44:17,990

 

1355
00:44:17,990 --> 00:44:19,910

 

1356
00:44:19,910 --> 00:44:22,390

 

1357
00:44:22,390 --> 00:44:27,589

 

1358
00:44:27,589 --> 00:44:30,550

 

1359
00:44:30,550 --> 00:44:31,670

 

1360
00:44:31,670 --> 00:44:34,710

 

1361
00:44:34,710 --> 00:44:38,470

 

1362
00:44:38,470 --> 00:44:40,550

 

1363
00:44:40,550 --> 00:44:42,069

 

1364
00:44:42,069 --> 00:44:44,390

 

1365
00:44:44,390 --> 00:44:45,910

 

1366
00:44:45,910 --> 00:44:47,109

 

1367
00:44:47,109 --> 00:44:47,119

 

1368
00:44:47,119 --> 00:44:47,589


1369
00:44:47,589 --> 00:44:49,510

 

1370
00:44:49,510 --> 00:44:51,030

 

1371
00:44:51,030 --> 00:44:52,790

 

1372
00:44:52,790 --> 00:44:54,790

 

1373
00:44:54,790 --> 00:44:59,589

 

1374
00:44:59,589 --> 00:45:12,150

 

1375
00:45:12,150 --> 00:45:14,790

 

1376
00:45:14,790 --> 00:45:18,150

 

1377
00:45:18,150 --> 00:45:19,910

 

1378
00:45:19,910 --> 00:45:21,349

 

1379
00:45:21,349 --> 00:45:23,589

 

1380
00:45:23,589 --> 00:45:26,150

 

1381
00:45:26,150 --> 00:45:27,829

 

1382
00:45:27,829 --> 00:45:29,670

 

1383
00:45:29,670 --> 00:45:30,870

 

1384
00:45:30,870 --> 00:45:32,790

 

1385
00:45:32,790 --> 00:45:34,069

 

1386
00:45:34,069 --> 00:45:37,510

 

1387
00:45:37,510 --> 00:45:39,910

 

1388
00:45:39,910 --> 00:45:41,750

 

1389
00:45:41,750 --> 00:45:43,750

 

1390
00:45:43,750 --> 00:45:45,190

 

1391
00:45:45,190 --> 00:45:47,670

 

1392
00:45:47,670 --> 00:45:49,270

 

1393
00:45:49,270 --> 00:45:50,950

 

1394
00:45:50,950 --> 00:45:53,510

 

1395
00:45:53,510 --> 00:45:54,630

 

1396
00:45:54,630 --> 00:45:56,150

 

1397
00:45:56,150 --> 00:45:58,790

 

1398
00:45:58,790 --> 00:46:01,109

 

1399
00:46:01,109 --> 00:46:01,119

 

1400
00:46:01,119 --> 00:46:02,309


1401
00:46:02,309 --> 00:46:05,670

 

1402
00:46:05,670 --> 00:46:08,230

 

1403
00:46:08,230 --> 00:46:09,910

 

1404
00:46:09,910 --> 00:46:12,790

 

1405
00:46:12,790 --> 00:46:12,800

 

1406
00:46:12,800 --> 00:46:13,510


1407
00:46:13,510 --> 00:46:17,270

 

1408
00:46:17,270 --> 00:46:17,280

 

1409
00:46:17,280 --> 00:46:20,790


1410
00:46:20,790 --> 00:46:24,790

 

1411
00:46:24,790 --> 00:46:26,870

 

1412
00:46:26,870 --> 00:46:28,870

 

1413
00:46:28,870 --> 00:46:30,630

 

1414
00:46:30,630 --> 00:46:32,309

 

1415
00:46:32,309 --> 00:46:33,510

 

1416
00:46:33,510 --> 00:46:34,230

 

1417
00:46:34,230 --> 00:46:36,309

 

1418
00:46:36,309 --> 00:46:38,390

 

1419
00:46:38,390 --> 00:46:41,430

 

1420
00:46:41,430 --> 00:46:43,349

 

1421
00:46:43,349 --> 00:46:44,950

 

1422
00:46:44,950 --> 00:46:45,829

 

1423
00:46:45,829 --> 00:46:49,349

 

1424
00:46:49,349 --> 00:46:53,270

 

1425
00:46:53,270 --> 00:46:56,550

 

1426
00:46:56,550 --> 00:46:58,230

 

1427
00:46:58,230 --> 00:47:30,710

 

1428
00:47:30,710 --> 00:47:33,190

 

1429
00:47:33,190 --> 00:47:35,109

 

1430
00:47:35,109 --> 00:47:36,870

 

1431
00:47:36,870 --> 00:47:36,880

 

1432
00:47:36,880 --> 00:47:37,430


1433
00:47:37,430 --> 00:47:40,630

 

1434
00:47:40,630 --> 00:47:43,030

 

1435
00:47:43,030 --> 00:47:45,109

 

1436
00:47:45,109 --> 00:47:47,430

 

1437
00:47:47,430 --> 00:47:49,910

 

1438
00:47:49,910 --> 00:47:51,430

 

1439
00:47:51,430 --> 00:47:52,069

 

1440
00:47:52,069 --> 00:47:55,510

 

1441
00:47:55,510 --> 00:47:58,710

 

1442
00:47:58,710 --> 00:47:58,720

 

1443
00:47:58,720 --> 00:47:59,589


1444
00:47:59,589 --> 00:48:01,589

 

1445
00:48:01,589 --> 00:48:03,750

 

1446
00:48:03,750 --> 00:48:04,549

 

1447
00:48:04,549 --> 00:48:07,990

 

1448
00:48:07,990 --> 00:48:10,870

 

1449
00:48:10,870 --> 00:48:12,790

 

1450
00:48:12,790 --> 00:48:15,430

 

1451
00:48:15,430 --> 00:48:17,030

 

1452
00:48:17,030 --> 00:48:19,349

 

1453
00:48:19,349 --> 00:48:20,790

 

1454
00:48:20,790 --> 00:48:20,800

 

1455
00:48:20,800 --> 00:48:21,270


1456
00:48:21,270 --> 00:48:23,829

 

1457
00:48:23,829 --> 00:48:25,589

 

1458
00:48:25,589 --> 00:48:25,599

 

1459
00:48:25,599 --> 00:48:25,990


1460
00:48:25,990 --> 00:48:27,510

 

1461
00:48:27,510 --> 00:48:29,750

 

1462
00:48:29,750 --> 00:48:29,760

 

1463
00:48:29,760 --> 00:48:30,309


1464
00:48:30,309 --> 00:48:33,030

 

1465
00:48:33,030 --> 00:48:35,109

 

1466
00:48:35,109 --> 00:48:37,589

 

1467
00:48:37,589 --> 00:48:39,670

 

1468
00:48:39,670 --> 00:48:42,470

 

1469
00:48:42,470 --> 00:48:42,480

 

1470
00:48:42,480 --> 00:48:43,349


1471
00:48:43,349 --> 00:48:45,030

 

1472
00:48:45,030 --> 00:48:47,030

 

1473
00:48:47,030 --> 00:48:48,870

 

1474
00:48:48,870 --> 00:48:48,880

 

1475
00:48:48,880 --> 00:48:49,670


1476
00:48:49,670 --> 00:48:51,589

 

1477
00:48:51,589 --> 00:48:52,710

 

1478
00:48:52,710 --> 00:48:55,589

 

1479
00:48:55,589 --> 00:48:56,950

 

1480
00:48:56,950 --> 00:49:00,150

 

1481
00:49:00,150 --> 00:49:02,309

 

1482
00:49:02,309 --> 00:49:04,069

 

1483
00:49:04,069 --> 00:49:07,829

 

1484
00:49:07,829 --> 00:49:09,349

 

1485
00:49:09,349 --> 00:49:11,349

 

1486
00:49:11,349 --> 00:49:13,589

 

1487
00:49:13,589 --> 00:49:15,430

 

1488
00:49:15,430 --> 00:49:17,510

 

1489
00:49:17,510 --> 00:49:19,990

 

1490
00:49:19,990 --> 00:49:22,150

 

1491
00:49:22,150 --> 00:49:23,910

 

1492
00:49:23,910 --> 00:49:24,870

 

1493
00:49:24,870 --> 00:49:27,510

 

1494
00:49:27,510 --> 00:49:29,990

 

1495
00:49:29,990 --> 00:49:34,630

 

1496
00:49:34,630 --> 00:49:36,870

 

1497
00:49:36,870 --> 00:49:38,630

 

1498
00:49:38,630 --> 00:49:38,640

 

1499
00:49:38,640 --> 00:49:39,109


1500
00:49:39,109 --> 00:49:41,349

 

1501
00:49:41,349 --> 00:49:42,390

 

1502
00:49:42,390 --> 00:49:45,349

 

1503
00:49:45,349 --> 00:49:46,630

 

1504
00:49:46,630 --> 00:49:49,349

 

1505
00:49:49,349 --> 00:49:49,359

 

1506
00:49:49,359 --> 00:49:50,069


1507
00:49:50,069 --> 00:49:51,670

 

1508
00:49:51,670 --> 00:49:53,510

 

1509
00:49:53,510 --> 00:49:55,270

 

1510
00:49:55,270 --> 00:49:57,430

 

1511
00:49:57,430 --> 00:49:58,870

 

1512
00:49:58,870 --> 00:50:01,109

 

1513
00:50:01,109 --> 00:50:02,230

 

1514
00:50:02,230 --> 00:50:05,030

 

1515
00:50:05,030 --> 00:50:06,230

 

1516
00:50:06,230 --> 00:50:07,910

 

1517
00:50:07,910 --> 00:50:09,430

 

1518
00:50:09,430 --> 00:50:11,670

 

1519
00:50:11,670 --> 00:50:13,430

 

1520
00:50:13,430 --> 00:50:14,470

 

1521
00:50:14,470 --> 00:50:16,549

 

1522
00:50:16,549 --> 00:50:17,670

 

1523
00:50:17,670 --> 00:50:20,950

 

1524
00:50:20,950 --> 00:50:23,589

 

1525
00:50:23,589 --> 00:50:24,630

 

1526
00:50:24,630 --> 00:50:26,470

 

1527
00:50:26,470 --> 00:50:28,790

 

1528
00:50:28,790 --> 00:50:29,910

 

1529
00:50:29,910 --> 00:50:31,190

 

1530
00:50:31,190 --> 00:50:34,230

 

1531
00:50:34,230 --> 00:50:38,829

 

1532
00:50:38,829 --> 00:50:38,839

 

1533
00:50:38,839 --> 00:50:51,589


1534
00:50:51,589 --> 00:50:51,599

 

1535
00:50:51,599 --> 00:50:53,680