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all right lecture 14 was about two main  topics I guess we had slender adorned  chains sort of fatter linkages and then  hinge dissection most of our time was  actually spent with the slender  adornments and proving that that works  but most of our questions today are  about hinge dissections because that's  kind of the most fun and there's a lot  more to say about them so first question  is is there any software for hinge  dissections and short answer is no  surprisingly so this would definitely be  a cool project possibility there are a  bunch of examples let me switch to these  on the web just sort of random examples  cool dissections people thought were so  neat they wanted to animate them and so  they basically constructed where the  coordinates were over time in  Mathematica and then put it on the web  as a illustration of that so this is a  equilateral triangle - a hexagon regular  hexagon this is a hinge dissection by  Grieg Fredrickson and then is drawn by  rick Mabrey here's another one for an  equilateral triangle - a pentagon pretty  cool they're hinged in a tree-like  fashion even just kind of unusual and  these are Gregg Fredrickson is one of  the Masters of hinge dissections and  dissections in general it's probably the  master of dissections in general as he  has three books or different kinds of  dissections this is actually a hinge  dissection on the cover here of the the  purple and pink pieces hinged like this  into into a smaller star from a the  outline of a big star to the interior of  a smaller star and then this star fits  nicely inside this is another hinge  dissection this this book is entirely  about hinge dissection so they're not  just the kinds we've seen another kind  called twist hinging which I think this  is a twist Inge piece flips around the  other side and then there's a third book  about a different kind of hinge  dissection that's more of a surface  hinge dissection where you've got to  you've got the front and back of this  surface and you and you fold them  like piano hinges with hinges in the  plane all very cool books you should  check them out if you want to know more  about dissections they're more about  here cool examples some design  techniques for how to make them I'll  show you one such design technique later  on today and but not a ton of Theory  here in particular because there wasn't  a ton of Theory when these books were  written so that's some hinge dissections  as I said cool project would be to make  a general tool for animating hinge  dissections there's only a handful out  there greg has digital files of lots of  his hinge dissections probably be  willing to share them I haven't talked  to him about it if there was a good  engine for animating them I think would  be cool even cooler would be to  implement the slender an orangeade  business take one of these hinge  dissections maybe they just hinge but  sometimes there's collisions and but we  already know if you refine these guys to  be slender which you can do if they're  triangulated you can do it with only  losing a factor of three in the number  of pieces it cool to implement that and  then you can do the slender and orange  folding via CDR which I have an  implementation of or it's not that hard  to build one if you have a LP solver so  various project possibilities another  cool project would be to just design  more hinge dissections there's still  interesting questions either use fewer  pieces or just make elegant designs  related to the implementation idea a  particular family of hinge dissections  that could be fun to implement are  embodied by this alphabet I showed this  in lecture you can take the letter six  and convert into a square convert into  an eight and convert into four and four  into a nine via these 128 pieces I  didn't talk much about this theorem thus  I thought I'd give you a little sketch  of how this works it's actually very  simple to construct the folded states of  these hinge dissections and it could be  an interesting thing to implement and  it's also just kind of fun this is way  earlier 1999 way before we knew that  everything was possible we could at  least do all polyominoes of the given  size so let's let's just think about  polyominoes are the Polly hexes Polly  aemon's we have equilateral triangles  and these are called Polly abalos  for silly reasons basically by analogy  to a Diablo which is a juggling device  you can hinge dissect any of them here  you take each square and you cut it into  two half squares then you hinge them  together like this  this is one two three four five six  seven eight so this will make any four  square object any Tetris piece and  generally take two n pieces and you can  make any and omona and the way you prove  that that is universal that it can fold  into anything it's not so clear from  this picture but it's actually really  easy to prove by induction so the first  thing to do in this inductive proof is  to check that you can do it for N equals  1 ok that may sound trivial but this is  actually core the key property you need  in a hinge dissection of a single square  into your general family is that there's  a hinge visible on every edge of the  object so here this hinge kind of covers  this edge it covers that edge so both of  these edges have hinges on them and the  other two a it just have a hinge on them  they happen to be shared hinges but  that's ok and each of these that's true  the triangle it's a little more awkward  do you actually need two hinges to cover  the three sides but you can but you only  need these two pieces one of those  non-convex it's may be hard to fold  continuously but you'd refine it if you  wanted to do slender adornments so let's  not worry about continuous motion yet so  that's the base case of the induction  how do I do for N equals 1 now  inductively if I have some shape I want  to build I'll take what I call the dual  graph of that shape so make a vertex for  every square connect them together if  they share an edge the square square an  edge and then look at a spanning tree of  that shape so just cut some of these  edges until you have tree connectivity  among among those squares every tree has  at least two leaves except in the fall  but every mathematical tree always has  to at least two leaves like this is a  leaf if I cut here this would also be a  leaf leaf is a degree 1 vertex  so that's the square that only shares  one side so pluck off that leaf remove  that square the resulting n minus 1  squares by assumption can be made by  this hinge dissection with 2 times n  minus 1 pieces so now we just have to  attach this guy on and here's a figure  for that down at the bottom this is the  same thing for triangles if you and and  poly abalos are in the upper right so  you have some existing hinge dissection  you don't really know what it's like and  you want to add this leaf back on so it  shares one edge with one guy now this  guy could be oriented this way or it  could be oriented this way but it's the  same by reflection so let's say it's  oriented this way we we know the square  is made up by two half squares by  induction and so we know that there's a  hinge here now this hinge connects to  some things in this case it's some to  some T prime could be here could be up  here and all we do is stick s on here  now s can rotate we have our solution  for one guy and there's two different  orientations for him we're going to  choose this orientation because it puts  this hinge right there and so once we do  that normally this would be a cycle and  this thing would be a cycle through here  but we just redo the hinges in here so  that this the cycle gets bigger and the  thing the important thing to verify is  that the orientations the triangles are  the same just like the hinge dissection  picture I showed we always go from the  base edge to the next base edge to the  next base edge of these right isosceles  triangles and all the triangles are on  the outside of the cycle so we actually  construct a cyclic hinge dissection then  at the end you could break it and make  it a path but and this one is even  slender remember right triangles are  slender barely you can look at all the  inward normals they hit the base edge so  this will even move continuously if it's  an open chain for closed chains we don't  know  so that's polyominoes poly aemon's are  similar  pretty much the same thing you just in  this case you might have hinges on both  sides but you rotate this thing so one  of the hinges lines up and you just  reconnect the hinges it's not hard to  show you can always do that the hinges  will never cross and this proves that  these folded states exist and then we  use the slender stuff to do and  continuous motions actually when this  paper was written we didn't have slender  adornments at back at 99 even in 2005  when the journal version appeared so  it's only now that we know that motions  are possible by in this case directly in  this case with some refinement so I  thought that would just be fun to see  you can do some other crazy things so  this is a hinge dissection from any four  aemond this of four equilateral  triangles drawn together to any four AMA  no this is a Tetris piece it's  essentially a superposition of this idea  with you see and here these these four  lines make the hinge dissection of  dudeney from 1902 from an equilateral  triangle to a square and with some extra  stuff added in this is maybe a  foreshadowing of the idea of refinement  although we didn't really realize it at  the time you adds we want to add some  hinges so that we have hinges on the  midpoints of the edges instead of the  corners that turns out to be a bit more  efficient in this case so we add some  hinging still hinge Abul individually  but now we have hinges at the corners  and so at the midpoints and and we'll  have the same property over here and and  it causes it allows you to to hinge  these together actually here it looks  like there some of them are at the  corners not the midpoints so it's a bit  messy in in general we can prove if you  have any shape and you want to make poly  that shape so this is called a shape X  you want to make poly X's you can do it  as long as the copies of the X are only  rotated and they're joined at  corresponding edges so if you check this  guy's just been rotated 180 degrees  generally you can join these things  together at matching edges and the basic  technique is just subdivide the thing  triangulate  drawn the dual of the triangulation and  then connect to the midpoints of the  edges and you can show basically instead  of the hinge dissection going around  like this you can just make it go around  like this  and come back this way and if you check  the sequence of pieces they could visit  it's identical if you go around this way  or if you go around this way and that's  enough to show that any folded state is  valid with the triangles and squares  were essentially exploiting the symmetry  of these pieces so you can rotate them  to make them compatible here they're  forced to be compatible by assuming we  only joined matching edges so that was  the 2d poly form paper see Fredrickson  was one of the authors and 3d here's  easy way to generalize that if you take  for example a tetrahedron regular  tetrahedron you take the centroid and  cut everything to the centroid and you  end up cutting your tetrahedron has four  sides into four of these more slender  tetrahedra and then you take four of  them and join them together in this way  you do have to be careful in the way  that you joined them because again on  every face we want to have an incident  hinge so it's we've got to take care in  the way that you manage them together to  make sure that that is the case but it's  also cyclically hinged this gets joined  to that and basically the same inductive  proof works you just pluck off a leaf  show that you can turn the thing so that  one of the hinges aligns with the  inductive construction and then just  join the hinges across instead of within  the cycles so pretty easy what are we  talking about hinge dissections software  I guess those are those would be fun  things to implement they've never been  implemented and especially to see them  folding I thought I'd show you a little  bit about hinge dissection Hardware  different ways you can make them  physically real this is a kind of meso  scale call it this is a one centimeter  bar so not super tiny but I think this  could scale down quite a bit we have a  petri dish here with some liquid in it  she can read up there maybe this is the  coolest example we have a square  made up of four pieces and you add a  little bit of salt to that liquid and it  pops into the equilateral triangle  configuration so it's sort of  spontaneously folding hinging  essentially these pieces are slanted a  little bit and they have they prefer one  weighting causes them to fold one way  but when you add the salt they end up  flopping the other way you could see  they're a little bit inexact because of  that but pretty awesome the kinds of  intersections you can get them all to  actuate even without much room to do so  this is done at Harvard George  Whitesides  it's a group chemistry kind of related  it's not exactly hinge dissections but I  feel like it's the same spirit is this  idea of DNA origami it's called where  you take one big strand of DNA and you  force it to fold into a particular shape  here or folding it into a happy face the  way that's done is you add in a bunch of  little pieces of DNA so this string  basically has a this DNA strand has a  random string on it written on it  basically and you identify oh I want  these guys to glue together so you take  this piece of the random string and this  piece of the random string you construct  a piece of DNA it has both of those like  a little zipper to cause those to zip up  you do that all over the place there's  now automatic tools to do this it's  really easy to make DNA origami it  basically always works there's a limit  to how big this thing can be because  it's this the main strand here is a  single piece of DNA and those are hard  to make super big at least currently but  you get some really nice happy faces and  mass-produce them hundred nanometer  scale it's kind of like hinge dissection  because that strand of DNA is moving  it's actually more like a fixed angle  chain kind of like a hinge dissection  and we're essentially using here  universality of hinge dissections of  something like polyominoes though the  shapes are a little bit more awkward and  they've made a maps of the world you can  do two color patterns make snowflakes  the word DNA and crazy stuff started by  Paul Rothman but a lot of people do DNA  origami these days  cool  next paper I wanted to show you this is  fairly recent and it's about getting  continuous motions in particular in 3d  of hinge dissection like things so here  we have a chain of balls these are more  like ball and socket joints so you can  maybe see them better here there's a  there's a member going in from the green  guy into the center of the red guy and  there's a slot and the red guy can fold  around the the blue guy can fold around  the red guy and question is okay this is  great you can prove universality you can  make any shape you just subdivide your  dog or whatever into two by two by two  square let's and then we know how to  connect those together to make a nice  Hamiltonian cycle visits everything but  can you actually fold a chain of balls  like this into that dog and the answer  is always yes essentially you you feed a  big string of these balls into that's  actually what's happening in this  animation here though it's a little hard  to tell you're feeding in say well at  one of the legs one of the extreme  points in some direction this chain of  balls and as they go in they just start  tracking along the path and you just  need to check that you can track along  the path as this guy goes into a corner  for example you can actually navigate  the corner well at all time staying  within the tube if you can stay within  the tube you know you won't collide with  the rest of the chain because these this  tube is non-self intersecting and so  this the 2d version is fairly easy it's  just circles a little trickier to check  that it actually is possible with just  one turn with a u-turn and with kind of  I don't know you call this not a u-turn  we change in two directions two  dimensions all at once all of these are  possible with this particular mechanism  whatever mechanism you have if it can do  this then you can make anything another  way to prove motions exists for this  kind of poly form special case why do we  care about this for building robots so  these are somewhat different mechanisms  but these are have two examples built  here at the MIT Center for bits and  atoms  over in the Media Lab with Neil  Gershenfeld and many many people so you  get some idea this is a fairly small guy  I mean the actual size is about this big  you see some feet in the background give  you some sense of scale it's not very  many pieces but if you made a really  long chain it would really be able to  fold into anything you want just servos  to make the turns here this is a much  larger one the right version is folding  and you get some idea of scale here this  is when it's fully extended a hundred  and forty four  should that be feet or inches it's  really big so a little bit slower of  course because it has to move a lot more  and it's also quite a bit longer this is  built in particular by a Skylar Tibbits  here so that's the idea of robots in  general we'd like to make robots that  can change their shape we've seen sheet  folding robots but these are more Shane  folding robots inspired by proteins and  DNA and things like that sort of big  versions of DNA origami what's cool  about them is that they stay connected  throughout the motion you can keep the  your wiring and all the you can keep  your batteries and whatnot and your  communication channels connected in this  kind of scenario this is by contrast to  more common approaches to reconfigurable  robots you have individual units and  they can attach and detach from each  other you can see like these guys  picking up blocks moving stuff around  it's definitely cool but in practice  it's it's a lot harder to build these  kinds of robots because the attach  detach mechanism it's hard to get them  to align perfectly it's hard to get the  electrical connectivity every piece has  to have a battery instead of like every  tenth piece or one battery to drive  everything or tethering or whatever you  can do some very cool things and there's  a lot of algorithms around for doing  this daniela roofs here at MIT built  this robot and a bunch of others there's  also a very cool theory about these I've  worked on them you can prove for example  that well all of these models can  simulate each other  two constant factors in scale so you can  take your favorite robot a tamale cube  and simulate a crystalline robot or vice  versa  and then there's efficient algorithms -  this crystalline robots they can just  expand and contract and and detach and  attach and you can prove that given two  configurations you can change it from  one to the other up to some scale factor  you can even do it extremely fast in the  log end time if all the robots are  activating all at once anyway there's  cool stuff about reconfigure robots but  the hinge dissections offers an  alternative where everything stays  connected at all times but closely  related I think that was the hardware  story so we go back to our proof of  hinge dissections and why it works and  the one of the kind of surprised I  didn't show this in lecture but I don't  remember why I didn't that's a one  missing piece was how do you go from a  rectangle of one size to rectangle of  another may recall we had a triangle we  triangulated our polygons we so we ended  up with some arbitrary triangles then we  cut parallel to the base halfway up got  this you can put this over here put this  over here you get a rectangle of some  unknown height and then to make it  Universal we wanted to convert  everything into a rectangle of height  Epsilon so that then we could just  string them together obviously area has  to be preserved here if we string  together all the epsilon height  rectangles we've got one super long  epsilon height rectangle and then we  overlay the two dissections this is how  we did dissections but how do you go how  do you do this step from one rectangle  to another and this is a very old  dissection at least 1778 it's not it  wasn't published by Montu club but it's  he's credited in this publication and  this is frederickson's diagram of it so  you take the fatter rectangle and then  you take the longer rectangle and you  first you make multiple copies of the  fat  rectangle just serve tile a strip of the  plain to the right and then you angle  the thin rectangle slightly first of all  you line up these corners to the top  left corners line up and then we want  the top right corner of the thin  rectangle to lie on this bottom line  turns out this always works it's not  totally obvious but essentially these  copies of the rectangle you can kind of  fold them up and when you go off the  right edge here you're essentially  coming back on the left edge here and  then you're going this way then you're  going this way and this little piece is  exactly the same as this little piece  and well from that you get a dissection  that's not hinged but you can see that  this big rectangle has the tiny piece  here which conveniently fits right over  there it's like a wrap around the other  direction and then this piece well  everything matches up here the only  other weird thing is this bottom when  you go below the bottom you also wrap  around to the top and just check all the  pieces match up and yeah you've got your  dissection yeah it's kind of crazy you  have to check this works for all  parameters but it does and in general of  course if you have a very long rectangle  you need many pieces to the fat one but  that's essentially optimal okay for fun  this is a general technique called the P  slide technique or superposing to to  tessellations of your shape this is you  can use that same technique for example  to get the hinge dissection from regular  square to equilateral triangle you just  angle it right so that for example this  midpoint it's this midpoint and various  other alignments happen like this  midpoint falls on that edge and if you  look at it right these cuts give you the  four pieces for the square to I guess  you can see it right here here are the  four pieces of the square and if you  check the everything matches up you can  also make equilateral triangle in this  case it happens to be hinged that  doesn't always happen  it's a little tricky to tell maybe but  with practice you can see it I mentioned  at some point that you could take this  and turn it into a table but one of the  that had there has four sides or has  three sides one of the annoying things  about the table is that you need legs on  each of the pieces so Frederickson was  playing around with this fairly recently  2008 and he came up with this  alternative way of essentially the same  technique but you end up with one big  piece and lots of smaller pieces so the  idea is you just have a big leg or a  bunch of legs under one piece of the  table and so this is what the dissection  looks like unfortunately it's not  Hannibal but if you add in a couple  pieces you can make it hinge Abul so at  this point the universality result was  probably known this is actually a lot  easier than the way we do it  specialized to this kind of scenario  this hinges I think something like this  maybe even an animation of it yeah drawn  by Frederickson so you can see a careful  orchestration here just to make sure  that indeed the you can avoid collision  and so that's the proposed table no one  has built it another project would be to  build some intersections for example  this one as real furniture would be  pretty neat I have a couple examples  here of real furniture built this is the  dudeney dissection in the four piece  kind of a cabinet it's got lots of  shelves looks really practical uh-huh  and I don't know the bottom it looks  like there's a bunch of wheels down  there definitely you have to have a  bunch of table legs in this case but you  can really reconfigure it in all sorts  of ways it's a close-up  Alex vehicle is made by D house company  any German speakers anyone know what  house means same same an English house  so they actually built a house I can't  tell whether this is a real building or  a very good computer rendering I may be  real what's it looks like a rendering  yeah at some point later they have  people walking by but it could be a  composite anyway it's an idea of having  a house for any season you can  reconfigure it dynamically with these  tracks it's a pretty cool idea it would  be neat to experiment with anyway hinge  dissections in practice it's funny to  take a 2d dissection but I think in  architectural setting you can't change  where the floor is probably to do  dissection makes sense there's the real  maybe real version anyway so that was a  those rectangular rectangle okay  I'm cheating a little bit but another  question this is a very specific  question but for step three which is  where we did all the action of riehen  geing stuff I said number of pieces  roughly doubles I meant to say at least  roughly doubles so in the worst case the  point is it can be at least exponential  it definitely can be more because in  general remember it looks something like  this the point is you need you need at  least two triangles per edge here  because they need to fit together to  make these little kites so you at least  double for every edge that you visit in  the worst case you visit the hole all  the edges of the polygon so you end up  doubling everything but it can be worse  because sometimes if you don't have a  lot of room in this corner you've got to  divide into lots of very tiny triangles  I think that probably only happens  towards the beginning after you've cut  them small you won't have to cut them  even even smaller but I don't know for  sure but point is it's at least  exponential and this is the more  complicated diagram  but I claimed that you could get a  pseudo-polynomial bound how do you do  that this is a little trick wall and  still have time though so let me go over  the rough idea also what the claim is  so pseudo-polynomial Bend I'm not going  to claim this for arbitrary polygon so  that I think it's probably true what we  argue in the paper is that if the  vertices of the polygon lie on a grid  then we're okay just a little hard to  keep track of otherwise I will scale  things to make this the integer grid and  then the claim is the number of pieces  is polynomial and the number of vertices  n and r are usually some ratio of the  longest distance the smallest distance  in this case R is the grid size like an  r by r grid that's like the size of the  overall grid divided by the size of a  grid cell so basically the same thing so  how do we prove this the the general  idea so we have these these messy  constructions and essentially we're in  ducting removing one hinge and then  moving the next hinge or moving the next  hinge essentially all of those  inductions are nested inside each other  you completely refine to do one thing  then you have to refine to do the next  one  in the existing refinement so we have a  very deep recursion one way to think of  it order n depth recursion so we end up  with exponential and n but instead what  we can do is only recurse to constant  depth and if you're just more careful in  the overall construction  this is possible how let me give you  some of the steps you need more gadgets  and you need to follow so before I said  oh there's some dissection out there  it's known you triangulate you convert  triangle to square triangle to rectangle  rectangle to rectangle then superpose  that was the dissection then we hinge it  arbitrarily then we fixed the hinges one  at a time here I want to actually follow  those steps and keep hinge dissection as  much as possible so we're going to  triangulate the polygons but in this  case we're gonna subdivide further and  also triangulate with all the grid  points as vertices so it's a little hard  to draw here's the grid let's draw a  polygon to make a very exciting polygon  with so few vertices but maybe something  like that okay if I triangulate this  thing and all all the interior points  are very many interior points this  example maybe I'll make a slightly  different one there's two interior  points I want to triangulate with those  as vertices of the triangles so maybe  I'll do something like this okay a  couple of different shapes of triangles  here but they all have the same area  this is called pix theorem special case  of pix theorem so here they're all 1/2  square even though this one spans a  weird shape it's 1/2 square of area so  the nice thing is if I do this in  polygon a and in polygon B the triangles  there's equal number of triangles of  same size because they have matching  areas originally so  that's probably a way to do this for  general polygons I think this is the  only step that requires grids except  it's also a lot easier to analyze this  bound with grits so that's I guess an  open problem to work out without grids  okay the next thing is we'd really like  a chain of triangles right now we just  have a blob of triangles and we can  chain if I the triangles this is a step  that was I don't know if I showed the  figure last time this is what we do to  slender if I everything we have some  general hinge dissection don't know what  it looks like we just take each of the  triangles subdivide at there in center  cut and then you hinge around the  outside and you'll get one in this case  one cycle of slender triangles in this  case all we care about is that it's a  chain so we have some general thing here  we subdivide each of them like this and  then you hinge around and so now I've  got a hinged collection of triangles for  a and I've got a hinge collection of  triangles for B I'm just gonna do a to  be here I should probably say that to  two shapes and conveniently these  triangles will still have matching areas  they're all now 1/6 I do it right so get  a chain of area 1/6 triangles and I have  the same number for a and for B so this  is kind of cool of course the triangles  could be different shapes but I  basically have a chain of various  triangles they're all the same area I'm  going to draw for a I have a similar  chain for B and I just need to convert  basically triangle per triangle from A  to B so now my problem is a lot easier I  have these hinges which I need to  preserve that's a little trickier  this is actually an idea suggested by  Epstein before the universality is  it's like all we need to do is do  triangle to triangle while preserving  two hinges then we could do anything to  anything so we're following that plan  and now we're going to use all the fancy  gadgets we have to do triangle to  triangle while preserving these hinges  and not not blowing up the number pieces  too much have it definitely simpler  we're down to triangle to triangle next  step ok next problem yeah this is  slightly annoying I said oh great these  triangles are matching up but I'm not  going to be able to do triangle to  triangle and get exactly the hinges I  want where I want them so I'm gonna have  to end up for example moving this hinge  to another corner so we're gonna use a  new gadget actually for fixing which  vertices connect to which triangles this  is maybe not obvious yet that we need  this but we will and we're gonna use a  slightly somewhat more efficient version  of essentially the same idea so we've  got we've got a hinge here in the middle  basically can't control where the hinge  goes but it's supposed to go to one of  the corners so we're gonna reconfigure  it in this way so we assume we have some  way of doing it and here's the thing we  assumed that a maybe this has already  happened to a we don't want to recurse  into a because you know then we get  exponential blow-up I'm gonna have to do  this for every single triangle here  there's n of them that's a lot I don't  want to get deep recursion I don't want  to get depth and recursion so if but if  I cut up in this way in fact I only need  to cut up B and if B hasn't been touched  yet this is okay and then I'll do it the  next way in the next triangle next  triangle and they won't interact that's  the good news  so how do we do it well we cut up a  little we call it tight fan I believe  here there here there's two kites we get  these triangles to match these two these  triangles to match these two cut up this  little piece along the side and either  the green stays in here green is  attached to the pink so then our magenta  so if we keep the green in here triangle  stays there if we pull everything out  and there's a little hole made here to  make that more plausible but in reality  we have to subdivide to get slender so  if we instead reconfigure the green to  lie along the edge and the blue blue can  turn around here and fit inside because  it has exactly the same shape these two  chains are identical  I can also fit in here and then we've  moved the magenta over to that side so  that's cool that works and it doesn't  touch a so it's a slight variation of  what we had before and it's good  so that's pseudopolynomial and they  don't interact and so we can move these  things however we need to according to  what step four produces for us so it's  maybe slightly out of order I could have  called that step four in the step three  but get to the more exciting part  finally we do triangle the triangle it's  a little crazy I'm gonna give you three  constructions they'd give us what we  want and then I'm gonna claim I can  overlay them this is what we can't do  with intersections but I'm gonna do it  anyway bear with me the final gadget  will say how to overlay them but let's  start with the relatively simple goal of  triangle to rectangle well this I  already showed you and the nice thing  about triangle to rectangle it's three  piece dissection is you can hinge it  here and here  and it works just fine okay so that's  already a hinged dissection that's the  easy step then we want to take that  rectangle and convert it into a tiny all  right not tiny same same area but an  epsilon height rectangle because  remember we have two triangles they're  different shapes so they have different  heights this one will end up being half  the height but it won't match what we  for this triangle so I'm going to do it  steps a and B for each of the triangles  and then I have 2 epsilon height  rectangles and then the challenge is to  convert one into the other this is a  challenge because they have hinges on  them so with dissections you just  overlay these two cut-ups but hinge  dissections there's hinges you have to  preserve we can't do that okay  first part is well I already showed this  first part is step B which I showed you  already going from one rectangle to  another here's another diagram of it it  turns out it's almost hinged you can  essentially just flop back and forth and  back and forth except at the end you  might be in trouble so there's one step  here and depending on parity exactly  this piece of the rectangle is hinged  here but I really want it to be hinged  here so I'm just gonna move it over here  I have tools for moving hinges around so  it turns out you have to check that this  is safe but you just do one hinge moving  and then you're okay so in this case  this should actually go a little bit  deeper the bottom figure shows when you  go too deep you can cut cut and this is  just like the previous diagram of  triangle to rectangle you do that at the  bottom you'll be fine as a couple  different cases and exactly the parity  and how you end up three cases I guess  but in all cases the rest can be hinged  you just need this one step in the  middle to fix fix it so most of it is  just swinging back and forth so it's  almost hinged which is good news because  we have things that are almost we have  tools to make almost hinge things  actually hinged so that's cool so  basically we've covered every and B at  this point but the last part is C or you  know how do we superpose all of these  things and this is using another gadget  called pseudo cuts and essentially you  have some nice hinge dissection already  and you want to add a cut and a hinge  just imagine cutting all  way through here and adding a hinge I  guess on the yellow side here and  somehow I want it I want this thing to  fold in all the ways that you Steve  would be able to fold so it can fold  into a but then I also want it to be  able to fold at this hinge and they're  eventually fold into B and it's  complicated but again the same idea so  we've got these yellow guys should  normally live in here and so it's  everything yellow is yellow we these are  triangles these are triangles -  triangles so they're like little quads  they have holes just the right size for  the yellow these guys have holes just  the right side sorry how's it go okay I  see it's purple then blue then yellow I  believe it's the yellow fits into the  blue anyway whatever whatever works  these guys nest together and when they  nest together they fill these little  holes and then there's matching patterns  out here that all fit how does it go  actually sorry I think they're all  triangles this just looks multicolored  so it looks like purple here is going  into the cyan one at the next level  the yellow guys are going into the  purple I see so there's a triangle and a  quad here lovely  and then the these guys stretch across  so definitely a little more complicated  but it and this you know you lose factor  of two or whatever but if you apply  these pseudo cuts in the right order and  these are fairly simple cuttings that we  have to do we know these cuts are you're  mostly a striping so if you just apply  them in order you don't get blow up I'll  just wave my hands at that it's a little  hard to draw the picture obviously but  that's how it goes that's  pseudo-polynomial hinge dissection this  is why I had this it was intentional I  didn't cover in lecture because it's  pretty complicated there wasn't time  any questions last topic is higher  dimensions can we get a brief overview  of 3d dissections so this is more a  dissection question than a hinging  question although of course you could  ask does all this work for hinged  dissections pseudopolynomial we don't  necessarily know for straight-up proving  that hinge dissections exist the claim  is it hasn't been written up formally  yet the same techniques work you can  take any dissection and convert it into  a hinged dissection but in 3d turns out  dissections by themselves are not so  simple there's a lot of open problems  something some nice things are known so  let me tell you about 3d this section  okay if I want to convert one polyhedron  P into another polyhedron Q obviously  the volumes must be the same assuming  we're doing a reasonable cutting and not  some crazy axiom of choice thing so  volumes have to match just like four  polygons the areas have to match but  that turns out to be not enough and this  goes back to a Hilbert problem so you  may heard of David Hilbert he wrote this  paper of like twenty three open problems  at the turn of previous century in 1900  this is problem three and it's it's kind  of it wasn't directly about hinge  dissections a little bit or about  dissections rather a little bit  convoluted about some certain axioms and  proving certain things but in particular  is asking are there two tetrahedra of  equal base and altitude so equal volume  which can in no way be split up into  congruent tetrahedra so there's no way  to dissect one end to the other if  that's true it would show that certain  axioms are necessary in certain verbs  and it turns out it is true there are  tetrahedra of equal volume where you  cannot do this and that I don't have a  slide for it but this was proved by  guy named n and he came up with  something called the well that we now  called the denon variant he didn't call  it that himself and these things must  also match it's called an invariant  meaning that no matter how you cut the  things up and reassemble the denon  variant doesn't change and so if you  have any hope of going from P to Q those  two things must match and then Seidler  so this is this was in 1901 then proved  that this was a necessary condition so  like a year after that appeared in 1965  a little bit later Seidler proved that  this is all that's necessary so these  are sufficient conditions if P and Q had  the same volume the same Dennett  variants then there is actually a  dissection improved it's somewhat  algebraically somewhat constructively  I'm not sure exactly there's a simpler  proof by Jessen in 1968 and he proved  that actually in 4d the same is true and  4d you need the volumes to match and the  venom variants to match that's enough in  5d and higher no one knows what it takes  for a dissection pretty weird could be  interesting to study these more  carefully let me tell you briefly about  Denon variants a little awkward unless  you're familiar with tensor product  space how many people know that tensor  product space if you okay if you've done  quantum stuff I guess it's more common  I'm not familiar with tensor product  space but here we go  tensor you know I can read Wikipedia  with the best of them it's it's a fairly  simple notion just has somewhat weird  notation you can do things like take  something X and write tensor products  with Y and what this means is basically  don't mess with this product okay it's  it's a product really this is two things  x and y they're not interchangeable  they're in completely different worlds  different units whatever you can't like  multiply them they just hang out side by  side you also can't put them around  kent's it's not commutative okay  but some things hold like if you take I  don't know Z and add it to this product  you do have distributivity so you can  get X as this notice doesn't look very  correct if I have this then you can  multiply that out so you get X tensor  with y plus X tensor with C so that  holds it also holds on the left and the  other thing is that constants can come  out so if you have C times X tensor with  Y this is the same thing as C times X  tensor with Y so in the end I'm gonna  have a bunch of these pairs these tensor  pairs and I'm also able to add them  together and nothing happens when you  add them together to just hang out so in  general you can also have a constant  factor so it's you have a linear  combination of pairs basically what am I  doing this because here's the denim  variance then in variant says look with  polyhedra you've got two things it's  gonna be the X and the y over there  you've got edge links and you've got  dihedral angles so look at every edge  here's an edge of my polyhedron here it  has some length which I'll call L of E  and there's some angle here which we'll  call theta of E add this up over every  edge so the denon variant is going to be  the sum over all edges of the length  tensor with the angle angle is is the  angle between these two planes so that's  the dihedral angle yep  so that it's for every edge there's one  dihedral angle which is sort of the  interior solid angle there of the at the  edge so this is kind of what's going on  and these things have to match now it's  a little more complicated sorry it's not  really just the angle essentially if you  add  national multiples of Pi nothing happens  so you actually take this weird group  all rationals times pi and all this  means is if you have two angles and  their difference that you subtract them  and you get a rational multiple pi then  there those two angles are considered  the same so what this is really saying  is I only care about sort of the  irrational part of Pi roughly you add PI  over two that doesn't change anything  why this thing well if I take an edge  and for example I cut it in half  anywhere I could cut it at an irrational  fraction or whatever I will get two  lengths but they'll be ten surd with the  same angle I didn't change the angle and  so by distributivity once you get things  inside the same place so in this case  we'll get two lengths that add up they  they match okay so you can sort of as  long as you have matching angles you can  add the links together that's what  distributivity tells you similarly if I  tried to cut this angle in some piece  could be an irrational ratio between the  two pieces they will have the same edge  lengths so and I'm gonna have matching  edge lengths so I can use distributivity  and add the angles back together so  basically when you dissect this thing  will not change it's a little more  awkward when I cut here because this was  originally a PI and then I cut it into  some pieces and this is where you need  the arrest the rational multiples of Pi  not mattering but eventually you can  prove done invariant is invariant the  harder proof you can prove that it's  also sufficient if you have matching  volumes as recently proved like a few  years ago 2008 that the weather the  denon variant of one polyhedron and  another match is decidable so there is  an algorithm to tell whether two  polyhedra have the same this thing  decidable is a pretty weak statement  natural open problem is is there a good  algorithm to do it we don't know if it  does match is there good algorithm to  find the dissection we don't know these  may be easy if you really understand the  proofs deeply but at the time no one  cared about algorithms at this point  it's we need to go back and really  understand how to actually do 3d  dissections so that we could then do 3d  hinge dissections  that's it don't forget organic  convention is on Saturday should be fun  you

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all right lecture 14 was about two main  topics I guess we had slender adorned  chains sort of fatter linkages and then  hinge dissection most of our time was  actually spent with the slender  adornments and proving that that works  but most of our questions today are  about hinge dissections because that's  kind of the most fun and there's a lot  more to say about them so first question  is is there any software for hinge  dissections and short answer is no  surprisingly so this would definitely be  a cool project possibility there are a  bunch of examples let me switch to these  on the web just sort of random examples  cool dissections people thought were so  neat they wanted to animate them and so  they basically constructed where the  coordinates were over time in  Mathematica and then put it on the web  as a illustration of that so this is a  equilateral triangle - a hexagon regular  hexagon this is a hinge dissection by  Grieg Fredrickson and then is drawn by  rick Mabrey here's another one for an  equilateral triangle - a pentagon pretty  cool they're hinged in a tree-like  fashion even just kind of unusual and  these are Gregg Fredrickson is one of  the Masters of hinge dissections and  dissections in general it's probably the  master of dissections in general as he  has three books or different kinds of  dissections this is actually a hinge  dissection on the cover here of the the  purple and pink pieces hinged like this  into into a smaller star from a the  outline of a big star to the interior of  a smaller star and then this star fits  nicely inside this is another hinge  dissection this this book is entirely  about hinge dissection so they're not  just the kinds we've seen another kind  called twist hinging which I think this  is a twist Inge piece flips around the  other side and then there's a third book  about a different kind of hinge  dissection that's more of a surface  hinge dissection where you've got to  you've got the front and back of this  surface and you and you fold them  like piano hinges with hinges in the  plane all very cool books you should  check them out if you want to know more  about dissections they're more about  here cool examples some design  techniques for how to make them I'll  show you one such design technique later  on today and but not a ton of Theory  here in particular because there wasn't  a ton of Theory when these books were  written so that's some hinge dissections  as I said cool project would be to make  a general tool for animating hinge  dissections there's only a handful out  there greg has digital files of lots of  his hinge dissections probably be  willing to share them I haven't talked  to him about it if there was a good  engine for animating them I think would  be cool even cooler would be to  implement the slender an orangeade  business take one of these hinge  dissections maybe they just hinge but  sometimes there's collisions and but we  already know if you refine these guys to  be slender which you can do if they're  triangulated you can do it with only  losing a factor of three in the number  of pieces it cool to implement that and  then you can do the slender and orange  folding via CDR which I have an  implementation of or it's not that hard  to build one if you have a LP solver so  various project possibilities another  cool project would be to just design  more hinge dissections there's still  interesting questions either use fewer  pieces or just make elegant designs  related to the implementation idea a  particular family of hinge dissections  that could be fun to implement are  embodied by this alphabet I showed this  in lecture you can take the letter six  and convert into a square convert into  an eight and convert into four and four  into a nine via these 128 pieces I  didn't talk much about this theorem thus  I thought I'd give you a little sketch  of how this works it's actually very  simple to construct the folded states of  these hinge dissections and it could be  an interesting thing to implement and  it's also just kind of fun this is way  earlier 1999 way before we knew that  everything was possible we could at  least do all polyominoes of the given  size so let's let's just think about  polyominoes are the Polly hexes Polly  aemon's we have equilateral triangles  and these are called Polly abalos  for silly reasons basically by analogy  to a Diablo which is a juggling device  you can hinge dissect any of them here  you take each square and you cut it into  two half squares then you hinge them  together like this  this is one two three four five six  seven eight so this will make any four  square object any Tetris piece and  generally take two n pieces and you can  make any and omona and the way you prove  that that is universal that it can fold  into anything it's not so clear from  this picture but it's actually really  easy to prove by induction so the first  thing to do in this inductive proof is  to check that you can do it for N equals  1 ok that may sound trivial but this is  actually core the key property you need  in a hinge dissection of a single square  into your general family is that there's  a hinge visible on every edge of the  object so here this hinge kind of covers  this edge it covers that edge so both of  these edges have hinges on them and the  other two a it just have a hinge on them  they happen to be shared hinges but  that's ok and each of these that's true  the triangle it's a little more awkward  do you actually need two hinges to cover  the three sides but you can but you only  need these two pieces one of those  non-convex it's may be hard to fold  continuously but you'd refine it if you  wanted to do slender adornments so let's  not worry about continuous motion yet so  that's the base case of the induction  how do I do for N equals 1 now  inductively if I have some shape I want  to build I'll take what I call the dual  graph of that shape so make a vertex for  every square connect them together if  they share an edge the square square an  edge and then look at a spanning tree of  that shape so just cut some of these  edges until you have tree connectivity  among among those squares every tree has  at least two leaves except in the fall  but every mathematical tree always has  to at least two leaves like this is a  leaf if I cut here this would also be a  leaf leaf is a degree 1 vertex  so that's the square that only shares  one side so pluck off that leaf remove  that square the resulting n minus 1  squares by assumption can be made by  this hinge dissection with 2 times n  minus 1 pieces so now we just have to  attach this guy on and here's a figure  for that down at the bottom this is the  same thing for triangles if you and and  poly abalos are in the upper right so  you have some existing hinge dissection  you don't really know what it's like and  you want to add this leaf back on so it  shares one edge with one guy now this  guy could be oriented this way or it  could be oriented this way but it's the  same by reflection so let's say it's  oriented this way we we know the square  is made up by two half squares by  induction and so we know that there's a  hinge here now this hinge connects to  some things in this case it's some to  some T prime could be here could be up  here and all we do is stick s on here  now s can rotate we have our solution  for one guy and there's two different  orientations for him we're going to  choose this orientation because it puts  this hinge right there and so once we do  that normally this would be a cycle and  this thing would be a cycle through here  but we just redo the hinges in here so  that this the cycle gets bigger and the  thing the important thing to verify is  that the orientations the triangles are  the same just like the hinge dissection  picture I showed we always go from the  base edge to the next base edge to the  next base edge of these right isosceles  triangles and all the triangles are on  the outside of the cycle so we actually  construct a cyclic hinge dissection then  at the end you could break it and make  it a path but and this one is even  slender remember right triangles are  slender barely you can look at all the  inward normals they hit the base edge so  this will even move continuously if it's  an open chain for closed chains we don't  know  so that's polyominoes poly aemon's are  similar  pretty much the same thing you just in  this case you might have hinges on both  sides but you rotate this thing so one  of the hinges lines up and you just  reconnect the hinges it's not hard to  show you can always do that the hinges  will never cross and this proves that  these folded states exist and then we  use the slender stuff to do and  continuous motions actually when this  paper was written we didn't have slender  adornments at back at 99 even in 2005  when the journal version appeared so  it's only now that we know that motions  are possible by in this case directly in  this case with some refinement so I  thought that would just be fun to see  you can do some other crazy things so  this is a hinge dissection from any four  aemond this of four equilateral  triangles drawn together to any four AMA  no this is a Tetris piece it's  essentially a superposition of this idea  with you see and here these these four  lines make the hinge dissection of  dudeney from 1902 from an equilateral  triangle to a square and with some extra  stuff added in this is maybe a  foreshadowing of the idea of refinement  although we didn't really realize it at  the time you adds we want to add some  hinges so that we have hinges on the  midpoints of the edges instead of the  corners that turns out to be a bit more  efficient in this case so we add some  hinging still hinge Abul individually  but now we have hinges at the corners  and so at the midpoints and and we'll  have the same property over here and and  it causes it allows you to to hinge  these together actually here it looks  like there some of them are at the  corners not the midpoints so it's a bit  messy in in general we can prove if you  have any shape and you want to make poly  that shape so this is called a shape X  you want to make poly X's you can do it  as long as the copies of the X are only  rotated and they're joined at  corresponding edges so if you check this  guy's just been rotated 180 degrees  generally you can join these things  together at matching edges and the basic  technique is just subdivide the thing  triangulate  drawn the dual of the triangulation and  then connect to the midpoints of the  edges and you can show basically instead  of the hinge dissection going around  like this you can just make it go around  like this  and come back this way and if you check  the sequence of pieces they could visit  it's identical if you go around this way  or if you go around this way and that's  enough to show that any folded state is  valid with the triangles and squares  were essentially exploiting the symmetry  of these pieces so you can rotate them  to make them compatible here they're  forced to be compatible by assuming we  only joined matching edges so that was  the 2d poly form paper see Fredrickson  was one of the authors and 3d here's  easy way to generalize that if you take  for example a tetrahedron regular  tetrahedron you take the centroid and  cut everything to the centroid and you  end up cutting your tetrahedron has four  sides into four of these more slender  tetrahedra and then you take four of  them and join them together in this way  you do have to be careful in the way  that you joined them because again on  every face we want to have an incident  hinge so it's we've got to take care in  the way that you manage them together to  make sure that that is the case but it's  also cyclically hinged this gets joined  to that and basically the same inductive  proof works you just pluck off a leaf  show that you can turn the thing so that  one of the hinges aligns with the  inductive construction and then just  join the hinges across instead of within  the cycles so pretty easy what are we  talking about hinge dissections software  I guess those are those would be fun  things to implement they've never been  implemented and especially to see them  folding I thought I'd show you a little  bit about hinge dissection Hardware  different ways you can make them  physically real this is a kind of meso  scale call it this is a one centimeter  bar so not super tiny but I think this  could scale down quite a bit we have a  petri dish here with some liquid in it  she can read up there maybe this is the  coolest example we have a square  made up of four pieces and you add a  little bit of salt to that liquid and it  pops into the equilateral triangle  configuration so it's sort of  spontaneously folding hinging  essentially these pieces are slanted a  little bit and they have they prefer one  weighting causes them to fold one way  but when you add the salt they end up  flopping the other way you could see  they're a little bit inexact because of  that but pretty awesome the kinds of  intersections you can get them all to  actuate even without much room to do so  this is done at Harvard George  Whitesides  it's a group chemistry kind of related  it's not exactly hinge dissections but I  feel like it's the same spirit is this  idea of DNA origami it's called where  you take one big strand of DNA and you  force it to fold into a particular shape  here or folding it into a happy face the  way that's done is you add in a bunch of  little pieces of DNA so this string  basically has a this DNA strand has a  random string on it written on it  basically and you identify oh I want  these guys to glue together so you take  this piece of the random string and this  piece of the random string you construct  a piece of DNA it has both of those like  a little zipper to cause those to zip up  you do that all over the place there's  now automatic tools to do this it's  really easy to make DNA origami it  basically always works there's a limit  to how big this thing can be because  it's this the main strand here is a  single piece of DNA and those are hard  to make super big at least currently but  you get some really nice happy faces and  mass-produce them hundred nanometer  scale it's kind of like hinge dissection  because that strand of DNA is moving  it's actually more like a fixed angle  chain kind of like a hinge dissection  and we're essentially using here  universality of hinge dissections of  something like polyominoes though the  shapes are a little bit more awkward and  they've made a maps of the world you can  do two color patterns make snowflakes  the word DNA and crazy stuff started by  Paul Rothman but a lot of people do DNA  origami these days  cool  next paper I wanted to show you this is  fairly recent and it's about getting  continuous motions in particular in 3d  of hinge dissection like things so here  we have a chain of balls these are more  like ball and socket joints so you can  maybe see them better here there's a  there's a member going in from the green  guy into the center of the red guy and  there's a slot and the red guy can fold  around the the blue guy can fold around  the red guy and question is okay this is  great you can prove universality you can  make any shape you just subdivide your  dog or whatever into two by two by two  square let's and then we know how to  connect those together to make a nice  Hamiltonian cycle visits everything but  can you actually fold a chain of balls  like this into that dog and the answer  is always yes essentially you you feed a  big string of these balls into that's  actually what's happening in this  animation here though it's a little hard  to tell you're feeding in say well at  one of the legs one of the extreme  points in some direction this chain of  balls and as they go in they just start  tracking along the path and you just  need to check that you can track along  the path as this guy goes into a corner  for example you can actually navigate  the corner well at all time staying  within the tube if you can stay within  the tube you know you won't collide with  the rest of the chain because these this  tube is non-self intersecting and so  this the 2d version is fairly easy it's  just circles a little trickier to check  that it actually is possible with just  one turn with a u-turn and with kind of  I don't know you call this not a u-turn  we change in two directions two  dimensions all at once all of these are  possible with this particular mechanism  whatever mechanism you have if it can do  this then you can make anything another  way to prove motions exists for this  kind of poly form special case why do we  care about this for building robots so  these are somewhat different mechanisms  but these are have two examples built  here at the MIT Center for bits and  atoms  over in the Media Lab with Neil  Gershenfeld and many many people so you  get some idea this is a fairly small guy  I mean the actual size is about this big  you see some feet in the background give  you some sense of scale it's not very  many pieces but if you made a really  long chain it would really be able to  fold into anything you want just servos  to make the turns here this is a much  larger one the right version is folding  and you get some idea of scale here this  is when it's fully extended a hundred  and forty four  should that be feet or inches it's  really big so a little bit slower of  course because it has to move a lot more  and it's also quite a bit longer this is  built in particular by a Skylar Tibbits  here so that's the idea of robots in  general we'd like to make robots that  can change their shape we've seen sheet  folding robots but these are more Shane  folding robots inspired by proteins and  DNA and things like that sort of big  versions of DNA origami what's cool  about them is that they stay connected  throughout the motion you can keep the  your wiring and all the you can keep  your batteries and whatnot and your  communication channels connected in this  kind of scenario this is by contrast to  more common approaches to reconfigurable  robots you have individual units and  they can attach and detach from each  other you can see like these guys  picking up blocks moving stuff around  it's definitely cool but in practice  it's it's a lot harder to build these  kinds of robots because the attach  detach mechanism it's hard to get them  to align perfectly it's hard to get the  electrical connectivity every piece has  to have a battery instead of like every  tenth piece or one battery to drive  everything or tethering or whatever you  can do some very cool things and there's  a lot of algorithms around for doing  this daniela roofs here at MIT built  this robot and a bunch of others there's  also a very cool theory about these I've  worked on them you can prove for example  that well all of these models can  simulate each other  two constant factors in scale so you can  take your favorite robot a tamale cube  and simulate a crystalline robot or vice  versa  and then there's efficient algorithms -  this crystalline robots they can just  expand and contract and and detach and  attach and you can prove that given two  configurations you can change it from  one to the other up to some scale factor  you can even do it extremely fast in the  log end time if all the robots are  activating all at once anyway there's  cool stuff about reconfigure robots but  the hinge dissections offers an  alternative where everything stays  connected at all times but closely  related I think that was the hardware  story so we go back to our proof of  hinge dissections and why it works and  the one of the kind of surprised I  didn't show this in lecture but I don't  remember why I didn't that's a one  missing piece was how do you go from a  rectangle of one size to rectangle of  another may recall we had a triangle we  triangulated our polygons we so we ended  up with some arbitrary triangles then we  cut parallel to the base halfway up got  this you can put this over here put this  over here you get a rectangle of some  unknown height and then to make it  Universal we wanted to convert  everything into a rectangle of height  Epsilon so that then we could just  string them together obviously area has  to be preserved here if we string  together all the epsilon height  rectangles we've got one super long  epsilon height rectangle and then we  overlay the two dissections this is how  we did dissections but how do you go how  do you do this step from one rectangle  to another and this is a very old  dissection at least 1778 it's not it  wasn't published by Montu club but it's  he's credited in this publication and  this is frederickson's diagram of it so  you take the fatter rectangle and then  you take the longer rectangle and you  first you make multiple copies of the  fat  rectangle just serve tile a strip of the  plain to the right and then you angle  the thin rectangle slightly first of all  you line up these corners to the top  left corners line up and then we want  the top right corner of the thin  rectangle to lie on this bottom line  turns out this always works it's not  totally obvious but essentially these  copies of the rectangle you can kind of  fold them up and when you go off the  right edge here you're essentially  coming back on the left edge here and  then you're going this way then you're  going this way and this little piece is  exactly the same as this little piece  and well from that you get a dissection  that's not hinged but you can see that  this big rectangle has the tiny piece  here which conveniently fits right over  there it's like a wrap around the other  direction and then this piece well  everything matches up here the only  other weird thing is this bottom when  you go below the bottom you also wrap  around to the top and just check all the  pieces match up and yeah you've got your  dissection yeah it's kind of crazy you  have to check this works for all  parameters but it does and in general of  course if you have a very long rectangle  you need many pieces to the fat one but  that's essentially optimal okay for fun  this is a general technique called the P  slide technique or superposing to to  tessellations of your shape this is you  can use that same technique for example  to get the hinge dissection from regular  square to equilateral triangle you just  angle it right so that for example this  midpoint it's this midpoint and various  other alignments happen like this  midpoint falls on that edge and if you  look at it right these cuts give you the  four pieces for the square to I guess  you can see it right here here are the  four pieces of the square and if you  check the everything matches up you can  also make equilateral triangle in this  case it happens to be hinged that  doesn't always happen  it's a little tricky to tell maybe but  with practice you can see it I mentioned  at some point that you could take this  and turn it into a table but one of the  that had there has four sides or has  three sides one of the annoying things  about the table is that you need legs on  each of the pieces so Frederickson was  playing around with this fairly recently  2008 and he came up with this  alternative way of essentially the same  technique but you end up with one big  piece and lots of smaller pieces so the  idea is you just have a big leg or a  bunch of legs under one piece of the  table and so this is what the dissection  looks like unfortunately it's not  Hannibal but if you add in a couple  pieces you can make it hinge Abul so at  this point the universality result was  probably known this is actually a lot  easier than the way we do it  specialized to this kind of scenario  this hinges I think something like this  maybe even an animation of it yeah drawn  by Frederickson so you can see a careful  orchestration here just to make sure  that indeed the you can avoid collision  and so that's the proposed table no one  has built it another project would be to  build some intersections for example  this one as real furniture would be  pretty neat I have a couple examples  here of real furniture built this is the  dudeney dissection in the four piece  kind of a cabinet it's got lots of  shelves looks really practical uh-huh  and I don't know the bottom it looks  like there's a bunch of wheels down  there definitely you have to have a  bunch of table legs in this case but you  can really reconfigure it in all sorts  of ways it's a close-up  Alex vehicle is made by D house company  any German speakers anyone know what  house means same same an English house  so they actually built a house I can't  tell whether this is a real building or  a very good computer rendering I may be  real what's it looks like a rendering  yeah at some point later they have  people walking by but it could be a  composite anyway it's an idea of having  a house for any season you can  reconfigure it dynamically with these  tracks it's a pretty cool idea it would  be neat to experiment with anyway hinge  dissections in practice it's funny to  take a 2d dissection but I think in  architectural setting you can't change  where the floor is probably to do  dissection makes sense there's the real  maybe real version anyway so that was a  those rectangular rectangle okay  I'm cheating a little bit but another  question this is a very specific  question but for step three which is  where we did all the action of riehen  geing stuff I said number of pieces  roughly doubles I meant to say at least  roughly doubles so in the worst case the  point is it can be at least exponential  it definitely can be more because in  general remember it looks something like  this the point is you need you need at  least two triangles per edge here  because they need to fit together to  make these little kites so you at least  double for every edge that you visit in  the worst case you visit the hole all  the edges of the polygon so you end up  doubling everything but it can be worse  because sometimes if you don't have a  lot of room in this corner you've got to  divide into lots of very tiny triangles  I think that probably only happens  towards the beginning after you've cut  them small you won't have to cut them  even even smaller but I don't know for  sure but point is it's at least  exponential and this is the more  complicated diagram  but I claimed that you could get a  pseudo-polynomial bound how do you do  that this is a little trick wall and  still have time though so let me go over  the rough idea also what the claim is  so pseudo-polynomial Bend I'm not going  to claim this for arbitrary polygon so  that I think it's probably true what we  argue in the paper is that if the  vertices of the polygon lie on a grid  then we're okay just a little hard to  keep track of otherwise I will scale  things to make this the integer grid and  then the claim is the number of pieces  is polynomial and the number of vertices  n and r are usually some ratio of the  longest distance the smallest distance  in this case R is the grid size like an  r by r grid that's like the size of the  overall grid divided by the size of a  grid cell so basically the same thing so  how do we prove this the the general  idea so we have these these messy  constructions and essentially we're in  ducting removing one hinge and then  moving the next hinge or moving the next  hinge essentially all of those  inductions are nested inside each other  you completely refine to do one thing  then you have to refine to do the next  one  in the existing refinement so we have a  very deep recursion one way to think of  it order n depth recursion so we end up  with exponential and n but instead what  we can do is only recurse to constant  depth and if you're just more careful in  the overall construction  this is possible how let me give you  some of the steps you need more gadgets  and you need to follow so before I said  oh there's some dissection out there  it's known you triangulate you convert  triangle to square triangle to rectangle  rectangle to rectangle then superpose  that was the dissection then we hinge it  arbitrarily then we fixed the hinges one  at a time here I want to actually follow  those steps and keep hinge dissection as  much as possible so we're going to  triangulate the polygons but in this  case we're gonna subdivide further and  also triangulate with all the grid  points as vertices so it's a little hard  to draw here's the grid let's draw a  polygon to make a very exciting polygon  with so few vertices but maybe something  like that okay if I triangulate this  thing and all all the interior points  are very many interior points this  example maybe I'll make a slightly  different one there's two interior  points I want to triangulate with those  as vertices of the triangles so maybe  I'll do something like this okay a  couple of different shapes of triangles  here but they all have the same area  this is called pix theorem special case  of pix theorem so here they're all 1/2  square even though this one spans a  weird shape it's 1/2 square of area so  the nice thing is if I do this in  polygon a and in polygon B the triangles  there's equal number of triangles of  same size because they have matching  areas originally so  that's probably a way to do this for  general polygons I think this is the  only step that requires grids except  it's also a lot easier to analyze this  bound with grits so that's I guess an  open problem to work out without grids  okay the next thing is we'd really like  a chain of triangles right now we just  have a blob of triangles and we can  chain if I the triangles this is a step  that was I don't know if I showed the  figure last time this is what we do to  slender if I everything we have some  general hinge dissection don't know what  it looks like we just take each of the  triangles subdivide at there in center  cut and then you hinge around the  outside and you'll get one in this case  one cycle of slender triangles in this  case all we care about is that it's a  chain so we have some general thing here  we subdivide each of them like this and  then you hinge around and so now I've  got a hinged collection of triangles for  a and I've got a hinge collection of  triangles for B I'm just gonna do a to  be here I should probably say that to  two shapes and conveniently these  triangles will still have matching areas  they're all now 1/6 I do it right so get  a chain of area 1/6 triangles and I have  the same number for a and for B so this  is kind of cool of course the triangles  could be different shapes but I  basically have a chain of various  triangles they're all the same area I'm  going to draw for a I have a similar  chain for B and I just need to convert  basically triangle per triangle from A  to B so now my problem is a lot easier I  have these hinges which I need to  preserve that's a little trickier  this is actually an idea suggested by  Epstein before the universality is  it's like all we need to do is do  triangle to triangle while preserving  two hinges then we could do anything to  anything so we're following that plan  and now we're going to use all the fancy  gadgets we have to do triangle to  triangle while preserving these hinges  and not not blowing up the number pieces  too much have it definitely simpler  we're down to triangle to triangle next  step ok next problem yeah this is  slightly annoying I said oh great these  triangles are matching up but I'm not  going to be able to do triangle to  triangle and get exactly the hinges I  want where I want them so I'm gonna have  to end up for example moving this hinge  to another corner so we're gonna use a  new gadget actually for fixing which  vertices connect to which triangles this  is maybe not obvious yet that we need  this but we will and we're gonna use a  slightly somewhat more efficient version  of essentially the same idea so we've  got we've got a hinge here in the middle  basically can't control where the hinge  goes but it's supposed to go to one of  the corners so we're gonna reconfigure  it in this way so we assume we have some  way of doing it and here's the thing we  assumed that a maybe this has already  happened to a we don't want to recurse  into a because you know then we get  exponential blow-up I'm gonna have to do  this for every single triangle here  there's n of them that's a lot I don't  want to get deep recursion I don't want  to get depth and recursion so if but if  I cut up in this way in fact I only need  to cut up B and if B hasn't been touched  yet this is okay and then I'll do it the  next way in the next triangle next  triangle and they won't interact that's  the good news  so how do we do it well we cut up a  little we call it tight fan I believe  here there here there's two kites we get  these triangles to match these two these  triangles to match these two cut up this  little piece along the side and either  the green stays in here green is  attached to the pink so then our magenta  so if we keep the green in here triangle  stays there if we pull everything out  and there's a little hole made here to  make that more plausible but in reality  we have to subdivide to get slender so  if we instead reconfigure the green to  lie along the edge and the blue blue can  turn around here and fit inside because  it has exactly the same shape these two  chains are identical  I can also fit in here and then we've  moved the magenta over to that side so  that's cool that works and it doesn't  touch a so it's a slight variation of  what we had before and it's good  so that's pseudopolynomial and they  don't interact and so we can move these  things however we need to according to  what step four produces for us so it's  maybe slightly out of order I could have  called that step four in the step three  but get to the more exciting part  finally we do triangle the triangle it's  a little crazy I'm gonna give you three  constructions they'd give us what we  want and then I'm gonna claim I can  overlay them this is what we can't do  with intersections but I'm gonna do it  anyway bear with me the final gadget  will say how to overlay them but let's  start with the relatively simple goal of  triangle to rectangle well this I  already showed you and the nice thing  about triangle to rectangle it's three  piece dissection is you can hinge it  here and here  and it works just fine okay so that's  already a hinged dissection that's the  easy step then we want to take that  rectangle and convert it into a tiny all  right not tiny same same area but an  epsilon height rectangle because  remember we have two triangles they're  different shapes so they have different  heights this one will end up being half  the height but it won't match what we  for this triangle so I'm going to do it  steps a and B for each of the triangles  and then I have 2 epsilon height  rectangles and then the challenge is to  convert one into the other this is a  challenge because they have hinges on  them so with dissections you just  overlay these two cut-ups but hinge  dissections there's hinges you have to  preserve we can't do that okay  first part is well I already showed this  first part is step B which I showed you  already going from one rectangle to  another here's another diagram of it it  turns out it's almost hinged you can  essentially just flop back and forth and  back and forth except at the end you  might be in trouble so there's one step  here and depending on parity exactly  this piece of the rectangle is hinged  here but I really want it to be hinged  here so I'm just gonna move it over here  I have tools for moving hinges around so  it turns out you have to check that this  is safe but you just do one hinge moving  and then you're okay so in this case  this should actually go a little bit  deeper the bottom figure shows when you  go too deep you can cut cut and this is  just like the previous diagram of  triangle to rectangle you do that at the  bottom you'll be fine as a couple  different cases and exactly the parity  and how you end up three cases I guess  but in all cases the rest can be hinged  you just need this one step in the  middle to fix fix it so most of it is  just swinging back and forth so it's  almost hinged which is good news because  we have things that are almost we have  tools to make almost hinge things  actually hinged so that's cool so  basically we've covered every and B at  this point but the last part is C or you  know how do we superpose all of these  things and this is using another gadget  called pseudo cuts and essentially you  have some nice hinge dissection already  and you want to add a cut and a hinge  just imagine cutting all  way through here and adding a hinge I  guess on the yellow side here and  somehow I want it I want this thing to  fold in all the ways that you Steve  would be able to fold so it can fold  into a but then I also want it to be  able to fold at this hinge and they're  eventually fold into B and it's  complicated but again the same idea so  we've got these yellow guys should  normally live in here and so it's  everything yellow is yellow we these are  triangles these are triangles -  triangles so they're like little quads  they have holes just the right size for  the yellow these guys have holes just  the right side sorry how's it go okay I  see it's purple then blue then yellow I  believe it's the yellow fits into the  blue anyway whatever whatever works  these guys nest together and when they  nest together they fill these little  holes and then there's matching patterns  out here that all fit how does it go  actually sorry I think they're all  triangles this just looks multicolored  so it looks like purple here is going  into the cyan one at the next level  the yellow guys are going into the  purple I see so there's a triangle and a  quad here lovely  and then the these guys stretch across  so definitely a little more complicated  but it and this you know you lose factor  of two or whatever but if you apply  these pseudo cuts in the right order and  these are fairly simple cuttings that we  have to do we know these cuts are you're  mostly a striping so if you just apply  them in order you don't get blow up I'll  just wave my hands at that it's a little  hard to draw the picture obviously but  that's how it goes that's  pseudo-polynomial hinge dissection this  is why I had this it was intentional I  didn't cover in lecture because it's  pretty complicated there wasn't time  any questions last topic is higher  dimensions can we get a brief overview  of 3d dissections so this is more a  dissection question than a hinging  question although of course you could  ask does all this work for hinged  dissections pseudopolynomial we don't  necessarily know for straight-up proving  that hinge dissections exist the claim  is it hasn't been written up formally  yet the same techniques work you can  take any dissection and convert it into  a hinged dissection but in 3d turns out  dissections by themselves are not so  simple there's a lot of open problems  something some nice things are known so  let me tell you about 3d this section  okay if I want to convert one polyhedron  P into another polyhedron Q obviously  the volumes must be the same assuming  we're doing a reasonable cutting and not  some crazy axiom of choice thing so  volumes have to match just like four  polygons the areas have to match but  that turns out to be not enough and this  goes back to a Hilbert problem so you  may heard of David Hilbert he wrote this  paper of like twenty three open problems  at the turn of previous century in 1900  this is problem three and it's it's kind  of it wasn't directly about hinge  dissections a little bit or about  dissections rather a little bit  convoluted about some certain axioms and  proving certain things but in particular  is asking are there two tetrahedra of  equal base and altitude so equal volume  which can in no way be split up into  congruent tetrahedra so there's no way  to dissect one end to the other if  that's true it would show that certain  axioms are necessary in certain verbs  and it turns out it is true there are  tetrahedra of equal volume where you  cannot do this and that I don't have a  slide for it but this was proved by  guy named n and he came up with  something called the well that we now  called the denon variant he didn't call  it that himself and these things must  also match it's called an invariant  meaning that no matter how you cut the  things up and reassemble the denon  variant doesn't change and so if you  have any hope of going from P to Q those  two things must match and then Seidler  so this is this was in 1901 then proved  that this was a necessary condition so  like a year after that appeared in 1965  a little bit later Seidler proved that  this is all that's necessary so these  are sufficient conditions if P and Q had  the same volume the same Dennett  variants then there is actually a  dissection improved it's somewhat  algebraically somewhat constructively  I'm not sure exactly there's a simpler  proof by Jessen in 1968 and he proved  that actually in 4d the same is true and  4d you need the volumes to match and the  venom variants to match that's enough in  5d and higher no one knows what it takes  for a dissection pretty weird could be  interesting to study these more  carefully let me tell you briefly about  Denon variants a little awkward unless  you're familiar with tensor product  space how many people know that tensor  product space if you okay if you've done  quantum stuff I guess it's more common  I'm not familiar with tensor product  space but here we go  tensor you know I can read Wikipedia  with the best of them it's it's a fairly  simple notion just has somewhat weird  notation you can do things like take  something X and write tensor products  with Y and what this means is basically  don't mess with this product okay it's  it's a product really this is two things  x and y they're not interchangeable  they're in completely different worlds  different units whatever you can't like  multiply them they just hang out side by  side you also can't put them around  kent's it's not commutative okay  but some things hold like if you take I  don't know Z and add it to this product  you do have distributivity so you can  get X as this notice doesn't look very  correct if I have this then you can  multiply that out so you get X tensor  with y plus X tensor with C so that  holds it also holds on the left and the  other thing is that constants can come  out so if you have C times X tensor with  Y this is the same thing as C times X  tensor with Y so in the end I'm gonna  have a bunch of these pairs these tensor  pairs and I'm also able to add them  together and nothing happens when you  add them together to just hang out so in  general you can also have a constant  factor so it's you have a linear  combination of pairs basically what am I  doing this because here's the denim  variance then in variant says look with  polyhedra you've got two things it's  gonna be the X and the y over there  you've got edge links and you've got  dihedral angles so look at every edge  here's an edge of my polyhedron here it  has some length which I'll call L of E  and there's some angle here which we'll  call theta of E add this up over every  edge so the denon variant is going to be  the sum over all edges of the length  tensor with the angle angle is is the  angle between these two planes so that's  the dihedral angle yep  so that it's for every edge there's one  dihedral angle which is sort of the  interior solid angle there of the at the  edge so this is kind of what's going on  and these things have to match now it's  a little more complicated sorry it's not  really just the angle essentially if you  add  national multiples of Pi nothing happens  so you actually take this weird group  all rationals times pi and all this  means is if you have two angles and  their difference that you subtract them  and you get a rational multiple pi then  there those two angles are considered  the same so what this is really saying  is I only care about sort of the  irrational part of Pi roughly you add PI  over two that doesn't change anything  why this thing well if I take an edge  and for example I cut it in half  anywhere I could cut it at an irrational  fraction or whatever I will get two  lengths but they'll be ten surd with the  same angle I didn't change the angle and  so by distributivity once you get things  inside the same place so in this case  we'll get two lengths that add up they  they match okay so you can sort of as  long as you have matching angles you can  add the links together that's what  distributivity tells you similarly if I  tried to cut this angle in some piece  could be an irrational ratio between the  two pieces they will have the same edge  lengths so and I'm gonna have matching  edge lengths so I can use distributivity  and add the angles back together so  basically when you dissect this thing  will not change it's a little more  awkward when I cut here because this was  originally a PI and then I cut it into  some pieces and this is where you need  the arrest the rational multiples of Pi  not mattering but eventually you can  prove done invariant is invariant the  harder proof you can prove that it's  also sufficient if you have matching  volumes as recently proved like a few  years ago 2008 that the weather the  denon variant of one polyhedron and  another match is decidable so there is  an algorithm to tell whether two  polyhedra have the same this thing  decidable is a pretty weak statement  natural open problem is is there a good  algorithm to do it we don't know if it  does match is there good algorithm to  find the dissection we don't know these  may be easy if you really understand the  proofs deeply but at the time no one  cared about algorithms at this point  it's we need to go back and really  understand how to actually do 3d  dissections so that we could then do 3d  hinge dissections  that's it don't forget organic  convention is on Saturday should be fun  you
 

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315
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316
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319
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320
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321
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323
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326
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327
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328
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329
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330
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331
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332
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333
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335
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341
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342
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343
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344
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345
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346
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347
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348
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349
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350
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351
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352
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353
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354
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355
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356
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357
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358
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359
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360
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361
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362
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363
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364
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365
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366
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367
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368
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369
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370
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371
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372
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373
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374
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375
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376
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377
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378
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379
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380
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381
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382
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383
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384
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385
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386
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387
00:15:26,030 --> 00:15:26,040

 

388
00:15:26,040 --> 00:15:28,770


389
00:15:28,770 --> 00:15:32,250

 

390
00:15:32,250 --> 00:15:37,920

 

391
00:15:37,920 --> 00:15:39,930

 

392
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393
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394
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395
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396
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397
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398
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399
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400
00:15:58,440 --> 00:16:03,060

 

401
00:16:03,060 --> 00:16:04,710

 

402
00:16:04,710 --> 00:16:06,810

 

403
00:16:06,810 --> 00:16:08,700

 

404
00:16:08,700 --> 00:16:10,770

 

405
00:16:10,770 --> 00:16:12,300

 

406
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407
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408
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409
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410
00:16:25,470 --> 00:16:28,980

 

411
00:16:28,980 --> 00:16:30,420

 

412
00:16:30,420 --> 00:16:31,830

 

413
00:16:31,830 --> 00:16:34,920

 

414
00:16:34,920 --> 00:16:36,660

 

415
00:16:36,660 --> 00:16:39,450

 

416
00:16:39,450 --> 00:16:41,550

 

417
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418
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419
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420
00:16:47,490 --> 00:16:49,200

 

421
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422
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423
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425
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427
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428
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429
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430
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431
00:17:10,699 --> 00:17:13,199

 

432
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433
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434
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435
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436
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437
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438
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439
00:17:30,090 --> 00:17:32,040

 

440
00:17:32,040 --> 00:17:34,110

 

441
00:17:34,110 --> 00:17:36,270

 

442
00:17:36,270 --> 00:17:36,280

 

443
00:17:36,280 --> 00:17:36,870


444
00:17:36,870 --> 00:17:39,990

 

445
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446
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447
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448
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449
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450
00:17:57,029 --> 00:17:59,220

 

451
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452
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453
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454
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455
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456
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457
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458
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459
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460
00:18:26,490 --> 00:18:27,630

 

461
00:18:27,630 --> 00:18:31,919

 

462
00:18:31,919 --> 00:18:33,630

 

463
00:18:33,630 --> 00:18:38,399

 

464
00:18:38,399 --> 00:18:39,899

 

465
00:18:39,899 --> 00:18:41,610

 

466
00:18:41,610 --> 00:18:43,529

 

467
00:18:43,529 --> 00:18:46,770

 

468
00:18:46,770 --> 00:18:48,510

 

469
00:18:48,510 --> 00:18:51,330

 

470
00:18:51,330 --> 00:18:52,470

 

471
00:18:52,470 --> 00:18:54,750

 

472
00:18:54,750 --> 00:18:57,090

 

473
00:18:57,090 --> 00:18:59,220

 

474
00:18:59,220 --> 00:19:01,590

 

475
00:19:01,590 --> 00:19:04,529

 

476
00:19:04,529 --> 00:19:07,230

 

477
00:19:07,230 --> 00:19:09,600

 

478
00:19:09,600 --> 00:19:11,340

 

479
00:19:11,340 --> 00:19:12,480

 

480
00:19:12,480 --> 00:19:14,520

 

481
00:19:14,520 --> 00:19:17,130

 

482
00:19:17,130 --> 00:19:18,539

 

483
00:19:18,539 --> 00:19:21,120

 

484
00:19:21,120 --> 00:19:22,590

 

485
00:19:22,590 --> 00:19:24,510

 

486
00:19:24,510 --> 00:19:26,159

 

487
00:19:26,159 --> 00:19:28,380

 

488
00:19:28,380 --> 00:19:30,149

 

489
00:19:30,149 --> 00:19:33,539

 

490
00:19:33,539 --> 00:19:35,010

 

491
00:19:35,010 --> 00:19:37,260

 

492
00:19:37,260 --> 00:19:38,909

 

493
00:19:38,909 --> 00:19:41,820

 

494
00:19:41,820 --> 00:19:43,620

 

495
00:19:43,620 --> 00:19:45,450

 

496
00:19:45,450 --> 00:19:49,799

 

497
00:19:49,799 --> 00:19:50,780

 

498
00:19:50,780 --> 00:19:53,330

 

499
00:19:53,330 --> 00:19:55,880

 

500
00:19:55,880 --> 00:19:58,220

 

501
00:19:58,220 --> 00:19:58,230

 

502
00:19:58,230 --> 00:19:59,300


503
00:19:59,300 --> 00:20:01,190

 

504
00:20:01,190 --> 00:20:03,230

 

505
00:20:03,230 --> 00:20:06,050

 

506
00:20:06,050 --> 00:20:09,890

 

507
00:20:09,890 --> 00:20:12,200

 

508
00:20:12,200 --> 00:20:14,510

 

509
00:20:14,510 --> 00:20:16,850

 

510
00:20:16,850 --> 00:20:18,800

 

511
00:20:18,800 --> 00:20:20,720

 

512
00:20:20,720 --> 00:20:22,460

 

513
00:20:22,460 --> 00:20:23,840

 

514
00:20:23,840 --> 00:20:25,670

 

515
00:20:25,670 --> 00:20:29,150

 

516
00:20:29,150 --> 00:20:32,230

 

517
00:20:32,230 --> 00:20:37,010

 

518
00:20:37,010 --> 00:20:41,630

 

519
00:20:41,630 --> 00:20:44,630

 

520
00:20:44,630 --> 00:20:48,590

 

521
00:20:48,590 --> 00:20:50,810

 

522
00:20:50,810 --> 00:20:52,700

 

523
00:20:52,700 --> 00:20:54,410

 

524
00:20:54,410 --> 00:21:02,150

 

525
00:21:02,150 --> 00:21:04,520

 

526
00:21:04,520 --> 00:21:06,620

 

527
00:21:06,620 --> 00:21:10,070

 

528
00:21:10,070 --> 00:21:11,930

 

529
00:21:11,930 --> 00:21:14,030

 

530
00:21:14,030 --> 00:21:16,460

 

531
00:21:16,460 --> 00:21:18,680

 

532
00:21:18,680 --> 00:21:20,000

 

533
00:21:20,000 --> 00:21:22,100

 

534
00:21:22,100 --> 00:21:24,800

 

535
00:21:24,800 --> 00:21:27,050

 

536
00:21:27,050 --> 00:21:28,040

 

537
00:21:28,040 --> 00:21:29,590

 

538
00:21:29,590 --> 00:21:34,250

 

539
00:21:34,250 --> 00:21:36,710

 

540
00:21:36,710 --> 00:21:39,740

 

541
00:21:39,740 --> 00:21:41,360

 

542
00:21:41,360 --> 00:21:44,320

 

543
00:21:44,320 --> 00:21:47,930

 

544
00:21:47,930 --> 00:21:49,580

 

545
00:21:49,580 --> 00:21:52,070

 

546
00:21:52,070 --> 00:21:55,610

 

547
00:21:55,610 --> 00:21:59,270

 

548
00:21:59,270 --> 00:22:01,390

 

549
00:22:01,390 --> 00:22:04,310

 

550
00:22:04,310 --> 00:22:04,320

 

551
00:22:04,320 --> 00:22:04,580


552
00:22:04,580 --> 00:22:07,399

 

553
00:22:07,399 --> 00:22:10,549

 

554
00:22:10,549 --> 00:22:14,149

 

555
00:22:14,149 --> 00:22:15,560

 

556
00:22:15,560 --> 00:22:17,480

 

557
00:22:17,480 --> 00:22:19,519

 

558
00:22:19,519 --> 00:22:22,720

 

559
00:22:22,720 --> 00:22:25,340

 

560
00:22:25,340 --> 00:22:27,710

 

561
00:22:27,710 --> 00:22:30,860

 

562
00:22:30,860 --> 00:22:32,419

 

563
00:22:32,419 --> 00:22:33,620

 

564
00:22:33,620 --> 00:22:35,269

 

565
00:22:35,269 --> 00:22:38,090

 

566
00:22:38,090 --> 00:22:40,580

 

567
00:22:40,580 --> 00:22:43,039

 

568
00:22:43,039 --> 00:22:47,810

 

569
00:22:47,810 --> 00:22:52,700

 

570
00:22:52,700 --> 00:22:54,560

 

571
00:22:54,560 --> 00:22:56,690

 

572
00:22:56,690 --> 00:22:58,760

 

573
00:22:58,760 --> 00:23:03,080

 

574
00:23:03,080 --> 00:23:05,060

 

575
00:23:05,060 --> 00:23:06,620

 

576
00:23:06,620 --> 00:23:09,110

 

577
00:23:09,110 --> 00:23:12,500

 

578
00:23:12,500 --> 00:23:14,510

 

579
00:23:14,510 --> 00:23:16,850

 

580
00:23:16,850 --> 00:23:17,720

 

581
00:23:17,720 --> 00:23:22,610

 

582
00:23:22,610 --> 00:23:24,080

 

583
00:23:24,080 --> 00:23:26,899

 

584
00:23:26,899 --> 00:23:31,880

 

585
00:23:31,880 --> 00:23:33,649

 

586
00:23:33,649 --> 00:23:37,720

 

587
00:23:37,720 --> 00:23:41,539

 

588
00:23:41,539 --> 00:23:43,370

 

589
00:23:43,370 --> 00:23:47,330

 

590
00:23:47,330 --> 00:23:50,299

 

591
00:23:50,299 --> 00:23:52,909

 

592
00:23:52,909 --> 00:23:55,820

 

593
00:23:55,820 --> 00:23:57,440

 

594
00:23:57,440 --> 00:24:00,830

 

595
00:24:00,830 --> 00:24:03,560

 

596
00:24:03,560 --> 00:24:06,830

 

597
00:24:06,830 --> 00:24:08,000

 

598
00:24:08,000 --> 00:24:10,610

 

599
00:24:10,610 --> 00:24:12,320

 

600
00:24:12,320 --> 00:24:14,240

 

601
00:24:14,240 --> 00:24:15,500

 

602
00:24:15,500 --> 00:24:17,130

 

603
00:24:17,130 --> 00:24:19,380

 

604
00:24:19,380 --> 00:24:22,950

 

605
00:24:22,950 --> 00:24:24,960

 

606
00:24:24,960 --> 00:24:29,310

 

607
00:24:29,310 --> 00:24:31,050

 

608
00:24:31,050 --> 00:24:32,310

 

609
00:24:32,310 --> 00:24:34,740

 

610
00:24:34,740 --> 00:24:36,930

 

611
00:24:36,930 --> 00:24:39,080

 

612
00:24:39,080 --> 00:24:41,130

 

613
00:24:41,130 --> 00:24:42,660

 

614
00:24:42,660 --> 00:24:45,180

 

615
00:24:45,180 --> 00:24:48,570

 

616
00:24:48,570 --> 00:24:50,850

 

617
00:24:50,850 --> 00:24:53,160

 

618
00:24:53,160 --> 00:24:57,120

 

619
00:24:57,120 --> 00:24:59,280

 

620
00:24:59,280 --> 00:25:01,740

 

621
00:25:01,740 --> 00:25:05,400

 

622
00:25:05,400 --> 00:25:06,900

 

623
00:25:06,900 --> 00:25:07,980

 

624
00:25:07,980 --> 00:25:10,380

 

625
00:25:10,380 --> 00:25:11,940

 

626
00:25:11,940 --> 00:25:15,120

 

627
00:25:15,120 --> 00:25:17,880

 

628
00:25:17,880 --> 00:25:21,690

 

629
00:25:21,690 --> 00:25:22,950

 

630
00:25:22,950 --> 00:25:25,800

 

631
00:25:25,800 --> 00:25:29,160

 

632
00:25:29,160 --> 00:25:31,470

 

633
00:25:31,470 --> 00:25:33,090

 

634
00:25:33,090 --> 00:25:35,040

 

635
00:25:35,040 --> 00:25:38,070

 

636
00:25:38,070 --> 00:25:42,000

 

637
00:25:42,000 --> 00:25:43,640

 

638
00:25:43,640 --> 00:25:45,750

 

639
00:25:45,750 --> 00:25:47,880

 

640
00:25:47,880 --> 00:25:50,400

 

641
00:25:50,400 --> 00:25:51,450

 

642
00:25:51,450 --> 00:25:52,950

 

643
00:25:52,950 --> 00:25:56,070

 

644
00:25:56,070 --> 00:25:57,540

 

645
00:25:57,540 --> 00:26:02,320

 

646
00:26:02,320 --> 00:26:09,130

 

647
00:26:09,130 --> 00:26:11,630

 

648
00:26:11,630 --> 00:26:14,529

 

649
00:26:14,529 --> 00:26:18,019

 

650
00:26:18,019 --> 00:26:19,880

 

651
00:26:19,880 --> 00:26:23,180

 

652
00:26:23,180 --> 00:26:27,140

 

653
00:26:27,140 --> 00:26:28,760

 

654
00:26:28,760 --> 00:26:30,140

 

655
00:26:30,140 --> 00:26:32,720

 

656
00:26:32,720 --> 00:26:34,700

 

657
00:26:34,700 --> 00:26:37,370

 

658
00:26:37,370 --> 00:26:39,889

 

659
00:26:39,889 --> 00:26:47,149

 

660
00:26:47,149 --> 00:26:50,570

 

661
00:26:50,570 --> 00:26:52,159

 

662
00:26:52,159 --> 00:26:53,630

 

663
00:26:53,630 --> 00:26:55,220

 

664
00:26:55,220 --> 00:26:58,299

 

665
00:26:58,299 --> 00:27:03,880

 

666
00:27:03,880 --> 00:27:06,610

 

667
00:27:06,610 --> 00:27:09,889

 

668
00:27:09,889 --> 00:27:11,480

 

669
00:27:11,480 --> 00:27:14,990

 

670
00:27:14,990 --> 00:27:16,639

 

671
00:27:16,639 --> 00:27:19,220

 

672
00:27:19,220 --> 00:27:21,409

 

673
00:27:21,409 --> 00:27:24,649

 

674
00:27:24,649 --> 00:27:26,419

 

675
00:27:26,419 --> 00:27:29,000

 

676
00:27:29,000 --> 00:27:31,370

 

677
00:27:31,370 --> 00:27:33,320

 

678
00:27:33,320 --> 00:27:35,090

 

679
00:27:35,090 --> 00:27:37,039

 

680
00:27:37,039 --> 00:27:41,419

 

681
00:27:41,419 --> 00:27:44,450

 

682
00:27:44,450 --> 00:27:46,070

 

683
00:27:46,070 --> 00:27:48,799

 

684
00:27:48,799 --> 00:27:52,820

 

685
00:27:52,820 --> 00:27:54,590

 

686
00:27:54,590 --> 00:27:56,360

 

687
00:27:56,360 --> 00:27:58,159

 

688
00:27:58,159 --> 00:27:59,690

 

689
00:27:59,690 --> 00:28:01,279

 

690
00:28:01,279 --> 00:28:03,169

 

691
00:28:03,169 --> 00:28:05,960

 

692
00:28:05,960 --> 00:28:07,850

 

693
00:28:07,850 --> 00:28:10,669

 

694
00:28:10,669 --> 00:28:12,380

 

695
00:28:12,380 --> 00:28:14,750

 

696
00:28:14,750 --> 00:28:18,140

 

697
00:28:18,140 --> 00:28:21,310

 

698
00:28:21,310 --> 00:28:25,250

 

699
00:28:25,250 --> 00:28:28,570

 

700
00:28:28,570 --> 00:28:40,430

 

701
00:28:40,430 --> 00:28:42,140

 

702
00:28:42,140 --> 00:28:44,030

 

703
00:28:44,030 --> 00:28:46,760

 

704
00:28:46,760 --> 00:28:49,540

 

705
00:28:49,540 --> 00:28:56,840

 

706
00:28:56,840 --> 00:29:05,390

 

707
00:29:05,390 --> 00:29:10,150

 

708
00:29:10,150 --> 00:29:14,950

 

709
00:29:14,950 --> 00:29:19,280

 

710
00:29:19,280 --> 00:29:22,850

 

711
00:29:22,850 --> 00:29:24,290

 

712
00:29:24,290 --> 00:29:27,080

 

713
00:29:27,080 --> 00:29:30,440

 

714
00:29:30,440 --> 00:29:32,120

 

715
00:29:32,120 --> 00:29:37,190

 

716
00:29:37,190 --> 00:29:42,170

 

717
00:29:42,170 --> 00:29:48,040

 

718
00:29:48,040 --> 00:29:49,880

 

719
00:29:49,880 --> 00:29:52,070

 

720
00:29:52,070 --> 00:29:53,510

 

721
00:29:53,510 --> 00:29:55,100

 

722
00:29:55,100 --> 00:29:57,620

 

723
00:29:57,620 --> 00:30:00,020

 

724
00:30:00,020 --> 00:30:01,400

 

725
00:30:01,400 --> 00:30:01,410

 

726
00:30:01,410 --> 00:30:01,760


727
00:30:01,760 --> 00:30:03,830

 

728
00:30:03,830 --> 00:30:06,110

 

729
00:30:06,110 --> 00:30:09,500

 

730
00:30:09,500 --> 00:30:14,030

 

731
00:30:14,030 --> 00:30:18,470

 

732
00:30:18,470 --> 00:30:21,470

 

733
00:30:21,470 --> 00:30:25,239

 

734
00:30:25,239 --> 00:30:33,709

 

735
00:30:33,709 --> 00:30:39,709

 

736
00:30:39,709 --> 00:30:42,619

 

737
00:30:42,619 --> 00:30:44,149

 

738
00:30:44,149 --> 00:30:47,719

 

739
00:30:47,719 --> 00:30:49,669

 

740
00:30:49,669 --> 00:30:52,519

 

741
00:30:52,519 --> 00:30:54,649

 

742
00:30:54,649 --> 00:30:56,209

 

743
00:30:56,209 --> 00:30:57,739

 

744
00:30:57,739 --> 00:31:01,699

 

745
00:31:01,699 --> 00:31:06,139

 

746
00:31:06,139 --> 00:31:07,459

 

747
00:31:07,459 --> 00:31:09,489

 

748
00:31:09,489 --> 00:31:12,889

 

749
00:31:12,889 --> 00:31:17,989

 

750
00:31:17,989 --> 00:31:25,899

 

751
00:31:25,899 --> 00:31:31,249

 

752
00:31:31,249 --> 00:31:35,029

 

753
00:31:35,029 --> 00:31:38,389

 

754
00:31:38,389 --> 00:31:42,009

 

755
00:31:42,009 --> 00:31:44,149

 

756
00:31:44,149 --> 00:31:45,440

 

757
00:31:45,440 --> 00:31:51,829

 

758
00:31:51,829 --> 00:31:53,180

 

759
00:31:53,180 --> 00:31:55,219

 

760
00:31:55,219 --> 00:32:03,699

 

761
00:32:03,699 --> 00:32:05,749

 

762
00:32:05,749 --> 00:32:07,419

 

763
00:32:07,419 --> 00:32:10,159

 

764
00:32:10,159 --> 00:32:13,009

 

765
00:32:13,009 --> 00:32:14,989

 

766
00:32:14,989 --> 00:32:18,589

 

767
00:32:18,589 --> 00:32:19,879

 

768
00:32:19,879 --> 00:32:22,849

 

769
00:32:22,849 --> 00:32:24,709

 

770
00:32:24,709 --> 00:32:26,299

 

771
00:32:26,299 --> 00:32:36,009

 

772
00:32:36,009 --> 00:32:38,810

 

773
00:32:38,810 --> 00:32:40,219

 

774
00:32:40,219 --> 00:32:41,930

 

775
00:32:41,930 --> 00:32:44,359

 

776
00:32:44,359 --> 00:32:47,479

 

777
00:32:47,479 --> 00:32:52,209

 

778
00:32:52,209 --> 00:32:54,320

 

779
00:32:54,320 --> 00:32:55,849

 

780
00:32:55,849 --> 00:32:59,869

 

781
00:32:59,869 --> 00:33:06,739

 

782
00:33:06,739 --> 00:33:09,379

 

783
00:33:09,379 --> 00:33:13,609

 

784
00:33:13,609 --> 00:33:15,859

 

785
00:33:15,859 --> 00:33:18,019

 

786
00:33:18,019 --> 00:33:19,219

 

787
00:33:19,219 --> 00:33:21,009

 

788
00:33:21,009 --> 00:33:24,079

 

789
00:33:24,079 --> 00:33:27,200

 

790
00:33:27,200 --> 00:33:30,289

 

791
00:33:30,289 --> 00:33:32,180

 

792
00:33:32,180 --> 00:33:35,419

 

793
00:33:35,419 --> 00:33:38,539

 

794
00:33:38,539 --> 00:33:40,279

 

795
00:33:40,279 --> 00:33:42,499

 

796
00:33:42,499 --> 00:33:43,999

 

797
00:33:43,999 --> 00:33:45,680

 

798
00:33:45,680 --> 00:33:48,739

 

799
00:33:48,739 --> 00:33:53,869

 

800
00:33:53,869 --> 00:33:56,329

 

801
00:33:56,329 --> 00:34:04,399

 

802
00:34:04,399 --> 00:34:11,539

 

803
00:34:11,539 --> 00:34:14,720

 

804
00:34:14,720 --> 00:34:15,919

 

805
00:34:15,919 --> 00:34:17,690

 

806
00:34:17,690 --> 00:34:20,510

 

807
00:34:20,510 --> 00:34:23,960

 

808
00:34:23,960 --> 00:34:26,779

 

809
00:34:26,779 --> 00:34:31,789

 

810
00:34:31,789 --> 00:34:34,279

 

811
00:34:34,279 --> 00:34:36,260

 

812
00:34:36,260 --> 00:34:37,519

 

813
00:34:37,519 --> 00:34:39,769

 

814
00:34:39,769 --> 00:34:42,409

 

815
00:34:42,409 --> 00:34:44,659

 

816
00:34:44,659 --> 00:34:46,639

 

817
00:34:46,639 --> 00:34:47,839

 

818
00:34:47,839 --> 00:34:49,760

 

819
00:34:49,760 --> 00:34:52,240

 

820
00:34:52,240 --> 00:34:54,879

 

821
00:34:54,879 --> 00:34:57,230

 

822
00:34:57,230 --> 00:34:58,490

 

823
00:34:58,490 --> 00:35:01,549

 

824
00:35:01,549 --> 00:35:04,069

 

825
00:35:04,069 --> 00:35:06,650

 

826
00:35:06,650 --> 00:35:13,430

 

827
00:35:13,430 --> 00:35:15,589

 

828
00:35:15,589 --> 00:35:20,329

 

829
00:35:20,329 --> 00:35:21,410

 

830
00:35:21,410 --> 00:35:22,970

 

831
00:35:22,970 --> 00:35:24,920

 

832
00:35:24,920 --> 00:35:26,870

 

833
00:35:26,870 --> 00:35:30,950

 

834
00:35:30,950 --> 00:35:35,650

 

835
00:35:35,650 --> 00:35:44,359

 

836
00:35:44,359 --> 00:35:45,950

 

837
00:35:45,950 --> 00:35:48,859

 

838
00:35:48,859 --> 00:35:51,740

 

839
00:35:51,740 --> 00:35:54,770

 

840
00:35:54,770 --> 00:35:57,470

 

841
00:35:57,470 --> 00:35:58,880

 

842
00:35:58,880 --> 00:36:00,890

 

843
00:36:00,890 --> 00:36:02,930

 

844
00:36:02,930 --> 00:36:05,240

 

845
00:36:05,240 --> 00:36:07,940

 

846
00:36:07,940 --> 00:36:11,390

 

847
00:36:11,390 --> 00:36:13,579

 

848
00:36:13,579 --> 00:36:16,819

 

849
00:36:16,819 --> 00:36:19,309

 

850
00:36:19,309 --> 00:36:21,049

 

851
00:36:21,049 --> 00:36:22,430

 

852
00:36:22,430 --> 00:36:24,200

 

853
00:36:24,200 --> 00:36:27,020

 

854
00:36:27,020 --> 00:36:30,710

 

855
00:36:30,710 --> 00:36:33,079

 

856
00:36:33,079 --> 00:36:35,839

 

857
00:36:35,839 --> 00:36:37,309

 

858
00:36:37,309 --> 00:36:39,620

 

859
00:36:39,620 --> 00:36:40,960

 

860
00:36:40,960 --> 00:36:44,390

 

861
00:36:44,390 --> 00:36:49,160

 

862
00:36:49,160 --> 00:36:52,370

 

863
00:36:52,370 --> 00:36:53,960

 

864
00:36:53,960 --> 00:36:56,120

 

865
00:36:56,120 --> 00:36:58,130

 

866
00:36:58,130 --> 00:37:00,289

 

867
00:37:00,289 --> 00:37:03,620

 

868
00:37:03,620 --> 00:37:06,890

 

869
00:37:06,890 --> 00:37:09,259

 

870
00:37:09,259 --> 00:37:11,029

 

871
00:37:11,029 --> 00:37:12,920

 

872
00:37:12,920 --> 00:37:17,240

 

873
00:37:17,240 --> 00:37:19,460

 

874
00:37:19,460 --> 00:37:22,880

 

875
00:37:22,880 --> 00:37:25,220

 

876
00:37:25,220 --> 00:37:26,930

 

877
00:37:26,930 --> 00:37:28,099

 

878
00:37:28,099 --> 00:37:30,589

 

879
00:37:30,589 --> 00:37:33,680

 

880
00:37:33,680 --> 00:37:36,950

 

881
00:37:36,950 --> 00:37:38,660

 

882
00:37:38,660 --> 00:37:42,970

 

883
00:37:42,970 --> 00:37:45,920

 

884
00:37:45,920 --> 00:37:47,809

 

885
00:37:47,809 --> 00:37:50,120

 

886
00:37:50,120 --> 00:37:53,210

 

887
00:37:53,210 --> 00:37:55,220

 

888
00:37:55,220 --> 00:37:56,779

 

889
00:37:56,779 --> 00:38:01,039

 

890
00:38:01,039 --> 00:38:11,180

 

891
00:38:11,180 --> 00:38:14,140

 

892
00:38:14,140 --> 00:38:16,579

 

893
00:38:16,579 --> 00:38:18,289

 

894
00:38:18,289 --> 00:38:20,660

 

895
00:38:20,660 --> 00:38:22,430

 

896
00:38:22,430 --> 00:38:25,309

 

897
00:38:25,309 --> 00:38:28,370

 

898
00:38:28,370 --> 00:38:30,549

 

899
00:38:30,549 --> 00:38:34,130

 

900
00:38:34,130 --> 00:38:36,049

 

901
00:38:36,049 --> 00:38:38,240

 

902
00:38:38,240 --> 00:38:39,980

 

903
00:38:39,980 --> 00:38:40,700

 

904
00:38:40,700 --> 00:38:46,759

 

905
00:38:46,759 --> 00:38:48,230

 

906
00:38:48,230 --> 00:38:53,690

 

907
00:38:53,690 --> 00:38:58,849

 

908
00:38:58,849 --> 00:39:01,700

 

909
00:39:01,700 --> 00:39:04,519

 

910
00:39:04,519 --> 00:39:06,049

 

911
00:39:06,049 --> 00:39:07,249

 

912
00:39:07,249 --> 00:39:09,230

 

913
00:39:09,230 --> 00:39:11,420

 

914
00:39:11,420 --> 00:39:13,339

 

915
00:39:13,339 --> 00:39:14,780

 

916
00:39:14,780 --> 00:39:19,130

 

917
00:39:19,130 --> 00:39:20,900

 

918
00:39:20,900 --> 00:39:23,059

 

919
00:39:23,059 --> 00:39:24,829

 

920
00:39:24,829 --> 00:39:28,099

 

921
00:39:28,099 --> 00:39:31,250

 

922
00:39:31,250 --> 00:39:32,480

 

923
00:39:32,480 --> 00:39:36,430

 

924
00:39:36,430 --> 00:39:39,440

 

925
00:39:39,440 --> 00:39:46,069

 

926
00:39:46,069 --> 00:39:47,990

 

927
00:39:47,990 --> 00:39:50,089

 

928
00:39:50,089 --> 00:39:53,270

 

929
00:39:53,270 --> 00:39:56,000

 

930
00:39:56,000 --> 00:39:58,190

 

931
00:39:58,190 --> 00:40:01,849

 

932
00:40:01,849 --> 00:40:03,829

 

933
00:40:03,829 --> 00:40:06,710

 

934
00:40:06,710 --> 00:40:08,420

 

935
00:40:08,420 --> 00:40:10,819

 

936
00:40:10,819 --> 00:40:13,280

 

937
00:40:13,280 --> 00:40:14,540

 

938
00:40:14,540 --> 00:40:17,390

 

939
00:40:17,390 --> 00:40:20,720

 

940
00:40:20,720 --> 00:40:22,190

 

941
00:40:22,190 --> 00:40:25,309

 

942
00:40:25,309 --> 00:40:28,099

 

943
00:40:28,099 --> 00:40:29,900

 

944
00:40:29,900 --> 00:40:32,059

 

945
00:40:32,059 --> 00:40:34,099

 

946
00:40:34,099 --> 00:40:35,720

 

947
00:40:35,720 --> 00:40:38,319

 

948
00:40:38,319 --> 00:40:40,849

 

949
00:40:40,849 --> 00:40:42,410

 

950
00:40:42,410 --> 00:40:49,010

 

951
00:40:49,010 --> 00:40:50,150

 

952
00:40:50,150 --> 00:40:51,799

 

953
00:40:51,799 --> 00:40:53,390

 

954
00:40:53,390 --> 00:40:54,770

 

955
00:40:54,770 --> 00:40:59,740

 

956
00:40:59,740 --> 00:41:02,870

 

957
00:41:02,870 --> 00:41:07,130

 

958
00:41:07,130 --> 00:41:08,900

 

959
00:41:08,900 --> 00:41:11,829

 

960
00:41:11,829 --> 00:41:16,339

 

961
00:41:16,339 --> 00:41:18,770

 

962
00:41:18,770 --> 00:41:21,910

 

963
00:41:21,910 --> 00:41:25,160

 

964
00:41:25,160 --> 00:41:27,230

 

965
00:41:27,230 --> 00:41:28,730

 

966
00:41:28,730 --> 00:41:31,010

 

967
00:41:31,010 --> 00:41:32,390

 

968
00:41:32,390 --> 00:41:33,799

 

969
00:41:33,799 --> 00:41:36,049

 

970
00:41:36,049 --> 00:41:38,660

 

971
00:41:38,660 --> 00:41:42,049

 

972
00:41:42,049 --> 00:41:44,390

 

973
00:41:44,390 --> 00:41:46,549

 

974
00:41:46,549 --> 00:41:49,400

 

975
00:41:49,400 --> 00:41:52,789

 

976
00:41:52,789 --> 00:41:54,950

 

977
00:41:54,950 --> 00:41:56,630

 

978
00:41:56,630 --> 00:41:58,670

 

979
00:41:58,670 --> 00:42:01,460

 

980
00:42:01,460 --> 00:42:06,200

 

981
00:42:06,200 --> 00:42:10,099

 

982
00:42:10,099 --> 00:42:12,799

 

983
00:42:12,799 --> 00:42:15,970

 

984
00:42:15,970 --> 00:42:18,980

 

985
00:42:18,980 --> 00:42:20,180

 

986
00:42:20,180 --> 00:42:24,109

 

987
00:42:24,109 --> 00:42:28,250

 

988
00:42:28,250 --> 00:42:29,329

 

989
00:42:29,329 --> 00:42:31,930

 

990
00:42:31,930 --> 00:42:34,430

 

991
00:42:34,430 --> 00:42:37,370

 

992
00:42:37,370 --> 00:42:39,109

 

993
00:42:39,109 --> 00:42:42,380

 

994
00:42:42,380 --> 00:42:43,609

 

995
00:42:43,609 --> 00:42:48,880

 

996
00:42:48,880 --> 00:42:52,640

 

997
00:42:52,640 --> 00:42:55,760

 

998
00:42:55,760 --> 00:42:58,549

 

999
00:42:58,549 --> 00:43:02,980

 

1000
00:43:02,980 --> 00:43:05,450

 

1001
00:43:05,450 --> 00:43:07,910

 

1002
00:43:07,910 --> 00:43:11,270

 

1003
00:43:11,270 --> 00:43:14,690

 

1004
00:43:14,690 --> 00:43:17,599

 

1005
00:43:17,599 --> 00:43:19,849

 

1006
00:43:19,849 --> 00:43:22,130

 

1007
00:43:22,130 --> 00:43:24,530

 

1008
00:43:24,530 --> 00:43:26,150

 

1009
00:43:26,150 --> 00:43:27,319

 

1010
00:43:27,319 --> 00:43:31,710

 

1011
00:43:31,710 --> 00:43:38,880

 

1012
00:43:38,880 --> 00:43:41,490

 

1013
00:43:41,490 --> 00:43:46,050

 

1014
00:43:46,050 --> 00:43:47,670

 

1015
00:43:47,670 --> 00:43:48,839

 

1016
00:43:48,839 --> 00:43:50,040

 

1017
00:43:50,040 --> 00:43:52,440

 

1018
00:43:52,440 --> 00:43:56,670

 

1019
00:43:56,670 --> 00:43:59,460

 

1020
00:43:59,460 --> 00:44:01,349

 

1021
00:44:01,349 --> 00:44:05,250

 

1022
00:44:05,250 --> 00:44:07,349

 

1023
00:44:07,349 --> 00:44:10,200

 

1024
00:44:10,200 --> 00:44:12,300

 

1025
00:44:12,300 --> 00:44:14,540

 

1026
00:44:14,540 --> 00:44:16,849

 

1027
00:44:16,849 --> 00:44:22,520

 

1028
00:44:22,520 --> 00:44:25,050

 

1029
00:44:25,050 --> 00:44:29,040

 

1030
00:44:29,040 --> 00:44:39,270

 

1031
00:44:39,270 --> 00:44:41,460

 

1032
00:44:41,460 --> 00:44:44,490

 

1033
00:44:44,490 --> 00:44:46,829

 

1034
00:44:46,829 --> 00:44:48,420

 

1035
00:44:48,420 --> 00:44:51,720

 

1036
00:44:51,720 --> 00:44:54,270

 

1037
00:44:54,270 --> 00:44:56,880

 

1038
00:44:56,880 --> 00:45:00,569

 

1039
00:45:00,569 --> 00:45:03,680

 

1040
00:45:03,680 --> 00:45:08,069

 

1041
00:45:08,069 --> 00:45:10,349

 

1042
00:45:10,349 --> 00:45:12,089

 

1043
00:45:12,089 --> 00:45:13,620

 

1044
00:45:13,620 --> 00:45:16,620

 

1045
00:45:16,620 --> 00:45:19,020

 

1046
00:45:19,020 --> 00:45:21,380

 

1047
00:45:21,380 --> 00:45:24,180

 

1048
00:45:24,180 --> 00:45:26,190

 

1049
00:45:26,190 --> 00:45:28,200

 

1050
00:45:28,200 --> 00:45:30,450

 

1051
00:45:30,450 --> 00:45:31,890

 

1052
00:45:31,890 --> 00:45:33,390

 

1053
00:45:33,390 --> 00:45:36,059

 

1054
00:45:36,059 --> 00:45:38,640

 

1055
00:45:38,640 --> 00:45:41,430

 

1056
00:45:41,430 --> 00:45:44,610

 

1057
00:45:44,610 --> 00:45:48,300

 

1058
00:45:48,300 --> 00:45:50,340

 

1059
00:45:50,340 --> 00:45:51,960

 

1060
00:45:51,960 --> 00:45:55,980

 

1061
00:45:55,980 --> 00:45:57,900

 

1062
00:45:57,900 --> 00:45:59,250

 

1063
00:45:59,250 --> 00:46:01,200

 

1064
00:46:01,200 --> 00:46:03,330

 

1065
00:46:03,330 --> 00:46:05,250

 

1066
00:46:05,250 --> 00:46:11,400

 

1067
00:46:11,400 --> 00:46:15,060

 

1068
00:46:15,060 --> 00:46:17,280

 

1069
00:46:17,280 --> 00:46:21,690

 

1070
00:46:21,690 --> 00:46:24,180

 

1071
00:46:24,180 --> 00:46:26,550

 

1072
00:46:26,550 --> 00:46:28,950

 

1073
00:46:28,950 --> 00:46:30,480

 

1074
00:46:30,480 --> 00:46:31,530

 

1075
00:46:31,530 --> 00:46:33,720

 

1076
00:46:33,720 --> 00:46:35,490

 

1077
00:46:35,490 --> 00:46:37,470

 

1078
00:46:37,470 --> 00:46:41,940

 

1079
00:46:41,940 --> 00:46:43,860

 

1080
00:46:43,860 --> 00:46:46,140

 

1081
00:46:46,140 --> 00:46:47,850

 

1082
00:46:47,850 --> 00:46:51,630

 

1083
00:46:51,630 --> 00:46:54,420

 

1084
00:46:54,420 --> 00:46:55,650

 

1085
00:46:55,650 --> 00:46:58,560

 

1086
00:46:58,560 --> 00:47:02,910

 

1087
00:47:02,910 --> 00:47:04,380

 

1088
00:47:04,380 --> 00:47:06,960

 

1089
00:47:06,960 --> 00:47:09,330

 

1090
00:47:09,330 --> 00:47:11,160

 

1091
00:47:11,160 --> 00:47:12,630

 

1092
00:47:12,630 --> 00:47:15,530

 

1093
00:47:15,530 --> 00:47:25,650

 

1094
00:47:25,650 --> 00:47:29,940

 

1095
00:47:29,940 --> 00:47:31,950

 

1096
00:47:31,950 --> 00:47:33,600

 

1097
00:47:33,600 --> 00:47:36,180

 

1098
00:47:36,180 --> 00:47:38,940

 

1099
00:47:38,940 --> 00:47:41,340

 

1100
00:47:41,340 --> 00:47:43,830

 

1101
00:47:43,830 --> 00:47:46,200

 

1102
00:47:46,200 --> 00:47:47,910

 

1103
00:47:47,910 --> 00:47:50,400

 

1104
00:47:50,400 --> 00:47:52,440

 

1105
00:47:52,440 --> 00:47:53,550

 

1106
00:47:53,550 --> 00:47:57,830

 

1107
00:47:57,830 --> 00:48:01,020

 

1108
00:48:01,020 --> 00:48:04,560

 

1109
00:48:04,560 --> 00:48:07,440

 

1110
00:48:07,440 --> 00:48:11,670

 

1111
00:48:11,670 --> 00:48:15,600

 

1112
00:48:15,600 --> 00:48:18,510

 

1113
00:48:18,510 --> 00:48:24,690

 

1114
00:48:24,690 --> 00:48:29,820

 

1115
00:48:29,820 --> 00:48:31,440

 

1116
00:48:31,440 --> 00:48:35,490

 

1117
00:48:35,490 --> 00:48:41,340

 

1118
00:48:41,340 --> 00:48:46,260

 

1119
00:48:46,260 --> 00:48:48,060

 

1120
00:48:48,060 --> 00:48:49,950

 

1121
00:48:49,950 --> 00:48:51,480

 

1122
00:48:51,480 --> 00:48:52,980

 

1123
00:48:52,980 --> 00:48:54,750

 

1124
00:48:54,750 --> 00:48:55,890

 

1125
00:48:55,890 --> 00:48:59,820

 

1126
00:48:59,820 --> 00:49:03,420

 

1127
00:49:03,420 --> 00:49:05,910

 

1128
00:49:05,910 --> 00:49:08,040

 

1129
00:49:08,040 --> 00:49:09,420

 

1130
00:49:09,420 --> 00:49:11,610

 

1131
00:49:11,610 --> 00:49:14,780

 

1132
00:49:14,780 --> 00:49:20,400

 

1133
00:49:20,400 --> 00:49:24,330

 

1134
00:49:24,330 --> 00:49:26,400

 

1135
00:49:26,400 --> 00:49:30,060

 

1136
00:49:30,060 --> 00:49:32,250

 

1137
00:49:32,250 --> 00:49:36,200

 

1138
00:49:36,200 --> 00:49:45,390

 

1139
00:49:45,390 --> 00:49:48,000

 

1140
00:49:48,000 --> 00:49:49,890

 

1141
00:49:49,890 --> 00:49:52,860

 

1142
00:49:52,860 --> 00:49:54,450

 

1143
00:49:54,450 --> 00:49:57,810

 

1144
00:49:57,810 --> 00:50:00,090

 

1145
00:50:00,090 --> 00:50:02,640

 

1146
00:50:02,640 --> 00:50:06,510

 

1147
00:50:06,510 --> 00:50:10,200

 

1148
00:50:10,200 --> 00:50:10,210

 

1149
00:50:10,210 --> 00:50:10,890


1150
00:50:10,890 --> 00:50:12,690

 

1151
00:50:12,690 --> 00:50:17,690

 

1152
00:50:17,690 --> 00:50:21,480

 

1153
00:50:21,480 --> 00:50:23,010

 

1154
00:50:23,010 --> 00:50:25,230

 

1155
00:50:25,230 --> 00:50:27,359

 

1156
00:50:27,359 --> 00:50:28,680

 

1157
00:50:28,680 --> 00:50:30,750

 

1158
00:50:30,750 --> 00:50:32,069

 

1159
00:50:32,069 --> 00:50:34,650

 

1160
00:50:34,650 --> 00:50:36,120

 

1161
00:50:36,120 --> 00:50:39,480

 

1162
00:50:39,480 --> 00:50:41,059

 

1163
00:50:41,059 --> 00:50:43,170

 

1164
00:50:43,170 --> 00:50:45,779

 

1165
00:50:45,779 --> 00:50:47,609

 

1166
00:50:47,609 --> 00:50:49,319

 

1167
00:50:49,319 --> 00:50:53,819

 

1168
00:50:53,819 --> 00:50:56,130

 

1169
00:50:56,130 --> 00:50:58,980

 

1170
00:50:58,980 --> 00:51:01,289

 

1171
00:51:01,289 --> 00:51:02,700

 

1172
00:51:02,700 --> 00:51:04,620

 

1173
00:51:04,620 --> 00:51:06,960

 

1174
00:51:06,960 --> 00:51:09,240

 

1175
00:51:09,240 --> 00:51:11,609

 

1176
00:51:11,609 --> 00:51:13,920

 

1177
00:51:13,920 --> 00:51:16,200

 

1178
00:51:16,200 --> 00:51:17,549

 

1179
00:51:17,549 --> 00:51:19,740

 

1180
00:51:19,740 --> 00:51:21,960

 

1181
00:51:21,960 --> 00:51:24,839

 

1182
00:51:24,839 --> 00:51:27,569

 

1183
00:51:27,569 --> 00:51:29,220

 

1184
00:51:29,220 --> 00:51:30,630

 

1185
00:51:30,630 --> 00:51:32,250

 

1186
00:51:32,250 --> 00:51:34,589

 

1187
00:51:34,589 --> 00:51:36,539

 

1188
00:51:36,539 --> 00:51:38,339

 

1189
00:51:38,339 --> 00:51:40,019

 

1190
00:51:40,019 --> 00:51:43,079

 

1191
00:51:43,079 --> 00:51:47,849

 

1192
00:51:47,849 --> 00:51:49,319

 

1193
00:51:49,319 --> 00:51:52,620

 

1194
00:51:52,620 --> 00:51:53,940

 

1195
00:51:53,940 --> 00:51:56,660

 

1196
00:51:56,660 --> 00:51:58,470

 

1197
00:51:58,470 --> 00:52:00,180

 

1198
00:52:00,180 --> 00:52:02,730

 

1199
00:52:02,730 --> 00:52:04,410

 

1200
00:52:04,410 --> 00:52:07,260

 

1201
00:52:07,260 --> 00:52:09,240

 

1202
00:52:09,240 --> 00:52:10,859

 

1203
00:52:10,859 --> 00:52:13,079

 

1204
00:52:13,079 --> 00:52:14,970

 

1205
00:52:14,970 --> 00:52:17,160

 

1206
00:52:17,160 --> 00:52:19,049

 

1207
00:52:19,049 --> 00:52:22,789

 

1208
00:52:22,789 --> 00:52:27,559

 

1209
00:52:27,559 --> 00:52:37,470

 

1210
00:52:37,470 --> 00:52:37,480

 

1211
00:52:37,480 --> 00:52:39,540