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all right welcome back to six a49 today  we are going over tree maker folding  polyhedra folding checker boards and  organizer lots of stuff and can't fit  everything in today but we'll be doing  more Tree Maker in some sense from a  more practical perspective in the very  next lecture video and next class with a  guest lecture by Jason ku so today we  have a bunch of questions  we'll start with with Tree Maker so the  first question is wow you could make all  these crazy things with the tree method  are they really uniaxial and the answer  is yes  and I think it's probably easiest to  just go through some examples these are  all from origami design secrets this  book by Robert Lang which is on the  recommended reading list and I think  there's even an electronic copy  available through MIT libraries so you  can read it online or maybe one in the  library there's two editions these  figures are from the first edition so  here is a the goal is to design scorpion  scorpion barrel egg since they're all  Robert Lang designs you come up with  this stick figure your tree tree maker  makes this crease pattern you can also  see the rivers here and the and the  disks and then you get this base and  then it folds into that scorpion and so  while the Scorpion doesn't look like it  has a single axis and a scorpion maybe  there's kind of an axis down the center  but you know it's it's not just this  axis it branches and it branches again  you can make a tree even and still  uniaxial so uniaxial refers to this axis  in the base all the flaps are attached  to that one spot so in this picture it's  vertical usually in the we draw the  figures we think of it as the horizontal  plane or horizontal line so we imagine  this is the floor and then we have you  know flaps attached to the floor  something like this  so in that sense when you flatten the  space everything lies along this axis  even though in terms of branching  structure you can hit lots of branches  so that's to clarify what uniaxial means  maybe not the best term but it's have a  few more examples here this hat includes  a color reversal some crazy stick figure  and you get your pattern here's a  somewhat simpler one that makes them  more three-dimensional horse this alamo  stallion so you can go through and for  each of these i've just reverse  engineered from this crease pattern what  the stick figure must be and you're  going to do this on pset two which will  be released in a few minutes and so this  uniaxial is to contrast from something  with the kind of most famous non  uniaxial biaxial origami bases mont  rolls dog base which looks like this so  it's very easy to turn this into all  sorts of dogs this is a wiener dog but  you have kind of one axis here and one  axis here which is different it's not  the same kind of branching tree  structure of course you can make a dog  using uniaxial methods as well but it's  kind of a fun counter example other  things you might do so that's uniaxial  next question is do people use tree  maker and or organizer in practice and  the short answer is no I would say most  origamist don't use software to design  things but they use a lot of the ideas  in their own designs so in particularly  the tree method of origami design which  tree maker is implementing almost every  advanced origami designer uses would say  not everyone but most of the complicated  most complex origami that you see uses  the tree method at least to get started  organizer is much newer so it's not used  as much most of the cool designs I'll  show you a bunch of both are by Tomohiro  tachi who invented organizer but I think  it's still evolving and hopefully these  more advanced techniques will catch on  with time if and by now you've seen lots  of examples of the tree method  I just show a few more from Robert  Lang's website this is a local delicacy  uh-huh and these these are all on on  Lange origami calm on his website and  we'll see even more examples in the next  class Jason ku is going to give an  overview various artists and in some  cases they publish cruise patterns not  always this is not quite the tree method  so this example was sort of standard  tree method you've got disks and rivers  which aren't drawn here you get a crease  pattern this could be done directly from  Tree Maker I don't know for sure whether  it was this pattern is not done with  tree maker because it's not regular  disks this is the box pleating version  of the tree method which there's another  question about so we'll talk about that  more but you see all the creases here  are horizontal vertical or diagonal 45  degrees that makes it a lot easier to  follow a lot easier to find reference  points you see it falls nicely on a grid  but still you get an arbitrary tree  structure I think I have another example  this one also well this this has 22 and  1/2 degree folds it's a little bit more  general also a very non stick figure  like models that particularly it's  obviously called the fiddler crab no  sorry I have the wrong title here I will  fix that this I forget this guy's name  but if anyone remembers he wrote about  the division between scientists and  people who know literature as two  different types of intellectuals that  know almost nothing about each other at  least in the forget early 1900s when he  was writing so and one more example this  is based on a sixty degree grid so all  the creases I think lie at 60 degrees to  the axis or horizontal there might be  some 30 degrees  so we'll talk a little bit about this  alternate version of the tree method  that doesn't just use arbitrary circles  but tries to stay on these nice grids  here's just one more example this Jason  ku is the designer of this model called  pan and he'll be giving the guest  lecture so this is just one example but  tons and tons of advanced origami  artists use a tree method of origami  design and you get some pretty cool  crease patterns from it but I think most  people do it by hand by hand usually on  a computer using some drawing programs  that can compute intersections and do  things with high accuracy and then they  draw these pictures then they fold them  and then you get your base and then you  shape it into the model so next I want  to show you some practical examples of  organizer so as a bunch on Tomohiro  tachi z-- Flickr site this is one of the  earliest this is just making a negative  curvature surface that curves in both  ways like a saddle called a hyperbolic  paraboloid we'll see more of those in  the future but this is folded from one  square paper and there's tabs on the  backside that you don't see here here's  making a kind of 3d bell curve  here's a computer mouse apparently the  the scroll wheel works fine doesn't  support USB though only bluetooth aha  we've got a nice mask these are all  you're given a 3d model and you fold  exactly that 3d model this is a tetrapod  so the here's sort of shows Toma here is  background in architecture we haven't  seen tetrapods they're used to hold back  the ocean and things from or hold the  hold from erosion things like that so  usually made out of concrete but here  you can make them out of paper here's a  flat design so here the mesh the  triangles were all in a plane and of  course you still get the tabs on the  backside but you make this very exact  leaf you can get the triangulation edges  exactly where you want them to express  you know these veins of the leaf  here's what it looks like to fold one of  these in practice is not quite a square  paper but could be a square and this is  actually folded at csail you might  recognize some of the furniture and so  while telling here is folding he uses  various devices paperclips and and clips  and so on to hold it in shape because  until it's completely collapsed it kind  of want to opens back up so it's kind of  like having a hundred hands at once but  when he's done he'll take all the all of  them off and you'll get so this I think  was about ten hours you get your bunny  uh-huh easy uh-huh or this is one we did  a little bit later just last year this  is laser cut sheet of Steel and now here  we've cut out bigger holes too so there  are too many accumulation of layers but  essentially the same design somewhat  coarser mesh of the bunny but and you  have to wear gloves  otherwise you'll cut yourself so we're  perforating the metal at decreases and  then also takes about eight hours a lot  harder to fold steel and in the end you  get your bunny  so in principle out of any sheet of  material you can make any 3d shape you  want that's the exciting thing so I  think organizers really powerful  obviously it's hard to fold you need to  be pretty advanced but there's a lot of  potential for designing very nonstick  figure like models so that was true  maker and organizer in practice so next  question is about this box pleating  which is the horizontal vertical and  45-degree creases and treemaker is there  some is there some theory for this and  indeed there is and we started working  on it me Marty and Rob Lang I think  during his first visit here which was  probably 2000  for something long time ago and the best  write-up of it currently is in origami  design secrets especially second edition  so it has a chapter on basic tree theory  a chapter on box pleating in general  this is sort of classical box pleating  but then there's a chapter on mixing the  two uniaxial box bleeding and polygon  packing these kind of go together so if  you're interested in this stuff you want  to design something the way that the  experts to check out Oregon's origami  design secrets or go to an origami  convention where Rob is talking about  this stuff we are working on this giant  manuscript it's currently called the  mathematics of origami design which is  in particular trying to prove the tree  the tree method always works and will  generalize to things like this but it's  not finished so we don't have a complete  proof yet that it all works and  everything seems fine but it's tedious  to write it all down  so still working on it but I thought I'd  show you an example of box pleating  uniaxial origami design from origami  design secrets so this is sort of  typical tree method of origami design  this is if you're not necessarily using  treemaker but you're doing it by hand  you think about okay suppose I want to  make this kind of insect with this stick  figure maybe I realize oh it's kind of  centrally symmetric so I'd like to make  the left half of the paper same as the  right half you can also express that in  the tree maker software and then you  start thinking about where these leaves  correspond to disks then you've got to  have the corresponding rivers in between  them so you might start with this kind  of layout and then you kind of you try  to blow it up until things are can't be  expanded anymore and there's lots of  touching so that would be the usual  approach then from that you could apply  the tree method get the crease pattern  with the box pleated version essentially  instead of disks you have squares and  instead of rivers you have these  orthogonal channels of constant width  and so I'll just wave my hands and say  that happens now you get these weird  things of you get these gaps you tend to  get more gaps in this way but in this  case you instead of just having a square  you can extend it out to be a rectangle  so these guys are actually going to fill  in these holes  this and then from this you can start  constructing a crease pattern so you  start with these Ridge creases and this  is something like a generalization it's  actually something like straight  skeleton which we'll be covering in a  couple of lectures for a different  purpose where it was originally  developed for origami purposes but these  are kind of these are just lots of  bisectors and you'll have to see the  general version later and then you start  putting in the hinge creases these are  yeah these green lines and they're  perpendicular to these these blue  creases are going to be the floor and so  on you fill it in eventually you get  your complete crease pattern so in a  nutshell that's how it works  you tend to get tabs that are constant  width like this house in height I guess  and so it's a general technique it's a  little bit much to explain here but read  the book if you want to see it and I say  it's probably some of the most common  there's a lot of intuitive ways to do it  and then a lot of the details are worked  out in that book but it's quite common  to design bases in this way because it's  just so much easier to fold them you  don't have to construct really weird  angles a lot cleaner but a little bit  less efficient because of the instead of  using a disk which is kind of the  minimum amount of paper you need to make  a flap you're using a square so you're  kind of wasting those little corners but  not that much more inefficient cool so  that was that method next question I  have is about the triangulation  algorithm in the which I didn't even  cover in lecture so this was you you you  know you set up you have your piece of  paper and you assign each leaf in this  tree to some point in the piece of paper  and you satisfy the active path  condition so each of these distances  measured on the paper should be greater  than or equal to the distance measured  on the tree which is the floor of the  base  great and so if you happen to have some  equalities you draw inactive paths but  what if you don't have any or you don't  have enough the tree method in its  original form only works when the active  paths decompose the piece of paper into  convex polygons this is what you'd like  to have happen each of the phases here  is a convex polygon that happens great  you can use the universal molecule and  you get your folding if it doesn't  happen you have to modify your tree a  little bit there a couple ways to do  this in the in the tree maker you can  modify these edge lengths and try to  modify them as little as possible so  that things touch but the easy way to  prove that this that something is  possible because remember if we if we  add a little bit to the base you know we  get an extra flap that doesn't really  hurt us at the end we can hide the flap  just fold it against other flaps and  just pretend it wasn't there so as long  as we can add flaps in order to make  this crease pattern or in order to make  the active paths decomposed into convex  polygons we're happy and in fact what we  prove is that you can do it you can keep  adding flaps until you get into  triangles and triangles are always  convex and so they make us happy so we  will end up adding a bunch of flaps in  our tree in order to triangulate with  active paths and then in each of these  we just fill in a rabbit-ear molecule  okay so that's the goal now let me tell  you how the triangulation actually works  originally described by Lang it's also  in the textbook for this class I'll try  to give an abbreviated version here so a  little bit technical here's the idea  so suppose alright so we've said we've  assigned some leaves and suppose we have  some region that's not convex it's not a  triangle has more than three sides now  some of these edges may come from active  paths and some of them may be from the  boundary of the paper so maybe the paper  is here so this edge is not active just  have it just the boundary of this region  because boundary the paper okay so there  are two types of edges but there should  be I guess at least one active edge and  there's at least four edges total so  what I'm going to do is look at any of  the active edges and I'm going to  imagine okay so that active edge this is  in in the piece of paper but now I'm  going to think about it in the tree so  the tree looks like something we don't  really know what any active active path  here corresponds to an active path in  the tree of exactly the same length so  maybe it's from this leaf to this leaf  so it corresponds to color this guy  corresponds to this path and that some  of those lengths should be equal to that  length it isn't obviously but there's a  scale factor in between here lambda okay  so here's what I'm going to do I'm going  to modify the tree by adding a new leaf  somewhere off of this path okay where ah  let's for convenience let's assume that  that this this path length here is one  then I'm going to measure out some  distance X here which will leave a  different distance 1 minus X here and  then I'm going to make this length  I guess L and I'm going to call this  leaf L capital L okay this is a  modification to a tree I can do it I can  do it for any value of x between 0 & 1  and I can do it for any value of L  greater or equal to 0 so what I'd like  to do is design the tree so that this  point ends up in an interesting place on  the piece of paper in fact I claim that  no matter where I draw L here capital L  as my desired place now what I'd really  like is for I should give these guys  names this is called you are say in the  notes I call it V and W so here we have  W and V what I would like is for these  two paths to also be active meaning the  lengths here match the lengths in the  tree I claim that no matter where I put  L in the plane anywhere in the plane I  can as choose X and choose L so that  these two lengths are exactly correct do  you believe me not really how many  people think this is obvious good couple  maybe let me give you a quick sketch  it's not that interesting so I just want  to do it very briefly if you get it  right if you let X be a free parameter  but fix L little L then what this  corresponds to is in fact a an ellipse  with V and W as Fossey of the ellipse  and because you have to hold the sum of  these lengths fixed if you fix if you  let X vary you know this length Plus  this length is always going to be the  same if you fix L and so this is called  the major axis the sum of these two  lengths is going to be  like 1 plus 2l and so if I have a point  basically there's some ellipse of the  appropriate size it passes through L  that will let me choose little L and  then as I vary X I walk around the  ellipse so I just choose the appropriate  value of x that gets me the desired  point on the ellipse that's it  ok so for any point at the point is the  set of all ellipses with these two fuzzy  spans the entire plane and so wherever  you want to put L you can make those two  pans active ok this is good because it  makes a little triangle but of course if  I just add an arbitrary triangle not  very interesting the good thing is I'm  free to put L wherever I want  I'd really like to put L say here and  draw active paths like that  because then I'm kind of decomposing my  polygon triangles that's not always  possible so let's just go through the  cases in all cases we're going to  simplify a polygon make it have fewer  sides and that's a next page so the  claim is well Bo there's sort of three  cases to reorganize this a little bit  one thing that would be nice is I can  place L they all draw another version of  this so I've got you V I'll draw the  active pads in red so we know UV is  active got L if I could somehow place L  so that's active with two other points  two other leaves sorry this is V and W  this is U and this is T if I can do this  and the region  it looks something like that then I'm  happy I'm going to put L there and add  in these four edges and essentially if  you look at any one of these regions  that remains there's a triangle here  that's definitely okay these regions  each of them will have fewer edges than  the original region that's pretty easy  to prove because you have because you're  connecting to these four vertices this  region won't have this vertex or this  vertex so it has one added vertex and to  remove vertices at least W and T this  one will be smaller and it's a symmetric  for all of them okay so this would be a  good case so what I'm going to do is try  to move L around I'm going to start very  close to V W and just start moving off  the edge initially nothing else will be  active because it's basically right on  top of V W but as I move around I might  get another active path okay maybe this  is case one a before we get there is one  is you get L u active suppose you just  have to get one one additional active  path the other case is nothing else  becomes active so here's V here's W  here's L here's u so this is active  active active active okay what I'm going  to do in this situation is if I want to  keep Lu active I can actually move L on  a circle as long as L stays on this  circle Lu remains active okay so just  move it along a circle now there are two  possibilities it could be becomes active  with something else then I'm done or it  could be it doesn't which means it will  hit the boundary so we'll hit a boundary  point so case one a so this is case one  you can make something active either you  when you move along the circle either  you make another thing active or you  won't you hit the boundary  case 1 B is you hit the boundary and  then I claim you're also happy so in  that situation one more time we've got L  on the boundary and we've got some path  here here here here these are all active  so here's L okay I claim again each of  these regions has fewer vertices than  before should be pretty obvious this one  is emitting W and this was you yeah  meaning W and it's emitting whatever is  down here if anything yeah I think it's  pretty obvious kind of check all the  cases is a few details here but should  be each of these regions has strictly  fewer vertices than it did before and so  you're making progress if you started  with all phases having some number of  edges each time you do one of these  steps you decrease it in the last case  case two is you can't make anything  active this can happen for example if  you're in the corner of the piece of  paper and you have an active path I  guess that's a little less exciting  maybe your piece of paper is this shape  a little bit weirder so this will work  for any convex piece of paper so it's  not quite a triangle and you're moving L  around here and you just can't get  another active path there's sort of no  no vertices around this is W this is U  so has V can you write some day  and in this case L can go anywhere here  it's free in that case I'm going to put  L on one of the vertices of the paper so  I get this active and this active  and then I've decomposed that region  into pieces by this is what we call a  diagonal we're adding a vertex to vertex  edge in this polygon and that always  decreases the number of sides on in each  of the region sub regions so if I just  keep doing this taking the any any  region of size larger than three I will  eventually reduce them all to half size  three so they'll all be triangles so  that's a triangulation algorithm in a  nutshell you can look at the textbook if  you want to see it inst in more detail  but it's pretty simple this proves that  something is possible  I don't think treemaker actually  implements this algorithm specifically  but you can kind of do it by hand so I  have a little example here this is  treemaker  I drew this really weird tree we call it  a caterpillar tree yes you would  actually use it to make caterpillars and  if you just plug this into Tree Maker  with all the lengths unit it will give  you this error message I couldn't  construct all the polygons because  things weren't tight enough and if you  look carefully it's a little hard to see  these colors but the light green edges  those are the active paths and here so  this was a triangle and I was happy if  you look at this green polygon maybe  actually I'll draw with the tablet so  this thing is inactive but that's a  region bounded by active paths and  probably the top edge is not an active  path that's just a boundary but it's not  convex and so you're unhappy and so I  just kind of eyeball this and said okay  well probably I should add another leaf  here that that would maybe add a disk  that will probably fill that in so you  don't have to be super precise here we  add you know very carefully place to  disk to make things touch but because  Tree Maker is always just trying to blow  things up and make them touch it's quite  a bit easier in practice you just say  okay I will add an extra leaf here like  that to the tree and then I hit optimize  and boom it works in this case I was  lucky in general I might get a  non-convex region I'll just guess where  to add another leaf  and it works you compare I just do it  did that leaf have to be that big or  could I get away with a smaller one but  eventually you will get a base and if  you want you could carefully monitor  this proof and actually always succeed  but in practice is usually not that hard  questions about that so that was the  triangulation method in a nutshell next  question is about the universal molecule  and we might spend some more time on  this next class but I thought we could  actually look at this example because  it's got a bunch of different Universal  molecules so I will continue to draw so  let me pick let's say this universal  molecule here it's a quadrilateral  all right so those are four active paths  in this case and that corresponds to  some tree so I don't I don't have handy  here but it's going to be a tree with  four edges so truth  so if with four leaves sorry truth for  it leaves is going to look something  like this this is a piece of the bigger  tree which you recall looks something  like this and we add it's one somewhere  but this particular quadrilateral is  doing some particular sub tree of on  four leaves and particularly their four  leaves here correspond to four leaves  here I don't know which ones I don't  really need to know tree Mack true maker  keeps track of it for me and I happen to  know that these four paths are active  meaning the lengths in the plain here  match exactly the lengths as measured  along the tree so what I do to make this  work this is basically the floor of the  molecule these those four paths will all  lie on the ground level what I'm trying  to do is slice higher and higher in the  base to see what happens when I slice  and if you're if you think about it as  you move up here  in the plane this corresponds to moving  parallel to these edges so I'm shrinking  this polygon by parallel offsets so let  me draw one parallel offset  approximately parallel okay after I  shrink a little bit might look like that  an interesting moment in this case is  here fellas carefully should go sorry  should go to here  whoo I'm going to start over this is  actually already drawn for us this guy  okay this is a parallel offset and in  this case something interesting happens  which is this path so it turns out this  middle edge in this case becomes active  so what that means is originally if you  looked at these two points these two  leaves they had the wrong length okay so  that's going to correspond to something  like these two leaves they're opposite  corners of the quad and you measure the  length along the tree you get something  and the claim was in the plane it was  too big now as you do this parallel  offset as you shrink the polygon at some  point it will be just the right length  this could happen doesn't have to happen  if it happens you have to stop because  if you kept going it would become too  short and we know every length here must  be greater than or equal to length over  here so when it becomes equal we have to  stop we have to make this a crease  making it a crease basically says ah  this is exactly the right length I have  to split here and I shot guy and I have  to sort of grow now two different flaps  from that point the crease sort of makes  it horizontal and then we end up  shrinking into different parts we shrink  in this triangle which gives us a  rabbit-ear and we shrink up there in  that triangle gives us another rabbit  ear so that it will end up giving these  angular bisectors in general as you do  the shrinking you watch where the  vertices go those are your Ridge creases  so this one went along an angular  bisector that one did as well but once  we do this split operation because of  this newly active path which should also  call a gusset in the universal molecule  then these direct these vertices  actually split and go in two directions  one for this angular bisector one for  that angular bisector  so in general for a universal molecule  there are two things that can happen you  can have you can be shrinking your  polygons and suddenly discover there's a  newly active path and then you have to  divide it into two polygons and start  shrinking those separately or vertices  can disappear so it could be you're just  shrinking shrinking shrinking merrily  along the way and then suddenly these  two vertices collide with each other so  you get to ridge creases here in this  case you just treat those two vertices  now as one so you'll start shrinking  like this and so these vertices will now  start going that way so in general as  you're shrinking these the only two  types of events that can happen either  two vertices merge or you get a new  diagonal here that becomes active and  that happens you just divide this will  work for any convex polygon and you get  your molecule it's hard to do it's hard  to draw a picture of this without using  a computer tool or really keeping track  of what you're doing because how do you  tell when something becomes active you  have to look at your tree if to measure  lengths that's tricky to do without a  computer tool and that's why I tree  maker was made but it can be done  probably even easier with physical paper  if you know exactly what lengths you're  trying to match so those quick overview  of how universal molecule works in more  detail next I want to talk about a few  different open problems so one of them  was these are called gift wrapping  problems so we have things like given a  square a unit square let's say what's  the largest regular tetrahedron you can  wrap that's still open all these  problems are still open except for the  square to cube and which talked about in  lecture and I realize equilateral  triangle to tetrahedron that's also  really easy if you can do it without any  paper wastage is clearly off  at all but it would be very cool to  studies I think is to make a good good  project or a problem for the open  problem session folding a given  rectangle will give an aspect ratio into  a cube also open it pretty much any  version you can think of is open except  the ones that we've already seen also  wanted to mention for checkerboards  fun problem kind of in this spirit is  what would be the best way to fold a 2x2  checkerboard even for a 2x2 we have no  idea how to argue any kind of lower  bound about how bad you must do that you  can't just take a square and fold a 2x2  checkerboard of the same size surely you  can't do that but we don't know how to  prove that so it would be a nice target  we obviously have kind of upper bounds  we have constructions that make 2x2  checker boards think from a 3x3 grid you  can make a 2x2 checkerboard but is that  optimal we have no idea ah so a couple  questions about the checkerboard folding  so we had this picture of how you would  take a square of paper you start with a  square paper you fold it into this shape  with these long slits and then these  guys fold over and then there are also  tabs sticking up here which are not  drawn and they fall over and give you  the color reversal question was how do  you actually build this thing this is  not looking the axial and indeed we do  not use uniaxial techniques for this we  use a separate set of gadgets which are  kind of based on pleating and this will  actually be very similar to something we  see in a couple of lectures when we do  box pleating to make cubes and stuff but  you just compose these gadgets so this  is kind of a general tab gadget you  collapse this crease pattern and do a  couple more folds and you end up with  this tab sticking out here and you can  flip it up or down and you see that it  has these pleats running off to the side  this Mountain Valley Mountain Valley and  then Valley Mountain Valley Mountain and  so those have to go all the way through  the paper and they come in to this  construction which basically lets you  make a big slit in the paper or you can  also turn a corner and so you just prove  that these gadgets compose I mean it's  pretty easy that they compose you just  have to analyze how much of the paper  you're using and turns out to be really  good  that's a short version you want to try  it out fold some gadgets fun fun project  idea from from you guys is given an  image sample at low resolution make it  black and white and then come up with  the crease pattern that would fold via  this checkerboard technique into exactly  that two color pixel pattern you could  make all sorts of useful things like  space invaders stuff like that you'd  never want to fold them I think but  maybe with low enough resolution you  could fold them but I think they'd be  cool just as crease patterns by  themselves you input a different pattern  I'd like you know web applet who changed  the pixel pattern and boom it gives you  the crease pattern I think it would be a  fun project if you're into implementing  things it's essentially the algorithm we  already have but it needs to be just  written out in detail and cut it up okay  here's what it looks like to fold an  eight by eight checkerboard with this  method these are all by Robert Lang so a  lot of precreasing along this huge grid  I think Flea 48 by 48 and then some tape  to hold things shut mostly for  photographing and you collapse now we're  down by a factor of two or so now you  can see the slits in one direction  I think this method all the particularly  being use here only has slits in one one  direction oh no here we got slits in the  vertical direction as well you can see  the tabs sticking out here they're going  to fall down there's some more of the  tabs start folding over and boom you've  got your 8 by 8 checkerboard Robert Lang  says wow that was not one of the easier  things I've done and this is folded from  a 48 by 42 checkerboard in principle  this method can go down to 36 plus  epsilon by 36 plus epsilon but it's a  little bit gets Messier crease pattern  wise and 36 would be better than the  best known 8 by board if you want  seamless notice these are seamless  squares no crease lines through them  this one's not necessarily this is not  better than known techniques but it  follows our algorithm and that's what  the crease pattern looks like  cool  next we go to organizer so as I  mentioned in lecture there's two  versions of organizer there's the  version that's implemented in the  software and there's a version that  we're proving is always correct these  are different because the version the  software doesn't always work and it  remains that way because the software  version is more practical it's and also  the the theoretical version is still a  moving target we change it every few  weeks to fix part of the proof  it's almost done but it's been almost  done for a couple of years so we're  still working on it tell me here was  just visiting a couple weeks ago as I  mentioned and we're closing in I  actually have with me the current draft  of the paper and it's not nearly as long  as this one with the tree method but  it's still growing and working out  making sure all the details check out so  that the theoretical version is kind of  complicated so I thought I'm  particularly giving this question I  talked about the software version  because it's actually a lot simpler only  catches it doesn't always work and it's  described in this paper if you want to  read it by Tomohiro about organizer 2010  and I have a few figures from that paper  well this is still from Tommy Harris  flicker so we had this example where you  wanted to fold this hyperbolic  paraboloid and here's what the crease  pattern looks like it's actually pretty  simple you've got these white polygons  which are polygons from the surface okay  that's what you need to fold and the  goal is to lay them out on the piece of  paper this square is the piece of paper  so that I can for example I need to  bring this polygon to touch this polygon  this edge has to touch this edge  wouldn't it be great if I could just  fold the bisector of those two edges and  this would come right on to here  sometimes that happens but for example  if this polygon is way down here  that won't happen it will fold over and  they won't align you have there's two  issues first you have to get the angles  to match then also there's this vertical  shifting if you place the polygons in  the plane in a good way this will work  you always get alignment  and what organizer implements is a  non-convex optimization a constraint  projection to make all of the edges work  sometimes that's possible it's actually  possible fairly often and that's when  the organizer software works if it's not  possible you're screwed this this  particular method won't work but when  it's possible things are great  so essentially you have to bring the  edges together then you also have to  bring the vertices together like these  four vertices come together to a point  and so you get two kinds of gadgets  which are the vertex tucking molecule to  bring vertices together and edge tucking  molecule to bring two edges together  edge tucking molecule is a single crease  trivial in this construction vertex  tucking molecule is complicated and in  general what happens is for these points  we construct what's called a Voronoi  diagram which is essentially for each of  these points if you grow a disc at equal  speeds around all of them and when the  discs meet you stop them so then these  two discs will meet along this  perpendicular bisector and they'll keep  growing until they kind of all die out  and you'll get this tree structure in  general those are your main creases you  follow that structure and you do some  stuff essentially what happens when you  when you fold along this Voronoi diagram  you also have to add in these creases  from the Voronoi diagram to the points  when you do that you'll get a kind of  mushroomy structure that has too much  material if you set it up right all the  angles of paper that you have are larger  than what you need in the folded state  which would look like this and so what  you do are lots of little pleats to  reduce the angle if you've got a big  angle of paper you can do a mountain in  a valley and make it a smaller angle and  that's what all these creases are doing  you've got these pleats to reduce the  angle here reduce the amount of material  here and here and to in order to match  the 3d structure so overall what the  algorithm does is first given the  surface it constructs a suitable 3d  structure of where to put  all the extra tabs then it designs  things so that the angles are bigger  here than what they need to be there  this is again a constraint projection  also constrains this these edges to meet  up perfectly and then it just applies  all these crimps which are it's easier  in a computer than to see it here to  make all the angles correct to match 3d  model yeah question right so for the  sheetmetal bunny we just cut out these  polygons because otherwise it's a mess  and you get lots of layers and the point  was to make it out of one sheet of  material but not necessarily a square  yeah so if you have really thick  material recommend cutting those out and  we were laser cutting you know to score  the lines anyway so why not cut out some  holes as well yeah so that way you just  have edge talking molecules which are  really easy Mountain Valley Mountain and  so it actually is pretty practical to  fold these things out of generalised  sheets with holes so the the non  software version of organizer works the  same way except it does the full  generality of constructing this tuck  proxy that where the tab should go and  that can get very messy in general it's  just tedious to explain there all these  crazy spherical diagrams that you saw in  lecture but it's not that exciting the  more interesting part is that the edge  tucking molecules can't just be a single  crease anymore  they have to in general kind of follow  some paths to get to where they need to  go and and but it's similar you just do  a bunch of different folds kind of you  can sweep them one way or the other in  order to navigate this edge to be where  it needs to be so again you place the  faces in the plane in the mathematical  version you can place them anywhere you  want doesn't matter and then you have to  shrink them until they're small enough  that things work then you route these  paths to get from each edge to each  corresponding edge then  the edge talking molecules are fairly  straightforward the vertex tucking  molecules become even Messier we still  use a Voronoi diagram in the end it's  just there's a lot more points here  we're just using the corners of the  triangles as your the source points for  growing stuff in the general case you  have to add lots of points in the middle  too but it turns out not to matter  you can throw in tons of points you can  always fold the Voronoi diagram you get  this mushroomy thing and then you can  just fold away the parts you don't need  you have to make them really small in  order to make these tabs not too big so  if the tabs are too big they collide  with each other that already happened in  the software version even in the  software version this might not work  because this tab may be huge but you can  in this case you just add a few more and  make this pleat up and down and you'll  avoid collision so that was a quick but  a little bit more clear overview of  organizer and how it works  any more questions cool that's it for  today  you

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all right welcome back to six a49 today  we are going over tree maker folding  polyhedra folding checker boards and  organizer lots of stuff and can't fit  everything in today but we'll be doing  more Tree Maker in some sense from a  more practical perspective in the very  next lecture video and next class with a  guest lecture by Jason ku so today we  have a bunch of questions  we'll start with with Tree Maker so the  first question is wow you could make all  these crazy things with the tree method  are they really uniaxial and the answer  is yes  and I think it's probably easiest to  just go through some examples these are  all from origami design secrets this  book by Robert Lang which is on the  recommended reading list and I think  there's even an electronic copy  available through MIT libraries so you  can read it online or maybe one in the  library there's two editions these  figures are from the first edition so  here is a the goal is to design scorpion  scorpion barrel egg since they're all  Robert Lang designs you come up with  this stick figure your tree tree maker  makes this crease pattern you can also  see the rivers here and the and the  disks and then you get this base and  then it folds into that scorpion and so  while the Scorpion doesn't look like it  has a single axis and a scorpion maybe  there's kind of an axis down the center  but you know it's it's not just this  axis it branches and it branches again  you can make a tree even and still  uniaxial so uniaxial refers to this axis  in the base all the flaps are attached  to that one spot so in this picture it's  vertical usually in the we draw the  figures we think of it as the horizontal  plane or horizontal line so we imagine  this is the floor and then we have you  know flaps attached to the floor  something like this  so in that sense when you flatten the  space everything lies along this axis  even though in terms of branching  structure you can hit lots of branches  so that's to clarify what uniaxial means  maybe not the best term but it's have a  few more examples here this hat includes  a color reversal some crazy stick figure  and you get your pattern here's a  somewhat simpler one that makes them  more three-dimensional horse this alamo  stallion so you can go through and for  each of these i've just reverse  engineered from this crease pattern what  the stick figure must be and you're  going to do this on pset two which will  be released in a few minutes and so this  uniaxial is to contrast from something  with the kind of most famous non  uniaxial biaxial origami bases mont  rolls dog base which looks like this so  it's very easy to turn this into all  sorts of dogs this is a wiener dog but  you have kind of one axis here and one  axis here which is different it's not  the same kind of branching tree  structure of course you can make a dog  using uniaxial methods as well but it's  kind of a fun counter example other  things you might do so that's uniaxial  next question is do people use tree  maker and or organizer in practice and  the short answer is no I would say most  origamist don't use software to design  things but they use a lot of the ideas  in their own designs so in particularly  the tree method of origami design which  tree maker is implementing almost every  advanced origami designer uses would say  not everyone but most of the complicated  most complex origami that you see uses  the tree method at least to get started  organizer is much newer so it's not used  as much most of the cool designs I'll  show you a bunch of both are by Tomohiro  tachi who invented organizer but I think  it's still evolving and hopefully these  more advanced techniques will catch on  with time if and by now you've seen lots  of examples of the tree method  I just show a few more from Robert  Lang's website this is a local delicacy  uh-huh and these these are all on on  Lange origami calm on his website and  we'll see even more examples in the next  class Jason ku is going to give an  overview various artists and in some  cases they publish cruise patterns not  always this is not quite the tree method  so this example was sort of standard  tree method you've got disks and rivers  which aren't drawn here you get a crease  pattern this could be done directly from  Tree Maker I don't know for sure whether  it was this pattern is not done with  tree maker because it's not regular  disks this is the box pleating version  of the tree method which there's another  question about so we'll talk about that  more but you see all the creases here  are horizontal vertical or diagonal 45  degrees that makes it a lot easier to  follow a lot easier to find reference  points you see it falls nicely on a grid  but still you get an arbitrary tree  structure I think I have another example  this one also well this this has 22 and  1/2 degree folds it's a little bit more  general also a very non stick figure  like models that particularly it's  obviously called the fiddler crab no  sorry I have the wrong title here I will  fix that this I forget this guy's name  but if anyone remembers he wrote about  the division between scientists and  people who know literature as two  different types of intellectuals that  know almost nothing about each other at  least in the forget early 1900s when he  was writing so and one more example this  is based on a sixty degree grid so all  the creases I think lie at 60 degrees to  the axis or horizontal there might be  some 30 degrees  so we'll talk a little bit about this  alternate version of the tree method  that doesn't just use arbitrary circles  but tries to stay on these nice grids  here's just one more example this Jason  ku is the designer of this model called  pan and he'll be giving the guest  lecture so this is just one example but  tons and tons of advanced origami  artists use a tree method of origami  design and you get some pretty cool  crease patterns from it but I think most  people do it by hand by hand usually on  a computer using some drawing programs  that can compute intersections and do  things with high accuracy and then they  draw these pictures then they fold them  and then you get your base and then you  shape it into the model so next I want  to show you some practical examples of  organizer so as a bunch on Tomohiro  tachi z-- Flickr site this is one of the  earliest this is just making a negative  curvature surface that curves in both  ways like a saddle called a hyperbolic  paraboloid we'll see more of those in  the future but this is folded from one  square paper and there's tabs on the  backside that you don't see here here's  making a kind of 3d bell curve  here's a computer mouse apparently the  the scroll wheel works fine doesn't  support USB though only bluetooth aha  we've got a nice mask these are all  you're given a 3d model and you fold  exactly that 3d model this is a tetrapod  so the here's sort of shows Toma here is  background in architecture we haven't  seen tetrapods they're used to hold back  the ocean and things from or hold the  hold from erosion things like that so  usually made out of concrete but here  you can make them out of paper here's a  flat design so here the mesh the  triangles were all in a plane and of  course you still get the tabs on the  backside but you make this very exact  leaf you can get the triangulation edges  exactly where you want them to express  you know these veins of the leaf  here's what it looks like to fold one of  these in practice is not quite a square  paper but could be a square and this is  actually folded at csail you might  recognize some of the furniture and so  while telling here is folding he uses  various devices paperclips and and clips  and so on to hold it in shape because  until it's completely collapsed it kind  of want to opens back up so it's kind of  like having a hundred hands at once but  when he's done he'll take all the all of  them off and you'll get so this I think  was about ten hours you get your bunny  uh-huh easy uh-huh or this is one we did  a little bit later just last year this  is laser cut sheet of Steel and now here  we've cut out bigger holes too so there  are too many accumulation of layers but  essentially the same design somewhat  coarser mesh of the bunny but and you  have to wear gloves  otherwise you'll cut yourself so we're  perforating the metal at decreases and  then also takes about eight hours a lot  harder to fold steel and in the end you  get your bunny  so in principle out of any sheet of  material you can make any 3d shape you  want that's the exciting thing so I  think organizers really powerful  obviously it's hard to fold you need to  be pretty advanced but there's a lot of  potential for designing very nonstick  figure like models so that was true  maker and organizer in practice so next  question is about this box pleating  which is the horizontal vertical and  45-degree creases and treemaker is there  some is there some theory for this and  indeed there is and we started working  on it me Marty and Rob Lang I think  during his first visit here which was  probably 2000  for something long time ago and the best  write-up of it currently is in origami  design secrets especially second edition  so it has a chapter on basic tree theory  a chapter on box pleating in general  this is sort of classical box pleating  but then there's a chapter on mixing the  two uniaxial box bleeding and polygon  packing these kind of go together so if  you're interested in this stuff you want  to design something the way that the  experts to check out Oregon's origami  design secrets or go to an origami  convention where Rob is talking about  this stuff we are working on this giant  manuscript it's currently called the  mathematics of origami design which is  in particular trying to prove the tree  the tree method always works and will  generalize to things like this but it's  not finished so we don't have a complete  proof yet that it all works and  everything seems fine but it's tedious  to write it all down  so still working on it but I thought I'd  show you an example of box pleating  uniaxial origami design from origami  design secrets so this is sort of  typical tree method of origami design  this is if you're not necessarily using  treemaker but you're doing it by hand  you think about okay suppose I want to  make this kind of insect with this stick  figure maybe I realize oh it's kind of  centrally symmetric so I'd like to make  the left half of the paper same as the  right half you can also express that in  the tree maker software and then you  start thinking about where these leaves  correspond to disks then you've got to  have the corresponding rivers in between  them so you might start with this kind  of layout and then you kind of you try  to blow it up until things are can't be  expanded anymore and there's lots of  touching so that would be the usual  approach then from that you could apply  the tree method get the crease pattern  with the box pleated version essentially  instead of disks you have squares and  instead of rivers you have these  orthogonal channels of constant width  and so I'll just wave my hands and say  that happens now you get these weird  things of you get these gaps you tend to  get more gaps in this way but in this  case you instead of just having a square  you can extend it out to be a rectangle  so these guys are actually going to fill  in these holes  this and then from this you can start  constructing a crease pattern so you  start with these Ridge creases and this  is something like a generalization it's  actually something like straight  skeleton which we'll be covering in a  couple of lectures for a different  purpose where it was originally  developed for origami purposes but these  are kind of these are just lots of  bisectors and you'll have to see the  general version later and then you start  putting in the hinge creases these are  yeah these green lines and they're  perpendicular to these these blue  creases are going to be the floor and so  on you fill it in eventually you get  your complete crease pattern so in a  nutshell that's how it works  you tend to get tabs that are constant  width like this house in height I guess  and so it's a general technique it's a  little bit much to explain here but read  the book if you want to see it and I say  it's probably some of the most common  there's a lot of intuitive ways to do it  and then a lot of the details are worked  out in that book but it's quite common  to design bases in this way because it's  just so much easier to fold them you  don't have to construct really weird  angles a lot cleaner but a little bit  less efficient because of the instead of  using a disk which is kind of the  minimum amount of paper you need to make  a flap you're using a square so you're  kind of wasting those little corners but  not that much more inefficient cool so  that was that method next question I  have is about the triangulation  algorithm in the which I didn't even  cover in lecture so this was you you you  know you set up you have your piece of  paper and you assign each leaf in this  tree to some point in the piece of paper  and you satisfy the active path  condition so each of these distances  measured on the paper should be greater  than or equal to the distance measured  on the tree which is the floor of the  base  great and so if you happen to have some  equalities you draw inactive paths but  what if you don't have any or you don't  have enough the tree method in its  original form only works when the active  paths decompose the piece of paper into  convex polygons this is what you'd like  to have happen each of the phases here  is a convex polygon that happens great  you can use the universal molecule and  you get your folding if it doesn't  happen you have to modify your tree a  little bit there a couple ways to do  this in the in the tree maker you can  modify these edge lengths and try to  modify them as little as possible so  that things touch but the easy way to  prove that this that something is  possible because remember if we if we  add a little bit to the base you know we  get an extra flap that doesn't really  hurt us at the end we can hide the flap  just fold it against other flaps and  just pretend it wasn't there so as long  as we can add flaps in order to make  this crease pattern or in order to make  the active paths decomposed into convex  polygons we're happy and in fact what we  prove is that you can do it you can keep  adding flaps until you get into  triangles and triangles are always  convex and so they make us happy so we  will end up adding a bunch of flaps in  our tree in order to triangulate with  active paths and then in each of these  we just fill in a rabbit-ear molecule  okay so that's the goal now let me tell  you how the triangulation actually works  originally described by Lang it's also  in the textbook for this class I'll try  to give an abbreviated version here so a  little bit technical here's the idea  so suppose alright so we've said we've  assigned some leaves and suppose we have  some region that's not convex it's not a  triangle has more than three sides now  some of these edges may come from active  paths and some of them may be from the  boundary of the paper so maybe the paper  is here so this edge is not active just  have it just the boundary of this region  because boundary the paper okay so there  are two types of edges but there should  be I guess at least one active edge and  there's at least four edges total so  what I'm going to do is look at any of  the active edges and I'm going to  imagine okay so that active edge this is  in in the piece of paper but now I'm  going to think about it in the tree so  the tree looks like something we don't  really know what any active active path  here corresponds to an active path in  the tree of exactly the same length so  maybe it's from this leaf to this leaf  so it corresponds to color this guy  corresponds to this path and that some  of those lengths should be equal to that  length it isn't obviously but there's a  scale factor in between here lambda okay  so here's what I'm going to do I'm going  to modify the tree by adding a new leaf  somewhere off of this path okay where ah  let's for convenience let's assume that  that this this path length here is one  then I'm going to measure out some  distance X here which will leave a  different distance 1 minus X here and  then I'm going to make this length  I guess L and I'm going to call this  leaf L capital L okay this is a  modification to a tree I can do it I can  do it for any value of x between 0 & 1  and I can do it for any value of L  greater or equal to 0 so what I'd like  to do is design the tree so that this  point ends up in an interesting place on  the piece of paper in fact I claim that  no matter where I draw L here capital L  as my desired place now what I'd really  like is for I should give these guys  names this is called you are say in the  notes I call it V and W so here we have  W and V what I would like is for these  two paths to also be active meaning the  lengths here match the lengths in the  tree I claim that no matter where I put  L in the plane anywhere in the plane I  can as choose X and choose L so that  these two lengths are exactly correct do  you believe me not really how many  people think this is obvious good couple  maybe let me give you a quick sketch  it's not that interesting so I just want  to do it very briefly if you get it  right if you let X be a free parameter  but fix L little L then what this  corresponds to is in fact a an ellipse  with V and W as Fossey of the ellipse  and because you have to hold the sum of  these lengths fixed if you fix if you  let X vary you know this length Plus  this length is always going to be the  same if you fix L and so this is called  the major axis the sum of these two  lengths is going to be  like 1 plus 2l and so if I have a point  basically there's some ellipse of the  appropriate size it passes through L  that will let me choose little L and  then as I vary X I walk around the  ellipse so I just choose the appropriate  value of x that gets me the desired  point on the ellipse that's it  ok so for any point at the point is the  set of all ellipses with these two fuzzy  spans the entire plane and so wherever  you want to put L you can make those two  pans active ok this is good because it  makes a little triangle but of course if  I just add an arbitrary triangle not  very interesting the good thing is I'm  free to put L wherever I want  I'd really like to put L say here and  draw active paths like that  because then I'm kind of decomposing my  polygon triangles that's not always  possible so let's just go through the  cases in all cases we're going to  simplify a polygon make it have fewer  sides and that's a next page so the  claim is well Bo there's sort of three  cases to reorganize this a little bit  one thing that would be nice is I can  place L they all draw another version of  this so I've got you V I'll draw the  active pads in red so we know UV is  active got L if I could somehow place L  so that's active with two other points  two other leaves sorry this is V and W  this is U and this is T if I can do this  and the region  it looks something like that then I'm  happy I'm going to put L there and add  in these four edges and essentially if  you look at any one of these regions  that remains there's a triangle here  that's definitely okay these regions  each of them will have fewer edges than  the original region that's pretty easy  to prove because you have because you're  connecting to these four vertices this  region won't have this vertex or this  vertex so it has one added vertex and to  remove vertices at least W and T this  one will be smaller and it's a symmetric  for all of them okay so this would be a  good case so what I'm going to do is try  to move L around I'm going to start very  close to V W and just start moving off  the edge initially nothing else will be  active because it's basically right on  top of V W but as I move around I might  get another active path okay maybe this  is case one a before we get there is one  is you get L u active suppose you just  have to get one one additional active  path the other case is nothing else  becomes active so here's V here's W  here's L here's u so this is active  active active active okay what I'm going  to do in this situation is if I want to  keep Lu active I can actually move L on  a circle as long as L stays on this  circle Lu remains active okay so just  move it along a circle now there are two  possibilities it could be becomes active  with something else then I'm done or it  could be it doesn't which means it will  hit the boundary so we'll hit a boundary  point so case one a so this is case one  you can make something active either you  when you move along the circle either  you make another thing active or you  won't you hit the boundary  case 1 B is you hit the boundary and  then I claim you're also happy so in  that situation one more time we've got L  on the boundary and we've got some path  here here here here these are all active  so here's L okay I claim again each of  these regions has fewer vertices than  before should be pretty obvious this one  is emitting W and this was you yeah  meaning W and it's emitting whatever is  down here if anything yeah I think it's  pretty obvious kind of check all the  cases is a few details here but should  be each of these regions has strictly  fewer vertices than it did before and so  you're making progress if you started  with all phases having some number of  edges each time you do one of these  steps you decrease it in the last case  case two is you can't make anything  active this can happen for example if  you're in the corner of the piece of  paper and you have an active path I  guess that's a little less exciting  maybe your piece of paper is this shape  a little bit weirder so this will work  for any convex piece of paper so it's  not quite a triangle and you're moving L  around here and you just can't get  another active path there's sort of no  no vertices around this is W this is U  so has V can you write some day  and in this case L can go anywhere here  it's free in that case I'm going to put  L on one of the vertices of the paper so  I get this active and this active  and then I've decomposed that region  into pieces by this is what we call a  diagonal we're adding a vertex to vertex  edge in this polygon and that always  decreases the number of sides on in each  of the region sub regions so if I just  keep doing this taking the any any  region of size larger than three I will  eventually reduce them all to half size  three so they'll all be triangles so  that's a triangulation algorithm in a  nutshell you can look at the textbook if  you want to see it inst in more detail  but it's pretty simple this proves that  something is possible  I don't think treemaker actually  implements this algorithm specifically  but you can kind of do it by hand so I  have a little example here this is  treemaker  I drew this really weird tree we call it  a caterpillar tree yes you would  actually use it to make caterpillars and  if you just plug this into Tree Maker  with all the lengths unit it will give  you this error message I couldn't  construct all the polygons because  things weren't tight enough and if you  look carefully it's a little hard to see  these colors but the light green edges  those are the active paths and here so  this was a triangle and I was happy if  you look at this green polygon maybe  actually I'll draw with the tablet so  this thing is inactive but that's a  region bounded by active paths and  probably the top edge is not an active  path that's just a boundary but it's not  convex and so you're unhappy and so I  just kind of eyeball this and said okay  well probably I should add another leaf  here that that would maybe add a disk  that will probably fill that in so you  don't have to be super precise here we  add you know very carefully place to  disk to make things touch but because  Tree Maker is always just trying to blow  things up and make them touch it's quite  a bit easier in practice you just say  okay I will add an extra leaf here like  that to the tree and then I hit optimize  and boom it works in this case I was  lucky in general I might get a  non-convex region I'll just guess where  to add another leaf  and it works you compare I just do it  did that leaf have to be that big or  could I get away with a smaller one but  eventually you will get a base and if  you want you could carefully monitor  this proof and actually always succeed  but in practice is usually not that hard  questions about that so that was the  triangulation method in a nutshell next  question is about the universal molecule  and we might spend some more time on  this next class but I thought we could  actually look at this example because  it's got a bunch of different Universal  molecules so I will continue to draw so  let me pick let's say this universal  molecule here it's a quadrilateral  all right so those are four active paths  in this case and that corresponds to  some tree so I don't I don't have handy  here but it's going to be a tree with  four edges so truth  so if with four leaves sorry truth for  it leaves is going to look something  like this this is a piece of the bigger  tree which you recall looks something  like this and we add it's one somewhere  but this particular quadrilateral is  doing some particular sub tree of on  four leaves and particularly their four  leaves here correspond to four leaves  here I don't know which ones I don't  really need to know tree Mack true maker  keeps track of it for me and I happen to  know that these four paths are active  meaning the lengths in the plain here  match exactly the lengths as measured  along the tree so what I do to make this  work this is basically the floor of the  molecule these those four paths will all  lie on the ground level what I'm trying  to do is slice higher and higher in the  base to see what happens when I slice  and if you're if you think about it as  you move up here  in the plane this corresponds to moving  parallel to these edges so I'm shrinking  this polygon by parallel offsets so let  me draw one parallel offset  approximately parallel okay after I  shrink a little bit might look like that  an interesting moment in this case is  here fellas carefully should go sorry  should go to here  whoo I'm going to start over this is  actually already drawn for us this guy  okay this is a parallel offset and in  this case something interesting happens  which is this path so it turns out this  middle edge in this case becomes active  so what that means is originally if you  looked at these two points these two  leaves they had the wrong length okay so  that's going to correspond to something  like these two leaves they're opposite  corners of the quad and you measure the  length along the tree you get something  and the claim was in the plane it was  too big now as you do this parallel  offset as you shrink the polygon at some  point it will be just the right length  this could happen doesn't have to happen  if it happens you have to stop because  if you kept going it would become too  short and we know every length here must  be greater than or equal to length over  here so when it becomes equal we have to  stop we have to make this a crease  making it a crease basically says ah  this is exactly the right length I have  to split here and I shot guy and I have  to sort of grow now two different flaps  from that point the crease sort of makes  it horizontal and then we end up  shrinking into different parts we shrink  in this triangle which gives us a  rabbit-ear and we shrink up there in  that triangle gives us another rabbit  ear so that it will end up giving these  angular bisectors in general as you do  the shrinking you watch where the  vertices go those are your Ridge creases  so this one went along an angular  bisector that one did as well but once  we do this split operation because of  this newly active path which should also  call a gusset in the universal molecule  then these direct these vertices  actually split and go in two directions  one for this angular bisector one for  that angular bisector  so in general for a universal molecule  there are two things that can happen you  can have you can be shrinking your  polygons and suddenly discover there's a  newly active path and then you have to  divide it into two polygons and start  shrinking those separately or vertices  can disappear so it could be you're just  shrinking shrinking shrinking merrily  along the way and then suddenly these  two vertices collide with each other so  you get to ridge creases here in this  case you just treat those two vertices  now as one so you'll start shrinking  like this and so these vertices will now  start going that way so in general as  you're shrinking these the only two  types of events that can happen either  two vertices merge or you get a new  diagonal here that becomes active and  that happens you just divide this will  work for any convex polygon and you get  your molecule it's hard to do it's hard  to draw a picture of this without using  a computer tool or really keeping track  of what you're doing because how do you  tell when something becomes active you  have to look at your tree if to measure  lengths that's tricky to do without a  computer tool and that's why I tree  maker was made but it can be done  probably even easier with physical paper  if you know exactly what lengths you're  trying to match so those quick overview  of how universal molecule works in more  detail next I want to talk about a few  different open problems so one of them  was these are called gift wrapping  problems so we have things like given a  square a unit square let's say what's  the largest regular tetrahedron you can  wrap that's still open all these  problems are still open except for the  square to cube and which talked about in  lecture and I realize equilateral  triangle to tetrahedron that's also  really easy if you can do it without any  paper wastage is clearly off  at all but it would be very cool to  studies I think is to make a good good  project or a problem for the open  problem session folding a given  rectangle will give an aspect ratio into  a cube also open it pretty much any  version you can think of is open except  the ones that we've already seen also  wanted to mention for checkerboards  fun problem kind of in this spirit is  what would be the best way to fold a 2x2  checkerboard even for a 2x2 we have no  idea how to argue any kind of lower  bound about how bad you must do that you  can't just take a square and fold a 2x2  checkerboard of the same size surely you  can't do that but we don't know how to  prove that so it would be a nice target  we obviously have kind of upper bounds  we have constructions that make 2x2  checker boards think from a 3x3 grid you  can make a 2x2 checkerboard but is that  optimal we have no idea ah so a couple  questions about the checkerboard folding  so we had this picture of how you would  take a square of paper you start with a  square paper you fold it into this shape  with these long slits and then these  guys fold over and then there are also  tabs sticking up here which are not  drawn and they fall over and give you  the color reversal question was how do  you actually build this thing this is  not looking the axial and indeed we do  not use uniaxial techniques for this we  use a separate set of gadgets which are  kind of based on pleating and this will  actually be very similar to something we  see in a couple of lectures when we do  box pleating to make cubes and stuff but  you just compose these gadgets so this  is kind of a general tab gadget you  collapse this crease pattern and do a  couple more folds and you end up with  this tab sticking out here and you can  flip it up or down and you see that it  has these pleats running off to the side  this Mountain Valley Mountain Valley and  then Valley Mountain Valley Mountain and  so those have to go all the way through  the paper and they come in to this  construction which basically lets you  make a big slit in the paper or you can  also turn a corner and so you just prove  that these gadgets compose I mean it's  pretty easy that they compose you just  have to analyze how much of the paper  you're using and turns out to be really  good  that's a short version you want to try  it out fold some gadgets fun fun project  idea from from you guys is given an  image sample at low resolution make it  black and white and then come up with  the crease pattern that would fold via  this checkerboard technique into exactly  that two color pixel pattern you could  make all sorts of useful things like  space invaders stuff like that you'd  never want to fold them I think but  maybe with low enough resolution you  could fold them but I think they'd be  cool just as crease patterns by  themselves you input a different pattern  I'd like you know web applet who changed  the pixel pattern and boom it gives you  the crease pattern I think it would be a  fun project if you're into implementing  things it's essentially the algorithm we  already have but it needs to be just  written out in detail and cut it up okay  here's what it looks like to fold an  eight by eight checkerboard with this  method these are all by Robert Lang so a  lot of precreasing along this huge grid  I think Flea 48 by 48 and then some tape  to hold things shut mostly for  photographing and you collapse now we're  down by a factor of two or so now you  can see the slits in one direction  I think this method all the particularly  being use here only has slits in one one  direction oh no here we got slits in the  vertical direction as well you can see  the tabs sticking out here they're going  to fall down there's some more of the  tabs start folding over and boom you've  got your 8 by 8 checkerboard Robert Lang  says wow that was not one of the easier  things I've done and this is folded from  a 48 by 42 checkerboard in principle  this method can go down to 36 plus  epsilon by 36 plus epsilon but it's a  little bit gets Messier crease pattern  wise and 36 would be better than the  best known 8 by board if you want  seamless notice these are seamless  squares no crease lines through them  this one's not necessarily this is not  better than known techniques but it  follows our algorithm and that's what  the crease pattern looks like  cool  next we go to organizer so as I  mentioned in lecture there's two  versions of organizer there's the  version that's implemented in the  software and there's a version that  we're proving is always correct these  are different because the version the  software doesn't always work and it  remains that way because the software  version is more practical it's and also  the the theoretical version is still a  moving target we change it every few  weeks to fix part of the proof  it's almost done but it's been almost  done for a couple of years so we're  still working on it tell me here was  just visiting a couple weeks ago as I  mentioned and we're closing in I  actually have with me the current draft  of the paper and it's not nearly as long  as this one with the tree method but  it's still growing and working out  making sure all the details check out so  that the theoretical version is kind of  complicated so I thought I'm  particularly giving this question I  talked about the software version  because it's actually a lot simpler only  catches it doesn't always work and it's  described in this paper if you want to  read it by Tomohiro about organizer 2010  and I have a few figures from that paper  well this is still from Tommy Harris  flicker so we had this example where you  wanted to fold this hyperbolic  paraboloid and here's what the crease  pattern looks like it's actually pretty  simple you've got these white polygons  which are polygons from the surface okay  that's what you need to fold and the  goal is to lay them out on the piece of  paper this square is the piece of paper  so that I can for example I need to  bring this polygon to touch this polygon  this edge has to touch this edge  wouldn't it be great if I could just  fold the bisector of those two edges and  this would come right on to here  sometimes that happens but for example  if this polygon is way down here  that won't happen it will fold over and  they won't align you have there's two  issues first you have to get the angles  to match then also there's this vertical  shifting if you place the polygons in  the plane in a good way this will work  you always get alignment  and what organizer implements is a  non-convex optimization a constraint  projection to make all of the edges work  sometimes that's possible it's actually  possible fairly often and that's when  the organizer software works if it's not  possible you're screwed this this  particular method won't work but when  it's possible things are great  so essentially you have to bring the  edges together then you also have to  bring the vertices together like these  four vertices come together to a point  and so you get two kinds of gadgets  which are the vertex tucking molecule to  bring vertices together and edge tucking  molecule to bring two edges together  edge tucking molecule is a single crease  trivial in this construction vertex  tucking molecule is complicated and in  general what happens is for these points  we construct what's called a Voronoi  diagram which is essentially for each of  these points if you grow a disc at equal  speeds around all of them and when the  discs meet you stop them so then these  two discs will meet along this  perpendicular bisector and they'll keep  growing until they kind of all die out  and you'll get this tree structure in  general those are your main creases you  follow that structure and you do some  stuff essentially what happens when you  when you fold along this Voronoi diagram  you also have to add in these creases  from the Voronoi diagram to the points  when you do that you'll get a kind of  mushroomy structure that has too much  material if you set it up right all the  angles of paper that you have are larger  than what you need in the folded state  which would look like this and so what  you do are lots of little pleats to  reduce the angle if you've got a big  angle of paper you can do a mountain in  a valley and make it a smaller angle and  that's what all these creases are doing  you've got these pleats to reduce the  angle here reduce the amount of material  here and here and to in order to match  the 3d structure so overall what the  algorithm does is first given the  surface it constructs a suitable 3d  structure of where to put  all the extra tabs then it designs  things so that the angles are bigger  here than what they need to be there  this is again a constraint projection  also constrains this these edges to meet  up perfectly and then it just applies  all these crimps which are it's easier  in a computer than to see it here to  make all the angles correct to match 3d  model yeah question right so for the  sheetmetal bunny we just cut out these  polygons because otherwise it's a mess  and you get lots of layers and the point  was to make it out of one sheet of  material but not necessarily a square  yeah so if you have really thick  material recommend cutting those out and  we were laser cutting you know to score  the lines anyway so why not cut out some  holes as well yeah so that way you just  have edge talking molecules which are  really easy Mountain Valley Mountain and  so it actually is pretty practical to  fold these things out of generalised  sheets with holes so the the non  software version of organizer works the  same way except it does the full  generality of constructing this tuck  proxy that where the tab should go and  that can get very messy in general it's  just tedious to explain there all these  crazy spherical diagrams that you saw in  lecture but it's not that exciting the  more interesting part is that the edge  tucking molecules can't just be a single  crease anymore  they have to in general kind of follow  some paths to get to where they need to  go and and but it's similar you just do  a bunch of different folds kind of you  can sweep them one way or the other in  order to navigate this edge to be where  it needs to be so again you place the  faces in the plane in the mathematical  version you can place them anywhere you  want doesn't matter and then you have to  shrink them until they're small enough  that things work then you route these  paths to get from each edge to each  corresponding edge then  the edge talking molecules are fairly  straightforward the vertex tucking  molecules become even Messier we still  use a Voronoi diagram in the end it's  just there's a lot more points here  we're just using the corners of the  triangles as your the source points for  growing stuff in the general case you  have to add lots of points in the middle  too but it turns out not to matter  you can throw in tons of points you can  always fold the Voronoi diagram you get  this mushroomy thing and then you can  just fold away the parts you don't need  you have to make them really small in  order to make these tabs not too big so  if the tabs are too big they collide  with each other that already happened in  the software version even in the  software version this might not work  because this tab may be huge but you can  in this case you just add a few more and  make this pleat up and down and you'll  avoid collision so that was a quick but  a little bit more clear overview of  organizer and how it works  any more questions cool that's it for  today  you
 

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00:07:29,810 --> 00:07:33,110

 

171
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172
00:07:36,320 --> 00:07:44,370

 

173
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174
00:07:47,100 --> 00:07:51,510

 

175
00:07:51,510 --> 00:07:55,040

 

176
00:07:55,040 --> 00:08:00,170

 

177
00:08:00,170 --> 00:08:02,790

 

178
00:08:02,790 --> 00:08:08,040

 

179
00:08:08,040 --> 00:08:11,250

 

180
00:08:11,250 --> 00:08:13,320

 

181
00:08:13,320 --> 00:08:15,210

 

182
00:08:15,210 --> 00:08:18,320

 

183
00:08:18,320 --> 00:08:20,880

 

184
00:08:20,880 --> 00:08:22,710

 

185
00:08:22,710 --> 00:08:26,430

 

186
00:08:26,430 --> 00:08:29,460

 

187
00:08:29,460 --> 00:08:33,450

 

188
00:08:33,450 --> 00:08:34,980

 

189
00:08:34,980 --> 00:08:38,370

 

190
00:08:38,370 --> 00:08:40,380

 

191
00:08:40,380 --> 00:08:42,690

 

192
00:08:42,690 --> 00:08:50,050

 

193
00:08:50,050 --> 00:08:52,990

 

194
00:08:52,990 --> 00:08:54,580

 

195
00:08:54,580 --> 00:08:57,730

 

196
00:08:57,730 --> 00:08:59,770

 

197
00:08:59,770 --> 00:09:04,420

 

198
00:09:04,420 --> 00:09:06,480

 

199
00:09:06,480 --> 00:09:11,620

 

200
00:09:11,620 --> 00:09:14,410

 

201
00:09:14,410 --> 00:09:16,060

 

202
00:09:16,060 --> 00:09:18,040

 

203
00:09:18,040 --> 00:09:21,490

 

204
00:09:21,490 --> 00:09:24,340

 

205
00:09:24,340 --> 00:09:27,550

 

206
00:09:27,550 --> 00:09:30,870

 

207
00:09:30,870 --> 00:09:37,000

 

208
00:09:37,000 --> 00:09:39,550

 

209
00:09:39,550 --> 00:09:44,200

 

210
00:09:44,200 --> 00:09:47,110

 

211
00:09:47,110 --> 00:09:49,720

 

212
00:09:49,720 --> 00:09:52,000

 

213
00:09:52,000 --> 00:09:56,470

 

214
00:09:56,470 --> 00:09:57,250

 

215
00:09:57,250 --> 00:09:59,350

 

216
00:09:59,350 --> 00:10:01,210

 

217
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218
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219
00:10:10,750 --> 00:10:13,800

 

220
00:10:13,800 --> 00:10:16,360

 

221
00:10:16,360 --> 00:10:18,160

 

222
00:10:18,160 --> 00:10:19,810

 

223
00:10:19,810 --> 00:10:21,280

 

224
00:10:21,280 --> 00:10:24,250

 

225
00:10:24,250 --> 00:10:26,710

 

226
00:10:26,710 --> 00:10:29,230

 

227
00:10:29,230 --> 00:10:34,450

 

228
00:10:34,450 --> 00:10:38,380

 

229
00:10:38,380 --> 00:10:40,960

 

230
00:10:40,960 --> 00:10:42,520

 

231
00:10:42,520 --> 00:10:45,910

 

232
00:10:45,910 --> 00:10:48,640

 

233
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234
00:10:50,650 --> 00:10:54,520

 

235
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236
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237
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238
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239
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240
00:11:06,970 --> 00:11:09,580

 

241
00:11:09,580 --> 00:11:11,920

 

242
00:11:11,920 --> 00:11:13,780

 

243
00:11:13,780 --> 00:11:15,430

 

244
00:11:15,430 --> 00:11:17,920

 

245
00:11:17,920 --> 00:11:20,800

 

246
00:11:20,800 --> 00:11:22,120

 

247
00:11:22,120 --> 00:11:25,120

 

248
00:11:25,120 --> 00:11:27,640

 

249
00:11:27,640 --> 00:11:29,140

 

250
00:11:29,140 --> 00:11:30,640

 

251
00:11:30,640 --> 00:11:34,930

 

252
00:11:34,930 --> 00:11:36,790

 

253
00:11:36,790 --> 00:11:39,400

 

254
00:11:39,400 --> 00:11:41,050

 

255
00:11:41,050 --> 00:11:44,830

 

256
00:11:44,830 --> 00:11:47,590

 

257
00:11:47,590 --> 00:11:49,630

 

258
00:11:49,630 --> 00:11:51,280

 

259
00:11:51,280 --> 00:11:53,800

 

260
00:11:53,800 --> 00:11:54,760

 

261
00:11:54,760 --> 00:11:57,040

 

262
00:11:57,040 --> 00:11:58,660

 

263
00:11:58,660 --> 00:12:01,450

 

264
00:12:01,450 --> 00:12:03,610

 

265
00:12:03,610 --> 00:12:05,620

 

266
00:12:05,620 --> 00:12:07,240

 

267
00:12:07,240 --> 00:12:08,770

 

268
00:12:08,770 --> 00:12:11,200

 

269
00:12:11,200 --> 00:12:13,870

 

270
00:12:13,870 --> 00:12:17,110

 

271
00:12:17,110 --> 00:12:18,790

 

272
00:12:18,790 --> 00:12:20,200

 

273
00:12:20,200 --> 00:12:21,760

 

274
00:12:21,760 --> 00:12:23,890

 

275
00:12:23,890 --> 00:12:25,840

 

276
00:12:25,840 --> 00:12:27,550

 

277
00:12:27,550 --> 00:12:29,080

 

278
00:12:29,080 --> 00:12:30,640

 

279
00:12:30,640 --> 00:12:32,470

 

280
00:12:32,470 --> 00:12:34,930

 

281
00:12:34,930 --> 00:12:36,460

 

282
00:12:36,460 --> 00:12:37,930

 

283
00:12:37,930 --> 00:12:40,030

 

284
00:12:40,030 --> 00:12:41,670

 

285
00:12:41,670 --> 00:12:44,770

 

286
00:12:44,770 --> 00:12:46,930

 

287
00:12:46,930 --> 00:12:48,790

 

288
00:12:48,790 --> 00:12:51,970

 

289
00:12:51,970 --> 00:12:54,970

 

290
00:12:54,970 --> 00:12:56,890

 

291
00:12:56,890 --> 00:12:59,590

 

292
00:12:59,590 --> 00:13:02,260

 

293
00:13:02,260 --> 00:13:03,970

 

294
00:13:03,970 --> 00:13:05,380

 

295
00:13:05,380 --> 00:13:07,000

 

296
00:13:07,000 --> 00:13:09,140

 

297
00:13:09,140 --> 00:13:13,190

 

298
00:13:13,190 --> 00:13:14,600

 

299
00:13:14,600 --> 00:13:19,700

 

300
00:13:19,700 --> 00:13:22,100

 

301
00:13:22,100 --> 00:13:23,540

 

302
00:13:23,540 --> 00:13:24,890

 

303
00:13:24,890 --> 00:13:26,330

 

304
00:13:26,330 --> 00:13:27,590

 

305
00:13:27,590 --> 00:13:31,370

 

306
00:13:31,370 --> 00:13:32,780

 

307
00:13:32,780 --> 00:13:35,060

 

308
00:13:35,060 --> 00:13:38,030

 

309
00:13:38,030 --> 00:13:40,150

 

310
00:13:40,150 --> 00:13:45,920

 

311
00:13:45,920 --> 00:13:48,560

 

312
00:13:48,560 --> 00:13:51,230

 

313
00:13:51,230 --> 00:13:53,180

 

314
00:13:53,180 --> 00:13:56,090

 

315
00:13:56,090 --> 00:13:57,380

 

316
00:13:57,380 --> 00:14:00,440

 

317
00:14:00,440 --> 00:14:05,320

 

318
00:14:05,320 --> 00:14:08,840

 

319
00:14:08,840 --> 00:14:10,880

 

320
00:14:10,880 --> 00:14:13,190

 

321
00:14:13,190 --> 00:14:14,900

 

322
00:14:14,900 --> 00:14:16,280

 

323
00:14:16,280 --> 00:14:17,990

 

324
00:14:17,990 --> 00:14:21,860

 

325
00:14:21,860 --> 00:14:23,660

 

326
00:14:23,660 --> 00:14:25,010

 

327
00:14:25,010 --> 00:14:26,570

 

328
00:14:26,570 --> 00:14:29,510

 

329
00:14:29,510 --> 00:14:32,960

 

330
00:14:32,960 --> 00:14:34,250

 

331
00:14:34,250 --> 00:14:35,840

 

332
00:14:35,840 --> 00:14:37,940

 

333
00:14:37,940 --> 00:14:39,740

 

334
00:14:39,740 --> 00:14:43,610

 

335
00:14:43,610 --> 00:14:47,360

 

336
00:14:47,360 --> 00:14:48,650

 

337
00:14:48,650 --> 00:14:51,650

 

338
00:14:51,650 --> 00:14:55,400

 

339
00:14:55,400 --> 00:14:57,230

 

340
00:14:57,230 --> 00:15:01,010

 

341
00:15:01,010 --> 00:15:02,900

 

342
00:15:02,900 --> 00:15:05,690

 

343
00:15:05,690 --> 00:15:07,940

 

344
00:15:07,940 --> 00:15:10,460

 

345
00:15:10,460 --> 00:15:12,770

 

346
00:15:12,770 --> 00:15:15,350

 

347
00:15:15,350 --> 00:15:15,360

 

348
00:15:15,360 --> 00:15:17,660


349
00:15:17,660 --> 00:15:20,790

 

350
00:15:20,790 --> 00:15:24,420

 

351
00:15:24,420 --> 00:15:26,430

 

352
00:15:26,430 --> 00:15:29,760

 

353
00:15:29,760 --> 00:15:33,780

 

354
00:15:33,780 --> 00:15:36,360

 

355
00:15:36,360 --> 00:15:39,210

 

356
00:15:39,210 --> 00:15:42,000

 

357
00:15:42,000 --> 00:15:44,250

 

358
00:15:44,250 --> 00:15:47,070

 

359
00:15:47,070 --> 00:15:48,930

 

360
00:15:48,930 --> 00:15:50,790

 

361
00:15:50,790 --> 00:15:53,070

 

362
00:15:53,070 --> 00:15:55,530

 

363
00:15:55,530 --> 00:15:57,060

 

364
00:15:57,060 --> 00:15:59,280

 

365
00:15:59,280 --> 00:16:01,740

 

366
00:16:01,740 --> 00:16:03,360

 

367
00:16:03,360 --> 00:16:05,640

 

368
00:16:05,640 --> 00:16:08,250

 

369
00:16:08,250 --> 00:16:10,110

 

370
00:16:10,110 --> 00:16:12,780

 

371
00:16:12,780 --> 00:16:14,400

 

372
00:16:14,400 --> 00:16:16,800

 

373
00:16:16,800 --> 00:16:19,620

 

374
00:16:19,620 --> 00:16:22,410

 

375
00:16:22,410 --> 00:16:24,270

 

376
00:16:24,270 --> 00:16:26,700

 

377
00:16:26,700 --> 00:16:28,710

 

378
00:16:28,710 --> 00:16:31,380

 

379
00:16:31,380 --> 00:16:33,300

 

380
00:16:33,300 --> 00:16:37,530

 

381
00:16:37,530 --> 00:16:40,050

 

382
00:16:40,050 --> 00:16:42,720

 

383
00:16:42,720 --> 00:16:44,430

 

384
00:16:44,430 --> 00:16:47,330

 

385
00:16:47,330 --> 00:16:50,250

 

386
00:16:50,250 --> 00:16:56,560

 

387
00:16:56,560 --> 00:16:58,850

 

388
00:16:58,850 --> 00:17:03,980

 

389
00:17:03,980 --> 00:17:06,560

 

390
00:17:06,560 --> 00:17:09,470

 

391
00:17:09,470 --> 00:17:16,430

 

392
00:17:16,430 --> 00:17:18,710

 

393
00:17:18,710 --> 00:17:22,670

 

394
00:17:22,670 --> 00:17:25,550

 

395
00:17:25,550 --> 00:17:29,060

 

396
00:17:29,060 --> 00:17:31,250

 

397
00:17:31,250 --> 00:17:32,930

 

398
00:17:32,930 --> 00:17:37,550

 

399
00:17:37,550 --> 00:17:39,350

 

400
00:17:39,350 --> 00:17:41,330

 

401
00:17:41,330 --> 00:17:43,670

 

402
00:17:43,670 --> 00:17:46,870

 

403
00:17:46,870 --> 00:17:50,810

 

404
00:17:50,810 --> 00:17:52,580

 

405
00:17:52,580 --> 00:17:56,500

 

406
00:17:56,500 --> 00:18:00,080

 

407
00:18:00,080 --> 00:18:02,150

 

408
00:18:02,150 --> 00:18:04,040

 

409
00:18:04,040 --> 00:18:05,750

 

410
00:18:05,750 --> 00:18:10,490

 

411
00:18:10,490 --> 00:18:14,600

 

412
00:18:14,600 --> 00:18:16,340

 

413
00:18:16,340 --> 00:18:18,620

 

414
00:18:18,620 --> 00:18:24,010

 

415
00:18:24,010 --> 00:18:27,890

 

416
00:18:27,890 --> 00:18:29,600

 

417
00:18:29,600 --> 00:18:31,160

 

418
00:18:31,160 --> 00:18:37,850

 

419
00:18:37,850 --> 00:18:39,470

 

420
00:18:39,470 --> 00:18:44,090

 

421
00:18:44,090 --> 00:18:49,330

 

422
00:18:49,330 --> 00:18:51,650

 

423
00:18:51,650 --> 00:18:57,570

 

424
00:18:57,570 --> 00:19:01,530

 

425
00:19:01,530 --> 00:19:05,160

 

426
00:19:05,160 --> 00:19:08,370

 

427
00:19:08,370 --> 00:19:11,340

 

428
00:19:11,340 --> 00:19:15,150

 

429
00:19:15,150 --> 00:19:19,740

 

430
00:19:19,740 --> 00:19:21,870

 

431
00:19:21,870 --> 00:19:23,490

 

432
00:19:23,490 --> 00:19:24,900

 

433
00:19:24,900 --> 00:19:28,320

 

434
00:19:28,320 --> 00:19:30,510

 

435
00:19:30,510 --> 00:19:32,580

 

436
00:19:32,580 --> 00:19:35,490

 

437
00:19:35,490 --> 00:19:39,420

 

438
00:19:39,420 --> 00:19:42,390

 

439
00:19:42,390 --> 00:19:45,180

 

440
00:19:45,180 --> 00:19:48,060

 

441
00:19:48,060 --> 00:19:52,850

 

442
00:19:52,850 --> 00:19:58,830

 

443
00:19:58,830 --> 00:20:03,060

 

444
00:20:03,060 --> 00:20:05,400

 

445
00:20:05,400 --> 00:20:08,190

 

446
00:20:08,190 --> 00:20:11,420

 

447
00:20:11,420 --> 00:20:14,760

 

448
00:20:14,760 --> 00:20:18,180

 

449
00:20:18,180 --> 00:20:23,580

 

450
00:20:23,580 --> 00:20:27,300

 

451
00:20:27,300 --> 00:20:30,480

 

452
00:20:30,480 --> 00:20:32,880

 

453
00:20:32,880 --> 00:20:35,400

 

454
00:20:35,400 --> 00:20:39,060

 

455
00:20:39,060 --> 00:20:43,740

 

456
00:20:43,740 --> 00:20:50,240

 

457
00:20:50,240 --> 00:20:54,390

 

458
00:20:54,390 --> 00:20:57,090

 

459
00:20:57,090 --> 00:21:00,630

 

460
00:21:00,630 --> 00:21:02,970

 

461
00:21:02,970 --> 00:21:04,470

 

462
00:21:04,470 --> 00:21:09,330

 

463
00:21:09,330 --> 00:21:10,890

 

464
00:21:10,890 --> 00:21:11,790

 

465
00:21:11,790 --> 00:21:16,290

 

466
00:21:16,290 --> 00:21:18,480

 

467
00:21:18,480 --> 00:21:20,010

 

468
00:21:20,010 --> 00:21:21,860

 

469
00:21:21,860 --> 00:21:25,260

 

470
00:21:25,260 --> 00:21:26,670

 

471
00:21:26,670 --> 00:21:28,470

 

472
00:21:28,470 --> 00:21:29,970

 

473
00:21:29,970 --> 00:21:32,910

 

474
00:21:32,910 --> 00:21:34,290

 

475
00:21:34,290 --> 00:21:36,390

 

476
00:21:36,390 --> 00:21:38,310

 

477
00:21:38,310 --> 00:21:40,200

 

478
00:21:40,200 --> 00:21:42,450

 

479
00:21:42,450 --> 00:21:44,280

 

480
00:21:44,280 --> 00:21:46,020

 

481
00:21:46,020 --> 00:21:47,730

 

482
00:21:47,730 --> 00:21:50,060

 

483
00:21:50,060 --> 00:21:52,470

 

484
00:21:52,470 --> 00:21:54,600

 

485
00:21:54,600 --> 00:21:56,520

 

486
00:21:56,520 --> 00:21:59,070

 

487
00:21:59,070 --> 00:22:01,200

 

488
00:22:01,200 --> 00:22:03,390

 

489
00:22:03,390 --> 00:22:09,870

 

490
00:22:09,870 --> 00:22:22,830

 

491
00:22:22,830 --> 00:22:38,540

 

492
00:22:38,540 --> 00:22:40,890

 

493
00:22:40,890 --> 00:22:43,920

 

494
00:22:43,920 --> 00:22:49,590

 

495
00:22:49,590 --> 00:22:53,190

 

496
00:22:53,190 --> 00:22:56,580

 

497
00:22:56,580 --> 00:22:59,030

 

498
00:22:59,030 --> 00:23:04,460

 

499
00:23:04,460 --> 00:23:10,950

 

500
00:23:10,950 --> 00:23:14,830

 

501
00:23:14,830 --> 00:23:17,350

 

502
00:23:17,350 --> 00:23:21,730

 

503
00:23:21,730 --> 00:23:25,390

 

504
00:23:25,390 --> 00:23:26,980

 

505
00:23:26,980 --> 00:23:29,800

 

506
00:23:29,800 --> 00:23:31,600

 

507
00:23:31,600 --> 00:23:33,550

 

508
00:23:33,550 --> 00:23:35,470

 

509
00:23:35,470 --> 00:23:37,720

 

510
00:23:37,720 --> 00:23:39,910

 

511
00:23:39,910 --> 00:23:42,760

 

512
00:23:42,760 --> 00:23:45,640

 

513
00:23:45,640 --> 00:23:48,040

 

514
00:23:48,040 --> 00:23:49,810

 

515
00:23:49,810 --> 00:23:52,030

 

516
00:23:52,030 --> 00:23:54,370

 

517
00:23:54,370 --> 00:23:56,350

 

518
00:23:56,350 --> 00:23:59,080

 

519
00:23:59,080 --> 00:24:01,990

 

520
00:24:01,990 --> 00:24:03,730

 

521
00:24:03,730 --> 00:24:07,150

 

522
00:24:07,150 --> 00:24:12,370

 

523
00:24:12,370 --> 00:24:16,600

 

524
00:24:16,600 --> 00:24:21,220

 

525
00:24:21,220 --> 00:24:24,130

 

526
00:24:24,130 --> 00:24:28,090

 

527
00:24:28,090 --> 00:24:32,130

 

528
00:24:32,130 --> 00:24:36,910

 

529
00:24:36,910 --> 00:24:40,600

 

530
00:24:40,600 --> 00:24:43,270

 

531
00:24:43,270 --> 00:24:47,140

 

532
00:24:47,140 --> 00:24:52,900

 

533
00:24:52,900 --> 00:24:58,750

 

534
00:24:58,750 --> 00:25:00,730

 

535
00:25:00,730 --> 00:25:02,980

 

536
00:25:02,980 --> 00:25:07,000

 

537
00:25:07,000 --> 00:25:08,980

 

538
00:25:08,980 --> 00:25:11,620

 

539
00:25:11,620 --> 00:25:15,160

 

540
00:25:15,160 --> 00:25:17,170

 

541
00:25:17,170 --> 00:25:18,610

 

542
00:25:18,610 --> 00:25:20,950

 

543
00:25:20,950 --> 00:25:24,230

 

544
00:25:24,230 --> 00:25:30,050

 

545
00:25:30,050 --> 00:25:35,010

 

546
00:25:35,010 --> 00:25:43,830

 

547
00:25:43,830 --> 00:25:48,870

 

548
00:25:48,870 --> 00:25:53,270

 

549
00:25:53,270 --> 00:25:58,290

 

550
00:25:58,290 --> 00:26:00,510

 

551
00:26:00,510 --> 00:26:05,700

 

552
00:26:05,700 --> 00:26:17,420

 

553
00:26:17,420 --> 00:26:19,890

 

554
00:26:19,890 --> 00:26:22,980

 

555
00:26:22,980 --> 00:26:24,390

 

556
00:26:24,390 --> 00:26:27,060

 

557
00:26:27,060 --> 00:26:28,950

 

558
00:26:28,950 --> 00:26:32,250

 

559
00:26:32,250 --> 00:26:33,990

 

560
00:26:33,990 --> 00:26:37,050

 

561
00:26:37,050 --> 00:26:39,030

 

562
00:26:39,030 --> 00:26:42,090

 

563
00:26:42,090 --> 00:26:48,090

 

564
00:26:48,090 --> 00:26:55,860

 

565
00:26:55,860 --> 00:26:57,120

 

566
00:26:57,120 --> 00:27:03,380

 

567
00:27:03,380 --> 00:27:05,460

 

568
00:27:05,460 --> 00:27:07,170

 

569
00:27:07,170 --> 00:27:09,630

 

570
00:27:09,630 --> 00:27:12,330

 

571
00:27:12,330 --> 00:27:14,610

 

572
00:27:14,610 --> 00:27:15,990

 

573
00:27:15,990 --> 00:27:17,190

 

574
00:27:17,190 --> 00:27:21,110

 

575
00:27:21,110 --> 00:27:24,660

 

576
00:27:24,660 --> 00:27:27,270

 

577
00:27:27,270 --> 00:27:29,640

 

578
00:27:29,640 --> 00:27:33,360

 

579
00:27:33,360 --> 00:27:35,670

 

580
00:27:35,670 --> 00:27:37,590

 

581
00:27:37,590 --> 00:27:40,890

 

582
00:27:40,890 --> 00:27:42,420

 

583
00:27:42,420 --> 00:27:44,820

 

584
00:27:44,820 --> 00:27:48,240

 

585
00:27:48,240 --> 00:27:50,310

 

586
00:27:50,310 --> 00:27:52,830

 

587
00:27:52,830 --> 00:27:54,960

 

588
00:27:54,960 --> 00:27:57,170

 

589
00:27:57,170 --> 00:28:00,570

 

590
00:28:00,570 --> 00:28:01,950

 

591
00:28:01,950 --> 00:28:05,550

 

592
00:28:05,550 --> 00:28:07,350

 

593
00:28:07,350 --> 00:28:11,100

 

594
00:28:11,100 --> 00:28:12,780

 

595
00:28:12,780 --> 00:28:14,430

 

596
00:28:14,430 --> 00:28:16,290

 

597
00:28:16,290 --> 00:28:18,270

 

598
00:28:18,270 --> 00:28:19,710

 

599
00:28:19,710 --> 00:28:19,720

 

600
00:28:19,720 --> 00:28:20,460


601
00:28:20,460 --> 00:28:22,830

 

602
00:28:22,830 --> 00:28:24,930

 

603
00:28:24,930 --> 00:28:27,710

 

604
00:28:27,710 --> 00:28:30,810

 

605
00:28:30,810 --> 00:28:33,540

 

606
00:28:33,540 --> 00:28:34,920

 

607
00:28:34,920 --> 00:28:36,180

 

608
00:28:36,180 --> 00:28:38,340

 

609
00:28:38,340 --> 00:28:39,810

 

610
00:28:39,810 --> 00:28:42,030

 

611
00:28:42,030 --> 00:28:45,660

 

612
00:28:45,660 --> 00:28:48,030

 

613
00:28:48,030 --> 00:28:50,910

 

614
00:28:50,910 --> 00:28:59,480

 

615
00:28:59,480 --> 00:29:05,580

 

616
00:29:05,580 --> 00:29:07,470

 

617
00:29:07,470 --> 00:29:09,240

 

618
00:29:09,240 --> 00:29:10,860

 

619
00:29:10,860 --> 00:29:14,130

 

620
00:29:14,130 --> 00:29:15,810

 

621
00:29:15,810 --> 00:29:18,210

 

622
00:29:18,210 --> 00:29:20,220

 

623
00:29:20,220 --> 00:29:22,350

 

624
00:29:22,350 --> 00:29:23,790

 

625
00:29:23,790 --> 00:29:25,230

 

626
00:29:25,230 --> 00:29:27,210

 

627
00:29:27,210 --> 00:29:28,560

 

628
00:29:28,560 --> 00:29:31,020

 

629
00:29:31,020 --> 00:29:32,970

 

630
00:29:32,970 --> 00:29:38,220

 

631
00:29:38,220 --> 00:29:40,740

 

632
00:29:40,740 --> 00:29:44,130

 

633
00:29:44,130 --> 00:29:45,420

 

634
00:29:45,420 --> 00:29:47,760

 

635
00:29:47,760 --> 00:29:49,110

 

636
00:29:49,110 --> 00:29:51,870

 

637
00:29:51,870 --> 00:29:53,580

 

638
00:29:53,580 --> 00:29:55,080

 

639
00:29:55,080 --> 00:29:57,780

 

640
00:29:57,780 --> 00:29:59,130

 

641
00:29:59,130 --> 00:30:01,560

 

642
00:30:01,560 --> 00:30:05,660

 

643
00:30:05,660 --> 00:30:10,080

 

644
00:30:10,080 --> 00:30:14,130

 

645
00:30:14,130 --> 00:30:16,500

 

646
00:30:16,500 --> 00:30:18,360

 

647
00:30:18,360 --> 00:30:21,000

 

648
00:30:21,000 --> 00:30:22,590

 

649
00:30:22,590 --> 00:30:24,900

 

650
00:30:24,900 --> 00:30:35,850

 

651
00:30:35,850 --> 00:30:39,660

 

652
00:30:39,660 --> 00:30:49,350

 

653
00:30:49,350 --> 00:30:53,260

 

654
00:30:53,260 --> 00:30:56,710

 

655
00:30:56,710 --> 00:31:00,880

 

656
00:31:00,880 --> 00:31:02,470

 

657
00:31:02,470 --> 00:31:07,270

 

658
00:31:07,270 --> 00:31:09,460

 

659
00:31:09,460 --> 00:31:10,870

 

660
00:31:10,870 --> 00:31:13,210

 

661
00:31:13,210 --> 00:31:14,770

 

662
00:31:14,770 --> 00:31:19,890

 

663
00:31:19,890 --> 00:31:22,960

 

664
00:31:22,960 --> 00:31:26,230

 

665
00:31:26,230 --> 00:31:28,300

 

666
00:31:28,300 --> 00:31:31,150

 

667
00:31:31,150 --> 00:31:33,490

 

668
00:31:33,490 --> 00:31:35,620

 

669
00:31:35,620 --> 00:31:39,580

 

670
00:31:39,580 --> 00:31:41,170

 

671
00:31:41,170 --> 00:31:42,970

 

672
00:31:42,970 --> 00:31:45,940

 

673
00:31:45,940 --> 00:31:50,500

 

674
00:31:50,500 --> 00:31:53,020

 

675
00:31:53,020 --> 00:31:56,170

 

676
00:31:56,170 --> 00:31:59,860

 

677
00:31:59,860 --> 00:32:02,400

 

678
00:32:02,400 --> 00:32:05,530

 

679
00:32:05,530 --> 00:32:07,390

 

680
00:32:07,390 --> 00:32:09,160

 

681
00:32:09,160 --> 00:32:12,810

 

682
00:32:12,810 --> 00:32:16,500

 

683
00:32:16,500 --> 00:32:21,790

 

684
00:32:21,790 --> 00:32:28,220

 

685
00:32:28,220 --> 00:32:34,020

 

686
00:32:34,020 --> 00:32:35,990

 

687
00:32:35,990 --> 00:32:38,820

 

688
00:32:38,820 --> 00:32:46,020

 

689
00:32:46,020 --> 00:32:49,850

 

690
00:32:49,850 --> 00:33:00,750

 

691
00:33:00,750 --> 00:33:07,549

 

692
00:33:07,549 --> 00:33:10,500

 

693
00:33:10,500 --> 00:33:12,750

 

694
00:33:12,750 --> 00:33:17,250

 

695
00:33:17,250 --> 00:33:20,629

 

696
00:33:20,629 --> 00:33:22,980

 

697
00:33:22,980 --> 00:33:24,419

 

698
00:33:24,419 --> 00:33:30,180

 

699
00:33:30,180 --> 00:33:31,379

 

700
00:33:31,379 --> 00:33:32,789

 

701
00:33:32,789 --> 00:33:35,159

 

702
00:33:35,159 --> 00:33:38,190

 

703
00:33:38,190 --> 00:33:40,289

 

704
00:33:40,289 --> 00:33:43,289

 

705
00:33:43,289 --> 00:33:45,779

 

706
00:33:45,779 --> 00:33:47,930

 

707
00:33:47,930 --> 00:33:50,250

 

708
00:33:50,250 --> 00:33:53,100

 

709
00:33:53,100 --> 00:33:54,539

 

710
00:33:54,539 --> 00:33:56,940

 

711
00:33:56,940 --> 00:33:59,279

 

712
00:33:59,279 --> 00:34:03,930

 

713
00:34:03,930 --> 00:34:05,580

 

714
00:34:05,580 --> 00:34:07,830

 

715
00:34:07,830 --> 00:34:10,409

 

716
00:34:10,409 --> 00:34:16,320

 

717
00:34:16,320 --> 00:34:20,270

 

718
00:34:20,270 --> 00:34:22,950

 

719
00:34:22,950 --> 00:34:25,290

 

720
00:34:25,290 --> 00:34:27,300

 

721
00:34:27,300 --> 00:34:29,190

 

722
00:34:29,190 --> 00:34:31,079

 

723
00:34:31,079 --> 00:34:32,790

 

724
00:34:32,790 --> 00:34:34,440

 

725
00:34:34,440 --> 00:34:36,359

 

726
00:34:36,359 --> 00:34:37,950

 

727
00:34:37,950 --> 00:34:40,230

 

728
00:34:40,230 --> 00:34:41,820

 

729
00:34:41,820 --> 00:34:44,579

 

730
00:34:44,579 --> 00:34:46,530

 

731
00:34:46,530 --> 00:34:49,230

 

732
00:34:49,230 --> 00:34:52,169

 

733
00:34:52,169 --> 00:34:53,879

 

734
00:34:53,879 --> 00:34:55,290

 

735
00:34:55,290 --> 00:34:57,270

 

736
00:34:57,270 --> 00:34:59,170

 

737
00:34:59,170 --> 00:35:03,260

 

738
00:35:03,260 --> 00:35:05,020

 

739
00:35:05,020 --> 00:35:19,069

 

740
00:35:19,069 --> 00:35:21,260

 

741
00:35:21,260 --> 00:35:22,940

 

742
00:35:22,940 --> 00:35:24,920

 

743
00:35:24,920 --> 00:35:31,940

 

744
00:35:31,940 --> 00:35:34,190

 

745
00:35:34,190 --> 00:35:35,780

 

746
00:35:35,780 --> 00:35:37,819

 

747
00:35:37,819 --> 00:35:41,180

 

748
00:35:41,180 --> 00:35:43,250

 

749
00:35:43,250 --> 00:35:44,780

 

750
00:35:44,780 --> 00:35:46,490

 

751
00:35:46,490 --> 00:35:49,940

 

752
00:35:49,940 --> 00:35:53,329

 

753
00:35:53,329 --> 00:35:54,349

 

754
00:35:54,349 --> 00:35:55,790

 

755
00:35:55,790 --> 00:35:57,790

 

756
00:35:57,790 --> 00:36:03,530

 

757
00:36:03,530 --> 00:36:05,809

 

758
00:36:05,809 --> 00:36:08,540

 

759
00:36:08,540 --> 00:36:11,990

 

760
00:36:11,990 --> 00:36:14,059

 

761
00:36:14,059 --> 00:36:15,770

 

762
00:36:15,770 --> 00:36:17,329

 

763
00:36:17,329 --> 00:36:19,460

 

764
00:36:19,460 --> 00:36:20,839

 

765
00:36:20,839 --> 00:36:22,490

 

766
00:36:22,490 --> 00:36:23,870

 

767
00:36:23,870 --> 00:36:26,319

 

768
00:36:26,319 --> 00:36:28,730

 

769
00:36:28,730 --> 00:36:30,109

 

770
00:36:30,109 --> 00:36:34,520

 

771
00:36:34,520 --> 00:36:37,309

 

772
00:36:37,309 --> 00:36:43,760

 

773
00:36:43,760 --> 00:36:47,380

 

774
00:36:47,380 --> 00:36:50,300

 

775
00:36:50,300 --> 00:36:53,420

 

776
00:36:53,420 --> 00:36:55,970

 

777
00:36:55,970 --> 00:36:58,309

 

778
00:36:58,309 --> 00:37:00,260

 

779
00:37:00,260 --> 00:37:01,550

 

780
00:37:01,550 --> 00:37:04,430

 

781
00:37:04,430 --> 00:37:06,440

 

782
00:37:06,440 --> 00:37:08,540

 

783
00:37:08,540 --> 00:37:11,059

 

784
00:37:11,059 --> 00:37:12,650

 

785
00:37:12,650 --> 00:37:14,000

 

786
00:37:14,000 --> 00:37:16,130

 

787
00:37:16,130 --> 00:37:17,750

 

788
00:37:17,750 --> 00:37:21,160

 

789
00:37:21,160 --> 00:37:23,390

 

790
00:37:23,390 --> 00:37:26,450

 

791
00:37:26,450 --> 00:37:28,100

 

792
00:37:28,100 --> 00:37:32,390

 

793
00:37:32,390 --> 00:37:34,360

 

794
00:37:34,360 --> 00:37:37,190

 

795
00:37:37,190 --> 00:37:39,620

 

796
00:37:39,620 --> 00:37:42,710

 

797
00:37:42,710 --> 00:37:44,450

 

798
00:37:44,450 --> 00:37:47,750

 

799
00:37:47,750 --> 00:37:50,360

 

800
00:37:50,360 --> 00:37:53,030

 

801
00:37:53,030 --> 00:37:55,100

 

802
00:37:55,100 --> 00:37:57,110

 

803
00:37:57,110 --> 00:37:58,670

 

804
00:37:58,670 --> 00:38:00,140

 

805
00:38:00,140 --> 00:38:02,600

 

806
00:38:02,600 --> 00:38:06,020

 

807
00:38:06,020 --> 00:38:11,330

 

808
00:38:11,330 --> 00:38:13,160

 

809
00:38:13,160 --> 00:38:14,690

 

810
00:38:14,690 --> 00:38:17,030

 

811
00:38:17,030 --> 00:38:18,650

 

812
00:38:18,650 --> 00:38:21,410

 

813
00:38:21,410 --> 00:38:22,940

 

814
00:38:22,940 --> 00:38:24,590

 

815
00:38:24,590 --> 00:38:26,540

 

816
00:38:26,540 --> 00:38:28,850

 

817
00:38:28,850 --> 00:38:30,110

 

818
00:38:30,110 --> 00:38:31,730

 

819
00:38:31,730 --> 00:38:34,220

 

820
00:38:34,220 --> 00:38:36,290

 

821
00:38:36,290 --> 00:38:38,180

 

822
00:38:38,180 --> 00:38:39,860

 

823
00:38:39,860 --> 00:38:41,360

 

824
00:38:41,360 --> 00:38:45,500

 

825
00:38:45,500 --> 00:38:48,200

 

826
00:38:48,200 --> 00:38:50,360

 

827
00:38:50,360 --> 00:38:52,760

 

828
00:38:52,760 --> 00:38:54,230

 

829
00:38:54,230 --> 00:38:56,000

 

830
00:38:56,000 --> 00:39:00,230

 

831
00:39:00,230 --> 00:39:02,330

 

832
00:39:02,330 --> 00:39:03,890

 

833
00:39:03,890 --> 00:39:06,500

 

834
00:39:06,500 --> 00:39:08,330

 

835
00:39:08,330 --> 00:39:12,230

 

836
00:39:12,230 --> 00:39:14,330

 

837
00:39:14,330 --> 00:39:17,000

 

838
00:39:17,000 --> 00:39:19,460

 

839
00:39:19,460 --> 00:39:21,320

 

840
00:39:21,320 --> 00:39:22,700

 

841
00:39:22,700 --> 00:39:24,410

 

842
00:39:24,410 --> 00:39:26,390

 

843
00:39:26,390 --> 00:39:26,400

 

844
00:39:26,400 --> 00:39:27,020


845
00:39:27,020 --> 00:39:29,120

 

846
00:39:29,120 --> 00:39:33,710

 

847
00:39:33,710 --> 00:39:38,150

 

848
00:39:38,150 --> 00:39:40,340

 

849
00:39:40,340 --> 00:39:42,500

 

850
00:39:42,500 --> 00:39:44,540

 

851
00:39:44,540 --> 00:39:47,540

 

852
00:39:47,540 --> 00:39:50,480

 

853
00:39:50,480 --> 00:39:52,580

 

854
00:39:52,580 --> 00:39:55,700

 

855
00:39:55,700 --> 00:39:57,350

 

856
00:39:57,350 --> 00:39:58,580

 

857
00:39:58,580 --> 00:39:59,990

 

858
00:39:59,990 --> 00:40:01,580

 

859
00:40:01,580 --> 00:40:03,560

 

860
00:40:03,560 --> 00:40:05,960

 

861
00:40:05,960 --> 00:40:07,400

 

862
00:40:07,400 --> 00:40:09,230

 

863
00:40:09,230 --> 00:40:10,670

 

864
00:40:10,670 --> 00:40:12,020

 

865
00:40:12,020 --> 00:40:14,270

 

866
00:40:14,270 --> 00:40:20,450

 

867
00:40:20,450 --> 00:40:22,250

 

868
00:40:22,250 --> 00:40:23,390

 

869
00:40:23,390 --> 00:40:27,470

 

870
00:40:27,470 --> 00:40:29,990

 

871
00:40:29,990 --> 00:40:33,620

 

872
00:40:33,620 --> 00:40:34,850

 

873
00:40:34,850 --> 00:40:38,960

 

874
00:40:38,960 --> 00:40:40,940

 

875
00:40:40,940 --> 00:40:44,690

 

876
00:40:44,690 --> 00:40:47,030

 

877
00:40:47,030 --> 00:40:48,950

 

878
00:40:48,950 --> 00:40:50,720

 

879
00:40:50,720 --> 00:40:52,520

 

880
00:40:52,520 --> 00:40:54,680

 

881
00:40:54,680 --> 00:40:56,990

 

882
00:40:56,990 --> 00:41:01,010

 

883
00:41:01,010 --> 00:41:04,580

 

884
00:41:04,580 --> 00:41:06,380

 

885
00:41:06,380 --> 00:41:10,040

 

886
00:41:10,040 --> 00:41:14,060

 

887
00:41:14,060 --> 00:41:16,070

 

888
00:41:16,070 --> 00:41:18,050

 

889
00:41:18,050 --> 00:41:19,610

 

890
00:41:19,610 --> 00:41:22,970

 

891
00:41:22,970 --> 00:41:25,610

 

892
00:41:25,610 --> 00:41:27,230

 

893
00:41:27,230 --> 00:41:29,830

 

894
00:41:29,830 --> 00:41:32,210

 

895
00:41:32,210 --> 00:41:34,160

 

896
00:41:34,160 --> 00:41:36,080

 

897
00:41:36,080 --> 00:41:38,740

 

898
00:41:38,740 --> 00:41:38,750

 

899
00:41:38,750 --> 00:41:40,579


900
00:41:40,579 --> 00:41:44,480

 

901
00:41:44,480 --> 00:41:45,890

 

902
00:41:45,890 --> 00:41:48,410

 

903
00:41:48,410 --> 00:41:49,370

 

904
00:41:49,370 --> 00:41:51,170

 

905
00:41:51,170 --> 00:41:52,849

 

906
00:41:52,849 --> 00:41:54,200

 

907
00:41:54,200 --> 00:41:57,440

 

908
00:41:57,440 --> 00:41:59,390

 

909
00:41:59,390 --> 00:42:03,859

 

910
00:42:03,859 --> 00:42:05,839

 

911
00:42:05,839 --> 00:42:07,910

 

912
00:42:07,910 --> 00:42:10,670

 

913
00:42:10,670 --> 00:42:12,349

 

914
00:42:12,349 --> 00:42:14,240

 

915
00:42:14,240 --> 00:42:15,319

 

916
00:42:15,319 --> 00:42:16,760

 

917
00:42:16,760 --> 00:42:19,609

 

918
00:42:19,609 --> 00:42:22,220

 

919
00:42:22,220 --> 00:42:26,240

 

920
00:42:26,240 --> 00:42:29,240

 

921
00:42:29,240 --> 00:42:30,859

 

922
00:42:30,859 --> 00:42:34,250

 

923
00:42:34,250 --> 00:42:36,349

 

924
00:42:36,349 --> 00:42:37,640

 

925
00:42:37,640 --> 00:42:38,870

 

926
00:42:38,870 --> 00:42:40,280

 

927
00:42:40,280 --> 00:42:42,289

 

928
00:42:42,289 --> 00:42:44,210

 

929
00:42:44,210 --> 00:42:45,710

 

930
00:42:45,710 --> 00:42:50,569

 

931
00:42:50,569 --> 00:42:53,450

 

932
00:42:53,450 --> 00:42:55,339

 

933
00:42:55,339 --> 00:42:57,680

 

934
00:42:57,680 --> 00:42:58,940

 

935
00:42:58,940 --> 00:43:00,710

 

936
00:43:00,710 --> 00:43:02,180

 

937
00:43:02,180 --> 00:43:04,280

 

938
00:43:04,280 --> 00:43:07,730

 

939
00:43:07,730 --> 00:43:09,859

 

940
00:43:09,859 --> 00:43:11,930

 

941
00:43:11,930 --> 00:43:14,319

 

942
00:43:14,319 --> 00:43:18,260

 

943
00:43:18,260 --> 00:43:20,270

 

944
00:43:20,270 --> 00:43:21,650

 

945
00:43:21,650 --> 00:43:23,510

 

946
00:43:23,510 --> 00:43:25,520

 

947
00:43:25,520 --> 00:43:27,700

 

948
00:43:27,700 --> 00:43:29,690

 

949
00:43:29,690 --> 00:43:31,549

 

950
00:43:31,549 --> 00:43:33,980

 

951
00:43:33,980 --> 00:43:35,960

 

952
00:43:35,960 --> 00:43:37,280

 

953
00:43:37,280 --> 00:43:39,410

 

954
00:43:39,410 --> 00:43:42,680

 

955
00:43:42,680 --> 00:43:47,569

 

956
00:43:47,569 --> 00:43:49,460

 

957
00:43:49,460 --> 00:43:51,970

 

958
00:43:51,970 --> 00:43:55,220

 

959
00:43:55,220 --> 00:43:59,050

 

960
00:43:59,050 --> 00:44:01,160

 

961
00:44:01,160 --> 00:44:03,290

 

962
00:44:03,290 --> 00:44:05,450

 

963
00:44:05,450 --> 00:44:09,380

 

964
00:44:09,380 --> 00:44:11,150

 

965
00:44:11,150 --> 00:44:13,040

 

966
00:44:13,040 --> 00:44:14,690

 

967
00:44:14,690 --> 00:44:16,460

 

968
00:44:16,460 --> 00:44:17,810

 

969
00:44:17,810 --> 00:44:20,140

 

970
00:44:20,140 --> 00:44:23,680

 

971
00:44:23,680 --> 00:44:26,330

 

972
00:44:26,330 --> 00:44:29,030

 

973
00:44:29,030 --> 00:44:30,589

 

974
00:44:30,589 --> 00:44:32,480

 

975
00:44:32,480 --> 00:44:36,500

 

976
00:44:36,500 --> 00:44:38,750

 

977
00:44:38,750 --> 00:44:42,859

 

978
00:44:42,859 --> 00:44:44,150

 

979
00:44:44,150 --> 00:44:47,540

 

980
00:44:47,540 --> 00:44:50,510

 

981
00:44:50,510 --> 00:44:52,490

 

982
00:44:52,490 --> 00:44:55,700

 

983
00:44:55,700 --> 00:44:57,079

 

984
00:44:57,079 --> 00:44:59,390

 

985
00:44:59,390 --> 00:45:02,329

 

986
00:45:02,329 --> 00:45:04,339

 

987
00:45:04,339 --> 00:45:06,680

 

988
00:45:06,680 --> 00:45:09,950

 

989
00:45:09,950 --> 00:45:13,550

 

990
00:45:13,550 --> 00:45:15,200

 

991
00:45:15,200 --> 00:45:17,589

 

992
00:45:17,589 --> 00:45:20,180

 

993
00:45:20,180 --> 00:45:23,320

 

994
00:45:23,320 --> 00:45:26,180

 

995
00:45:26,180 --> 00:45:30,470

 

996
00:45:30,470 --> 00:45:32,839

 

997
00:45:32,839 --> 00:45:35,030

 

998
00:45:35,030 --> 00:45:37,940

 

999
00:45:37,940 --> 00:45:40,579

 

1000
00:45:40,579 --> 00:45:42,320

 

1001
00:45:42,320 --> 00:45:44,180

 

1002
00:45:44,180 --> 00:45:45,950

 

1003
00:45:45,950 --> 00:45:47,990

 

1004
00:45:47,990 --> 00:45:50,630

 

1005
00:45:50,630 --> 00:45:52,339

 

1006
00:45:52,339 --> 00:45:55,700

 

1007
00:45:55,700 --> 00:45:58,190

 

1008
00:45:58,190 --> 00:45:59,930

 

1009
00:45:59,930 --> 00:46:02,329

 

1010
00:46:02,329 --> 00:46:03,320

 

1011
00:46:03,320 --> 00:46:05,810

 

1012
00:46:05,810 --> 00:46:08,570

 

1013
00:46:08,570 --> 00:46:09,890

 

1014
00:46:09,890 --> 00:46:11,870

 

1015
00:46:11,870 --> 00:46:14,960

 

1016
00:46:14,960 --> 00:46:18,740

 

1017
00:46:18,740 --> 00:46:20,960

 

1018
00:46:20,960 --> 00:46:23,690

 

1019
00:46:23,690 --> 00:46:25,610

 

1020
00:46:25,610 --> 00:46:32,510

 

1021
00:46:32,510 --> 00:46:35,000

 

1022
00:46:35,000 --> 00:46:38,000

 

1023
00:46:38,000 --> 00:46:40,790

 

1024
00:46:40,790 --> 00:46:41,960

 

1025
00:46:41,960 --> 00:46:43,460

 

1026
00:46:43,460 --> 00:46:45,440

 

1027
00:46:45,440 --> 00:46:48,200

 

1028
00:46:48,200 --> 00:46:49,550

 

1029
00:46:49,550 --> 00:46:51,230

 

1030
00:46:51,230 --> 00:46:54,410

 

1031
00:46:54,410 --> 00:46:55,820

 

1032
00:46:55,820 --> 00:46:57,860

 

1033
00:46:57,860 --> 00:47:00,170

 

1034
00:47:00,170 --> 00:47:02,480

 

1035
00:47:02,480 --> 00:47:06,530

 

1036
00:47:06,530 --> 00:47:09,020

 

1037
00:47:09,020 --> 00:47:13,040

 

1038
00:47:13,040 --> 00:47:14,540

 

1039
00:47:14,540 --> 00:47:16,520

 

1040
00:47:16,520 --> 00:47:18,680

 

1041
00:47:18,680 --> 00:47:20,480

 

1042
00:47:20,480 --> 00:47:23,150

 

1043
00:47:23,150 --> 00:47:26,840

 

1044
00:47:26,840 --> 00:47:28,490

 

1045
00:47:28,490 --> 00:47:30,230

 

1046
00:47:30,230 --> 00:47:31,760

 

1047
00:47:31,760 --> 00:47:33,980

 

1048
00:47:33,980 --> 00:47:36,140

 

1049
00:47:36,140 --> 00:47:38,180

 

1050
00:47:38,180 --> 00:47:41,150

 

1051
00:47:41,150 --> 00:47:42,890

 

1052
00:47:42,890 --> 00:47:45,230

 

1053
00:47:45,230 --> 00:47:47,300

 

1054
00:47:47,300 --> 00:47:50,930

 

1055
00:47:50,930 --> 00:47:52,490

 

1056
00:47:52,490 --> 00:47:54,980

 

1057
00:47:54,980 --> 00:47:56,360

 

1058
00:47:56,360 --> 00:47:59,570

 

1059
00:47:59,570 --> 00:48:02,030

 

1060
00:48:02,030 --> 00:48:05,349

 

1061
00:48:05,349 --> 00:48:07,609

 

1062
00:48:07,609 --> 00:48:09,170

 

1063
00:48:09,170 --> 00:48:11,750

 

1064
00:48:11,750 --> 00:48:14,900

 

1065
00:48:14,900 --> 00:48:16,490

 

1066
00:48:16,490 --> 00:48:18,770

 

1067
00:48:18,770 --> 00:48:20,960

 

1068
00:48:20,960 --> 00:48:22,640

 

1069
00:48:22,640 --> 00:48:24,020

 

1070
00:48:24,020 --> 00:48:26,300

 

1071
00:48:26,300 --> 00:48:28,130

 

1072
00:48:28,130 --> 00:48:29,750

 

1073
00:48:29,750 --> 00:48:32,000

 

1074
00:48:32,000 --> 00:48:33,550

 

1075
00:48:33,550 --> 00:48:36,050

 

1076
00:48:36,050 --> 00:48:39,290

 

1077
00:48:39,290 --> 00:48:40,970

 

1078
00:48:40,970 --> 00:48:42,410

 

1079
00:48:42,410 --> 00:48:43,700

 

1080
00:48:43,700 --> 00:48:44,930

 

1081
00:48:44,930 --> 00:48:47,900

 

1082
00:48:47,900 --> 00:48:50,120

 

1083
00:48:50,120 --> 00:48:52,940

 

1084
00:48:52,940 --> 00:48:56,809

 

1085
00:48:56,809 --> 00:48:58,849

 

1086
00:48:58,849 --> 00:49:01,510

 

1087
00:49:01,510 --> 00:49:09,800

 

1088
00:49:09,800 --> 00:49:09,810

 

1089
00:49:09,810 --> 00:49:17,499


1090
00:49:17,499 --> 00:49:17,509

 

1091
00:49:17,509 --> 00:49:19,569