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alright so this lecture we talked about  fold in one cut two methods and a little  bit about polyhedron flattening so most  of the questions are about fold in one  cut I'll stick to that first question is  is there any cool software for doing  fold and cut now and the answer is yes  there's some software maybe not the  coolest possible yet but getting there  there's two saw pieces of software one  is from final project when this after  this lecture was given 2010 and another  one is in a sourceforge project called  jour egami or j origami both are written  in Java I believe I have this one this  is the Swan I've seen before I have it  here we can try it out all right it's  not online yet because I could use some  improvements but it's already pretty  cool so you can take something like the  angelfish here and if you like you can  you have it has an editor so you can  move your polygon around then you can  say okay please give me the straight  skeleton first straight skeleton only  and it will update on the fly and gives  you some nice intuition about how that  works I know that looks weird that in  turn goes up that way but that is  correct because it's bisecting this edge  and and some edge this one I guess yeah  they meet out here and then you go that  way so you can play with that and then  you can add in the perpendiculars to fix  a little bit longer so the Refresh may  not be as immediate but it works get  some really complicated behavior in  there there's some spiraling and  spiraling out but it's fairly well  behaved and you can here's one of the  simpler spiraling examples pretty much  however you this one is stable under  perturbation more or less if you move  the verge  he's all a little bit they should it  should continue spirelings like that  one's a little bit degenerate so we'll  go around and as you go get bigger and  bigger pieces of paper you'll get more  and more creases the this I should  mention you can if you you can add new  vertices as well and draw polygons you  can delete edges I'm holding all sorts  of crazy modifier keys to make this  happen but it does work the one other  thing you can do is snap to a grid  that's probably hard to see but you  there's a square grid underneath if you  hold down alt it snaps to a square grid  I have one other example I prepare it  has save and load it's pretty cool and  this example is one one of the dense  instances although this one has  rationale multiple so it doesn't  actually go forever because I drew it on  a grid so that I could get all the  horizontal and vertical and you can see  in particular it stops here because it  at some point it gives up and reflecting  perpendicular says that's enough won't  draw anymore so it's fairly robust in  that sense occasionally as using some  straight skeleton code they did not  write and it has some issues when you  have really degenerate situations which  they tried to mitigate but occasionally  it's not perfect but there are lots of  possible follow on projects to this work  improving the user interface actually  putting it on the web for people to play  with I think would be super cool  alternatively could port it to  JavaScript right now it's in Java and  make it even more accessible run on  iPhones and things like that and I think  still the one of the questions I posed  in lecture was can you make a nice  interface that would let you force  degeneracies make like in I think the  Swan has instances of this where you'd  like more than two more than three  skeleton edges to come together at a  point here they almost do and be nice to  be able to say effort to like snap to a  position where many things  come together because that in general  reduces number of creases substantially  you could also try to compute the folded  state be another interesting project  based on what we're going to talk about  in a little bit of folding the  underlying structure of corridors and  then you can compute a crease pattern  there you have various choices and that  lets you throw away some of these  creases like this is much messier than  the Swan crease pattern that's on my web  page because you don't need all of these  folds if you choose the right set of  subset of folds you can save a lot of  time so still lots of cool projects to  do here and be great to get this  software online but that's its current  state next question is what about odd  degree vertices this is actually a  pretty natural question even degree  vertices seem nice because you can kind  of well it relates to a page that I  didn't really talk very much about in  the in lecture but it was on in the  lecture notes of this idea of a side  assignment so in general you have  something like a swan there's the inside  of the Swan the outside of Swan and  general you have a bunch of regions and  for each region you'd like to know is it  above the cut line or below the cut line  if you imagine the cut line as  horizontal and in general the side  assignment would specify do I want my  region above or below and you can do  whatever you want now with even degree  vertices this is great you can just  alternate around and say above below  above below with odd degree vertices you  can't so you're going to have two  regions that are adjacent to each other  which are both above or both below so  what does that mean it means it's a  little bit hard to cut and there's  actually two models of cuts there's  scissor cuts which are in particular  what the question posed are had in mind  and probably what you might have had in  mind in general where the cut you're  going to make separates material from  above and below the line so this is the  cut line here and it would be really  nice if you have material on both sides  because that's usually how scissors work  Thank tear apart material an alternative  is that you have  a mathematical cut and mathematical cut  can cut right along a crease line so it  could be you have two polygons there's a  fold here between them and you can cut  right along that line maybe I shouldn't  draw scissors imagine a laser beam which  can zap right along the line so there's  no material on the left side of the line  just material on the right but yet it  separates the two things on the right so  this is this is I call it we call that  mathematical cut because it is the  natural definition mathematically you're  erasing a line but practically it's a  little hard to do so scissor cuts are  also nice this is a more restrictive  model this is general model so what I  was talking about in lecture implicitly  used mathematical cuts and that's when  you can say you can make anything with  one cut but and it would be nice to get  scissor cuts when possible and when  possible is basically when you have even  degree at every vertex and one fun  example of that is checkerboard this is  actually an old magic trick you have a  piece of usually tissue paper so it's  really thin pre colored as checkered  squares on both sides matching and you  can fold this because every vertex here  has degree 3 sorry it has degree 4 so  even degree you can assign the black  squares to be on one side or the blue  squares to be on one side the white  squares to be on the other side and so  you can make one cut and simultaneously  cut out all the white squares and all  the black squares which is kind of cool  this is old decades old so in general  you can do something with scissor cuts  our scissor cuts are going to be  possible if and only if you can find a  side assignment the sort of alternates  between above and below the cut line  meaning when you have two regions you  don't want them to that are adjacent you  don't want them both to be above and  below you'd like one to be above one to  be below and this is what's called a  face to coloring in planar graphs you  want to color the faces the regions of  your graph with two colors such that no  two adjacent cells have the same color  two adjacent faces at the same color and  that turns out to be equivalent to  having all vertices of even degree so  this is why that question was asking  about odd degree because indeed with  scissor cuts you can't do odd degree  there is a fun fact though that relates  these two things I mean so you have even  degree like polygons which is typical  case scissor cuts should be possible for  mathematical cuts can do everything what  if i have something has odd degree  vertices but i still want to use scissor  cuts well then it turns out two cuts are  enough pretty much that if you have i'll  call it two edge connected  planar graph equals the union of two  even graphs I should say even sub graphs  much matter so there's one exception  which is if you have an edge in your  desired set of cuts that partitions the  graph into two parts so if you could  delete an edge and disconnect the graph  into into more parts that there's no  hope of decomposing this into even  graphs I think check that but as long as  you do not have such these are called  bridges typically so bridges are  forbidden if you have no bridges then  you're what's called a two edge  connected planar graph and then you can  decompose your graph into two parts each  of which is even so you can fold make  one straight cut and get one even graph  fold make one stray cat get the other  even graph good luck folding at the  second time if it decompose but in  theory at least you with to scissor cuts  you can make anything that doesn't have  bridges so that's just kind of an aside  some graph theory that tells you a  little bit more about what you can do  with to scissor cuts pretty much  everything any question or questions  about that so that was odd degree  vertices next questions are about  folding and how exactly so I kind of  briefly sketched the proof for linear  corridors how you would fold or how you  would prove that this thing actually  folds this is that where skeleton method  first we'll go to dis packing afterwards  so how do you convert a set of linear  quarters into a tree then once you have  that correspondence how does trees  folding flat relate two corridors  folding flat and so i redrew this figure  to make it  bit clearer so this is one of the images  in the textbook where I've just faced  colored in this case with three colors  the corridors those are the regions of  constant width bounded by perpendiculars  this is for making a turtle that sort of  doesn't really matter and this is the  corresponding tree and this is the  folded state of this blue guy here  between B and C and to really illustrate  what's going on here I folded this thing  it's very it's tricky to fold let's say  and you added a little bit of a few  paper clips to make it really stay but  if you hold it right a little bit  challenging the projection of this  structure is exactly this tree maybe let  me show you some examples out here for  example this this flap is labeled a B so  it corresponds to this flap and it  corresponds to the material here between  a and B and this a part is at the very  tip here and attached to this is this  guy that would be this unlabeled pink  edge which corresponds so this one it's  unlabeled because it goes off to  infinity in principle so this would just  keep going this one turns around right  here then there's this edge this bit of  material so it's probably easier for you  see from that side and that corresponds  to this bc part which is exactly this  folding and if I laser-cut or even  scissor cut down these two lines and  just cut out that little folded part it  would look exactly like this so  conversely i mean this I was sort of  begging the question by assuming I could  fold this thing if you don't know how to  fold this thing you can go in reverse  direction and figure out how to fold it  by first modeling these corridors as a  tree how do you do that it's just like  in the tree method going from crease  pattern from the tree method to the  shadow tree except there we were given  the shadow tree as input here we have to  compute it so but it's like what you  solved in the  problem set is it one or two for each of  these corridors it has fixed width so  you make an edge of that length it's not  quite drawn to scale here this has been  scaled up by roughly a factor of two so  this length becomes this length and you  label it the same thing this set  connected component of perpendiculars a  goes to that point the connected  component of perpendiculars here this  thing which branches all of that is B  it's like a hinge in the tree method so  that's going to be a hinge between this  flap and two other flaps whatever  touches the be perpendicular so there's  this pink one and there's a cyan one  sign one is here it's attached to  whatever c is attached to so here's see  it branches off here bounces around all  that c and it's jason to the blue thing  we just did the yellow and the pink  that's this yellow and this pink  connecting to d and g and so on through  the thing so it's actually really easy  to map from here to the tree just every  corridor maps to an edge every connected  component of the perpendicular graph  maps to a point and what that mean this  is the projection it will look it will  always look exactly like this you'll be  able to independently manipulate each of  these corridors as a flap and  individually it's pretty easy to fold  each corridor it just looks like this  accordion thing so the the faces will be  stacked linearly in order from back to  front and be really clean they'll all  line up in this nice vertical strip and  then all you have so that's easy to  prove it exists I'm sketching the proof  now and then all you have to do is  attach them together so basically you  take first you take this tree view you  fold it flat here I can fold it flat  just by collapsing the top and bottom  then I replace each of these edges is  one of these vertical accordion strips  and then I just need to check that at  each vertex I can actually attach  everything that needs to attach without  getting crossings that's the one part of  the proof i'm not going to show you  because it's kind of tedious but  basically because this structure was a  was planar because it came from one  sheet you can show there's no not going  to be any crossings they're all of these  layers each of these edges expands to be  many layers and the layers will nicely  connect together so that's the sketch of  that proof any questions this is for  just linear corridors not circular ones  okay cool next question is about the bad  example that makes a dense set of  creases that completely fills the plane  their force completely unfolded and the  question was is it really unlikely or is  it actually very likely that this  happens and there are two things going  on here's the example again from the  textbook so it's got the dark blue is  the desire cut graph then the black  lines are the straight skeleton and then  the dashed lines are the beginning of  the perpendiculars and the point was to  make this corridor width versus this  corridor width versus this corridor with  vs. this quarter wits make those all  irrational multiples of each other and  then as this thing spirals around it  never finishes never hits itself and so  it just keeps going I think this is  where its ones currently going so indeed  and so the question is well irrational  multiples are actually very common if  you for example randomly perturb all  these vertices and you measure the sizes  of those corridors they will be with  probability one they will be irrational  multiples of each other because the  rationals are much more common than  rational numbers and that's true but the  other thing that this example requires  is this outer boundary we need that none  of these perpendiculars can escape out  to infinity if they do they won't stay  in there eventually that if there's a  tiny tiny gap here if these guys didn't  quite match up eventually because this  thing is dense it will find that little  gap because it has positive length when  it finds the gap it's going to spiral  around and around  around and go out to infinity instead of  being trapped inside so the unlikely  thing in this example is actually this  outer pink polygon that the  perpendicular coming out this way  bounced around did lots of things and  eventually hit the same vertex that's  actually unlikely and definitely in this  example and in general we claim that we  do not get cycles of perpendiculars  except in one kind of scenario which I  guess I could draw in the software let's  try it so whenever you have a vertex of  degree more than more than two so let's  do something like that a little too far  away so let me just add a little guy  like this and bring this a little closer  yeah this has got a lot of spiraling  let's make it a little bit cleaner here  that's actually kind of degenerate but  the point is if here i have three cut  edges I've kind of made them roughly  equal so this doesn't get huge but in  general if you have any straight  skeleton vertex and you reflect it  around those these bisectors you will  always come back to where you started so  whenever you this this requires that  each of these angles is strictly convex  so this is why you need at least three  cottages to come together here but once  you have at least three you will always  cycle around you always come back to  where you start it actually happens  again here this guy cycles around and  comes back so that's what sorry that  one's actually a little bit special  because of what I did with the dragging  in general it's going to be more like  this a little hard to see what's going  on here is that this innermost guy  cycles around indeed but everyone else  spirals around and keeps keeps going so  that's actually the perpendicular that  comes from here and goes that way and  then it spirals around it hasn't  finished yet but it would actually  spiral all the way out to infinity so  the claim is in general the  this is conjecture  with probability one if you randomly  perturb the vertices slightly the the  only cycles of perpendiculars that you  get are around a single vertex so  something like this example which we saw  where there was a straight skeleton edge  here here and then this guy I should  draw the skeleton edges it's a good test  of how accurate I drew it because I know  if this is perpendicular this is  perpendicular and this is perpendicular  this will always come back to its  starting point it's just property of  reflections basically because we satisfy  Kawasaki here this must be true because  these are bisectors of these guys so  this has to happen the claim is that's  the only situation it happens and if  this is all that happens you can prove  there's no density and in fact it folds  flat so this is the good case  unfortunately when we make real examples  we like degeneracies because they reduce  the number of folds so the theory says  avoid degeneracies to perturb things  slightly then it's guaranteed to fold if  you but in practice you want to add  degeneracies just carefully so that you  don't get dense behavior like the weird  example I showed you so that's  clarification why yeah why we think this  does not happen oh sorry why we think  with probability 1 things are good but  sadly we can't prove this maybe we'll  work on in the open problem session I  think it's tractable just haven't worked  on in a long time questions  okay so right the claim is if you  perturb this example everybody will  escape out to infinity like in this  spiral okay so i think the that's the  end of the skeleton method but before we  want go on I want to show some examples  so problem set we're up to four which  will go out later today one of the  questions is design your own your own  fold and cut and draw it so you probably  want to draw it in program like inkscape  or Adobe Illustrator it has good  snapping so you can find intersections  and things like that and compute angular  bisectors as in ruler and compass and  these are some examples from 2010 same  question there are lots of them but I  chose three they're particularly cool  this one has a line of symmetry you get  fish bone by G who's a PhD student in  the media lab this is by Sarah  eisenstaedt who's a PhD student in csail  working on folding things which is hat  looks pretty cool pretty fairly simple  all you have to fold your examples I  can't be too complicated and then Jason  KU who saw the guest lecture by it  wasn't impressed sufficiently impressed  by my jackal entrance he made a really  complicated one and folded it so those  are some inspiration points and i  thought i'd show you a magic trick which  is one of the sources for inspiration  for the fold and one cut problem so  piece of paper so this is a magic trick  of unknown origin it was described by  Martin Gardner I think probably in the  60s and the book appeared in the 90s and  it's a story of two politicians and  let's just be generic say one politician  was very much liked by the people on the  other politician was disliked by the  people imagine a simple world where it  was so simple and by freak accident both  politicians died at the same time and  both happen to be Christian so they go  to the gates of heaven so the story goes  and they arrive the gates of heaven I  guess st. Peter's the keeper of the  gates and st. Peter says well not just  anyone can get into heaven you have to  have a ticket and the good politician  being liked by the people has a ticket  which he folds flat and bad politician  has no ticket so the bad politician says  well you know we've had our  disagreements but maybe you could put in  a good word to Saint Peter or do  something to help me out I here heavens  a nice place maybe we could go there  together we resolve our differences  whatever so the good politician having a  ticket and the pair scissors like any  respecting politician takes this ticket  and makes one complete straight cut and  it's going to get a lot of pieces that  are going to be hold this carefully so  put that down and he hands all these  pieces to the bad politician says here  you go pet politician has no idea what  they're for so he hands them the st.  Peter st. Peter starts unfolding the  pieces I wonder what shapes i get i hear  there's a cool-looking problem about  this uh-huh so it's a little hard to do  without a table so I'm going to have to  use the board first we get the letter H  then we get to it right  put these down here then we get the  letter e okay then we get the letter L  and the letter L and if it's not a table  at arrange it and it clearly st. Peter  is not happy and the bad politician gets  sent straight to hell the good  politician hung on to one piece cleverly  and this ticket is still more or less  intact and he gets into it okay that's  the magic trick it's very simple full  bang it's easy to remember I mean known  this remember as this even for years and  you get exactly those pieces so pretty  cool of course from a mathematical  standpoint this is a little unsatisfying  so come on there's three pieces for the  h three pieces for the e surely you  could do better and from a rectangle you  can kind of do better uh-huh it's a  little awkward because the pieces are  not all the same size but is one scissor  cut so all the vertices are even degree  you cut it and you get all these pieces  which if I were more practiced at this I  could immediately pick out which one's  the H looks like okay I'll just do them  out of order here is the cross kind of a  cross not quite perfect proportions but  good enough here is this is the letter e  this one's actually more impressive when  you don't have a table because you can't  tell that they're all different sizes at  the letter e we've got the letter these  look like else kind of small else but  there you go through different  orientations and then we've got the  letter H there you go okay so that's our  new and improved version using  universality of folding cut so I gives  you some idea of a we're folding cut  came from and in recent times is the  magic community I mentioned Harry  Houdini did some tricks and Gerald low  but there's a bunch of these tricks  around and kind of cool saw the  checkerboard trick  and the the previous he'll trick so  those are some examples now we move on  to the disk packing method so I have one  question about this which is how exactly  do we go from the disk packing to yeah  cool proof the cool proof bro so how do  we go from the disk packing to the  decomposition into triangles and  quadrilaterals so I thought I'd just  review this slide we start with our  graph we offset it by some tiny Epsilon  that's to get things off the line  basically then we do this disk packing  and remember roughly how the disk  packing work so we put some disks at  each of the centers here it's a little  awkward cuz of the offset but you put  one at each of these four corners also  the corners of the piece of paper why  not then you also pack small enough  distal ong the edge so that you cover  the edge by diameters of the disks i do  that the same on all the sides basically  you try to put a big one in but if that  big one intersects you decompose it into  half that's how the algorithm works so  now you've covered the vertices in the  edges but you may have big gaps like  this too many too many sides on this gap  and so then you just greedily put the  largest disk you can in those gaps until  all you're left with and these yellow  gaps are triangles and quadrilaterals or  three sided gaps and for side of gaps  and then all we do is draw this red  graph by putting a vertex at the center  of each disk this is what the question  was about and then whenever two disks  touch we call this kissing disks for  historical reasons because they're just  barely touching I guess at their lips  I've got a lot of lips for disks uh-huh  we're in flatland so then whenever they  touch we draw the edge between the two  centers so that decomposes this piece of  paper into parts and because the gaps  are three sides those will correspond to  triangles or four sides those will  correspond to quadrilaterals so that's  all and then we put rabbit ears in here  and line Universal quad molecules in  there and it folds flat it will align  all of the red edges plus the  black edges on the outside and it will  align all these inner red edges with  these black edges and then we do the  sink folds that we mentioned and that  will get one of these out of the way of  the other and we'll end up aligning just  these cut edges so that's the quick  review that method so one question is  how do you allocate the disks is there  sort of a best way and I guess there  could be several measures maybe you  don't want really tiny discs because  that tends to lead to very tiny folds  but the standard measure here is how  many disks do you need because that will  reduce the number of folds so can you  minimize the number of discs and in  general it's known roughly how many  discs you need and the algorithm I just  told you achieves that number of discs  so first I'll give you the number number  of this is proportional to the integral  and say it was an easy bound but it is  the right answer okay there's a lot of  notation in here and this one you should  definitely not know let's you've taken a  computational geometry class even then  you probably wouldn't know it it's not  that common mostly comes up in meshing  and disk packing so it's pretty specific  it's called local feature size and this  is a once i define it is actually fairly  intuitive this bound local feature size  at X so we're imagining some kind of  polygon here and at every point of the  polygon let's look at a vertex here but  it works for any point along the  boundary actually maybe not even on the  boundary but we're interested here this  this notation del P is the boundary of  the polygon so this is Del P that just  means you know edges and vertices all of  those points so we take some point X on  the boundary and we look at the smallest  disk centered at that point that touches  another feature  so that's the feature size features here  are edges of the polygon that's all we  have now of course no matter what size  disk you hit this feature you hit this  feature so those don't count those are  incident features so in general it is  the smallest radius disk that hits a non  incident feature on incident feature  which is an edge so here as soon as the  disk gets this big it hits this edge and  so that's the local feature size is this  this radius okay in general for every  point you can draw some size disk so  over here it's going to be a little bit  bigger you can maybe get up to here  actually I guess it should be non  adjacent it was going to be a little bit  awkward I also don't want to count this  edge because if I'm right here the  feature size is super tiny and I don't I  don't want out lfs ever to be zero so  you also have to skip the the adjacent  edges which means the local feature size  at this point is going to be more like  hard to draw circles it's going to be  more like that distance that's when you  hit a non adjacent feature sorry so you  have to exclude this edge that you're on  and the two adjacent ones otherwise this  definition doesn't quite work out so  then you measure this for all points  just infinitely many of course continue  them along the boundary that's the  integral over X in the boundary of P and  you take one over the local feature size  of X and you integrate that for all X it  gives you a number and it turns out some  constant times that number is the right  answer the algorithm I achieve which I  described which is put the biggest dis  you can or divide in half if you can't  we'll get within a constant factor of  the best possible bound of all possible  dis packings so that so this is pretty  much known you can get within a constant  factor of the optimal dis packing if you  want to you're counting disks not a  great number I can't say  and this or n squared discs because you  just can't do that if you have if you  have a piece of paper and some really  long here's your cut graph doesn't quite  go to the ends though then you've got to  have a tiny disc here and that's going  to require you to have a bunch of disks  even though this has constant size  you're going to need I don't know how  many dis here especially if your paper  is really narrow you're going to need a  lot of disks that's the best you can  hope for for disk packing actually  brought a little example of something  like that where you have your goal is  just to cut out a single segment in the  middle of the paper this is to  illustrate odd degree vertices these are  degree 1 vertices you might think wow  it's impossible for me to cut this thing  without I don't know I'm just cutting I  could cut with my exacto knife from here  to here if I want to make a complete  straight cut you can fold it like this  this is what the skeleton method would  give you you fold it like this and then  you use your laser to cut right along  the line with scissor cuts it's not so  bad right I cut along I cut slightly in  its pretty much cutting along a line  just not quite tiny mistake so I cut off  a line slightly thicker line and I get  my slit in the paper so I mean  mathematical cuts are not really that  bad you're just cutting a slightly  larger version than the single slit that  you wanted to make have to cut very  carefully okay that was number of disks  next question is how does all of this  relate to the tree method this is kind  of a neat question both methods skeleton  method and the dis packing method relate  to the tree method in different ways I  thought I would talk about that so first  let's compare the disk packing method  which we just talked about to the tree  method both of them use Universal  molecules this one only uses universal  molecules for triangles and quads we  know in general Universal molecule works  for any convex polygon this is an  example of a pentagon and it works it  has a gusset to get the tree lengths  right here we use gussets to kind of get  the tree lengths right as you may recall  we want the perpendicular  to go to the disk kissing points those  tangen sees because then they align with  the others and then we never get  perpendicular spiraling or doing other  bad things so number of creases becomes  proportional to the number of disks  number this is this messy thing so  that's that let's see what else here we  only use disks over here we could just  use disk but we there's also the ability  to use rivers so this is more general it  in that we can do bigger bigger polygons  in just quads and we can do rivers not  just circles here the input though is a  tree with specified lengths and it's  actually copy it's NP hard to place the  disks correctly we proved that last time  over here the input is a polygon it's  already been embedded on the sheet of  paper there's no really hard decisions  here except you have to fill the paper  with disks but that's not so hard so you  can do this in polynomial time  polynomial in even almost roughly this  time times a log factor you can find  this placement of disks so that's not  too hard whereas over here it's NP hard  to place a distally at least find a  displacement you could do so basically  the inputs differ and it's using the  same backbone or the same what do you  call it so front end the same back end I  suppose where we're decomposed after  we've decomposed into a bunch of  molecules you just fold the molecules  and you're basically done here we have  to do some sinks at the end but it's  using it for different purposes  otherwise they are very similar and  indeed this is inspired by by that okay  so that was the disk packing method then  there's the straight skeleton method  these look much more different but in  fact they are quite related if you do  fold and cut on a convex polygon it  gives you roughly the universal molecule  just with no gussets so here neither one  is more general than the other the  universal molecule is more general in  that it has gussets which lets it  control the tree topology you get and  control the lengths you get in the tree  over here we saw that the tree just  happens you can't control the tree it's  just a tool to prove the shadow tree  that's just a tool to prove that it  folds flat but you can do non convex  polygons which is much more general than  convex polygons so they each have their  own sort of unique claims to fame  natural question is can you combine them  and in fact you can and this was you may  remember a few bunch of lectures ago I  think lecture class for so I showed this  slide of the the meat of origami design  secrets volume 2 all right second  edition rather there's tree theory which  we talked about there was a box pleating  tree theory which should briefly showed  and there's this more general thing  called polygon packing and this is in  some sense the fusion of straight  skeleton method of fold and cut  basically the straight skeleton plus  tree theory and this is done jointly  between Robert Lang and the domains and  here's an example of kind of what it  looks like a little hard to see here but  I will point out that if you look at the  blue lines which are kind of the cut  lines here of course they're the active  paths this guy is non convex and if you  look at the red stuff in there it is  exactly the straight skeleton of that  polygon so those red lines are the  straight skeleton of this blue thing so  there you go we've got the folding cot  straight skeleton being used in that  setting you've got perpendicular all  over the place which is a annoying  feature of the skeleton method it is  inherited by this method and if you look  at something like this rectangle its  convex but you have gussets this is a  gusset this is a gusset gusset guess  it's a little hard to see but there's a  we basically split the rectangle into  this rectangle this square this  rectangle the square this rectangle so  this is really the fusion of the two  it's not yet been mathematically  formalized because we're first proving  that the tree method works then we will  finally get to this but if you want  these slightly more informal but  practical version read origami design  secrets and they'll explain all the  different cases and what we believe  Eve is a correct algorithm we just  haven't proved all the cases yet and you  can use it to design cool things in the  Box pleating setting and get more  efficient than if you required  rectangles or some convex shape cool so  that's that else do we have I have one  almost done one cool open question posed  by you is what if instead of cutting  with a straight line I cut with a  constant curvature line so a piece of a  circular arc then of course the things I  get will have circular arc boundaries  but can I get any circular arc boundary  with the with that fixed curvature or  are there limitations and at first I  thought the answer is clearly yes but uh  I do wonder about situations like this  can you align these two by folding I've  thought about enough to be sure but not  not totally sure where's if I have  things like this I think this is this is  good i mean i could fold here and in  general i could just kind of pretend  that there's a segment and the segment i  saw a fold and cut on the segments and  that will also align the circular arcs  that won't work in general but i feel  comfortable if they have the same  orientation it looks good if when they  flip orientations i'm not sure so neat  problem to think about and one final  thing is about flattening this is about  this question this picture so we said oh  great because we can solve fold and cut  we can also take the boundary that  polygon and flatten it like this of  course the folding of the interior here  which we're ignoring in general this  whole thing goes through three  dimensions until we end up here one  dimension up if we solve the 3d folding  cot problem where you have a polyhedron  you fold it through 4d back into 3d  which is flat so that you align all the  boundaries of the faces of that  polyhedron and then just take the  boundary of that motion it's like  flattening a polyhedron like cereal box  or whatever  and indeed that will find a folded state  of a polyhedron there's flat but of  course that motion you can't you can't  really use that motion because it goes  through 4d this one will go through 3d  so it's not if you want it to flatten  this linkage that's also not valid so  you get a folded state but you don't get  the folding motion if you could solve 3d  folding cut our current state is we  don't know how to solve 3d folding cat  we do know how to solve polyhedron  flattening so this this relation is just  kind of interest not doesn't imply  anything super useful an open problem is  so we know how to sell polyhedron  flattening as a folded state what we  don't know so much about is folding  motions there is a recent paper from  just last year which will continuously  flattened by continuous motion any  convex polyhedron but non-convex  polyhedra we still don't know these are  roughly hand-drawn figures an  interesting project would be to actually  animate their motion I think it would  look pretty cool and be much more  convincing to that it works I mean I am  convinced that it works but it's a lot  harder to see visually from these  pictures be really cool to see the  emotions the actual continuous animation  and a couple last project ideas one  would be to make a fold and cut alphabet  like the maze software that you just use  it would be fun if you just type in a  message and out comes the crease pattern  this is sort of like a special case of a  fold and cut design tool but in letters  it's a little bit easier and a simp  different challenge would be if I just  want to be able to fold a single letter  at a time can you always do it with  simple folds so can you design an  alphabet that is simply folding cuttable  that would be even nicer and there's  some magicians who worked on that but  they couldn't get all the letters so be  nice to do with just a few folds any  letter of the alphabet and any digit  this is one we did for our conference in  honor martin gardner gathering for  Gardner five back in 2002 and here's  another possible idea for project  although mostly it's an excuse to show  you really cool pictures this is Peter  callison and he takes usually a4 sheets  of paper cuts them with some probably an  exacto knife or something and then from  that material that  septic he acquires builds a 3d structure  on that sheet so you see all the  material that was used and these are  just amazing and I don't think you could  necessarily do this level of fidelity  from a folding cut but it would be cool  to do a folding cut and then build a  sculpture out of the slits that you made  and I mean it's really cool when he can  use the cut out parts to be shadows of  the things that he builds especially  when those shadows are more complicated  than the chief he actually assembles it  has a really cool effect of the monsters  in the closet Pandora's box or boxing  box in a box you can't see that here but  you see it unfolding uh-huh bones inside  the hand the old versus new cities using  the positive and the negative is  actually a little boat up there as you  can see so pretty pretty amazing stuff  and for free versus caged and so on  interesting project would be to try to  make art at a folding cot just doing one  cut instead of a zillion cuts that's it  any more questions yeah in the circle  the backing and mess with other  assignment  okay and the circle packing method how'd  you assign mountains and valleys that  was briefly covered in lecture there was  this first you take a spanning tree of  all of the molecules and you basically  fold each molecule into your parent in  the tree so you each one hat each one  has a direction and from that you can  extract them out in valley assignment  you go into the parent it's a mountain  there everyone else's Valley basically  you're reversing one perpendicular fold  per molecule I think maybe two triangles  is just one for quads you might be  reversing too so you start with all the  all of the straight skeleton being  mountains all the perpendiculars being  valleys and you have to flip a few to  just get them to fit together a little  more complicated than what I said but I  would need a slide to show you the  details it's in the textbook anything  else alright that's folding cut  you

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alright so this lecture we talked about  fold in one cut two methods and a little  bit about polyhedron flattening so most  of the questions are about fold in one  cut I'll stick to that first question is  is there any cool software for doing  fold and cut now and the answer is yes  there's some software maybe not the  coolest possible yet but getting there  there's two saw pieces of software one  is from final project when this after  this lecture was given 2010 and another  one is in a sourceforge project called  jour egami or j origami both are written  in Java I believe I have this one this  is the Swan I've seen before I have it  here we can try it out all right it's  not online yet because I could use some  improvements but it's already pretty  cool so you can take something like the  angelfish here and if you like you can  you have it has an editor so you can  move your polygon around then you can  say okay please give me the straight  skeleton first straight skeleton only  and it will update on the fly and gives  you some nice intuition about how that  works I know that looks weird that in  turn goes up that way but that is  correct because it's bisecting this edge  and and some edge this one I guess yeah  they meet out here and then you go that  way so you can play with that and then  you can add in the perpendiculars to fix  a little bit longer so the Refresh may  not be as immediate but it works get  some really complicated behavior in  there there's some spiraling and  spiraling out but it's fairly well  behaved and you can here's one of the  simpler spiraling examples pretty much  however you this one is stable under  perturbation more or less if you move  the verge  he's all a little bit they should it  should continue spirelings like that  one's a little bit degenerate so we'll  go around and as you go get bigger and  bigger pieces of paper you'll get more  and more creases the this I should  mention you can if you you can add new  vertices as well and draw polygons you  can delete edges I'm holding all sorts  of crazy modifier keys to make this  happen but it does work the one other  thing you can do is snap to a grid  that's probably hard to see but you  there's a square grid underneath if you  hold down alt it snaps to a square grid  I have one other example I prepare it  has save and load it's pretty cool and  this example is one one of the dense  instances although this one has  rationale multiple so it doesn't  actually go forever because I drew it on  a grid so that I could get all the  horizontal and vertical and you can see  in particular it stops here because it  at some point it gives up and reflecting  perpendicular says that's enough won't  draw anymore so it's fairly robust in  that sense occasionally as using some  straight skeleton code they did not  write and it has some issues when you  have really degenerate situations which  they tried to mitigate but occasionally  it's not perfect but there are lots of  possible follow on projects to this work  improving the user interface actually  putting it on the web for people to play  with I think would be super cool  alternatively could port it to  JavaScript right now it's in Java and  make it even more accessible run on  iPhones and things like that and I think  still the one of the questions I posed  in lecture was can you make a nice  interface that would let you force  degeneracies make like in I think the  Swan has instances of this where you'd  like more than two more than three  skeleton edges to come together at a  point here they almost do and be nice to  be able to say effort to like snap to a  position where many things  come together because that in general  reduces number of creases substantially  you could also try to compute the folded  state be another interesting project  based on what we're going to talk about  in a little bit of folding the  underlying structure of corridors and  then you can compute a crease pattern  there you have various choices and that  lets you throw away some of these  creases like this is much messier than  the Swan crease pattern that's on my web  page because you don't need all of these  folds if you choose the right set of  subset of folds you can save a lot of  time so still lots of cool projects to  do here and be great to get this  software online but that's its current  state next question is what about odd  degree vertices this is actually a  pretty natural question even degree  vertices seem nice because you can kind  of well it relates to a page that I  didn't really talk very much about in  the in lecture but it was on in the  lecture notes of this idea of a side  assignment so in general you have  something like a swan there's the inside  of the Swan the outside of Swan and  general you have a bunch of regions and  for each region you'd like to know is it  above the cut line or below the cut line  if you imagine the cut line as  horizontal and in general the side  assignment would specify do I want my  region above or below and you can do  whatever you want now with even degree  vertices this is great you can just  alternate around and say above below  above below with odd degree vertices you  can't so you're going to have two  regions that are adjacent to each other  which are both above or both below so  what does that mean it means it's a  little bit hard to cut and there's  actually two models of cuts there's  scissor cuts which are in particular  what the question posed are had in mind  and probably what you might have had in  mind in general where the cut you're  going to make separates material from  above and below the line so this is the  cut line here and it would be really  nice if you have material on both sides  because that's usually how scissors work  Thank tear apart material an alternative  is that you have  a mathematical cut and mathematical cut  can cut right along a crease line so it  could be you have two polygons there's a  fold here between them and you can cut  right along that line maybe I shouldn't  draw scissors imagine a laser beam which  can zap right along the line so there's  no material on the left side of the line  just material on the right but yet it  separates the two things on the right so  this is this is I call it we call that  mathematical cut because it is the  natural definition mathematically you're  erasing a line but practically it's a  little hard to do so scissor cuts are  also nice this is a more restrictive  model this is general model so what I  was talking about in lecture implicitly  used mathematical cuts and that's when  you can say you can make anything with  one cut but and it would be nice to get  scissor cuts when possible and when  possible is basically when you have even  degree at every vertex and one fun  example of that is checkerboard this is  actually an old magic trick you have a  piece of usually tissue paper so it's  really thin pre colored as checkered  squares on both sides matching and you  can fold this because every vertex here  has degree 3 sorry it has degree 4 so  even degree you can assign the black  squares to be on one side or the blue  squares to be on one side the white  squares to be on the other side and so  you can make one cut and simultaneously  cut out all the white squares and all  the black squares which is kind of cool  this is old decades old so in general  you can do something with scissor cuts  our scissor cuts are going to be  possible if and only if you can find a  side assignment the sort of alternates  between above and below the cut line  meaning when you have two regions you  don't want them to that are adjacent you  don't want them both to be above and  below you'd like one to be above one to  be below and this is what's called a  face to coloring in planar graphs you  want to color the faces the regions of  your graph with two colors such that no  two adjacent cells have the same color  two adjacent faces at the same color and  that turns out to be equivalent to  having all vertices of even degree so  this is why that question was asking  about odd degree because indeed with  scissor cuts you can't do odd degree  there is a fun fact though that relates  these two things I mean so you have even  degree like polygons which is typical  case scissor cuts should be possible for  mathematical cuts can do everything what  if i have something has odd degree  vertices but i still want to use scissor  cuts well then it turns out two cuts are  enough pretty much that if you have i'll  call it two edge connected  planar graph equals the union of two  even graphs I should say even sub graphs  much matter so there's one exception  which is if you have an edge in your  desired set of cuts that partitions the  graph into two parts so if you could  delete an edge and disconnect the graph  into into more parts that there's no  hope of decomposing this into even  graphs I think check that but as long as  you do not have such these are called  bridges typically so bridges are  forbidden if you have no bridges then  you're what's called a two edge  connected planar graph and then you can  decompose your graph into two parts each  of which is even so you can fold make  one straight cut and get one even graph  fold make one stray cat get the other  even graph good luck folding at the  second time if it decompose but in  theory at least you with to scissor cuts  you can make anything that doesn't have  bridges so that's just kind of an aside  some graph theory that tells you a  little bit more about what you can do  with to scissor cuts pretty much  everything any question or questions  about that so that was odd degree  vertices next questions are about  folding and how exactly so I kind of  briefly sketched the proof for linear  corridors how you would fold or how you  would prove that this thing actually  folds this is that where skeleton method  first we'll go to dis packing afterwards  so how do you convert a set of linear  quarters into a tree then once you have  that correspondence how does trees  folding flat relate two corridors  folding flat and so i redrew this figure  to make it  bit clearer so this is one of the images  in the textbook where I've just faced  colored in this case with three colors  the corridors those are the regions of  constant width bounded by perpendiculars  this is for making a turtle that sort of  doesn't really matter and this is the  corresponding tree and this is the  folded state of this blue guy here  between B and C and to really illustrate  what's going on here I folded this thing  it's very it's tricky to fold let's say  and you added a little bit of a few  paper clips to make it really stay but  if you hold it right a little bit  challenging the projection of this  structure is exactly this tree maybe let  me show you some examples out here for  example this this flap is labeled a B so  it corresponds to this flap and it  corresponds to the material here between  a and B and this a part is at the very  tip here and attached to this is this  guy that would be this unlabeled pink  edge which corresponds so this one it's  unlabeled because it goes off to  infinity in principle so this would just  keep going this one turns around right  here then there's this edge this bit of  material so it's probably easier for you  see from that side and that corresponds  to this bc part which is exactly this  folding and if I laser-cut or even  scissor cut down these two lines and  just cut out that little folded part it  would look exactly like this so  conversely i mean this I was sort of  begging the question by assuming I could  fold this thing if you don't know how to  fold this thing you can go in reverse  direction and figure out how to fold it  by first modeling these corridors as a  tree how do you do that it's just like  in the tree method going from crease  pattern from the tree method to the  shadow tree except there we were given  the shadow tree as input here we have to  compute it so but it's like what you  solved in the  problem set is it one or two for each of  these corridors it has fixed width so  you make an edge of that length it's not  quite drawn to scale here this has been  scaled up by roughly a factor of two so  this length becomes this length and you  label it the same thing this set  connected component of perpendiculars a  goes to that point the connected  component of perpendiculars here this  thing which branches all of that is B  it's like a hinge in the tree method so  that's going to be a hinge between this  flap and two other flaps whatever  touches the be perpendicular so there's  this pink one and there's a cyan one  sign one is here it's attached to  whatever c is attached to so here's see  it branches off here bounces around all  that c and it's jason to the blue thing  we just did the yellow and the pink  that's this yellow and this pink  connecting to d and g and so on through  the thing so it's actually really easy  to map from here to the tree just every  corridor maps to an edge every connected  component of the perpendicular graph  maps to a point and what that mean this  is the projection it will look it will  always look exactly like this you'll be  able to independently manipulate each of  these corridors as a flap and  individually it's pretty easy to fold  each corridor it just looks like this  accordion thing so the the faces will be  stacked linearly in order from back to  front and be really clean they'll all  line up in this nice vertical strip and  then all you have so that's easy to  prove it exists I'm sketching the proof  now and then all you have to do is  attach them together so basically you  take first you take this tree view you  fold it flat here I can fold it flat  just by collapsing the top and bottom  then I replace each of these edges is  one of these vertical accordion strips  and then I just need to check that at  each vertex I can actually attach  everything that needs to attach without  getting crossings that's the one part of  the proof i'm not going to show you  because it's kind of tedious but  basically because this structure was a  was planar because it came from one  sheet you can show there's no not going  to be any crossings they're all of these  layers each of these edges expands to be  many layers and the layers will nicely  connect together so that's the sketch of  that proof any questions this is for  just linear corridors not circular ones  okay cool next question is about the bad  example that makes a dense set of  creases that completely fills the plane  their force completely unfolded and the  question was is it really unlikely or is  it actually very likely that this  happens and there are two things going  on here's the example again from the  textbook so it's got the dark blue is  the desire cut graph then the black  lines are the straight skeleton and then  the dashed lines are the beginning of  the perpendiculars and the point was to  make this corridor width versus this  corridor width versus this corridor with  vs. this quarter wits make those all  irrational multiples of each other and  then as this thing spirals around it  never finishes never hits itself and so  it just keeps going I think this is  where its ones currently going so indeed  and so the question is well irrational  multiples are actually very common if  you for example randomly perturb all  these vertices and you measure the sizes  of those corridors they will be with  probability one they will be irrational  multiples of each other because the  rationals are much more common than  rational numbers and that's true but the  other thing that this example requires  is this outer boundary we need that none  of these perpendiculars can escape out  to infinity if they do they won't stay  in there eventually that if there's a  tiny tiny gap here if these guys didn't  quite match up eventually because this  thing is dense it will find that little  gap because it has positive length when  it finds the gap it's going to spiral  around and around  around and go out to infinity instead of  being trapped inside so the unlikely  thing in this example is actually this  outer pink polygon that the  perpendicular coming out this way  bounced around did lots of things and  eventually hit the same vertex that's  actually unlikely and definitely in this  example and in general we claim that we  do not get cycles of perpendiculars  except in one kind of scenario which I  guess I could draw in the software let's  try it so whenever you have a vertex of  degree more than more than two so let's  do something like that a little too far  away so let me just add a little guy  like this and bring this a little closer  yeah this has got a lot of spiraling  let's make it a little bit cleaner here  that's actually kind of degenerate but  the point is if here i have three cut  edges I've kind of made them roughly  equal so this doesn't get huge but in  general if you have any straight  skeleton vertex and you reflect it  around those these bisectors you will  always come back to where you started so  whenever you this this requires that  each of these angles is strictly convex  so this is why you need at least three  cottages to come together here but once  you have at least three you will always  cycle around you always come back to  where you start it actually happens  again here this guy cycles around and  comes back so that's what sorry that  one's actually a little bit special  because of what I did with the dragging  in general it's going to be more like  this a little hard to see what's going  on here is that this innermost guy  cycles around indeed but everyone else  spirals around and keeps keeps going so  that's actually the perpendicular that  comes from here and goes that way and  then it spirals around it hasn't  finished yet but it would actually  spiral all the way out to infinity so  the claim is in general the  this is conjecture  with probability one if you randomly  perturb the vertices slightly the the  only cycles of perpendiculars that you  get are around a single vertex so  something like this example which we saw  where there was a straight skeleton edge  here here and then this guy I should  draw the skeleton edges it's a good test  of how accurate I drew it because I know  if this is perpendicular this is  perpendicular and this is perpendicular  this will always come back to its  starting point it's just property of  reflections basically because we satisfy  Kawasaki here this must be true because  these are bisectors of these guys so  this has to happen the claim is that's  the only situation it happens and if  this is all that happens you can prove  there's no density and in fact it folds  flat so this is the good case  unfortunately when we make real examples  we like degeneracies because they reduce  the number of folds so the theory says  avoid degeneracies to perturb things  slightly then it's guaranteed to fold if  you but in practice you want to add  degeneracies just carefully so that you  don't get dense behavior like the weird  example I showed you so that's  clarification why yeah why we think this  does not happen oh sorry why we think  with probability 1 things are good but  sadly we can't prove this maybe we'll  work on in the open problem session I  think it's tractable just haven't worked  on in a long time questions  okay so right the claim is if you  perturb this example everybody will  escape out to infinity like in this  spiral okay so i think the that's the  end of the skeleton method but before we  want go on I want to show some examples  so problem set we're up to four which  will go out later today one of the  questions is design your own your own  fold and cut and draw it so you probably  want to draw it in program like inkscape  or Adobe Illustrator it has good  snapping so you can find intersections  and things like that and compute angular  bisectors as in ruler and compass and  these are some examples from 2010 same  question there are lots of them but I  chose three they're particularly cool  this one has a line of symmetry you get  fish bone by G who's a PhD student in  the media lab this is by Sarah  eisenstaedt who's a PhD student in csail  working on folding things which is hat  looks pretty cool pretty fairly simple  all you have to fold your examples I  can't be too complicated and then Jason  KU who saw the guest lecture by it  wasn't impressed sufficiently impressed  by my jackal entrance he made a really  complicated one and folded it so those  are some inspiration points and i  thought i'd show you a magic trick which  is one of the sources for inspiration  for the fold and one cut problem so  piece of paper so this is a magic trick  of unknown origin it was described by  Martin Gardner I think probably in the  60s and the book appeared in the 90s and  it's a story of two politicians and  let's just be generic say one politician  was very much liked by the people on the  other politician was disliked by the  people imagine a simple world where it  was so simple and by freak accident both  politicians died at the same time and  both happen to be Christian so they go  to the gates of heaven so the story goes  and they arrive the gates of heaven I  guess st. Peter's the keeper of the  gates and st. Peter says well not just  anyone can get into heaven you have to  have a ticket and the good politician  being liked by the people has a ticket  which he folds flat and bad politician  has no ticket so the bad politician says  well you know we've had our  disagreements but maybe you could put in  a good word to Saint Peter or do  something to help me out I here heavens  a nice place maybe we could go there  together we resolve our differences  whatever so the good politician having a  ticket and the pair scissors like any  respecting politician takes this ticket  and makes one complete straight cut and  it's going to get a lot of pieces that  are going to be hold this carefully so  put that down and he hands all these  pieces to the bad politician says here  you go pet politician has no idea what  they're for so he hands them the st.  Peter st. Peter starts unfolding the  pieces I wonder what shapes i get i hear  there's a cool-looking problem about  this uh-huh so it's a little hard to do  without a table so I'm going to have to  use the board first we get the letter H  then we get to it right  put these down here then we get the  letter e okay then we get the letter L  and the letter L and if it's not a table  at arrange it and it clearly st. Peter  is not happy and the bad politician gets  sent straight to hell the good  politician hung on to one piece cleverly  and this ticket is still more or less  intact and he gets into it okay that's  the magic trick it's very simple full  bang it's easy to remember I mean known  this remember as this even for years and  you get exactly those pieces so pretty  cool of course from a mathematical  standpoint this is a little unsatisfying  so come on there's three pieces for the  h three pieces for the e surely you  could do better and from a rectangle you  can kind of do better uh-huh it's a  little awkward because the pieces are  not all the same size but is one scissor  cut so all the vertices are even degree  you cut it and you get all these pieces  which if I were more practiced at this I  could immediately pick out which one's  the H looks like okay I'll just do them  out of order here is the cross kind of a  cross not quite perfect proportions but  good enough here is this is the letter e  this one's actually more impressive when  you don't have a table because you can't  tell that they're all different sizes at  the letter e we've got the letter these  look like else kind of small else but  there you go through different  orientations and then we've got the  letter H there you go okay so that's our  new and improved version using  universality of folding cut so I gives  you some idea of a we're folding cut  came from and in recent times is the  magic community I mentioned Harry  Houdini did some tricks and Gerald low  but there's a bunch of these tricks  around and kind of cool saw the  checkerboard trick  and the the previous he'll trick so  those are some examples now we move on  to the disk packing method so I have one  question about this which is how exactly  do we go from the disk packing to yeah  cool proof the cool proof bro so how do  we go from the disk packing to the  decomposition into triangles and  quadrilaterals so I thought I'd just  review this slide we start with our  graph we offset it by some tiny Epsilon  that's to get things off the line  basically then we do this disk packing  and remember roughly how the disk  packing work so we put some disks at  each of the centers here it's a little  awkward cuz of the offset but you put  one at each of these four corners also  the corners of the piece of paper why  not then you also pack small enough  distal ong the edge so that you cover  the edge by diameters of the disks i do  that the same on all the sides basically  you try to put a big one in but if that  big one intersects you decompose it into  half that's how the algorithm works so  now you've covered the vertices in the  edges but you may have big gaps like  this too many too many sides on this gap  and so then you just greedily put the  largest disk you can in those gaps until  all you're left with and these yellow  gaps are triangles and quadrilaterals or  three sided gaps and for side of gaps  and then all we do is draw this red  graph by putting a vertex at the center  of each disk this is what the question  was about and then whenever two disks  touch we call this kissing disks for  historical reasons because they're just  barely touching I guess at their lips  I've got a lot of lips for disks uh-huh  we're in flatland so then whenever they  touch we draw the edge between the two  centers so that decomposes this piece of  paper into parts and because the gaps  are three sides those will correspond to  triangles or four sides those will  correspond to quadrilaterals so that's  all and then we put rabbit ears in here  and line Universal quad molecules in  there and it folds flat it will align  all of the red edges plus the  black edges on the outside and it will  align all these inner red edges with  these black edges and then we do the  sink folds that we mentioned and that  will get one of these out of the way of  the other and we'll end up aligning just  these cut edges so that's the quick  review that method so one question is  how do you allocate the disks is there  sort of a best way and I guess there  could be several measures maybe you  don't want really tiny discs because  that tends to lead to very tiny folds  but the standard measure here is how  many disks do you need because that will  reduce the number of folds so can you  minimize the number of discs and in  general it's known roughly how many  discs you need and the algorithm I just  told you achieves that number of discs  so first I'll give you the number number  of this is proportional to the integral  and say it was an easy bound but it is  the right answer okay there's a lot of  notation in here and this one you should  definitely not know let's you've taken a  computational geometry class even then  you probably wouldn't know it it's not  that common mostly comes up in meshing  and disk packing so it's pretty specific  it's called local feature size and this  is a once i define it is actually fairly  intuitive this bound local feature size  at X so we're imagining some kind of  polygon here and at every point of the  polygon let's look at a vertex here but  it works for any point along the  boundary actually maybe not even on the  boundary but we're interested here this  this notation del P is the boundary of  the polygon so this is Del P that just  means you know edges and vertices all of  those points so we take some point X on  the boundary and we look at the smallest  disk centered at that point that touches  another feature  so that's the feature size features here  are edges of the polygon that's all we  have now of course no matter what size  disk you hit this feature you hit this  feature so those don't count those are  incident features so in general it is  the smallest radius disk that hits a non  incident feature on incident feature  which is an edge so here as soon as the  disk gets this big it hits this edge and  so that's the local feature size is this  this radius okay in general for every  point you can draw some size disk so  over here it's going to be a little bit  bigger you can maybe get up to here  actually I guess it should be non  adjacent it was going to be a little bit  awkward I also don't want to count this  edge because if I'm right here the  feature size is super tiny and I don't I  don't want out lfs ever to be zero so  you also have to skip the the adjacent  edges which means the local feature size  at this point is going to be more like  hard to draw circles it's going to be  more like that distance that's when you  hit a non adjacent feature sorry so you  have to exclude this edge that you're on  and the two adjacent ones otherwise this  definition doesn't quite work out so  then you measure this for all points  just infinitely many of course continue  them along the boundary that's the  integral over X in the boundary of P and  you take one over the local feature size  of X and you integrate that for all X it  gives you a number and it turns out some  constant times that number is the right  answer the algorithm I achieve which I  described which is put the biggest dis  you can or divide in half if you can't  we'll get within a constant factor of  the best possible bound of all possible  dis packings so that so this is pretty  much known you can get within a constant  factor of the optimal dis packing if you  want to you're counting disks not a  great number I can't say  and this or n squared discs because you  just can't do that if you have if you  have a piece of paper and some really  long here's your cut graph doesn't quite  go to the ends though then you've got to  have a tiny disc here and that's going  to require you to have a bunch of disks  even though this has constant size  you're going to need I don't know how  many dis here especially if your paper  is really narrow you're going to need a  lot of disks that's the best you can  hope for for disk packing actually  brought a little example of something  like that where you have your goal is  just to cut out a single segment in the  middle of the paper this is to  illustrate odd degree vertices these are  degree 1 vertices you might think wow  it's impossible for me to cut this thing  without I don't know I'm just cutting I  could cut with my exacto knife from here  to here if I want to make a complete  straight cut you can fold it like this  this is what the skeleton method would  give you you fold it like this and then  you use your laser to cut right along  the line with scissor cuts it's not so  bad right I cut along I cut slightly in  its pretty much cutting along a line  just not quite tiny mistake so I cut off  a line slightly thicker line and I get  my slit in the paper so I mean  mathematical cuts are not really that  bad you're just cutting a slightly  larger version than the single slit that  you wanted to make have to cut very  carefully okay that was number of disks  next question is how does all of this  relate to the tree method this is kind  of a neat question both methods skeleton  method and the dis packing method relate  to the tree method in different ways I  thought I would talk about that so first  let's compare the disk packing method  which we just talked about to the tree  method both of them use Universal  molecules this one only uses universal  molecules for triangles and quads we  know in general Universal molecule works  for any convex polygon this is an  example of a pentagon and it works it  has a gusset to get the tree lengths  right here we use gussets to kind of get  the tree lengths right as you may recall  we want the perpendicular  to go to the disk kissing points those  tangen sees because then they align with  the others and then we never get  perpendicular spiraling or doing other  bad things so number of creases becomes  proportional to the number of disks  number this is this messy thing so  that's that let's see what else here we  only use disks over here we could just  use disk but we there's also the ability  to use rivers so this is more general it  in that we can do bigger bigger polygons  in just quads and we can do rivers not  just circles here the input though is a  tree with specified lengths and it's  actually copy it's NP hard to place the  disks correctly we proved that last time  over here the input is a polygon it's  already been embedded on the sheet of  paper there's no really hard decisions  here except you have to fill the paper  with disks but that's not so hard so you  can do this in polynomial time  polynomial in even almost roughly this  time times a log factor you can find  this placement of disks so that's not  too hard whereas over here it's NP hard  to place a distally at least find a  displacement you could do so basically  the inputs differ and it's using the  same backbone or the same what do you  call it so front end the same back end I  suppose where we're decomposed after  we've decomposed into a bunch of  molecules you just fold the molecules  and you're basically done here we have  to do some sinks at the end but it's  using it for different purposes  otherwise they are very similar and  indeed this is inspired by by that okay  so that was the disk packing method then  there's the straight skeleton method  these look much more different but in  fact they are quite related if you do  fold and cut on a convex polygon it  gives you roughly the universal molecule  just with no gussets so here neither one  is more general than the other the  universal molecule is more general in  that it has gussets which lets it  control the tree topology you get and  control the lengths you get in the tree  over here we saw that the tree just  happens you can't control the tree it's  just a tool to prove the shadow tree  that's just a tool to prove that it  folds flat but you can do non convex  polygons which is much more general than  convex polygons so they each have their  own sort of unique claims to fame  natural question is can you combine them  and in fact you can and this was you may  remember a few bunch of lectures ago I  think lecture class for so I showed this  slide of the the meat of origami design  secrets volume 2 all right second  edition rather there's tree theory which  we talked about there was a box pleating  tree theory which should briefly showed  and there's this more general thing  called polygon packing and this is in  some sense the fusion of straight  skeleton method of fold and cut  basically the straight skeleton plus  tree theory and this is done jointly  between Robert Lang and the domains and  here's an example of kind of what it  looks like a little hard to see here but  I will point out that if you look at the  blue lines which are kind of the cut  lines here of course they're the active  paths this guy is non convex and if you  look at the red stuff in there it is  exactly the straight skeleton of that  polygon so those red lines are the  straight skeleton of this blue thing so  there you go we've got the folding cot  straight skeleton being used in that  setting you've got perpendicular all  over the place which is a annoying  feature of the skeleton method it is  inherited by this method and if you look  at something like this rectangle its  convex but you have gussets this is a  gusset this is a gusset gusset guess  it's a little hard to see but there's a  we basically split the rectangle into  this rectangle this square this  rectangle the square this rectangle so  this is really the fusion of the two  it's not yet been mathematically  formalized because we're first proving  that the tree method works then we will  finally get to this but if you want  these slightly more informal but  practical version read origami design  secrets and they'll explain all the  different cases and what we believe  Eve is a correct algorithm we just  haven't proved all the cases yet and you  can use it to design cool things in the  Box pleating setting and get more  efficient than if you required  rectangles or some convex shape cool so  that's that else do we have I have one  almost done one cool open question posed  by you is what if instead of cutting  with a straight line I cut with a  constant curvature line so a piece of a  circular arc then of course the things I  get will have circular arc boundaries  but can I get any circular arc boundary  with the with that fixed curvature or  are there limitations and at first I  thought the answer is clearly yes but uh  I do wonder about situations like this  can you align these two by folding I've  thought about enough to be sure but not  not totally sure where's if I have  things like this I think this is this is  good i mean i could fold here and in  general i could just kind of pretend  that there's a segment and the segment i  saw a fold and cut on the segments and  that will also align the circular arcs  that won't work in general but i feel  comfortable if they have the same  orientation it looks good if when they  flip orientations i'm not sure so neat  problem to think about and one final  thing is about flattening this is about  this question this picture so we said oh  great because we can solve fold and cut  we can also take the boundary that  polygon and flatten it like this of  course the folding of the interior here  which we're ignoring in general this  whole thing goes through three  dimensions until we end up here one  dimension up if we solve the 3d folding  cot problem where you have a polyhedron  you fold it through 4d back into 3d  which is flat so that you align all the  boundaries of the faces of that  polyhedron and then just take the  boundary of that motion it's like  flattening a polyhedron like cereal box  or whatever  and indeed that will find a folded state  of a polyhedron there's flat but of  course that motion you can't you can't  really use that motion because it goes  through 4d this one will go through 3d  so it's not if you want it to flatten  this linkage that's also not valid so  you get a folded state but you don't get  the folding motion if you could solve 3d  folding cut our current state is we  don't know how to solve 3d folding cat  we do know how to solve polyhedron  flattening so this this relation is just  kind of interest not doesn't imply  anything super useful an open problem is  so we know how to sell polyhedron  flattening as a folded state what we  don't know so much about is folding  motions there is a recent paper from  just last year which will continuously  flattened by continuous motion any  convex polyhedron but non-convex  polyhedra we still don't know these are  roughly hand-drawn figures an  interesting project would be to actually  animate their motion I think it would  look pretty cool and be much more  convincing to that it works I mean I am  convinced that it works but it's a lot  harder to see visually from these  pictures be really cool to see the  emotions the actual continuous animation  and a couple last project ideas one  would be to make a fold and cut alphabet  like the maze software that you just use  it would be fun if you just type in a  message and out comes the crease pattern  this is sort of like a special case of a  fold and cut design tool but in letters  it's a little bit easier and a simp  different challenge would be if I just  want to be able to fold a single letter  at a time can you always do it with  simple folds so can you design an  alphabet that is simply folding cuttable  that would be even nicer and there's  some magicians who worked on that but  they couldn't get all the letters so be  nice to do with just a few folds any  letter of the alphabet and any digit  this is one we did for our conference in  honor martin gardner gathering for  Gardner five back in 2002 and here's  another possible idea for project  although mostly it's an excuse to show  you really cool pictures this is Peter  callison and he takes usually a4 sheets  of paper cuts them with some probably an  exacto knife or something and then from  that material that  septic he acquires builds a 3d structure  on that sheet so you see all the  material that was used and these are  just amazing and I don't think you could  necessarily do this level of fidelity  from a folding cut but it would be cool  to do a folding cut and then build a  sculpture out of the slits that you made  and I mean it's really cool when he can  use the cut out parts to be shadows of  the things that he builds especially  when those shadows are more complicated  than the chief he actually assembles it  has a really cool effect of the monsters  in the closet Pandora's box or boxing  box in a box you can't see that here but  you see it unfolding uh-huh bones inside  the hand the old versus new cities using  the positive and the negative is  actually a little boat up there as you  can see so pretty pretty amazing stuff  and for free versus caged and so on  interesting project would be to try to  make art at a folding cot just doing one  cut instead of a zillion cuts that's it  any more questions yeah in the circle  the backing and mess with other  assignment  okay and the circle packing method how'd  you assign mountains and valleys that  was briefly covered in lecture there was  this first you take a spanning tree of  all of the molecules and you basically  fold each molecule into your parent in  the tree so you each one hat each one  has a direction and from that you can  extract them out in valley assignment  you go into the parent it's a mountain  there everyone else's Valley basically  you're reversing one perpendicular fold  per molecule I think maybe two triangles  is just one for quads you might be  reversing too so you start with all the  all of the straight skeleton being  mountains all the perpendiculars being  valleys and you have to flip a few to  just get them to fit together a little  more complicated than what I said but I  would need a slide to show you the  details it's in the textbook anything  else alright that's folding cut  you
 

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169
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170
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171
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172
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174
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175
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176
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177
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178
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179
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180
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181
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182
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183
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184
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185
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186
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187
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188
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189
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190
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191
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192
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193
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194
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195
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196
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197
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198
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199
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200
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201
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202
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203
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204
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205
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206
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207
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208
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209
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210
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211
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212
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213
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214
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215
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216
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217
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218
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219
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220
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221
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222
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223
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224
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225
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226
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227
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228
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229
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230
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231
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232
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233
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234
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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
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241
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242
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243
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244
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245
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246
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247
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248
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249
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250
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251
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252
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253
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254
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255
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256
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257
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258
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259
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260
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261
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262
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263
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264
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265
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266
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267
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268
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269
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270
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271
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272
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273
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274
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275
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276
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277
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278
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279
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280
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281
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282
00:13:22,060 --> 00:13:27,010

 

283
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284
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285
00:13:31,690 --> 00:13:36,220

 

286
00:13:36,220 --> 00:13:39,880

 

287
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288
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289
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290
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291
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292
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293
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294
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295
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296
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297
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298
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299
00:14:09,490 --> 00:14:12,130

 

300
00:14:12,130 --> 00:14:14,199

 

301
00:14:14,199 --> 00:14:15,970

 

302
00:14:15,970 --> 00:14:18,370

 

303
00:14:18,370 --> 00:14:20,380

 

304
00:14:20,380 --> 00:14:22,180

 

305
00:14:22,180 --> 00:14:24,819

 

306
00:14:24,819 --> 00:14:27,730

 

307
00:14:27,730 --> 00:14:31,360

 

308
00:14:31,360 --> 00:14:33,370

 

309
00:14:33,370 --> 00:14:37,449

 

310
00:14:37,449 --> 00:14:39,069

 

311
00:14:39,069 --> 00:14:42,579

 

312
00:14:42,579 --> 00:14:43,110

 

313
00:14:43,110 --> 00:14:47,070

 

314
00:14:47,070 --> 00:14:49,620

 

315
00:14:49,620 --> 00:14:52,110

 

316
00:14:52,110 --> 00:14:53,519

 

317
00:14:53,519 --> 00:14:56,190

 

318
00:14:56,190 --> 00:15:00,000

 

319
00:15:00,000 --> 00:15:02,490

 

320
00:15:02,490 --> 00:15:04,320

 

321
00:15:04,320 --> 00:15:06,630

 

322
00:15:06,630 --> 00:15:08,970

 

323
00:15:08,970 --> 00:15:11,090

 

324
00:15:11,090 --> 00:15:14,310

 

325
00:15:14,310 --> 00:15:16,170

 

326
00:15:16,170 --> 00:15:18,840

 

327
00:15:18,840 --> 00:15:22,110

 

328
00:15:22,110 --> 00:15:24,590

 

329
00:15:24,590 --> 00:15:28,670

 

330
00:15:28,670 --> 00:15:31,800

 

331
00:15:31,800 --> 00:15:34,680

 

332
00:15:34,680 --> 00:15:36,810

 

333
00:15:36,810 --> 00:15:38,820

 

334
00:15:38,820 --> 00:15:41,870

 

335
00:15:41,870 --> 00:15:45,120

 

336
00:15:45,120 --> 00:15:46,320

 

337
00:15:46,320 --> 00:15:48,690

 

338
00:15:48,690 --> 00:15:51,630

 

339
00:15:51,630 --> 00:15:53,250

 

340
00:15:53,250 --> 00:15:56,880

 

341
00:15:56,880 --> 00:15:58,500

 

342
00:15:58,500 --> 00:15:59,940

 

343
00:15:59,940 --> 00:16:02,100

 

344
00:16:02,100 --> 00:16:04,579

 

345
00:16:04,579 --> 00:16:06,780

 

346
00:16:06,780 --> 00:16:10,140

 

347
00:16:10,140 --> 00:16:14,130

 

348
00:16:14,130 --> 00:16:17,610

 

349
00:16:17,610 --> 00:16:22,079

 

350
00:16:22,079 --> 00:16:26,910

 

351
00:16:26,910 --> 00:16:28,890

 

352
00:16:28,890 --> 00:16:31,110

 

353
00:16:31,110 --> 00:16:33,329

 

354
00:16:33,329 --> 00:16:35,640

 

355
00:16:35,640 --> 00:16:38,040

 

356
00:16:38,040 --> 00:16:39,900

 

357
00:16:39,900 --> 00:16:41,400

 

358
00:16:41,400 --> 00:16:43,800

 

359
00:16:43,800 --> 00:16:46,019

 

360
00:16:46,019 --> 00:16:48,360

 

361
00:16:48,360 --> 00:16:49,829

 

362
00:16:49,829 --> 00:16:51,540

 

363
00:16:51,540 --> 00:16:53,250

 

364
00:16:53,250 --> 00:16:54,210

 

365
00:16:54,210 --> 00:16:56,430

 

366
00:16:56,430 --> 00:16:58,560

 

367
00:16:58,560 --> 00:17:00,600

 

368
00:17:00,600 --> 00:17:02,910

 

369
00:17:02,910 --> 00:17:05,189

 

370
00:17:05,189 --> 00:17:07,679

 

371
00:17:07,679 --> 00:17:09,300

 

372
00:17:09,300 --> 00:17:13,260

 

373
00:17:13,260 --> 00:17:14,699

 

374
00:17:14,699 --> 00:17:17,630

 

375
00:17:17,630 --> 00:17:29,550

 

376
00:17:29,550 --> 00:17:33,510

 

377
00:17:33,510 --> 00:17:35,250

 

378
00:17:35,250 --> 00:17:41,190

 

379
00:17:41,190 --> 00:17:43,470

 

380
00:17:43,470 --> 00:17:45,480

 

381
00:17:45,480 --> 00:17:47,070

 

382
00:17:47,070 --> 00:17:50,640

 

383
00:17:50,640 --> 00:17:52,770

 

384
00:17:52,770 --> 00:17:54,720

 

385
00:17:54,720 --> 00:17:56,760

 

386
00:17:56,760 --> 00:17:59,280

 

387
00:17:59,280 --> 00:18:02,250

 

388
00:18:02,250 --> 00:18:07,110

 

389
00:18:07,110 --> 00:18:09,480

 

390
00:18:09,480 --> 00:18:11,100

 

391
00:18:11,100 --> 00:18:12,780

 

392
00:18:12,780 --> 00:18:15,240

 

393
00:18:15,240 --> 00:18:17,610

 

394
00:18:17,610 --> 00:18:19,260

 

395
00:18:19,260 --> 00:18:24,530

 

396
00:18:24,530 --> 00:18:29,190

 

397
00:18:29,190 --> 00:18:30,690

 

398
00:18:30,690 --> 00:18:32,580

 

399
00:18:32,580 --> 00:18:34,800

 

400
00:18:34,800 --> 00:18:37,590

 

401
00:18:37,590 --> 00:18:39,570

 

402
00:18:39,570 --> 00:18:40,890

 

403
00:18:40,890 --> 00:18:42,690

 

404
00:18:42,690 --> 00:18:47,370

 

405
00:18:47,370 --> 00:18:49,170

 

406
00:18:49,170 --> 00:18:52,890

 

407
00:18:52,890 --> 00:18:54,630

 

408
00:18:54,630 --> 00:18:57,180

 

409
00:18:57,180 --> 00:18:58,680

 

410
00:18:58,680 --> 00:19:01,350

 

411
00:19:01,350 --> 00:19:03,690

 

412
00:19:03,690 --> 00:19:05,669

 

413
00:19:05,669 --> 00:19:07,350

 

414
00:19:07,350 --> 00:19:08,760

 

415
00:19:08,760 --> 00:19:09,720

 

416
00:19:09,720 --> 00:19:11,610

 

417
00:19:11,610 --> 00:19:14,280

 

418
00:19:14,280 --> 00:19:15,870

 

419
00:19:15,870 --> 00:19:18,900

 

420
00:19:18,900 --> 00:19:20,550

 

421
00:19:20,550 --> 00:19:22,920

 

422
00:19:22,920 --> 00:19:25,140

 

423
00:19:25,140 --> 00:19:28,170

 

424
00:19:28,170 --> 00:19:30,810

 

425
00:19:30,810 --> 00:19:32,900

 

426
00:19:32,900 --> 00:19:36,840

 

427
00:19:36,840 --> 00:19:39,750

 

428
00:19:39,750 --> 00:19:45,420

 

429
00:19:45,420 --> 00:19:53,220

 

430
00:19:53,220 --> 00:19:59,310

 

431
00:19:59,310 --> 00:20:01,110

 

432
00:20:01,110 --> 00:20:16,560

 

433
00:20:16,560 --> 00:20:18,419

 

434
00:20:18,419 --> 00:20:20,749

 

435
00:20:20,749 --> 00:20:23,810

 

436
00:20:23,810 --> 00:20:27,690

 

437
00:20:27,690 --> 00:20:29,999

 

438
00:20:29,999 --> 00:20:32,700

 

439
00:20:32,700 --> 00:20:34,019

 

440
00:20:34,019 --> 00:20:36,930

 

441
00:20:36,930 --> 00:20:40,080

 

442
00:20:40,080 --> 00:20:43,740

 

443
00:20:43,740 --> 00:20:45,360

 

444
00:20:45,360 --> 00:20:46,799

 

445
00:20:46,799 --> 00:20:48,659

 

446
00:20:48,659 --> 00:20:51,690

 

447
00:20:51,690 --> 00:20:55,049

 

448
00:20:55,049 --> 00:20:56,639

 

449
00:20:56,639 --> 00:20:57,690

 

450
00:20:57,690 --> 00:21:00,090

 

451
00:21:00,090 --> 00:21:04,169

 

452
00:21:04,169 --> 00:21:05,190

 

453
00:21:05,190 --> 00:21:09,090

 

454
00:21:09,090 --> 00:21:11,460

 

455
00:21:11,460 --> 00:21:16,980

 

456
00:21:16,980 --> 00:21:19,100

 

457
00:21:19,100 --> 00:21:21,889

 

458
00:21:21,889 --> 00:21:26,460

 

459
00:21:26,460 --> 00:21:27,539

 

460
00:21:27,539 --> 00:21:29,100

 

461
00:21:29,100 --> 00:21:31,259

 

462
00:21:31,259 --> 00:21:32,279

 

463
00:21:32,279 --> 00:21:36,210

 

464
00:21:36,210 --> 00:21:43,510

 

465
00:21:43,510 --> 00:21:54,150

 

466
00:21:54,150 --> 00:21:57,070

 

467
00:21:57,070 --> 00:22:02,980

 

468
00:22:02,980 --> 00:22:05,380

 

469
00:22:05,380 --> 00:22:25,600

 

470
00:22:25,600 --> 00:22:28,600

 

471
00:22:28,600 --> 00:22:30,640

 

472
00:22:30,640 --> 00:22:35,890

 

473
00:22:35,890 --> 00:22:39,460

 

474
00:22:39,460 --> 00:22:41,620

 

475
00:22:41,620 --> 00:22:43,570

 

476
00:22:43,570 --> 00:22:46,720

 

477
00:22:46,720 --> 00:22:48,730

 

478
00:22:48,730 --> 00:22:50,680

 

479
00:22:50,680 --> 00:22:52,510

 

480
00:22:52,510 --> 00:22:56,799

 

481
00:22:56,799 --> 00:22:59,799

 

482
00:22:59,799 --> 00:23:02,409

 

483
00:23:02,409 --> 00:23:04,750

 

484
00:23:04,750 --> 00:23:06,130

 

485
00:23:06,130 --> 00:23:07,870

 

486
00:23:07,870 --> 00:23:09,960

 

487
00:23:09,960 --> 00:23:12,010

 

488
00:23:12,010 --> 00:23:13,990

 

489
00:23:13,990 --> 00:23:15,850

 

490
00:23:15,850 --> 00:23:18,520

 

491
00:23:18,520 --> 00:23:20,440

 

492
00:23:20,440 --> 00:23:23,110

 

493
00:23:23,110 --> 00:23:25,960

 

494
00:23:25,960 --> 00:23:27,610

 

495
00:23:27,610 --> 00:23:31,919

 

496
00:23:31,919 --> 00:23:37,600

 

497
00:23:37,600 --> 00:23:40,810

 

498
00:23:40,810 --> 00:23:43,720

 

499
00:23:43,720 --> 00:23:45,039

 

500
00:23:45,039 --> 00:23:46,360

 

501
00:23:46,360 --> 00:23:48,549

 

502
00:23:48,549 --> 00:23:55,330

 

503
00:23:55,330 --> 00:23:59,330

 

504
00:23:59,330 --> 00:24:02,389

 

505
00:24:02,389 --> 00:24:03,950

 

506
00:24:03,950 --> 00:24:07,580

 

507
00:24:07,580 --> 00:24:10,549

 

508
00:24:10,549 --> 00:24:12,560

 

509
00:24:12,560 --> 00:24:17,180

 

510
00:24:17,180 --> 00:24:18,799

 

511
00:24:18,799 --> 00:24:20,330

 

512
00:24:20,330 --> 00:24:23,720

 

513
00:24:23,720 --> 00:24:25,759

 

514
00:24:25,759 --> 00:24:28,330

 

515
00:24:28,330 --> 00:24:30,830

 

516
00:24:30,830 --> 00:24:32,749

 

517
00:24:32,749 --> 00:24:37,249

 

518
00:24:37,249 --> 00:24:39,680

 

519
00:24:39,680 --> 00:24:41,749

 

520
00:24:41,749 --> 00:24:43,850

 

521
00:24:43,850 --> 00:24:45,580

 

522
00:24:45,580 --> 00:24:50,690

 

523
00:24:50,690 --> 00:24:54,440

 

524
00:24:54,440 --> 00:24:57,609

 

525
00:24:57,609 --> 00:25:00,259

 

526
00:25:00,259 --> 00:25:02,930

 

527
00:25:02,930 --> 00:25:04,879

 

528
00:25:04,879 --> 00:25:07,789

 

529
00:25:07,789 --> 00:25:09,549

 

530
00:25:09,549 --> 00:25:11,989

 

531
00:25:11,989 --> 00:25:13,879

 

532
00:25:13,879 --> 00:25:17,899

 

533
00:25:17,899 --> 00:25:21,200

 

534
00:25:21,200 --> 00:25:24,440

 

535
00:25:24,440 --> 00:25:27,470

 

536
00:25:27,470 --> 00:25:32,200

 

537
00:25:32,200 --> 00:25:35,359

 

538
00:25:35,359 --> 00:25:37,129

 

539
00:25:37,129 --> 00:25:40,659

 

540
00:25:40,659 --> 00:25:44,149

 

541
00:25:44,149 --> 00:25:47,090

 

542
00:25:47,090 --> 00:25:50,869

 

543
00:25:50,869 --> 00:25:52,970

 

544
00:25:52,970 --> 00:25:54,649

 

545
00:25:54,649 --> 00:25:56,810

 

546
00:25:56,810 --> 00:26:01,879

 

547
00:26:01,879 --> 00:26:04,780

 

548
00:26:04,780 --> 00:26:06,400

 

549
00:26:06,400 --> 00:26:08,350

 

550
00:26:08,350 --> 00:26:10,720

 

551
00:26:10,720 --> 00:26:12,280

 

552
00:26:12,280 --> 00:26:14,500

 

553
00:26:14,500 --> 00:26:16,750

 

554
00:26:16,750 --> 00:26:19,540

 

555
00:26:19,540 --> 00:26:20,920

 

556
00:26:20,920 --> 00:26:23,230

 

557
00:26:23,230 --> 00:26:25,900

 

558
00:26:25,900 --> 00:26:26,980

 

559
00:26:26,980 --> 00:26:28,720

 

560
00:26:28,720 --> 00:26:30,160

 

561
00:26:30,160 --> 00:26:32,110

 

562
00:26:32,110 --> 00:26:34,090

 

563
00:26:34,090 --> 00:26:35,890

 

564
00:26:35,890 --> 00:26:39,400

 

565
00:26:39,400 --> 00:26:41,140

 

566
00:26:41,140 --> 00:26:44,320

 

567
00:26:44,320 --> 00:26:46,570

 

568
00:26:46,570 --> 00:26:48,250

 

569
00:26:48,250 --> 00:26:53,380

 

570
00:26:53,380 --> 00:26:57,010

 

571
00:26:57,010 --> 00:27:00,580

 

572
00:27:00,580 --> 00:27:02,500

 

573
00:27:02,500 --> 00:27:04,270

 

574
00:27:04,270 --> 00:27:08,230

 

575
00:27:08,230 --> 00:27:10,570

 

576
00:27:10,570 --> 00:27:11,740

 

577
00:27:11,740 --> 00:27:15,250

 

578
00:27:15,250 --> 00:27:16,990

 

579
00:27:16,990 --> 00:27:21,450

 

580
00:27:21,450 --> 00:27:33,930

 

581
00:27:33,930 --> 00:27:39,660

 

582
00:27:39,660 --> 00:27:47,800

 

583
00:27:47,800 --> 00:27:50,350

 

584
00:27:50,350 --> 00:27:52,660

 

585
00:27:52,660 --> 00:27:54,430

 

586
00:27:54,430 --> 00:27:55,960

 

587
00:27:55,960 --> 00:27:58,480

 

588
00:27:58,480 --> 00:28:02,080

 

589
00:28:02,080 --> 00:28:05,170

 

590
00:28:05,170 --> 00:28:08,830

 

591
00:28:08,830 --> 00:28:10,930

 

592
00:28:10,930 --> 00:28:13,990

 

593
00:28:13,990 --> 00:28:15,880

 

594
00:28:15,880 --> 00:28:18,910

 

595
00:28:18,910 --> 00:28:21,610

 

596
00:28:21,610 --> 00:28:23,410

 

597
00:28:23,410 --> 00:28:26,170

 

598
00:28:26,170 --> 00:28:28,360

 

599
00:28:28,360 --> 00:28:31,330

 

600
00:28:31,330 --> 00:28:33,940

 

601
00:28:33,940 --> 00:28:36,040

 

602
00:28:36,040 --> 00:28:38,080

 

603
00:28:38,080 --> 00:28:42,630

 

604
00:28:42,630 --> 00:28:45,910

 

605
00:28:45,910 --> 00:28:47,890

 

606
00:28:47,890 --> 00:28:52,570

 

607
00:28:52,570 --> 00:28:56,890

 

608
00:28:56,890 --> 00:28:58,750

 

609
00:28:58,750 --> 00:29:02,800

 

610
00:29:02,800 --> 00:29:04,330

 

611
00:29:04,330 --> 00:29:05,830

 

612
00:29:05,830 --> 00:29:08,110

 

613
00:29:08,110 --> 00:29:12,430

 

614
00:29:12,430 --> 00:29:16,030

 

615
00:29:16,030 --> 00:29:17,680

 

616
00:29:17,680 --> 00:29:19,420

 

617
00:29:19,420 --> 00:29:25,000

 

618
00:29:25,000 --> 00:29:27,010

 

619
00:29:27,010 --> 00:29:29,620

 

620
00:29:29,620 --> 00:29:32,050

 

621
00:29:32,050 --> 00:29:34,720

 

622
00:29:34,720 --> 00:29:36,010

 

623
00:29:36,010 --> 00:29:37,810

 

624
00:29:37,810 --> 00:29:39,520

 

625
00:29:39,520 --> 00:29:41,890

 

626
00:29:41,890 --> 00:29:43,330

 

627
00:29:43,330 --> 00:29:47,230

 

628
00:29:47,230 --> 00:29:49,450

 

629
00:29:49,450 --> 00:29:53,470

 

630
00:29:53,470 --> 00:29:56,680

 

631
00:29:56,680 --> 00:30:00,160

 

632
00:30:00,160 --> 00:30:04,660

 

633
00:30:04,660 --> 00:30:06,460

 

634
00:30:06,460 --> 00:30:08,020

 

635
00:30:08,020 --> 00:30:10,030

 

636
00:30:10,030 --> 00:30:12,640

 

637
00:30:12,640 --> 00:30:15,790

 

638
00:30:15,790 --> 00:30:18,340

 

639
00:30:18,340 --> 00:30:20,620

 

640
00:30:20,620 --> 00:30:23,230

 

641
00:30:23,230 --> 00:30:25,390

 

642
00:30:25,390 --> 00:30:28,000

 

643
00:30:28,000 --> 00:30:29,500

 

644
00:30:29,500 --> 00:30:31,600

 

645
00:30:31,600 --> 00:30:33,040

 

646
00:30:33,040 --> 00:30:35,830

 

647
00:30:35,830 --> 00:30:37,930

 

648
00:30:37,930 --> 00:30:41,020

 

649
00:30:41,020 --> 00:30:43,090

 

650
00:30:43,090 --> 00:30:44,650

 

651
00:30:44,650 --> 00:30:46,690

 

652
00:30:46,690 --> 00:30:50,920

 

653
00:30:50,920 --> 00:30:52,330

 

654
00:30:52,330 --> 00:30:54,640

 

655
00:30:54,640 --> 00:30:58,420

 

656
00:30:58,420 --> 00:31:00,280

 

657
00:31:00,280 --> 00:31:02,440

 

658
00:31:02,440 --> 00:31:04,240

 

659
00:31:04,240 --> 00:31:07,030

 

660
00:31:07,030 --> 00:31:09,310

 

661
00:31:09,310 --> 00:31:12,280

 

662
00:31:12,280 --> 00:31:14,350

 

663
00:31:14,350 --> 00:31:16,690

 

664
00:31:16,690 --> 00:31:18,700

 

665
00:31:18,700 --> 00:31:22,000

 

666
00:31:22,000 --> 00:31:25,060

 

667
00:31:25,060 --> 00:31:27,340

 

668
00:31:27,340 --> 00:31:30,220

 

669
00:31:30,220 --> 00:31:33,850

 

670
00:31:33,850 --> 00:31:36,340

 

671
00:31:36,340 --> 00:31:39,250

 

672
00:31:39,250 --> 00:31:41,500

 

673
00:31:41,500 --> 00:31:43,270

 

674
00:31:43,270 --> 00:31:45,340

 

675
00:31:45,340 --> 00:31:47,770

 

676
00:31:47,770 --> 00:31:50,110

 

677
00:31:50,110 --> 00:31:52,930

 

678
00:31:52,930 --> 00:31:55,750

 

679
00:31:55,750 --> 00:31:57,520

 

680
00:31:57,520 --> 00:31:59,620

 

681
00:31:59,620 --> 00:32:02,410

 

682
00:32:02,410 --> 00:32:03,790

 

683
00:32:03,790 --> 00:32:06,130

 

684
00:32:06,130 --> 00:32:07,870

 

685
00:32:07,870 --> 00:32:10,150

 

686
00:32:10,150 --> 00:32:15,880

 

687
00:32:15,880 --> 00:32:20,260

 

688
00:32:20,260 --> 00:32:22,510

 

689
00:32:22,510 --> 00:32:25,420

 

690
00:32:25,420 --> 00:32:26,620

 

691
00:32:26,620 --> 00:32:27,910

 

692
00:32:27,910 --> 00:32:29,620

 

693
00:32:29,620 --> 00:32:32,320

 

694
00:32:32,320 --> 00:32:34,450

 

695
00:32:34,450 --> 00:32:36,940

 

696
00:32:36,940 --> 00:32:39,100

 

697
00:32:39,100 --> 00:32:41,260

 

698
00:32:41,260 --> 00:32:43,270

 

699
00:32:43,270 --> 00:32:46,380

 

700
00:32:46,380 --> 00:32:52,090

 

701
00:32:52,090 --> 00:32:57,780

 

702
00:32:57,780 --> 00:33:00,760

 

703
00:33:00,760 --> 00:33:07,480

 

704
00:33:07,480 --> 00:33:09,730

 

705
00:33:09,730 --> 00:33:12,040

 

706
00:33:12,040 --> 00:33:13,630

 

707
00:33:13,630 --> 00:33:15,580

 

708
00:33:15,580 --> 00:33:17,980

 

709
00:33:17,980 --> 00:33:19,750

 

710
00:33:19,750 --> 00:33:25,600

 

711
00:33:25,600 --> 00:33:28,030

 

712
00:33:28,030 --> 00:33:31,510

 

713
00:33:31,510 --> 00:33:35,140

 

714
00:33:35,140 --> 00:33:40,540

 

715
00:33:40,540 --> 00:33:42,910

 

716
00:33:42,910 --> 00:33:44,620

 

717
00:33:44,620 --> 00:33:46,450

 

718
00:33:46,450 --> 00:33:49,270

 

719
00:33:49,270 --> 00:33:52,630

 

720
00:33:52,630 --> 00:33:56,020

 

721
00:33:56,020 --> 00:33:58,180

 

722
00:33:58,180 --> 00:34:01,420

 

723
00:34:01,420 --> 00:34:03,760

 

724
00:34:03,760 --> 00:34:08,620

 

725
00:34:08,620 --> 00:34:10,770

 

726
00:34:10,770 --> 00:34:12,570

 

727
00:34:12,570 --> 00:34:14,370

 

728
00:34:14,370 --> 00:34:17,040

 

729
00:34:17,040 --> 00:34:18,899

 

730
00:34:18,899 --> 00:34:20,280

 

731
00:34:20,280 --> 00:34:22,800

 

732
00:34:22,800 --> 00:34:34,860

 

733
00:34:34,860 --> 00:34:41,070

 

734
00:34:41,070 --> 00:34:45,810

 

735
00:34:45,810 --> 00:34:48,450

 

736
00:34:48,450 --> 00:34:51,659

 

737
00:34:51,659 --> 00:34:55,740

 

738
00:34:55,740 --> 00:34:58,560

 

739
00:34:58,560 --> 00:35:00,510

 

740
00:35:00,510 --> 00:35:06,470

 

741
00:35:06,470 --> 00:35:08,250

 

742
00:35:08,250 --> 00:35:10,940

 

743
00:35:10,940 --> 00:35:13,830

 

744
00:35:13,830 --> 00:35:15,870

 

745
00:35:15,870 --> 00:35:17,850

 

746
00:35:17,850 --> 00:35:20,580

 

747
00:35:20,580 --> 00:35:24,090

 

748
00:35:24,090 --> 00:35:25,830

 

749
00:35:25,830 --> 00:35:29,090

 

750
00:35:29,090 --> 00:35:32,040

 

751
00:35:32,040 --> 00:35:34,740

 

752
00:35:34,740 --> 00:35:37,980

 

753
00:35:37,980 --> 00:35:39,450

 

754
00:35:39,450 --> 00:35:41,280

 

755
00:35:41,280 --> 00:35:44,220

 

756
00:35:44,220 --> 00:35:46,290

 

757
00:35:46,290 --> 00:35:48,840

 

758
00:35:48,840 --> 00:35:50,520

 

759
00:35:50,520 --> 00:35:53,550

 

760
00:35:53,550 --> 00:35:56,190

 

761
00:35:56,190 --> 00:35:59,220

 

762
00:35:59,220 --> 00:36:02,370

 

763
00:36:02,370 --> 00:36:04,080

 

764
00:36:04,080 --> 00:36:05,880

 

765
00:36:05,880 --> 00:36:07,800

 

766
00:36:07,800 --> 00:36:09,690

 

767
00:36:09,690 --> 00:36:11,670

 

768
00:36:11,670 --> 00:36:13,740

 

769
00:36:13,740 --> 00:36:17,100

 

770
00:36:17,100 --> 00:36:18,600

 

771
00:36:18,600 --> 00:36:20,490

 

772
00:36:20,490 --> 00:36:23,190

 

773
00:36:23,190 --> 00:36:24,390

 

774
00:36:24,390 --> 00:36:27,000

 

775
00:36:27,000 --> 00:36:30,599

 

776
00:36:30,599 --> 00:36:32,940

 

777
00:36:32,940 --> 00:36:35,430

 

778
00:36:35,430 --> 00:36:38,010

 

779
00:36:38,010 --> 00:36:39,660

 

780
00:36:39,660 --> 00:36:41,190

 

781
00:36:41,190 --> 00:36:42,720

 

782
00:36:42,720 --> 00:36:45,089

 

783
00:36:45,089 --> 00:36:47,640

 

784
00:36:47,640 --> 00:36:48,930

 

785
00:36:48,930 --> 00:36:51,960

 

786
00:36:51,960 --> 00:36:53,430

 

787
00:36:53,430 --> 00:36:55,200

 

788
00:36:55,200 --> 00:36:57,269

 

789
00:36:57,269 --> 00:36:58,950

 

790
00:36:58,950 --> 00:37:00,059

 

791
00:37:00,059 --> 00:37:02,609

 

792
00:37:02,609 --> 00:37:04,200

 

793
00:37:04,200 --> 00:37:06,510

 

794
00:37:06,510 --> 00:37:09,120

 

795
00:37:09,120 --> 00:37:11,190

 

796
00:37:11,190 --> 00:37:13,049

 

797
00:37:13,049 --> 00:37:16,220

 

798
00:37:16,220 --> 00:37:19,260

 

799
00:37:19,260 --> 00:37:21,750

 

800
00:37:21,750 --> 00:37:23,609

 

801
00:37:23,609 --> 00:37:25,440

 

802
00:37:25,440 --> 00:37:30,170

 

803
00:37:30,170 --> 00:37:33,319

 

804
00:37:33,319 --> 00:37:38,700

 

805
00:37:38,700 --> 00:37:41,970

 

806
00:37:41,970 --> 00:37:44,269

 

807
00:37:44,269 --> 00:37:46,710

 

808
00:37:46,710 --> 00:37:48,990

 

809
00:37:48,990 --> 00:37:51,390

 

810
00:37:51,390 --> 00:37:53,279

 

811
00:37:53,279 --> 00:37:58,819

 

812
00:37:58,819 --> 00:38:02,160

 

813
00:38:02,160 --> 00:38:03,839

 

814
00:38:03,839 --> 00:38:06,299

 

815
00:38:06,299 --> 00:38:08,039

 

816
00:38:08,039 --> 00:38:10,859

 

817
00:38:10,859 --> 00:38:12,870

 

818
00:38:12,870 --> 00:38:14,339

 

819
00:38:14,339 --> 00:38:16,440

 

820
00:38:16,440 --> 00:38:18,480

 

821
00:38:18,480 --> 00:38:21,329

 

822
00:38:21,329 --> 00:38:24,150

 

823
00:38:24,150 --> 00:38:25,799

 

824
00:38:25,799 --> 00:38:27,359

 

825
00:38:27,359 --> 00:38:30,809

 

826
00:38:30,809 --> 00:38:32,849

 

827
00:38:32,849 --> 00:38:35,400

 

828
00:38:35,400 --> 00:38:36,960

 

829
00:38:36,960 --> 00:38:38,140

 

830
00:38:38,140 --> 00:38:41,470

 

831
00:38:41,470 --> 00:38:44,079

 

832
00:38:44,079 --> 00:38:45,839

 

833
00:38:45,839 --> 00:38:47,980

 

834
00:38:47,980 --> 00:38:50,079

 

835
00:38:50,079 --> 00:38:51,430

 

836
00:38:51,430 --> 00:38:55,390

 

837
00:38:55,390 --> 00:39:00,039

 

838
00:39:00,039 --> 00:39:02,650

 

839
00:39:02,650 --> 00:39:04,089

 

840
00:39:04,089 --> 00:39:06,430

 

841
00:39:06,430 --> 00:39:09,519

 

842
00:39:09,519 --> 00:39:12,970

 

843
00:39:12,970 --> 00:39:15,849

 

844
00:39:15,849 --> 00:39:18,640

 

845
00:39:18,640 --> 00:39:22,120

 

846
00:39:22,120 --> 00:39:25,230

 

847
00:39:25,230 --> 00:39:29,410

 

848
00:39:29,410 --> 00:39:30,940

 

849
00:39:30,940 --> 00:39:32,740

 

850
00:39:32,740 --> 00:39:35,319

 

851
00:39:35,319 --> 00:39:39,220

 

852
00:39:39,220 --> 00:39:40,660

 

853
00:39:40,660 --> 00:39:44,740

 

854
00:39:44,740 --> 00:39:49,269

 

855
00:39:49,269 --> 00:39:51,400

 

856
00:39:51,400 --> 00:39:53,019

 

857
00:39:53,019 --> 00:39:56,079

 

858
00:39:56,079 --> 00:39:58,750

 

859
00:39:58,750 --> 00:40:00,640

 

860
00:40:00,640 --> 00:40:06,309

 

861
00:40:06,309 --> 00:40:09,220

 

862
00:40:09,220 --> 00:40:11,620

 

863
00:40:11,620 --> 00:40:12,849

 

864
00:40:12,849 --> 00:40:14,289

 

865
00:40:14,289 --> 00:40:15,609

 

866
00:40:15,609 --> 00:40:17,620

 

867
00:40:17,620 --> 00:40:18,490

 

868
00:40:18,490 --> 00:40:20,440

 

869
00:40:20,440 --> 00:40:25,329

 

870
00:40:25,329 --> 00:40:27,519

 

871
00:40:27,519 --> 00:40:28,930

 

872
00:40:28,930 --> 00:40:31,269

 

873
00:40:31,269 --> 00:40:33,760

 

874
00:40:33,760 --> 00:40:37,150

 

875
00:40:37,150 --> 00:40:39,069

 

876
00:40:39,069 --> 00:40:43,000

 

877
00:40:43,000 --> 00:40:44,260

 

878
00:40:44,260 --> 00:40:45,940

 

879
00:40:45,940 --> 00:40:47,349

 

880
00:40:47,349 --> 00:40:49,390

 

881
00:40:49,390 --> 00:40:51,700

 

882
00:40:51,700 --> 00:40:53,230

 

883
00:40:53,230 --> 00:40:56,440

 

884
00:40:56,440 --> 00:40:58,810

 

885
00:40:58,810 --> 00:41:00,040

 

886
00:41:00,040 --> 00:41:01,870

 

887
00:41:01,870 --> 00:41:03,640

 

888
00:41:03,640 --> 00:41:06,099

 

889
00:41:06,099 --> 00:41:09,150

 

890
00:41:09,150 --> 00:41:11,530

 

891
00:41:11,530 --> 00:41:14,859

 

892
00:41:14,859 --> 00:41:17,410

 

893
00:41:17,410 --> 00:41:20,410

 

894
00:41:20,410 --> 00:41:23,200

 

895
00:41:23,200 --> 00:41:25,599

 

896
00:41:25,599 --> 00:41:27,910

 

897
00:41:27,910 --> 00:41:29,950

 

898
00:41:29,950 --> 00:41:31,359

 

899
00:41:31,359 --> 00:41:32,890

 

900
00:41:32,890 --> 00:41:34,450

 

901
00:41:34,450 --> 00:41:37,359

 

902
00:41:37,359 --> 00:41:38,710

 

903
00:41:38,710 --> 00:41:41,790

 

904
00:41:41,790 --> 00:41:44,829

 

905
00:41:44,829 --> 00:41:47,589

 

906
00:41:47,589 --> 00:41:50,170

 

907
00:41:50,170 --> 00:41:51,640

 

908
00:41:51,640 --> 00:41:54,070

 

909
00:41:54,070 --> 00:41:55,900

 

910
00:41:55,900 --> 00:41:59,320

 

911
00:41:59,320 --> 00:42:05,950

 

912
00:42:05,950 --> 00:42:07,900

 

913
00:42:07,900 --> 00:42:10,480

 

914
00:42:10,480 --> 00:42:13,750

 

915
00:42:13,750 --> 00:42:16,000

 

916
00:42:16,000 --> 00:42:18,280

 

917
00:42:18,280 --> 00:42:21,130

 

918
00:42:21,130 --> 00:42:22,570

 

919
00:42:22,570 --> 00:42:25,060

 

920
00:42:25,060 --> 00:42:26,200

 

921
00:42:26,200 --> 00:42:28,329

 

922
00:42:28,329 --> 00:42:33,250

 

923
00:42:33,250 --> 00:42:36,190

 

924
00:42:36,190 --> 00:42:38,230

 

925
00:42:38,230 --> 00:42:39,910

 

926
00:42:39,910 --> 00:42:41,950

 

927
00:42:41,950 --> 00:42:43,990

 

928
00:42:43,990 --> 00:42:48,520

 

929
00:42:48,520 --> 00:42:50,070

 

930
00:42:50,070 --> 00:42:52,089

 

931
00:42:52,089 --> 00:42:54,099

 

932
00:42:54,099 --> 00:42:55,570

 

933
00:42:55,570 --> 00:42:57,490

 

934
00:42:57,490 --> 00:42:59,680

 

935
00:42:59,680 --> 00:43:02,650

 

936
00:43:02,650 --> 00:43:04,390

 

937
00:43:04,390 --> 00:43:05,650

 

938
00:43:05,650 --> 00:43:06,849

 

939
00:43:06,849 --> 00:43:10,120

 

940
00:43:10,120 --> 00:43:12,160

 

941
00:43:12,160 --> 00:43:14,109

 

942
00:43:14,109 --> 00:43:15,339

 

943
00:43:15,339 --> 00:43:20,410

 

944
00:43:20,410 --> 00:43:31,120

 

945
00:43:31,120 --> 00:43:33,640

 

946
00:43:33,640 --> 00:43:37,059

 

947
00:43:37,059 --> 00:43:39,460

 

948
00:43:39,460 --> 00:43:42,160

 

949
00:43:42,160 --> 00:43:45,039

 

950
00:43:45,039 --> 00:43:46,539

 

951
00:43:46,539 --> 00:43:48,339

 

952
00:43:48,339 --> 00:43:51,270

 

953
00:43:51,270 --> 00:43:54,490

 

954
00:43:54,490 --> 00:43:56,020

 

955
00:43:56,020 --> 00:44:03,930

 

956
00:44:03,930 --> 00:44:08,140

 

957
00:44:08,140 --> 00:44:10,960

 

958
00:44:10,960 --> 00:44:13,000

 

959
00:44:13,000 --> 00:44:18,490

 

960
00:44:18,490 --> 00:44:20,019

 

961
00:44:20,019 --> 00:44:21,549

 

962
00:44:21,549 --> 00:44:23,529

 

963
00:44:23,529 --> 00:44:24,970

 

964
00:44:24,970 --> 00:44:26,380

 

965
00:44:26,380 --> 00:44:28,329

 

966
00:44:28,329 --> 00:44:29,470

 

967
00:44:29,470 --> 00:44:30,789

 

968
00:44:30,789 --> 00:44:33,819

 

969
00:44:33,819 --> 00:44:39,880

 

970
00:44:39,880 --> 00:44:41,859

 

971
00:44:41,859 --> 00:44:44,740

 

972
00:44:44,740 --> 00:44:46,690

 

973
00:44:46,690 --> 00:44:48,279

 

974
00:44:48,279 --> 00:44:51,700

 

975
00:44:51,700 --> 00:44:54,339

 

976
00:44:54,339 --> 00:44:56,170

 

977
00:44:56,170 --> 00:44:57,519

 

978
00:44:57,519 --> 00:45:00,279

 

979
00:45:00,279 --> 00:45:02,319

 

980
00:45:02,319 --> 00:45:03,430

 

981
00:45:03,430 --> 00:45:06,880

 

982
00:45:06,880 --> 00:45:08,650

 

983
00:45:08,650 --> 00:45:11,650

 

984
00:45:11,650 --> 00:45:14,920

 

985
00:45:14,920 --> 00:45:16,359

 

986
00:45:16,359 --> 00:45:18,490

 

987
00:45:18,490 --> 00:45:19,089

 

988
00:45:19,089 --> 00:45:23,019

 

989
00:45:23,019 --> 00:45:27,819

 

990
00:45:27,819 --> 00:45:29,620

 

991
00:45:29,620 --> 00:45:30,940

 

992
00:45:30,940 --> 00:45:32,799

 

993
00:45:32,799 --> 00:45:34,630

 

994
00:45:34,630 --> 00:45:37,960

 

995
00:45:37,960 --> 00:45:39,400

 

996
00:45:39,400 --> 00:45:41,589

 

997
00:45:41,589 --> 00:45:44,019

 

998
00:45:44,019 --> 00:45:45,579

 

999
00:45:45,579 --> 00:45:47,589

 

1000
00:45:47,589 --> 00:45:49,989

 

1001
00:45:49,989 --> 00:45:52,329

 

1002
00:45:52,329 --> 00:45:56,019

 

1003
00:45:56,019 --> 00:45:57,880

 

1004
00:45:57,880 --> 00:46:00,130

 

1005
00:46:00,130 --> 00:46:01,660

 

1006
00:46:01,660 --> 00:46:04,210

 

1007
00:46:04,210 --> 00:46:08,440

 

1008
00:46:08,440 --> 00:46:10,859

 

1009
00:46:10,859 --> 00:46:13,120

 

1010
00:46:13,120 --> 00:46:15,249

 

1011
00:46:15,249 --> 00:46:17,710

 

1012
00:46:17,710 --> 00:46:19,150

 

1013
00:46:19,150 --> 00:46:20,950

 

1014
00:46:20,950 --> 00:46:22,450

 

1015
00:46:22,450 --> 00:46:25,120

 

1016
00:46:25,120 --> 00:46:26,289

 

1017
00:46:26,289 --> 00:46:27,549

 

1018
00:46:27,549 --> 00:46:29,370

 

1019
00:46:29,370 --> 00:46:32,650

 

1020
00:46:32,650 --> 00:46:38,109

 

1021
00:46:38,109 --> 00:46:40,089

 

1022
00:46:40,089 --> 00:46:41,950

 

1023
00:46:41,950 --> 00:46:43,150

 

1024
00:46:43,150 --> 00:46:45,249

 

1025
00:46:45,249 --> 00:46:46,930

 

1026
00:46:46,930 --> 00:46:49,120

 

1027
00:46:49,120 --> 00:46:51,910

 

1028
00:46:51,910 --> 00:46:53,950

 

1029
00:46:53,950 --> 00:46:55,420

 

1030
00:46:55,420 --> 00:46:57,430

 

1031
00:46:57,430 --> 00:46:59,319

 

1032
00:46:59,319 --> 00:47:01,059

 

1033
00:47:01,059 --> 00:47:03,430

 

1034
00:47:03,430 --> 00:47:04,719

 

1035
00:47:04,719 --> 00:47:06,579

 

1036
00:47:06,579 --> 00:47:08,410

 

1037
00:47:08,410 --> 00:47:09,999

 

1038
00:47:09,999 --> 00:47:11,920

 

1039
00:47:11,920 --> 00:47:13,420

 

1040
00:47:13,420 --> 00:47:18,069

 

1041
00:47:18,069 --> 00:47:19,900

 

1042
00:47:19,900 --> 00:47:21,579

 

1043
00:47:21,579 --> 00:47:23,739

 

1044
00:47:23,739 --> 00:47:26,410

 

1045
00:47:26,410 --> 00:47:29,529

 

1046
00:47:29,529 --> 00:47:31,930

 

1047
00:47:31,930 --> 00:47:33,009

 

1048
00:47:33,009 --> 00:47:36,880

 

1049
00:47:36,880 --> 00:47:38,229

 

1050
00:47:38,229 --> 00:47:39,909

 

1051
00:47:39,909 --> 00:47:43,899

 

1052
00:47:43,899 --> 00:47:45,459

 

1053
00:47:45,459 --> 00:47:47,620

 

1054
00:47:47,620 --> 00:47:49,359

 

1055
00:47:49,359 --> 00:47:52,559

 

1056
00:47:52,559 --> 00:47:55,839

 

1057
00:47:55,839 --> 00:47:58,059

 

1058
00:47:58,059 --> 00:48:00,729

 

1059
00:48:00,729 --> 00:48:02,169

 

1060
00:48:02,169 --> 00:48:04,479

 

1061
00:48:04,479 --> 00:48:06,789

 

1062
00:48:06,789 --> 00:48:11,589

 

1063
00:48:11,589 --> 00:48:13,659

 

1064
00:48:13,659 --> 00:48:17,439

 

1065
00:48:17,439 --> 00:48:26,079

 

1066
00:48:26,079 --> 00:48:27,399

 

1067
00:48:27,399 --> 00:48:30,159

 

1068
00:48:30,159 --> 00:48:33,999

 

1069
00:48:33,999 --> 00:48:39,120

 

1070
00:48:39,120 --> 00:48:40,929

 

1071
00:48:40,929 --> 00:48:43,149

 

1072
00:48:43,149 --> 00:48:46,989

 

1073
00:48:46,989 --> 00:48:51,609

 

1074
00:48:51,609 --> 00:48:54,249

 

1075
00:48:54,249 --> 00:48:54,259

 

1076
00:48:54,259 --> 00:48:57,180


1077
00:48:57,180 --> 00:48:59,530

 

1078
00:48:59,530 --> 00:49:00,640

 

1079
00:49:00,640 --> 00:49:03,160

 

1080
00:49:03,160 --> 00:49:04,810

 

1081
00:49:04,810 --> 00:49:07,240

 

1082
00:49:07,240 --> 00:49:10,090

 

1083
00:49:10,090 --> 00:49:12,760

 

1084
00:49:12,760 --> 00:49:14,320

 

1085
00:49:14,320 --> 00:49:15,640

 

1086
00:49:15,640 --> 00:49:17,650

 

1087
00:49:17,650 --> 00:49:19,510

 

1088
00:49:19,510 --> 00:49:21,010

 

1089
00:49:21,010 --> 00:49:25,870

 

1090
00:49:25,870 --> 00:49:27,880

 

1091
00:49:27,880 --> 00:49:29,980

 

1092
00:49:29,980 --> 00:49:32,020

 

1093
00:49:32,020 --> 00:49:33,340

 

1094
00:49:33,340 --> 00:49:35,050

 

1095
00:49:35,050 --> 00:49:38,170

 

1096
00:49:38,170 --> 00:49:39,490

 

1097
00:49:39,490 --> 00:49:41,980

 

1098
00:49:41,980 --> 00:49:45,730

 

1099
00:49:45,730 --> 00:49:56,440

 

1100
00:49:56,440 --> 00:49:56,450

 

1101
00:49:56,450 --> 00:49:58,510