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alright so lecture 13 was about a lot of  things we had algorithms for the  Carpenters rule theorem we had three of  those see there's the CDR stray new  pseudo triangulations which we're going  to talk about a bunch today there's the  energy method which we'll talk about a  little bit then there was locked trees  I'll give you there's one new result in  lock trees i'll tell you about and then  we had the last topic which was she oh  right last topic for today is four  dimensions which wasn't covered in class  at what we did talk about in class was  3d essentially no updates there all the  open problems are still open fairly  challenging but 40 we didn't get to talk  about it all and that's not too hard how  to unfold any 4d chain so that'll be fun  also 5d chains okay and 60 and 70 but  not eight now any any dimension first  question for today is why do we care so  much about expansiveness and this is  mostly will say I want to reference next  lecture which will use expansiveness but  indeed the the original point of  expansiveness was just a convenient way  to ensure self intersection essentially  before the idea of expansiveness we had  this idea of a really complicated  channel it's not very complicated but  for the Carpenters rule problem we had  this idea that well how do we  distinguish between a linkage that is  locked within epsilon I mean however you  draw it if it's if it's not self  touching it's going to jiggle a little  bit versus something that can eventually  unfold and there was no good way to  distinguish between just jiggling a  little bit and unfolding all the way  which was frustrating and expansiveness  gave us a way to guarantee that not only  did it avoid self intersection which is  nice if you expand who don't self  intersect but it also gave away that  things would actually be rigid if it was  impossible to do and so you couldn't  move at all that reduced it to a  rigidity theory problem or really a  tense a gritty theory problem which was  kind of what  this problem needed it's why it was open  for 25 years before before we could  prove it with tenseiga at ease so that's  one answer but in fact now that we know  expansive motions exist they're really  handy they let us prove that the energy  method worked for one and next class  we're going to use them to add thickness  to our edges add polygons or regions  attached to every edge and if you have  an expansive motion it turns out under  certain conditions this thing will still  work whereas with non expansive motions  like those produced by the energy method  it won't work in general so initially  was convenience but it turns out to be  handy mathematically to we will we've  seen one example just proving the energy  method that's actually our next topic we  will see another next lecture so the but  a few people asked about the proof of  the energy method so I wanted to review  it how do we know we don't get stuck in  a local minimum so this is a question to  ask if you're familiar with gradient  descent if not here's the kind of  picture imagine you have some energy  landscape in reality this is a surface  drawn over n times D dimensions but I  will assume n times d equals 1 because I  can draw it so here's configurations and  then on the x-axis or in general all the  other axes and then the y-axis which  will always be one thing is the energy  function so sometimes energy is high at  some points it's going to actually shoot  off to infinity that's where you'd self  intersect remember the energy function  was sum over all edges VW sum over all  vertices you won over the distance  between you and VW so when a distance  goes to zero the energy shoots to  infinity those are the points we want to  avoid so those are bad things but if we  start at some non self intersecting  configuration will have some finite  energy and the idea is well if we follow  the the negative gradient  that will decrease energy as fast as  possible and from here it's going to  basically follow this trajectory and it  would get stuck in this little well this  is called a local minimum but this local  minimum might be lower and so the worry  is that it depends where you start if I  start over here I'll fall down there but  if I start in this kind of well I'll end  up at this kind of accumulation point  and what I really want is maybe some  kind of global mental energy minima  that's the usual issue with gradient  descent is why people don't like to use  gradient descent sometimes because it  gets stuck however I Jason however with  the this particular setting we actually  proved that the energy method does not  get stuck in a local minimum and I'll  just review that proof because it went  by quickly in lecture so the idea in our  setting is we know that we know the  Carpenters rule theorem Carpenters rule  theorem tells us that any configuration  where we're not done and I'll start just  by thinking about a single chain maybe  an open chain or a closed chain any  chain that is not already straight or  convex not only can be unfolded but has  an expansive motion to unfolding  expensive motion tells us that if we  look at any distance between a vertex  and an edge this increases that's not  how we define expensive we defined it  between every pair of vertices but it  turns out if you have this is actually  something we needed for a non self  intersection you have a vertex on an  edge and you know that that well this  edges stays fixed length because it's a  bar we know that these distances  increase and this distance increases by  expansiveness then in fact  this point must go away from this bar  you can draw an ellipse if you held  these fixed it would move along an  ellipse or held the some fixed but  you've got to go outside that ellipse  this guy's got to get farther away from  every every vertex has to get farther  away from every bar if the pairwise  vertex vertex distances increase so if  you take this expanse of motion you know  that it is an energy decreasing motion  so what that tells you is that any  configuration that's not already  straighter convex has an energy  decreasing motion so this tells you you  never get stuck on a local minimum  because what's a local minimum in one  dimension it's a place where you cannot  go down from here and those are the  places we're worried about and the claim  is the only places where you don't have  a downhill motion our places where you  don't have expansive motions and if and  therefore the configuration must already  be straight and convex straight or  convex and then you're done so you don't  have to worry about getting stuck with  energy method there is one tricky part  which is if you have a chain if you have  a linkage that consists of multiple  pieces if you have some open chain some  clothes chains you can have some guys  stuck in here they won't necessarily  straighten out but there's a couple  kinds of things you could do you can  unfold this chain you can unfold this  chain or you could just let them fly  away from each other all of those things  will decrease energy and so the worry  here would be that these two linkages  fly apart and never really unfold  locally that would be bad for us because  you know we're always decreasing energy  there's always a decreasing flow but we  may not actually finish we might not  finally make this our goal in this case  remember is to make it outer convex  meaning that all the outermost chains  are straight all the outermost polygons  become convex so we want to argue this  and i'm just going to sketch the proof  here essentially if when you have  multiple  once they are very very very far away  their contribution to energy just by all  the pairwise distances across between  two components will be very very tiny  and the claim is actually the energy  method won't get you all the way to an  outer convex configuration but it will  get you within some epsilon of outer  convex you get to choose whatever  epsilon you want and so when these guys  are far enough away their contribution  to energy is much less than epsilon once  they're like N squared divided by  epsilon away or something this really  they're the contribution between them  will be far less than epsilon so what  really matters in energy is the all the  distances inside the the linkages inside  the components and we're following the  gradient motion which decreases energy  as fast as possible and so you can prove  that it's better if you want to decrease  energy as fast as possible it's better  at some point to unfold the pieces not  to just let them continue flying apart  so waving my hands a little bit there  but if you know that you're at least  epsilon away from being done then the  energy will be fairly high within the  components and therefore you it's worth  unfolding the components but it's  especially clear for one people one  component all right any questions about  that that's why the energy method works  the next set of topics is related to  pointed pseudo triangulations which are  indeed very cool there's dozens of  papers about them they've spawned their  whole little sub field of computational  geometry we touched on them a little bit  in lecture but there's a lot more to say  about it in particular why they even  work for the Carpenters rule problem I  didn't make clear but I would like to so  a couple things about them they're  actually originally invented in 1994 or  the first used I guess in 1994 at this  point they were called geodesic  triangulations for a particular reason  and  so for the Carpenters real problem we  had a polygon and then we pseudo  triangulated the inside and the outside  of the polygon here we're just a pseudo  triangulating the inside and that's  setting they're called geodesic  triangulations and it's for a particular  data structures problem actually this is  it some sometimes covered in 6851 which  is advanced data structures you can  check out online so if you the goal is  to decompose this polygon in a kind of  balanced way so first you choose a suit  of triangles kind of most in the middle  of the polygon then you recurse on the  sides actually I guess number one is  this one and if you do this it turns out  if you want to walk from any point in  this polygon to any other point in the  polygon you only visit log and pseudo  triangles which is a nice small number  and this is a particular problem called  ray shooting where you imagine you have  like a laser to shoot in some direction  the polygon you want to know when it  exits the polygon when it hits the  boundary and where it hits and to do  that one way to do that is to walk  through the pieces here the pseudo  triangles if you only have to walk  through log n of them turns out walking  through a single triangle only takes log  n time total amount of time is log  squared n so this is a pretty good way  to do ray shooting in polygons which is  nice so that's where they originally  come from but they weren't used very  much until until later so let me tell  you some nice properties about them  Suter triangulation has 2n minus 3 edges  and n minus 2 pseudo triangles always if  you draw i should say if you drive  pointed pseudo triangulation remember  pointed means at every vertex of the  suit of triangulation you have some  angle which is you have a reflex angle  bigger than 180 this is the pointed  property  sudo triangles mean that all the faces  look like this they only have three  convex vertices all the other vertices  are reflex so this is pseudo triangle  pointed pseudo triangulation has both of  these properties and it's pretty easy to  prove this by induction I won't go  through the proof here it's interesting  in the way that it is different from  triangulations so n here is the number  of points and I'm thinking about pointed  pseudo triangulations of the point set  meaning I take some points and I add  edges until I can't anymore subject to  the pointed property so I add this edge  I'm still pointed I add this edge I'm  still pointed I add this edge I'm still  pointed can't add this edge can't really  add that edge let's see I can add this  one and this one and this one anymore at  this one and now i'm done i can tell  that i'm done because at this point all  of the faces are pseudo triangles hf  three convex vertices and as soon as you  have that all the faces are pseudo  triangles you can show that you can't  adding will add any more edges while  still being pointed i'm still pointed  because i only added edges that preserve  pointed pneus and this is a sort of  greedy algorithm to construct point a  suitor triangulation is it always works  and we can count the number of edges I  won't bother but it will always be 2 n  minus 3 cool by contrast if you look at  regular triangulations so a  triangulation would go all the way to  the point where every face is a triangle  like this and in that case they have  more edges obviously in general they  will have here it's two more edges  because there are two interior vertices  so in general b to n minus 3 plus I  edges for a triangulation  or I is the number of interior points  and it will have n minus 2 plus I faces  triangles yeah so that's the pseudo  triangulations are quite minimal in the  number of edges in particular they're  minimally generically rigid right this  is the number of edges I want for a nice  rigid structure to n minus 3 is what I  should have and indeed pointed Souter  triangulations are minimally generically  rigid that's the next thing minimally  generically rigid this was the Lamont  condition and we saw a couple  characterisations but in particular you  can fairly easily prove that they  satisfied the Lamont condition why well  first thing to check is they have 2n  minus three edges that I just claimed  second thing to check is that if I look  at any subset of the vertices k of the  vertices they should only have at most  2k minus 3 edges among them they should  induce at most 2k minus 3 edges well  let's look at any K vertices what are  the edges that they induce well the  thing I know about them is that they  will induce I pointed set of edges  because this structure without the blue  edges is pointed if i remove edges from  it it will still be pointed right if you  have a big angle you will continue to  have a big angle if you remove edges  from it so matter which subset of edges  i look at in particular induce sub set  it will still be pointed if i have a  pointed set of edges on some k vertices  i can use that greedy algorithm to  finish the pseudo triangulation once if  as long as you're pointed you can add  edges  until you can't add a jizz subject to  the pointed constraint at all times  preserving pointed pneus this is what I  did to generate this triangulation the  first place if I take some subset of K  vertices I can do it again on those K  vertices when I'm finished I will have a  pseudo triangulation and that has 2n  minus three edges in this case 2 K minus  3 edges so I started with some pointed  set of edges i added stuff i ended up  with 2k minus 3 that means i started  with at most 2k minus 3 edges to k minus  3 and so that's how you prove the lemond  condition is kind of trivial because  pointed pneus is an inherited property  so if you believe in 2n minus 3 then you  believe minimally generically rigid cool  even cooler is that the converse is  roughly true some sense all lemond  graphs can be drawn as pseudo  triangulations it's one catch which is  that you need plain arity any plainer  lemond graph  has a pseudo triangulation as a  realization realization was just a way  to draw the graph in 2d now Souter  triangulations are planar meaning none  of the edges cross each other so you  definitely need a graph that can be  drawn in the plane without crossings  that's the planar constraint but if you  have a planar minimally generically  rigid graph you can always draw it as a  suit of triangulation so it's kind of a  converse of this theorem and a sense in  which suit of triangle ations are  universal for planar minimal generic  rigidity which is pretty neat I will not  prove that theorem but I will give you a  sense of our visual sense of what the  proof looks like it's based on the  henneberg construction so we know  minimally generically rigid graphs can  always be drawn in a henneberg way  either by adding new degree two vertices  or by adding new degree three vertices  and removing one of the existing edges  and the claim is you take any Henneberg  construction you can do it you can  implement it in a pseudo triangulation  so you start up with you know your  single edge that's the suitor  triangulation you keep adding things  preserving the fact that is it at all  times is a pointed pseudo triangulation  it's tricky but for example if you know  that you want to add this red vertex  next to these two you find a place that  will guarantee that this is a suitor  triangle this is the suit of triangle it  will be pointed for free because it's  just a degree to vertex so the challenge  is preserving pseudo triangles and same  for adding the degree three vertex this  is harder of course but you need to you  find a little patch here where if you  choose the point in there you get suit  of triangles on all three sides and also  that removing the edge still preserve  sort of triangulation it's definitely  not trivial to do this but it can be  done it's kind of neat that it can be  done for any Henneberg construction  cool the last thing I wanted to tell you  about sudo triangulations is why they  work from the Carpenters rule theorem  why do they give expansive motions well  they're rigid of course they don't give  you any motions but what we claimed is  that if you remove a convex hull edge so  if we take a pointed pseudo  triangulation and then we remove a  convex hull edge we of course get  something that's flexible because we  were minimally generically rigid so if  you remove an edge you will now be  generically flexible it turns out you're  not only generically flexible you are  actually flexible for at least a little  bit of time so appointed suit of  triangulation minus convex hull edge  it's not only as flexible but it flexes  expansively meaning if you look at all  pairwise distances expansively look at  all pairwise distances they either stay  the same they're connected by a bar or  they will increase and this is what we  wanted for Carpenters rule you can do  that motion for a little while then you  might have to switch to a different suit  of triangulation according to a flip  which I won't talk about we already did  in lecture but at all times the claim is  there is an expansive suitor  triangulation motion by removing a  single edge so why does this hold this  it turns out you can prove using things  similar to what we know from the our  proof of the Carpenters rule theorem  that we covered so it's kind of neat to  see the connection between the two  so why is this true well we already  argued that the thing is flexible  without the expansiveness constraint so  we want to add the expansiveness  constraint so just like before we're  going to add all pairwise struts so  between all pairs of vertices that say  they're not already connected by a bar  we add a strep between them meeting that  can only increase in length now we have  a 10 security we want to argue that  tensor gritty is infinitesimally  flexible so we're going to do that using  the duality just like CDR just like the  proof of the Carpenters rule theorem we  saw we say okay we want to prove this  thing as flexible by duality that's  equivalent to proving something about  the equilibrium stresses now in the case  of CDR on a single chain what we needed  to say was the equilibrium stresses were  all 0 and then we said ok if in order to  prove the equilibrium stresses we're all  0 we use Maxwell cremona theorem that's  equivalent to saying all polyhedral  liftings are flat and then we prove that  by looking at the maximum z-coordinate  in this case we can't argue that all the  stresses are zeros not true but if you  recall the duality claim I said if you  have a tense egra t10 cigarette is rigid  if and only if the underlying linkage is  rigid like when you replace struts with  bars and the which is it's going to be  true here and this there is a stress an  equilibrium stress that is nonzero on  every strut right that's equally  mistress being non zero and every strut  means that all the struts basically have  to stay fixed length they become VARs  and then you're done ok what I need to  prove the opposite I want to prove its  flexible I need to find a strut where  there's no stress  if I find a start where there's no  stress then basically that strut can  expand and there's a motion okay that's  roughly true huh it's a little bit more  complicated than that but I'm going to  just claim suffice is to prove there's a  strut that has 0 0 and every equilibrium  stress okay which strut am I going to  choose I am going to choose the one that  comes from this convex hull edge e so I  removed a convex hull edge as a bar it's  going to get replaced when I add all  pairwise struts by a strut so that I've  got the vertices on the corner here one  of these I replaced with a strut the  other guys are whatever they are it's a  suit of triangulation not a very  interesting ones strangulation in this  case a depart X here now it's a suitor  triangulation so this guy got replaced  with a strut for example I claim this  guy is 0 in every stress why is it zero  in every stress well claim  we've essentially proved this in any  equilibrium stress nonzero stresses must  be on or interior to convex polygons of  bars okay first believe this claim the  only place i have non-zero stresses when  I'm interior or on the boundary of a  convex polygon of bars and then you know  there might so there might be some  nonzero stresses in here these edges  might be nonzero stressed intuitively  what's happening here these guys are  going to serve as mountains in the  lifting and there's like a hole so you  go deep into the hole their claim is  that that is the picture so you've got  some convex polygons there's stuff in  here that could be bad could be stressed  but the stuff outside the convex  polygons whatever they are have to be  flat meaning they have zero stress I'm  jumping back and forth between the  lifting and the stresses by Maxwell  cremona those are the same thing now  look at edge E and G is a strut the  other edges here that I've drawn those  were the bars I haven't drawn all the  other struts I should really use a color  for this guy so this red edge there's  lots of other red edges i'm not drawing  between all pairs of vertices but the  only bars are the white edges and those  white edges well there's some convex  polygons here's a convex polygon here's  a convex polygon but none of the convex  polygons in close e because these on the  convex hull he's on the outside here so  it can't be stressed by this claim okay  so if you believe this claim then you  believe he has zero stress because it's  not interior to any convex bar polygon  because it's a strut not a bar ok now  let's improve the claim  it follows from the argument we gave in  that lecture 12 previous one will  reprove the Carpenters rule theorem but  it's not obvious that it follows let me  explain why so I want to look at the  maximum z-coordinate I want to look at  the region of all the points in the  plane that in the polyhedral lifting  given by Maxwell cabanas theorem they  are at maximum Z okay that is some  region it could be two dimensional it  could be one-dimensional can't be  three-dimensional as part of the plane  so I mean this region may have some  one-dimensional parts it may have some  two-dimensional parts like this whole  part all all of this stuff might lift to  maximum Z it could be you have some non  convex parts like this whatever this is  kind of generically what it might look  like I guess it could have cycles with  not filled interior all these things are  plausible except we had this lemma this  is like the key lemma the heart of the  proof of the Carpenters rule theorem if  you look at em and you look at a vertex  V of the boundary of ammo use this del  notation to mean I ignore the points in  here I want points along the boundary  here and in particular I want to look at  these vertices in this drawing ok look  at such a vertex V and suppose that V is  pointed we didn't use this terminology  because we didn't have it at the time  but maybe V looks like this and there's  an angle here this reflex that's a point  in vertex now i'm looking here just at  the bars ignoring the struts ok so  suppose vertex V  has a reflex angle among just the bars  then the claim is locally this entire  region must be an M must be locally at  the maximum z-coordinate okay how did we  prove that well a claim is this had to  be flat because there's no mountains  here so if you think of a reflex region  and V V is at the maximum z-coordinate  right it's on the boundary of the  maximum z-coordinate so it has maximum  possible Z so there's no way you can go  up these edges could be mountains but  all the other edges which are not drawn  here use blue there are some other edges  those are the struts coming to be those  all have to be valleys they'll have to  lift a valleys because stresses can only  carry a positive stress stress course  positive stress correspond to ballys  these guys are all valleys and this is  at maximum Z really can't use those  valleys right you're up here at maximum  Z if use valleys you go to higher Z  which is not allowed so these actually  i'll have to be flat out here which  means locally all this stuff is at  maximum Z ok that was the proof we had a  couple lectures ago now once you know  that that's at maximum Z you know that a  lot of these things are impossible as  look here's a reflex angle among bars I  mean there might be more bars here I  guess but we know that this is pointed  whatever is missing here there's there's  some reflex angle we're assuming we have  a pointed Souter triangulation and then  we remove an edge is still pointed so  that means all of this has to be in em  is it in this picture no contradiction  so you cannot have one dimensional parts  because wherever you have a one  dimensional part there's a reflex angle  that's not in em so that's bad in fact  they all have to be two-dimensional  parts like this can it be a  two-dimensional part like this no  because here's a reflex angle that  is not an M contradiction so the only  situation you can have is a convex  polygon with the interior all in M sorry  wrong the reverse the exterior should  all be an F because you have all these  reflex angles that has to contain the  pointed part and locally by this lemma  it has to be in em so all this stuff has  to locally be an M the only way is for  all of that to be an M the inside we  don't know could be stress is there and  this could be a hole where you go deeper  all of this is at the maximum  z-coordinate the boundary here the  inside who knows could go down so you  can have stresses interior to convex  polygons of bars but not exterior to the  convex polygon I mean M is either  everything and there's no boundary and  then there's no stress or it's not  everything and then it will have to have  stresses only interior to convex  polygons and that's what we're claiming  is that stresses can only be interior to  convex bar polygons and in particular  than E is not stressed the end this is  definitely harder than the case we  talked about for Carpenters rule theorem  when you have a single chain you're just  trying to prove there's the expanse of  motion it's a lot easier than this  because m is much simpler can basically  only be a path or cycle because you only  have you only have a single chain here M  could be more complicated but in the end  it's actually pretty darn simple any  questions about that  cool perfectly clear definitely bit  complicated but nice now it turns out  Souter triangulations are really at the  core of expansive motions just mention  one more theorem about them so you may  recall I mentioned if you look at the  space of infinitesimal motions of a  linkage or at integrity they form a cone  it's it lives in some high dimensional  space but I'll try to draw a cone here  these guys go off to infinity drawing it  in three dimensions I guess you have  some place here this is the zero zero  zero motion where nothing moves and  these are potential different motion  each of these points corresponds to a  set an assignment of velocity vectors  where you move and you preserve all the  constraints if I take such a motion I  can scale it up it's still a motion so  that's motions form a cone it's called a  convex cone every motion can be scaled  up or down all the way to zero so if we  look at the cone of expansive motions  it's also a cone it's you know it's a  cone of motions of a particular tense a  greater where we add all pairwise struts  then pseudo triangulations correspond to  these edges of the comb these edges  they're really raised right they go off  to infinity these things are called  extreme rays meaning their extreme in a  particular direction like if I want the  emotion that is the most this way then  it will be this this ray okay they're  kind of the corners of this polyhedron  the edges of the polyhedron if you will  although in higher dimensions is not it  is edges but they're called extreme  raised in the case of  okay the extreme raised in the cone of  expansive motions equal pointed Souter  triangulations minus one edge I think I  have a slide about it yeah so this is a  paper by rote dos santos and strain you  this is just a particular example for  five points these are all the different  suitor triangulations minus an edge in  some cases like here you get two  triangles that that becomes rigid so  they fill in the hole click to indicate  that's a rigid component but the claim  is 45 points these are the edges of the  expansive cone each of these guys has an  expense of motion like this guy would  rotate pretty simple this one would open  up a little like that in general all  these guys have an expansive motion  those are characterizing the cone those  are kind of the extreme raised in this  fairly high dimensional cone and has  something like 10 dimensions if it's 5  times 2 and you can prove that if you  interested to read the paper so in fact  pseudo triangulations are really core to  expansive motions if you wrote down the  linear program which is expect the  expansive motions then you are  essentially characterizing this as the  polyhedron and the linear program if you  know linear programming and you follow  like the simplex method simplex method  is all about finding where the edges are  and here they are so you would actually  if you implemented the linear program  ran it on a simplex solver you would get  sued appointed Souter triangulations  automatically they just pop out that's  actually originally how they were  discovered but then this is why they  were discovered that way  end of suit of triangulations alright  next question is have any of the open  problems been solved I get this almost  every lecture and usually the answer is  no so I don't tell you anything but this  time there is a nice result which is  related to this picture these are slides  from from the lecture we had this linear  lock tree which is which is minimal  among all linear lock trees minimum  number of edges we had this equilateral  ochiltree all the edges were the same  length was not strongly locked there was  some positive distances here but it is  locked I didn't mention this question  but an obvious open question is can you  get both at the same time is there a  linear equilateral lock tree if there  were probably be a lot easier to prove  lock than this mess so it'd be nice to  get both but in fact you cannot get both  there's this paper by a bunch of people  in the open problem session from two  years ago I finally got published last  last year and it proves a bunch of  things but one of the things that proves  there is no equilateral lock tree that  is also linear also lives in a single  dimension and the way it proves that  essentially so you take any linear  equilateral tree what does it look like  well it lives along a line here and all  its segmented into equal length chunks  so from a high level it looks like this  now in reality there could be lots of  edges along each of these segments and  there's you know interesting things  happening inside one of these vertices  could be I don't know something like  this that's one way to do a tree like  structure in there it could have parts  like this that go from side to side  without touching these guys who knows  what's happening here but from a high  level perspective it's a bunch of  segments in a path okay the idea is this  is basically just looking at five of the  vertices out of all n of them but if we  just look at those five vertices  everything looks good this is in  canonical form canonical form for a tree  is where all the edges point to the  right something like this is nice and  canonical so right now this is canonical  what we do is kind of look inside the  path here look for a break point where  things are not canonical and then we fix  it so here's just an example of that  here this is what it looks like  currently at a high level here's what it  looks like in reality ok so imagine so  the tree is doing this stuff locally at  all these vertices right now we're  coalescing this all into one big mega  vertex and then it looks like this but  suppose we now we realize oh actually  there's kind of a split here there's  stuff over to the right of this or stuff  over to write this or not directly  connected then what we'll do is pull it  apart treat these as two separate  vertices increase the number of vertices  in this picture by one so we end up  splitting w here according to whatever  is actually happening in the linkage and  then we say oh gosh this is not really  canonical right this edge is pointing to  the left these guys are pointing to the  right still but this one's pointing to  the left so it looks like this picture  we've got a rightward edge and then a  leftward edge we fix it by just rotating  this edge 180 degrees keeping at all  times this thing pointing to the right  so this kind of comes along for the ride  and then we will end up with this  picture now we're canonical again with  one more vertex  we repeat eventually will be canonical  with all the vertices and we're done so  if you take any two configurations you  canonicalize both of them you will end  up with essentially the same picture it  does matter which route you choose but  it's already known that it doesn't  matter which route you choose you can  change the route just by flopping the  tree with one motion or very small  number of motions so if you have two  configurations you can authorize both of  them then there's a motion between them  and you get a motion from one  configuration to the other therefore  there are no locked trees that are  equilateral and linear kind of nice the  last thing I want to talk about is 40 a  couple of people asked about 4d why is  it so different from 3d the essential  reason is we have one dimensional bars  but four dimensions of motion that's a  gap of three dimensions three dimensions  is a lot when you have one dimensional  chains in 3d or the gap of two  dimensions two dimensions turns out to  not be a lot so why is that let me prove  it to you here's actually an animation  of the motion just for fun this is in  the textbook this is the top row is if  you are zoomed out from the very  beginning this is mess and you basically  pull out a string it kind of actually  Wiggles around a lot here's what it  looks like zoomed in this is the  original mess and you end up flopping  this over here then flopping it over  there and it's hard to see because this  is a two-dimensional projection of a  four-dimensional thing but eventually  you pull open the whole thing so how  does this algorithm work it's actually  very simple for open chains it's very  simple closed chains a little more  complicated trees also very simple  unfolding  or D open chains this is spy Roxanna  caca and Joe Rourke back before the  Carpenters rule theorem I guess this  publication was later but the original  version was in 1998 I think 99 so what  do we do well what I do is look at the  end bar so let's say this is the end of  the chain from here it goes in some  direction there's no stuff what I'd like  to do is rotate this bar so that it  extends this bar so I have a 180 degree  angle here that's my goal if I can do  that i confuse that vertex never rotate  it again treat this as a longer edge  there for the remaining thing has n  minus 1 edges if I had NN geez  originally so I can just apply induction  in other words just repeat that  eventually the whole thing will be  straight so the challenge is can I make  a move can I fold this last bar to be to  extend the next the second bar it's two  issues to problems that could happen  here one is that there's no continuous  motion to get there the other problem is  that even instantaneously if I just you  know picked it up and dropped it into  the right location it might intersect  the rest of the chain maybe the chain  here comes and intersects that ray if it  intersects that ray I can just jiggle  the whole chain a little bit and it  won't is that clear let's see you've got  these one-dimensional bars this is a  one-dimensional target ray so if a bar  happens to be on this ray and I just  wiggle it a tiny bit it's going to come  out of that off of that place this will  actually even work in three dimensions  this step there's an even easier way to  do it is just to rotate this bar if you  rotate this bar you can basically point  the Ray wherever you want and if you  look at where that bar is pointing it's  kind of hard to draw the picture but you  just have to miss all of these  one-dimensional things so it's very easy  to miss okay so let's just assume for  now we'll kind of see this  again in a moment assume that this ray  happens to be clear there may be things  that come very close to it maybe they go  on the back side and then come on the  front but they never don't actually  touch the ring now think about where  this bar can go relative to this vertex  if I draw a little sphere here I'm going  to drop initially in three dimensions  but really it's in four dimensions so  have a little sphere where this bar is  currently corresponds to a single point  on the sphere so I'm currently here  where I want to be corresponds to  another point on the sphere my goal is  to find a path on the sphere that gets  from gets to the X gets to the buried  treasure okay this is essentially  treating this as a whole ray so I don't  even care that it's short if I just  imagine it going off to infinity can I  go from position a to position B now in  three dimensions this is a two  dimensional sphere and if you look at  the obstacles the things I have to avoid  those correspond to one dimensional bars  here which if you project them onto the  sphere correspond to great circular arcs  so the worry would be let me draw them  in red the obstacles the things that  will kill you if you hit them the worry  would be you have what we call a cage  something like this of one-dimensional  obstacles on the sphere that block this  X point block the target from the source  and in 3d this if you draw what happens  in the knitting needles this is what  happens you can't get you can't move the  last link to extend the previous one  because one dimensional barriers are a  problem on a two-dimensional sphere  because two minus one equals one but on  the three-dimensional sphere if you draw  one dimensional obstacles they have  basically no effect on the sphere they  do not disconnect the surface of a  three-dimensional sphere can you see  that the the analogy if you just keep  the difference in dimensions the same so  we have  3-dimensional sphere versus a  one-dimensional obstacle let's go down a  dimension of a two-dimensional sphere  and a zero-dimensional obstacle what's a  zero-dimensional obstacle a point so  imagine you have all these red points  which you can't touch and then you have  the initial position and the target  position B how do you get there no  problem and there's tons of paths okay  it's basically the same in three  dimension and four dimensions just  harder to see you have a three spear  which is the boundary of a four ball and  you have these one-dimensional arcs on  it and they just don't block your way at  all because they're so low dimensional  so that's intuitively why for dzz 5d is  also easy 60 and so on because the  obstacles are so tiny obstacles remain  one-dimensional on this high dimensional  sphere you can just fix one link at a  time eventually the whole thing will be  straight tah-dah 4d any final questions  yeah of holding a surface in four  dimensions roughly analogous holding a  two-dimensional surface in four  dimensions definitely this argument  breaks down and so you can get locked  things there if you'd asked earlier I  could show one example we have of a kind  of locks surface I don't know if there's  actually a lock to the surface in 4d  known though I suspect there is one that  could be an interesting question to work  on cool think about that all right let's  see you Thursday

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alright so lecture 13 was about a lot of  things we had algorithms for the  Carpenters rule theorem we had three of  those see there's the CDR stray new  pseudo triangulations which we're going  to talk about a bunch today there's the  energy method which we'll talk about a  little bit then there was locked trees  I'll give you there's one new result in  lock trees i'll tell you about and then  we had the last topic which was she oh  right last topic for today is four  dimensions which wasn't covered in class  at what we did talk about in class was  3d essentially no updates there all the  open problems are still open fairly  challenging but 40 we didn't get to talk  about it all and that's not too hard how  to unfold any 4d chain so that'll be fun  also 5d chains okay and 60 and 70 but  not eight now any any dimension first  question for today is why do we care so  much about expansiveness and this is  mostly will say I want to reference next  lecture which will use expansiveness but  indeed the the original point of  expansiveness was just a convenient way  to ensure self intersection essentially  before the idea of expansiveness we had  this idea of a really complicated  channel it's not very complicated but  for the Carpenters rule problem we had  this idea that well how do we  distinguish between a linkage that is  locked within epsilon I mean however you  draw it if it's if it's not self  touching it's going to jiggle a little  bit versus something that can eventually  unfold and there was no good way to  distinguish between just jiggling a  little bit and unfolding all the way  which was frustrating and expansiveness  gave us a way to guarantee that not only  did it avoid self intersection which is  nice if you expand who don't self  intersect but it also gave away that  things would actually be rigid if it was  impossible to do and so you couldn't  move at all that reduced it to a  rigidity theory problem or really a  tense a gritty theory problem which was  kind of what  this problem needed it's why it was open  for 25 years before before we could  prove it with tenseiga at ease so that's  one answer but in fact now that we know  expansive motions exist they're really  handy they let us prove that the energy  method worked for one and next class  we're going to use them to add thickness  to our edges add polygons or regions  attached to every edge and if you have  an expansive motion it turns out under  certain conditions this thing will still  work whereas with non expansive motions  like those produced by the energy method  it won't work in general so initially  was convenience but it turns out to be  handy mathematically to we will we've  seen one example just proving the energy  method that's actually our next topic we  will see another next lecture so the but  a few people asked about the proof of  the energy method so I wanted to review  it how do we know we don't get stuck in  a local minimum so this is a question to  ask if you're familiar with gradient  descent if not here's the kind of  picture imagine you have some energy  landscape in reality this is a surface  drawn over n times D dimensions but I  will assume n times d equals 1 because I  can draw it so here's configurations and  then on the x-axis or in general all the  other axes and then the y-axis which  will always be one thing is the energy  function so sometimes energy is high at  some points it's going to actually shoot  off to infinity that's where you'd self  intersect remember the energy function  was sum over all edges VW sum over all  vertices you won over the distance  between you and VW so when a distance  goes to zero the energy shoots to  infinity those are the points we want to  avoid so those are bad things but if we  start at some non self intersecting  configuration will have some finite  energy and the idea is well if we follow  the the negative gradient  that will decrease energy as fast as  possible and from here it's going to  basically follow this trajectory and it  would get stuck in this little well this  is called a local minimum but this local  minimum might be lower and so the worry  is that it depends where you start if I  start over here I'll fall down there but  if I start in this kind of well I'll end  up at this kind of accumulation point  and what I really want is maybe some  kind of global mental energy minima  that's the usual issue with gradient  descent is why people don't like to use  gradient descent sometimes because it  gets stuck however I Jason however with  the this particular setting we actually  proved that the energy method does not  get stuck in a local minimum and I'll  just review that proof because it went  by quickly in lecture so the idea in our  setting is we know that we know the  Carpenters rule theorem Carpenters rule  theorem tells us that any configuration  where we're not done and I'll start just  by thinking about a single chain maybe  an open chain or a closed chain any  chain that is not already straight or  convex not only can be unfolded but has  an expansive motion to unfolding  expensive motion tells us that if we  look at any distance between a vertex  and an edge this increases that's not  how we define expensive we defined it  between every pair of vertices but it  turns out if you have this is actually  something we needed for a non self  intersection you have a vertex on an  edge and you know that that well this  edges stays fixed length because it's a  bar we know that these distances  increase and this distance increases by  expansiveness then in fact  this point must go away from this bar  you can draw an ellipse if you held  these fixed it would move along an  ellipse or held the some fixed but  you've got to go outside that ellipse  this guy's got to get farther away from  every every vertex has to get farther  away from every bar if the pairwise  vertex vertex distances increase so if  you take this expanse of motion you know  that it is an energy decreasing motion  so what that tells you is that any  configuration that's not already  straighter convex has an energy  decreasing motion so this tells you you  never get stuck on a local minimum  because what's a local minimum in one  dimension it's a place where you cannot  go down from here and those are the  places we're worried about and the claim  is the only places where you don't have  a downhill motion our places where you  don't have expansive motions and if and  therefore the configuration must already  be straight and convex straight or  convex and then you're done so you don't  have to worry about getting stuck with  energy method there is one tricky part  which is if you have a chain if you have  a linkage that consists of multiple  pieces if you have some open chain some  clothes chains you can have some guys  stuck in here they won't necessarily  straighten out but there's a couple  kinds of things you could do you can  unfold this chain you can unfold this  chain or you could just let them fly  away from each other all of those things  will decrease energy and so the worry  here would be that these two linkages  fly apart and never really unfold  locally that would be bad for us because  you know we're always decreasing energy  there's always a decreasing flow but we  may not actually finish we might not  finally make this our goal in this case  remember is to make it outer convex  meaning that all the outermost chains  are straight all the outermost polygons  become convex so we want to argue this  and i'm just going to sketch the proof  here essentially if when you have  multiple  once they are very very very far away  their contribution to energy just by all  the pairwise distances across between  two components will be very very tiny  and the claim is actually the energy  method won't get you all the way to an  outer convex configuration but it will  get you within some epsilon of outer  convex you get to choose whatever  epsilon you want and so when these guys  are far enough away their contribution  to energy is much less than epsilon once  they're like N squared divided by  epsilon away or something this really  they're the contribution between them  will be far less than epsilon so what  really matters in energy is the all the  distances inside the the linkages inside  the components and we're following the  gradient motion which decreases energy  as fast as possible and so you can prove  that it's better if you want to decrease  energy as fast as possible it's better  at some point to unfold the pieces not  to just let them continue flying apart  so waving my hands a little bit there  but if you know that you're at least  epsilon away from being done then the  energy will be fairly high within the  components and therefore you it's worth  unfolding the components but it's  especially clear for one people one  component all right any questions about  that that's why the energy method works  the next set of topics is related to  pointed pseudo triangulations which are  indeed very cool there's dozens of  papers about them they've spawned their  whole little sub field of computational  geometry we touched on them a little bit  in lecture but there's a lot more to say  about it in particular why they even  work for the Carpenters rule problem I  didn't make clear but I would like to so  a couple things about them they're  actually originally invented in 1994 or  the first used I guess in 1994 at this  point they were called geodesic  triangulations for a particular reason  and  so for the Carpenters real problem we  had a polygon and then we pseudo  triangulated the inside and the outside  of the polygon here we're just a pseudo  triangulating the inside and that's  setting they're called geodesic  triangulations and it's for a particular  data structures problem actually this is  it some sometimes covered in 6851 which  is advanced data structures you can  check out online so if you the goal is  to decompose this polygon in a kind of  balanced way so first you choose a suit  of triangles kind of most in the middle  of the polygon then you recurse on the  sides actually I guess number one is  this one and if you do this it turns out  if you want to walk from any point in  this polygon to any other point in the  polygon you only visit log and pseudo  triangles which is a nice small number  and this is a particular problem called  ray shooting where you imagine you have  like a laser to shoot in some direction  the polygon you want to know when it  exits the polygon when it hits the  boundary and where it hits and to do  that one way to do that is to walk  through the pieces here the pseudo  triangles if you only have to walk  through log n of them turns out walking  through a single triangle only takes log  n time total amount of time is log  squared n so this is a pretty good way  to do ray shooting in polygons which is  nice so that's where they originally  come from but they weren't used very  much until until later so let me tell  you some nice properties about them  Suter triangulation has 2n minus 3 edges  and n minus 2 pseudo triangles always if  you draw i should say if you drive  pointed pseudo triangulation remember  pointed means at every vertex of the  suit of triangulation you have some  angle which is you have a reflex angle  bigger than 180 this is the pointed  property  sudo triangles mean that all the faces  look like this they only have three  convex vertices all the other vertices  are reflex so this is pseudo triangle  pointed pseudo triangulation has both of  these properties and it's pretty easy to  prove this by induction I won't go  through the proof here it's interesting  in the way that it is different from  triangulations so n here is the number  of points and I'm thinking about pointed  pseudo triangulations of the point set  meaning I take some points and I add  edges until I can't anymore subject to  the pointed property so I add this edge  I'm still pointed I add this edge I'm  still pointed I add this edge I'm still  pointed can't add this edge can't really  add that edge let's see I can add this  one and this one and this one anymore at  this one and now i'm done i can tell  that i'm done because at this point all  of the faces are pseudo triangles hf  three convex vertices and as soon as you  have that all the faces are pseudo  triangles you can show that you can't  adding will add any more edges while  still being pointed i'm still pointed  because i only added edges that preserve  pointed pneus and this is a sort of  greedy algorithm to construct point a  suitor triangulation is it always works  and we can count the number of edges I  won't bother but it will always be 2 n  minus 3 cool by contrast if you look at  regular triangulations so a  triangulation would go all the way to  the point where every face is a triangle  like this and in that case they have  more edges obviously in general they  will have here it's two more edges  because there are two interior vertices  so in general b to n minus 3 plus I  edges for a triangulation  or I is the number of interior points  and it will have n minus 2 plus I faces  triangles yeah so that's the pseudo  triangulations are quite minimal in the  number of edges in particular they're  minimally generically rigid right this  is the number of edges I want for a nice  rigid structure to n minus 3 is what I  should have and indeed pointed Souter  triangulations are minimally generically  rigid that's the next thing minimally  generically rigid this was the Lamont  condition and we saw a couple  characterisations but in particular you  can fairly easily prove that they  satisfied the Lamont condition why well  first thing to check is they have 2n  minus three edges that I just claimed  second thing to check is that if I look  at any subset of the vertices k of the  vertices they should only have at most  2k minus 3 edges among them they should  induce at most 2k minus 3 edges well  let's look at any K vertices what are  the edges that they induce well the  thing I know about them is that they  will induce I pointed set of edges  because this structure without the blue  edges is pointed if i remove edges from  it it will still be pointed right if you  have a big angle you will continue to  have a big angle if you remove edges  from it so matter which subset of edges  i look at in particular induce sub set  it will still be pointed if i have a  pointed set of edges on some k vertices  i can use that greedy algorithm to  finish the pseudo triangulation once if  as long as you're pointed you can add  edges  until you can't add a jizz subject to  the pointed constraint at all times  preserving pointed pneus this is what I  did to generate this triangulation the  first place if I take some subset of K  vertices I can do it again on those K  vertices when I'm finished I will have a  pseudo triangulation and that has 2n  minus three edges in this case 2 K minus  3 edges so I started with some pointed  set of edges i added stuff i ended up  with 2k minus 3 that means i started  with at most 2k minus 3 edges to k minus  3 and so that's how you prove the lemond  condition is kind of trivial because  pointed pneus is an inherited property  so if you believe in 2n minus 3 then you  believe minimally generically rigid cool  even cooler is that the converse is  roughly true some sense all lemond  graphs can be drawn as pseudo  triangulations it's one catch which is  that you need plain arity any plainer  lemond graph  has a pseudo triangulation as a  realization realization was just a way  to draw the graph in 2d now Souter  triangulations are planar meaning none  of the edges cross each other so you  definitely need a graph that can be  drawn in the plane without crossings  that's the planar constraint but if you  have a planar minimally generically  rigid graph you can always draw it as a  suit of triangulation so it's kind of a  converse of this theorem and a sense in  which suit of triangle ations are  universal for planar minimal generic  rigidity which is pretty neat I will not  prove that theorem but I will give you a  sense of our visual sense of what the  proof looks like it's based on the  henneberg construction so we know  minimally generically rigid graphs can  always be drawn in a henneberg way  either by adding new degree two vertices  or by adding new degree three vertices  and removing one of the existing edges  and the claim is you take any Henneberg  construction you can do it you can  implement it in a pseudo triangulation  so you start up with you know your  single edge that's the suitor  triangulation you keep adding things  preserving the fact that is it at all  times is a pointed pseudo triangulation  it's tricky but for example if you know  that you want to add this red vertex  next to these two you find a place that  will guarantee that this is a suitor  triangle this is the suit of triangle it  will be pointed for free because it's  just a degree to vertex so the challenge  is preserving pseudo triangles and same  for adding the degree three vertex this  is harder of course but you need to you  find a little patch here where if you  choose the point in there you get suit  of triangles on all three sides and also  that removing the edge still preserve  sort of triangulation it's definitely  not trivial to do this but it can be  done it's kind of neat that it can be  done for any Henneberg construction  cool the last thing I wanted to tell you  about sudo triangulations is why they  work from the Carpenters rule theorem  why do they give expansive motions well  they're rigid of course they don't give  you any motions but what we claimed is  that if you remove a convex hull edge so  if we take a pointed pseudo  triangulation and then we remove a  convex hull edge we of course get  something that's flexible because we  were minimally generically rigid so if  you remove an edge you will now be  generically flexible it turns out you're  not only generically flexible you are  actually flexible for at least a little  bit of time so appointed suit of  triangulation minus convex hull edge  it's not only as flexible but it flexes  expansively meaning if you look at all  pairwise distances expansively look at  all pairwise distances they either stay  the same they're connected by a bar or  they will increase and this is what we  wanted for Carpenters rule you can do  that motion for a little while then you  might have to switch to a different suit  of triangulation according to a flip  which I won't talk about we already did  in lecture but at all times the claim is  there is an expansive suitor  triangulation motion by removing a  single edge so why does this hold this  it turns out you can prove using things  similar to what we know from the our  proof of the Carpenters rule theorem  that we covered so it's kind of neat to  see the connection between the two  so why is this true well we already  argued that the thing is flexible  without the expansiveness constraint so  we want to add the expansiveness  constraint so just like before we're  going to add all pairwise struts so  between all pairs of vertices that say  they're not already connected by a bar  we add a strep between them meeting that  can only increase in length now we have  a 10 security we want to argue that  tensor gritty is infinitesimally  flexible so we're going to do that using  the duality just like CDR just like the  proof of the Carpenters rule theorem we  saw we say okay we want to prove this  thing as flexible by duality that's  equivalent to proving something about  the equilibrium stresses now in the case  of CDR on a single chain what we needed  to say was the equilibrium stresses were  all 0 and then we said ok if in order to  prove the equilibrium stresses we're all  0 we use Maxwell cremona theorem that's  equivalent to saying all polyhedral  liftings are flat and then we prove that  by looking at the maximum z-coordinate  in this case we can't argue that all the  stresses are zeros not true but if you  recall the duality claim I said if you  have a tense egra t10 cigarette is rigid  if and only if the underlying linkage is  rigid like when you replace struts with  bars and the which is it's going to be  true here and this there is a stress an  equilibrium stress that is nonzero on  every strut right that's equally  mistress being non zero and every strut  means that all the struts basically have  to stay fixed length they become VARs  and then you're done ok what I need to  prove the opposite I want to prove its  flexible I need to find a strut where  there's no stress  if I find a start where there's no  stress then basically that strut can  expand and there's a motion okay that's  roughly true huh it's a little bit more  complicated than that but I'm going to  just claim suffice is to prove there's a  strut that has 0 0 and every equilibrium  stress okay which strut am I going to  choose I am going to choose the one that  comes from this convex hull edge e so I  removed a convex hull edge as a bar it's  going to get replaced when I add all  pairwise struts by a strut so that I've  got the vertices on the corner here one  of these I replaced with a strut the  other guys are whatever they are it's a  suit of triangulation not a very  interesting ones strangulation in this  case a depart X here now it's a suitor  triangulation so this guy got replaced  with a strut for example I claim this  guy is 0 in every stress why is it zero  in every stress well claim  we've essentially proved this in any  equilibrium stress nonzero stresses must  be on or interior to convex polygons of  bars okay first believe this claim the  only place i have non-zero stresses when  I'm interior or on the boundary of a  convex polygon of bars and then you know  there might so there might be some  nonzero stresses in here these edges  might be nonzero stressed intuitively  what's happening here these guys are  going to serve as mountains in the  lifting and there's like a hole so you  go deep into the hole their claim is  that that is the picture so you've got  some convex polygons there's stuff in  here that could be bad could be stressed  but the stuff outside the convex  polygons whatever they are have to be  flat meaning they have zero stress I'm  jumping back and forth between the  lifting and the stresses by Maxwell  cremona those are the same thing now  look at edge E and G is a strut the  other edges here that I've drawn those  were the bars I haven't drawn all the  other struts I should really use a color  for this guy so this red edge there's  lots of other red edges i'm not drawing  between all pairs of vertices but the  only bars are the white edges and those  white edges well there's some convex  polygons here's a convex polygon here's  a convex polygon but none of the convex  polygons in close e because these on the  convex hull he's on the outside here so  it can't be stressed by this claim okay  so if you believe this claim then you  believe he has zero stress because it's  not interior to any convex bar polygon  because it's a strut not a bar ok now  let's improve the claim  it follows from the argument we gave in  that lecture 12 previous one will  reprove the Carpenters rule theorem but  it's not obvious that it follows let me  explain why so I want to look at the  maximum z-coordinate I want to look at  the region of all the points in the  plane that in the polyhedral lifting  given by Maxwell cabanas theorem they  are at maximum Z okay that is some  region it could be two dimensional it  could be one-dimensional can't be  three-dimensional as part of the plane  so I mean this region may have some  one-dimensional parts it may have some  two-dimensional parts like this whole  part all all of this stuff might lift to  maximum Z it could be you have some non  convex parts like this whatever this is  kind of generically what it might look  like I guess it could have cycles with  not filled interior all these things are  plausible except we had this lemma this  is like the key lemma the heart of the  proof of the Carpenters rule theorem if  you look at em and you look at a vertex  V of the boundary of ammo use this del  notation to mean I ignore the points in  here I want points along the boundary  here and in particular I want to look at  these vertices in this drawing ok look  at such a vertex V and suppose that V is  pointed we didn't use this terminology  because we didn't have it at the time  but maybe V looks like this and there's  an angle here this reflex that's a point  in vertex now i'm looking here just at  the bars ignoring the struts ok so  suppose vertex V  has a reflex angle among just the bars  then the claim is locally this entire  region must be an M must be locally at  the maximum z-coordinate okay how did we  prove that well a claim is this had to  be flat because there's no mountains  here so if you think of a reflex region  and V V is at the maximum z-coordinate  right it's on the boundary of the  maximum z-coordinate so it has maximum  possible Z so there's no way you can go  up these edges could be mountains but  all the other edges which are not drawn  here use blue there are some other edges  those are the struts coming to be those  all have to be valleys they'll have to  lift a valleys because stresses can only  carry a positive stress stress course  positive stress correspond to ballys  these guys are all valleys and this is  at maximum Z really can't use those  valleys right you're up here at maximum  Z if use valleys you go to higher Z  which is not allowed so these actually  i'll have to be flat out here which  means locally all this stuff is at  maximum Z ok that was the proof we had a  couple lectures ago now once you know  that that's at maximum Z you know that a  lot of these things are impossible as  look here's a reflex angle among bars I  mean there might be more bars here I  guess but we know that this is pointed  whatever is missing here there's there's  some reflex angle we're assuming we have  a pointed Souter triangulation and then  we remove an edge is still pointed so  that means all of this has to be in em  is it in this picture no contradiction  so you cannot have one dimensional parts  because wherever you have a one  dimensional part there's a reflex angle  that's not in em so that's bad in fact  they all have to be two-dimensional  parts like this can it be a  two-dimensional part like this no  because here's a reflex angle that  is not an M contradiction so the only  situation you can have is a convex  polygon with the interior all in M sorry  wrong the reverse the exterior should  all be an F because you have all these  reflex angles that has to contain the  pointed part and locally by this lemma  it has to be in em so all this stuff has  to locally be an M the only way is for  all of that to be an M the inside we  don't know could be stress is there and  this could be a hole where you go deeper  all of this is at the maximum  z-coordinate the boundary here the  inside who knows could go down so you  can have stresses interior to convex  polygons of bars but not exterior to the  convex polygon I mean M is either  everything and there's no boundary and  then there's no stress or it's not  everything and then it will have to have  stresses only interior to convex  polygons and that's what we're claiming  is that stresses can only be interior to  convex bar polygons and in particular  than E is not stressed the end this is  definitely harder than the case we  talked about for Carpenters rule theorem  when you have a single chain you're just  trying to prove there's the expanse of  motion it's a lot easier than this  because m is much simpler can basically  only be a path or cycle because you only  have you only have a single chain here M  could be more complicated but in the end  it's actually pretty darn simple any  questions about that  cool perfectly clear definitely bit  complicated but nice now it turns out  Souter triangulations are really at the  core of expansive motions just mention  one more theorem about them so you may  recall I mentioned if you look at the  space of infinitesimal motions of a  linkage or at integrity they form a cone  it's it lives in some high dimensional  space but I'll try to draw a cone here  these guys go off to infinity drawing it  in three dimensions I guess you have  some place here this is the zero zero  zero motion where nothing moves and  these are potential different motion  each of these points corresponds to a  set an assignment of velocity vectors  where you move and you preserve all the  constraints if I take such a motion I  can scale it up it's still a motion so  that's motions form a cone it's called a  convex cone every motion can be scaled  up or down all the way to zero so if we  look at the cone of expansive motions  it's also a cone it's you know it's a  cone of motions of a particular tense a  greater where we add all pairwise struts  then pseudo triangulations correspond to  these edges of the comb these edges  they're really raised right they go off  to infinity these things are called  extreme rays meaning their extreme in a  particular direction like if I want the  emotion that is the most this way then  it will be this this ray okay they're  kind of the corners of this polyhedron  the edges of the polyhedron if you will  although in higher dimensions is not it  is edges but they're called extreme  raised in the case of  okay the extreme raised in the cone of  expansive motions equal pointed Souter  triangulations minus one edge I think I  have a slide about it yeah so this is a  paper by rote dos santos and strain you  this is just a particular example for  five points these are all the different  suitor triangulations minus an edge in  some cases like here you get two  triangles that that becomes rigid so  they fill in the hole click to indicate  that's a rigid component but the claim  is 45 points these are the edges of the  expansive cone each of these guys has an  expense of motion like this guy would  rotate pretty simple this one would open  up a little like that in general all  these guys have an expansive motion  those are characterizing the cone those  are kind of the extreme raised in this  fairly high dimensional cone and has  something like 10 dimensions if it's 5  times 2 and you can prove that if you  interested to read the paper so in fact  pseudo triangulations are really core to  expansive motions if you wrote down the  linear program which is expect the  expansive motions then you are  essentially characterizing this as the  polyhedron and the linear program if you  know linear programming and you follow  like the simplex method simplex method  is all about finding where the edges are  and here they are so you would actually  if you implemented the linear program  ran it on a simplex solver you would get  sued appointed Souter triangulations  automatically they just pop out that's  actually originally how they were  discovered but then this is why they  were discovered that way  end of suit of triangulations alright  next question is have any of the open  problems been solved I get this almost  every lecture and usually the answer is  no so I don't tell you anything but this  time there is a nice result which is  related to this picture these are slides  from from the lecture we had this linear  lock tree which is which is minimal  among all linear lock trees minimum  number of edges we had this equilateral  ochiltree all the edges were the same  length was not strongly locked there was  some positive distances here but it is  locked I didn't mention this question  but an obvious open question is can you  get both at the same time is there a  linear equilateral lock tree if there  were probably be a lot easier to prove  lock than this mess so it'd be nice to  get both but in fact you cannot get both  there's this paper by a bunch of people  in the open problem session from two  years ago I finally got published last  last year and it proves a bunch of  things but one of the things that proves  there is no equilateral lock tree that  is also linear also lives in a single  dimension and the way it proves that  essentially so you take any linear  equilateral tree what does it look like  well it lives along a line here and all  its segmented into equal length chunks  so from a high level it looks like this  now in reality there could be lots of  edges along each of these segments and  there's you know interesting things  happening inside one of these vertices  could be I don't know something like  this that's one way to do a tree like  structure in there it could have parts  like this that go from side to side  without touching these guys who knows  what's happening here but from a high  level perspective it's a bunch of  segments in a path okay the idea is this  is basically just looking at five of the  vertices out of all n of them but if we  just look at those five vertices  everything looks good this is in  canonical form canonical form for a tree  is where all the edges point to the  right something like this is nice and  canonical so right now this is canonical  what we do is kind of look inside the  path here look for a break point where  things are not canonical and then we fix  it so here's just an example of that  here this is what it looks like  currently at a high level here's what it  looks like in reality ok so imagine so  the tree is doing this stuff locally at  all these vertices right now we're  coalescing this all into one big mega  vertex and then it looks like this but  suppose we now we realize oh actually  there's kind of a split here there's  stuff over to the right of this or stuff  over to write this or not directly  connected then what we'll do is pull it  apart treat these as two separate  vertices increase the number of vertices  in this picture by one so we end up  splitting w here according to whatever  is actually happening in the linkage and  then we say oh gosh this is not really  canonical right this edge is pointing to  the left these guys are pointing to the  right still but this one's pointing to  the left so it looks like this picture  we've got a rightward edge and then a  leftward edge we fix it by just rotating  this edge 180 degrees keeping at all  times this thing pointing to the right  so this kind of comes along for the ride  and then we will end up with this  picture now we're canonical again with  one more vertex  we repeat eventually will be canonical  with all the vertices and we're done so  if you take any two configurations you  canonicalize both of them you will end  up with essentially the same picture it  does matter which route you choose but  it's already known that it doesn't  matter which route you choose you can  change the route just by flopping the  tree with one motion or very small  number of motions so if you have two  configurations you can authorize both of  them then there's a motion between them  and you get a motion from one  configuration to the other therefore  there are no locked trees that are  equilateral and linear kind of nice the  last thing I want to talk about is 40 a  couple of people asked about 4d why is  it so different from 3d the essential  reason is we have one dimensional bars  but four dimensions of motion that's a  gap of three dimensions three dimensions  is a lot when you have one dimensional  chains in 3d or the gap of two  dimensions two dimensions turns out to  not be a lot so why is that let me prove  it to you here's actually an animation  of the motion just for fun this is in  the textbook this is the top row is if  you are zoomed out from the very  beginning this is mess and you basically  pull out a string it kind of actually  Wiggles around a lot here's what it  looks like zoomed in this is the  original mess and you end up flopping  this over here then flopping it over  there and it's hard to see because this  is a two-dimensional projection of a  four-dimensional thing but eventually  you pull open the whole thing so how  does this algorithm work it's actually  very simple for open chains it's very  simple closed chains a little more  complicated trees also very simple  unfolding  or D open chains this is spy Roxanna  caca and Joe Rourke back before the  Carpenters rule theorem I guess this  publication was later but the original  version was in 1998 I think 99 so what  do we do well what I do is look at the  end bar so let's say this is the end of  the chain from here it goes in some  direction there's no stuff what I'd like  to do is rotate this bar so that it  extends this bar so I have a 180 degree  angle here that's my goal if I can do  that i confuse that vertex never rotate  it again treat this as a longer edge  there for the remaining thing has n  minus 1 edges if I had NN geez  originally so I can just apply induction  in other words just repeat that  eventually the whole thing will be  straight so the challenge is can I make  a move can I fold this last bar to be to  extend the next the second bar it's two  issues to problems that could happen  here one is that there's no continuous  motion to get there the other problem is  that even instantaneously if I just you  know picked it up and dropped it into  the right location it might intersect  the rest of the chain maybe the chain  here comes and intersects that ray if it  intersects that ray I can just jiggle  the whole chain a little bit and it  won't is that clear let's see you've got  these one-dimensional bars this is a  one-dimensional target ray so if a bar  happens to be on this ray and I just  wiggle it a tiny bit it's going to come  out of that off of that place this will  actually even work in three dimensions  this step there's an even easier way to  do it is just to rotate this bar if you  rotate this bar you can basically point  the Ray wherever you want and if you  look at where that bar is pointing it's  kind of hard to draw the picture but you  just have to miss all of these  one-dimensional things so it's very easy  to miss okay so let's just assume for  now we'll kind of see this  again in a moment assume that this ray  happens to be clear there may be things  that come very close to it maybe they go  on the back side and then come on the  front but they never don't actually  touch the ring now think about where  this bar can go relative to this vertex  if I draw a little sphere here I'm going  to drop initially in three dimensions  but really it's in four dimensions so  have a little sphere where this bar is  currently corresponds to a single point  on the sphere so I'm currently here  where I want to be corresponds to  another point on the sphere my goal is  to find a path on the sphere that gets  from gets to the X gets to the buried  treasure okay this is essentially  treating this as a whole ray so I don't  even care that it's short if I just  imagine it going off to infinity can I  go from position a to position B now in  three dimensions this is a two  dimensional sphere and if you look at  the obstacles the things I have to avoid  those correspond to one dimensional bars  here which if you project them onto the  sphere correspond to great circular arcs  so the worry would be let me draw them  in red the obstacles the things that  will kill you if you hit them the worry  would be you have what we call a cage  something like this of one-dimensional  obstacles on the sphere that block this  X point block the target from the source  and in 3d this if you draw what happens  in the knitting needles this is what  happens you can't get you can't move the  last link to extend the previous one  because one dimensional barriers are a  problem on a two-dimensional sphere  because two minus one equals one but on  the three-dimensional sphere if you draw  one dimensional obstacles they have  basically no effect on the sphere they  do not disconnect the surface of a  three-dimensional sphere can you see  that the the analogy if you just keep  the difference in dimensions the same so  we have  3-dimensional sphere versus a  one-dimensional obstacle let's go down a  dimension of a two-dimensional sphere  and a zero-dimensional obstacle what's a  zero-dimensional obstacle a point so  imagine you have all these red points  which you can't touch and then you have  the initial position and the target  position B how do you get there no  problem and there's tons of paths okay  it's basically the same in three  dimension and four dimensions just  harder to see you have a three spear  which is the boundary of a four ball and  you have these one-dimensional arcs on  it and they just don't block your way at  all because they're so low dimensional  so that's intuitively why for dzz 5d is  also easy 60 and so on because the  obstacles are so tiny obstacles remain  one-dimensional on this high dimensional  sphere you can just fix one link at a  time eventually the whole thing will be  straight tah-dah 4d any final questions  yeah of holding a surface in four  dimensions roughly analogous holding a  two-dimensional surface in four  dimensions definitely this argument  breaks down and so you can get locked  things there if you'd asked earlier I  could show one example we have of a kind  of locks surface I don't know if there's  actually a lock to the surface in 4d  known though I suspect there is one that  could be an interesting question to work  on cool think about that all right let's  see you Thursday
 

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00:08:53,210 --> 00:08:55,790

 

196
00:08:55,790 --> 00:08:55,800

 

197
00:08:55,800 --> 00:08:56,480


198
00:08:56,480 --> 00:08:58,660

 

199
00:08:58,660 --> 00:09:02,150

 

200
00:09:02,150 --> 00:09:04,730

 

201
00:09:04,730 --> 00:09:06,829

 

202
00:09:06,829 --> 00:09:12,260

 

203
00:09:12,260 --> 00:09:13,910

 

204
00:09:13,910 --> 00:09:16,340

 

205
00:09:16,340 --> 00:09:21,290

 

206
00:09:21,290 --> 00:09:22,610

 

207
00:09:22,610 --> 00:09:25,910

 

208
00:09:25,910 --> 00:09:27,920

 

209
00:09:27,920 --> 00:09:30,350

 

210
00:09:30,350 --> 00:09:32,030

 

211
00:09:32,030 --> 00:09:34,190

 

212
00:09:34,190 --> 00:09:36,079

 

213
00:09:36,079 --> 00:09:39,920

 

214
00:09:39,920 --> 00:09:42,590

 

215
00:09:42,590 --> 00:09:45,710

 

216
00:09:45,710 --> 00:09:48,740

 

217
00:09:48,740 --> 00:09:50,480

 

218
00:09:50,480 --> 00:09:52,910

 

219
00:09:52,910 --> 00:09:55,040

 

220
00:09:55,040 --> 00:09:56,449

 

221
00:09:56,449 --> 00:09:58,550

 

222
00:09:58,550 --> 00:10:00,400

 

223
00:10:00,400 --> 00:10:02,660

 

224
00:10:02,660 --> 00:10:07,130

 

225
00:10:07,130 --> 00:10:09,560

 

226
00:10:09,560 --> 00:10:11,060

 

227
00:10:11,060 --> 00:10:13,190

 

228
00:10:13,190 --> 00:10:16,730

 

229
00:10:16,730 --> 00:10:18,350

 

230
00:10:18,350 --> 00:10:22,220

 

231
00:10:22,220 --> 00:10:28,000

 

232
00:10:28,000 --> 00:10:31,940

 

233
00:10:31,940 --> 00:10:34,010

 

234
00:10:34,010 --> 00:10:36,199

 

235
00:10:36,199 --> 00:10:38,090

 

236
00:10:38,090 --> 00:10:40,190

 

237
00:10:40,190 --> 00:10:43,850

 

238
00:10:43,850 --> 00:10:46,490

 

239
00:10:46,490 --> 00:10:48,740

 

240
00:10:48,740 --> 00:10:50,480

 

241
00:10:50,480 --> 00:10:54,230

 

242
00:10:54,230 --> 00:10:57,800

 

243
00:10:57,800 --> 00:11:00,800

 

244
00:11:00,800 --> 00:11:03,500

 

245
00:11:03,500 --> 00:11:04,880

 

246
00:11:04,880 --> 00:11:06,860

 

247
00:11:06,860 --> 00:11:06,870

 

248
00:11:06,870 --> 00:11:08,840


249
00:11:08,840 --> 00:11:10,790

 

250
00:11:10,790 --> 00:11:12,949

 

251
00:11:12,949 --> 00:11:14,480

 

252
00:11:14,480 --> 00:11:18,259

 

253
00:11:18,259 --> 00:11:19,879

 

254
00:11:19,879 --> 00:11:21,139

 

255
00:11:21,139 --> 00:11:25,069

 

256
00:11:25,069 --> 00:11:27,139

 

257
00:11:27,139 --> 00:11:29,660

 

258
00:11:29,660 --> 00:11:32,180

 

259
00:11:32,180 --> 00:11:36,230

 

260
00:11:36,230 --> 00:11:38,210

 

261
00:11:38,210 --> 00:11:40,370

 

262
00:11:40,370 --> 00:11:41,870

 

263
00:11:41,870 --> 00:11:43,790

 

264
00:11:43,790 --> 00:11:45,800

 

265
00:11:45,800 --> 00:11:50,300

 

266
00:11:50,300 --> 00:11:52,189

 

267
00:11:52,189 --> 00:11:53,720

 

268
00:11:53,720 --> 00:11:56,300

 

269
00:11:56,300 --> 00:11:58,370

 

270
00:11:58,370 --> 00:12:00,590

 

271
00:12:00,590 --> 00:12:04,790

 

272
00:12:04,790 --> 00:12:07,579

 

273
00:12:07,579 --> 00:12:09,019

 

274
00:12:09,019 --> 00:12:10,730

 

275
00:12:10,730 --> 00:12:13,129

 

276
00:12:13,129 --> 00:12:14,509

 

277
00:12:14,509 --> 00:12:16,309

 

278
00:12:16,309 --> 00:12:17,540

 

279
00:12:17,540 --> 00:12:20,150

 

280
00:12:20,150 --> 00:12:21,590

 

281
00:12:21,590 --> 00:12:23,150

 

282
00:12:23,150 --> 00:12:25,040

 

283
00:12:25,040 --> 00:12:28,999

 

284
00:12:28,999 --> 00:12:33,170

 

285
00:12:33,170 --> 00:12:35,360

 

286
00:12:35,360 --> 00:12:40,040

 

287
00:12:40,040 --> 00:12:42,970

 

288
00:12:42,970 --> 00:12:49,179

 

289
00:12:49,179 --> 00:13:00,379

 

290
00:13:00,379 --> 00:13:02,569

 

291
00:13:02,569 --> 00:13:04,370

 

292
00:13:04,370 --> 00:13:08,150

 

293
00:13:08,150 --> 00:13:09,740

 

294
00:13:09,740 --> 00:13:11,720

 

295
00:13:11,720 --> 00:13:14,120

 

296
00:13:14,120 --> 00:13:14,130

 

297
00:13:14,130 --> 00:13:15,890


298
00:13:15,890 --> 00:13:19,710

 

299
00:13:19,710 --> 00:13:21,240

 

300
00:13:21,240 --> 00:13:23,610

 

301
00:13:23,610 --> 00:13:29,510

 

302
00:13:29,510 --> 00:13:32,310

 

303
00:13:32,310 --> 00:13:34,680

 

304
00:13:34,680 --> 00:13:36,030

 

305
00:13:36,030 --> 00:13:38,370

 

306
00:13:38,370 --> 00:13:40,400

 

307
00:13:40,400 --> 00:13:42,960

 

308
00:13:42,960 --> 00:13:46,830

 

309
00:13:46,830 --> 00:13:48,330

 

310
00:13:48,330 --> 00:13:55,260

 

311
00:13:55,260 --> 00:13:59,970

 

312
00:13:59,970 --> 00:14:02,610

 

313
00:14:02,610 --> 00:14:04,680

 

314
00:14:04,680 --> 00:14:06,600

 

315
00:14:06,600 --> 00:14:09,330

 

316
00:14:09,330 --> 00:14:12,270

 

317
00:14:12,270 --> 00:14:20,880

 

318
00:14:20,880 --> 00:14:24,180

 

319
00:14:24,180 --> 00:14:25,560

 

320
00:14:25,560 --> 00:14:28,530

 

321
00:14:28,530 --> 00:14:31,740

 

322
00:14:31,740 --> 00:14:33,120

 

323
00:14:33,120 --> 00:14:35,520

 

324
00:14:35,520 --> 00:14:36,930

 

325
00:14:36,930 --> 00:14:38,400

 

326
00:14:38,400 --> 00:14:41,580

 

327
00:14:41,580 --> 00:14:43,710

 

328
00:14:43,710 --> 00:14:46,410

 

329
00:14:46,410 --> 00:14:49,430

 

330
00:14:49,430 --> 00:14:52,260

 

331
00:14:52,260 --> 00:14:53,970

 

332
00:14:53,970 --> 00:15:00,540

 

333
00:15:00,540 --> 00:15:02,310

 

334
00:15:02,310 --> 00:15:04,800

 

335
00:15:04,800 --> 00:15:09,030

 

336
00:15:09,030 --> 00:15:14,730

 

337
00:15:14,730 --> 00:15:16,920

 

338
00:15:16,920 --> 00:15:19,050

 

339
00:15:19,050 --> 00:15:22,610

 

340
00:15:22,610 --> 00:15:26,210

 

341
00:15:26,210 --> 00:15:28,800

 

342
00:15:28,800 --> 00:15:30,690

 

343
00:15:30,690 --> 00:15:37,820

 

344
00:15:37,820 --> 00:15:44,160

 

345
00:15:44,160 --> 00:15:45,900

 

346
00:15:45,900 --> 00:15:48,930

 

347
00:15:48,930 --> 00:15:51,750

 

348
00:15:51,750 --> 00:15:53,519

 

349
00:15:53,519 --> 00:15:55,920

 

350
00:15:55,920 --> 00:15:59,519

 

351
00:15:59,519 --> 00:16:01,470

 

352
00:16:01,470 --> 00:16:08,820

 

353
00:16:08,820 --> 00:16:16,530

 

354
00:16:16,530 --> 00:16:19,380

 

355
00:16:19,380 --> 00:16:21,630

 

356
00:16:21,630 --> 00:16:22,950

 

357
00:16:22,950 --> 00:16:25,740

 

358
00:16:25,740 --> 00:16:27,120

 

359
00:16:27,120 --> 00:16:28,760

 

360
00:16:28,760 --> 00:16:30,930

 

361
00:16:30,930 --> 00:16:32,820

 

362
00:16:32,820 --> 00:16:34,650

 

363
00:16:34,650 --> 00:16:37,410

 

364
00:16:37,410 --> 00:16:40,190

 

365
00:16:40,190 --> 00:16:49,560

 

366
00:16:49,560 --> 00:16:53,220

 

367
00:16:53,220 --> 00:16:54,600

 

368
00:16:54,600 --> 00:17:02,840

 

369
00:17:02,840 --> 00:17:05,669

 

370
00:17:05,669 --> 00:17:08,069

 

371
00:17:08,069 --> 00:17:10,319

 

372
00:17:10,319 --> 00:17:11,970

 

373
00:17:11,970 --> 00:17:13,350

 

374
00:17:13,350 --> 00:17:15,660

 

375
00:17:15,660 --> 00:17:18,090

 

376
00:17:18,090 --> 00:17:20,790

 

377
00:17:20,790 --> 00:17:24,840

 

378
00:17:24,840 --> 00:17:27,449

 

379
00:17:27,449 --> 00:17:32,160

 

380
00:17:32,160 --> 00:17:34,650

 

381
00:17:34,650 --> 00:17:34,660

 

382
00:17:34,660 --> 00:17:39,680


383
00:17:39,680 --> 00:17:45,480

 

384
00:17:45,480 --> 00:17:51,360

 

385
00:17:51,360 --> 00:17:53,880

 

386
00:17:53,880 --> 00:17:55,350

 

387
00:17:55,350 --> 00:17:57,240

 

388
00:17:57,240 --> 00:17:59,100

 

389
00:17:59,100 --> 00:18:03,140

 

390
00:18:03,140 --> 00:18:13,170

 

391
00:18:13,170 --> 00:18:15,240

 

392
00:18:15,240 --> 00:18:21,330

 

393
00:18:21,330 --> 00:18:23,190

 

394
00:18:23,190 --> 00:18:24,750

 

395
00:18:24,750 --> 00:18:32,760

 

396
00:18:32,760 --> 00:18:36,210

 

397
00:18:36,210 --> 00:18:38,010

 

398
00:18:38,010 --> 00:18:40,640

 

399
00:18:40,640 --> 00:18:43,890

 

400
00:18:43,890 --> 00:18:48,470

 

401
00:18:48,470 --> 00:18:52,170

 

402
00:18:52,170 --> 00:18:54,900

 

403
00:18:54,900 --> 00:18:57,360

 

404
00:18:57,360 --> 00:19:00,240

 

405
00:19:00,240 --> 00:19:06,650

 

406
00:19:06,650 --> 00:19:13,450

 

407
00:19:13,450 --> 00:19:22,419

 

408
00:19:22,419 --> 00:19:28,610

 

409
00:19:28,610 --> 00:19:32,870

 

410
00:19:32,870 --> 00:19:34,610

 

411
00:19:34,610 --> 00:19:36,710

 

412
00:19:36,710 --> 00:19:38,240

 

413
00:19:38,240 --> 00:19:39,950

 

414
00:19:39,950 --> 00:19:41,899

 

415
00:19:41,899 --> 00:19:43,070

 

416
00:19:43,070 --> 00:19:46,070

 

417
00:19:46,070 --> 00:19:48,169

 

418
00:19:48,169 --> 00:19:50,600

 

419
00:19:50,600 --> 00:19:51,590

 

420
00:19:51,590 --> 00:19:54,440

 

421
00:19:54,440 --> 00:19:59,419

 

422
00:19:59,419 --> 00:20:02,090

 

423
00:20:02,090 --> 00:20:05,180

 

424
00:20:05,180 --> 00:20:06,950

 

425
00:20:06,950 --> 00:20:08,869

 

426
00:20:08,869 --> 00:20:10,460

 

427
00:20:10,460 --> 00:20:12,860

 

428
00:20:12,860 --> 00:20:15,440

 

429
00:20:15,440 --> 00:20:18,590

 

430
00:20:18,590 --> 00:20:20,539

 

431
00:20:20,539 --> 00:20:23,029

 

432
00:20:23,029 --> 00:20:25,970

 

433
00:20:25,970 --> 00:20:27,980

 

434
00:20:27,980 --> 00:20:30,289

 

435
00:20:30,289 --> 00:20:31,310

 

436
00:20:31,310 --> 00:20:32,769

 

437
00:20:32,769 --> 00:20:35,180

 

438
00:20:35,180 --> 00:20:37,279

 

439
00:20:37,279 --> 00:20:39,889

 

440
00:20:39,889 --> 00:20:42,350

 

441
00:20:42,350 --> 00:20:46,070

 

442
00:20:46,070 --> 00:20:48,049

 

443
00:20:48,049 --> 00:20:50,450

 

444
00:20:50,450 --> 00:20:51,980

 

445
00:20:51,980 --> 00:20:55,009

 

446
00:20:55,009 --> 00:20:58,220

 

447
00:20:58,220 --> 00:20:59,899

 

448
00:20:59,899 --> 00:21:02,659

 

449
00:21:02,659 --> 00:21:05,029

 

450
00:21:05,029 --> 00:21:06,499

 

451
00:21:06,499 --> 00:21:09,320

 

452
00:21:09,320 --> 00:21:10,789

 

453
00:21:10,789 --> 00:21:13,850

 

454
00:21:13,850 --> 00:21:15,889

 

455
00:21:15,889 --> 00:21:17,210

 

456
00:21:17,210 --> 00:21:21,510

 

457
00:21:21,510 --> 00:21:25,740

 

458
00:21:25,740 --> 00:21:28,440

 

459
00:21:28,440 --> 00:21:29,670

 

460
00:21:29,670 --> 00:21:32,940

 

461
00:21:32,940 --> 00:21:34,050

 

462
00:21:34,050 --> 00:21:36,180

 

463
00:21:36,180 --> 00:21:39,750

 

464
00:21:39,750 --> 00:21:43,500

 

465
00:21:43,500 --> 00:21:54,000

 

466
00:21:54,000 --> 00:22:02,640

 

467
00:22:02,640 --> 00:22:04,530

 

468
00:22:04,530 --> 00:22:06,780

 

469
00:22:06,780 --> 00:22:08,100

 

470
00:22:08,100 --> 00:22:10,050

 

471
00:22:10,050 --> 00:22:11,250

 

472
00:22:11,250 --> 00:22:12,990

 

473
00:22:12,990 --> 00:22:15,510

 

474
00:22:15,510 --> 00:22:18,080

 

475
00:22:18,080 --> 00:22:25,670

 

476
00:22:25,670 --> 00:22:30,780

 

477
00:22:30,780 --> 00:22:36,090

 

478
00:22:36,090 --> 00:22:37,800

 

479
00:22:37,800 --> 00:22:39,660

 

480
00:22:39,660 --> 00:22:42,900

 

481
00:22:42,900 --> 00:22:44,760

 

482
00:22:44,760 --> 00:22:46,530

 

483
00:22:46,530 --> 00:22:47,700

 

484
00:22:47,700 --> 00:22:49,080

 

485
00:22:49,080 --> 00:22:51,990

 

486
00:22:51,990 --> 00:22:55,080

 

487
00:22:55,080 --> 00:22:56,400

 

488
00:22:56,400 --> 00:22:59,130

 

489
00:22:59,130 --> 00:23:01,620

 

490
00:23:01,620 --> 00:23:04,650

 

491
00:23:04,650 --> 00:23:08,250

 

492
00:23:08,250 --> 00:23:11,120

 

493
00:23:11,120 --> 00:23:16,200

 

494
00:23:16,200 --> 00:23:26,980

 

495
00:23:26,980 --> 00:23:34,279

 

496
00:23:34,279 --> 00:23:37,330

 

497
00:23:37,330 --> 00:23:41,270

 

498
00:23:41,270 --> 00:23:42,350

 

499
00:23:42,350 --> 00:23:44,419

 

500
00:23:44,419 --> 00:23:53,659

 

501
00:23:53,659 --> 00:23:55,460

 

502
00:23:55,460 --> 00:23:57,020

 

503
00:23:57,020 --> 00:23:59,299

 

504
00:23:59,299 --> 00:24:01,159

 

505
00:24:01,159 --> 00:24:02,779

 

506
00:24:02,779 --> 00:24:04,520

 

507
00:24:04,520 --> 00:24:16,190

 

508
00:24:16,190 --> 00:24:20,720

 

509
00:24:20,720 --> 00:24:22,279

 

510
00:24:22,279 --> 00:24:24,860

 

511
00:24:24,860 --> 00:24:27,350

 

512
00:24:27,350 --> 00:24:29,450

 

513
00:24:29,450 --> 00:24:32,659

 

514
00:24:32,659 --> 00:24:36,289

 

515
00:24:36,289 --> 00:24:37,789

 

516
00:24:37,789 --> 00:24:42,710

 

517
00:24:42,710 --> 00:24:43,970

 

518
00:24:43,970 --> 00:24:45,500

 

519
00:24:45,500 --> 00:24:46,760

 

520
00:24:46,760 --> 00:24:49,070

 

521
00:24:49,070 --> 00:24:50,750

 

522
00:24:50,750 --> 00:24:54,169

 

523
00:24:54,169 --> 00:24:57,830

 

524
00:24:57,830 --> 00:24:59,930

 

525
00:24:59,930 --> 00:25:02,330

 

526
00:25:02,330 --> 00:25:04,730

 

527
00:25:04,730 --> 00:25:06,350

 

528
00:25:06,350 --> 00:25:10,549

 

529
00:25:10,549 --> 00:25:14,419

 

530
00:25:14,419 --> 00:25:16,340

 

531
00:25:16,340 --> 00:25:19,010

 

532
00:25:19,010 --> 00:25:20,600

 

533
00:25:20,600 --> 00:25:22,310

 

534
00:25:22,310 --> 00:25:24,770

 

535
00:25:24,770 --> 00:25:28,310

 

536
00:25:28,310 --> 00:25:29,750

 

537
00:25:29,750 --> 00:25:32,960

 

538
00:25:32,960 --> 00:25:34,350

 

539
00:25:34,350 --> 00:25:35,640

 

540
00:25:35,640 --> 00:25:37,230

 

541
00:25:37,230 --> 00:25:39,900

 

542
00:25:39,900 --> 00:25:43,200

 

543
00:25:43,200 --> 00:25:44,490

 

544
00:25:44,490 --> 00:25:55,650

 

545
00:25:55,650 --> 00:26:10,410

 

546
00:26:10,410 --> 00:26:18,180

 

547
00:26:18,180 --> 00:26:20,490

 

548
00:26:20,490 --> 00:26:25,080

 

549
00:26:25,080 --> 00:26:27,510

 

550
00:26:27,510 --> 00:26:28,890

 

551
00:26:28,890 --> 00:26:33,810

 

552
00:26:33,810 --> 00:26:38,490

 

553
00:26:38,490 --> 00:26:41,580

 

554
00:26:41,580 --> 00:26:43,650

 

555
00:26:43,650 --> 00:26:45,480

 

556
00:26:45,480 --> 00:26:47,040

 

557
00:26:47,040 --> 00:26:50,280

 

558
00:26:50,280 --> 00:26:53,760

 

559
00:26:53,760 --> 00:26:57,810

 

560
00:26:57,810 --> 00:27:02,040

 

561
00:27:02,040 --> 00:27:15,430

 

562
00:27:15,430 --> 00:27:19,730

 

563
00:27:19,730 --> 00:27:26,690

 

564
00:27:26,690 --> 00:27:35,090

 

565
00:27:35,090 --> 00:27:47,690

 

566
00:27:47,690 --> 00:27:49,340

 

567
00:27:49,340 --> 00:27:53,180

 

568
00:27:53,180 --> 00:27:56,540

 

569
00:27:56,540 --> 00:27:57,860

 

570
00:27:57,860 --> 00:27:59,660

 

571
00:27:59,660 --> 00:28:01,490

 

572
00:28:01,490 --> 00:28:03,140

 

573
00:28:03,140 --> 00:28:04,310

 

574
00:28:04,310 --> 00:28:06,950

 

575
00:28:06,950 --> 00:28:10,460

 

576
00:28:10,460 --> 00:28:13,400

 

577
00:28:13,400 --> 00:28:17,750

 

578
00:28:17,750 --> 00:28:20,930

 

579
00:28:20,930 --> 00:28:22,430

 

580
00:28:22,430 --> 00:28:24,970

 

581
00:28:24,970 --> 00:28:28,820

 

582
00:28:28,820 --> 00:28:30,080

 

583
00:28:30,080 --> 00:28:31,610

 

584
00:28:31,610 --> 00:28:34,370

 

585
00:28:34,370 --> 00:28:38,090

 

586
00:28:38,090 --> 00:28:39,860

 

587
00:28:39,860 --> 00:28:41,270

 

588
00:28:41,270 --> 00:28:43,790

 

589
00:28:43,790 --> 00:28:46,790

 

590
00:28:46,790 --> 00:28:48,230

 

591
00:28:48,230 --> 00:28:50,240

 

592
00:28:50,240 --> 00:28:53,930

 

593
00:28:53,930 --> 00:28:57,290

 

594
00:28:57,290 --> 00:28:59,180

 

595
00:28:59,180 --> 00:29:01,790

 

596
00:29:01,790 --> 00:29:05,480

 

597
00:29:05,480 --> 00:29:08,660

 

598
00:29:08,660 --> 00:29:13,130

 

599
00:29:13,130 --> 00:29:14,900

 

600
00:29:14,900 --> 00:29:16,730

 

601
00:29:16,730 --> 00:29:18,860

 

602
00:29:18,860 --> 00:29:22,700

 

603
00:29:22,700 --> 00:29:30,910

 

604
00:29:30,910 --> 00:29:34,160

 

605
00:29:34,160 --> 00:29:38,180

 

606
00:29:38,180 --> 00:29:42,320

 

607
00:29:42,320 --> 00:29:44,290

 

608
00:29:44,290 --> 00:30:00,020

 

609
00:30:00,020 --> 00:30:01,760

 

610
00:30:01,760 --> 00:30:03,260

 

611
00:30:03,260 --> 00:30:08,860

 

612
00:30:08,860 --> 00:30:11,960

 

613
00:30:11,960 --> 00:30:18,200

 

614
00:30:18,200 --> 00:30:20,840

 

615
00:30:20,840 --> 00:30:23,180

 

616
00:30:23,180 --> 00:30:24,800

 

617
00:30:24,800 --> 00:30:27,950

 

618
00:30:27,950 --> 00:30:29,540

 

619
00:30:29,540 --> 00:30:32,600

 

620
00:30:32,600 --> 00:30:34,400

 

621
00:30:34,400 --> 00:30:37,610

 

622
00:30:37,610 --> 00:30:41,120

 

623
00:30:41,120 --> 00:30:42,380

 

624
00:30:42,380 --> 00:30:44,690

 

625
00:30:44,690 --> 00:30:47,240

 

626
00:30:47,240 --> 00:30:51,080

 

627
00:30:51,080 --> 00:30:53,600

 

628
00:30:53,600 --> 00:30:55,160

 

629
00:30:55,160 --> 00:30:59,830

 

630
00:30:59,830 --> 00:31:04,160

 

631
00:31:04,160 --> 00:31:07,340

 

632
00:31:07,340 --> 00:31:09,980

 

633
00:31:09,980 --> 00:31:11,210

 

634
00:31:11,210 --> 00:31:15,890

 

635
00:31:15,890 --> 00:31:18,790

 

636
00:31:18,790 --> 00:31:20,900

 

637
00:31:20,900 --> 00:31:22,430

 

638
00:31:22,430 --> 00:31:27,380

 

639
00:31:27,380 --> 00:31:33,290

 

640
00:31:33,290 --> 00:31:35,690

 

641
00:31:35,690 --> 00:31:38,590

 

642
00:31:38,590 --> 00:31:41,629

 

643
00:31:41,629 --> 00:31:53,229

 

644
00:31:53,229 --> 00:31:57,139

 

645
00:31:57,139 --> 00:32:07,310

 

646
00:32:07,310 --> 00:32:10,099

 

647
00:32:10,099 --> 00:32:12,979

 

648
00:32:12,979 --> 00:32:16,759

 

649
00:32:16,759 --> 00:32:20,169

 

650
00:32:20,169 --> 00:32:25,009

 

651
00:32:25,009 --> 00:32:26,839

 

652
00:32:26,839 --> 00:32:29,029

 

653
00:32:29,029 --> 00:32:31,579

 

654
00:32:31,579 --> 00:32:34,609

 

655
00:32:34,609 --> 00:32:36,319

 

656
00:32:36,319 --> 00:32:40,519

 

657
00:32:40,519 --> 00:32:44,029

 

658
00:32:44,029 --> 00:32:45,529

 

659
00:32:45,529 --> 00:32:46,969

 

660
00:32:46,969 --> 00:32:48,589

 

661
00:32:48,589 --> 00:32:50,509

 

662
00:32:50,509 --> 00:32:52,669

 

663
00:32:52,669 --> 00:32:54,560

 

664
00:32:54,560 --> 00:32:57,739

 

665
00:32:57,739 --> 00:33:00,229

 

666
00:33:00,229 --> 00:33:02,239

 

667
00:33:02,239 --> 00:33:03,979

 

668
00:33:03,979 --> 00:33:05,779

 

669
00:33:05,779 --> 00:33:08,690

 

670
00:33:08,690 --> 00:33:11,089

 

671
00:33:11,089 --> 00:33:13,789

 

672
00:33:13,789 --> 00:33:15,829

 

673
00:33:15,829 --> 00:33:19,869

 

674
00:33:19,869 --> 00:33:22,989

 

675
00:33:22,989 --> 00:33:26,499

 

676
00:33:26,499 --> 00:33:28,940

 

677
00:33:28,940 --> 00:33:30,709

 

678
00:33:30,709 --> 00:33:32,209

 

679
00:33:32,209 --> 00:33:34,249

 

680
00:33:34,249 --> 00:33:35,899

 

681
00:33:35,899 --> 00:33:38,419

 

682
00:33:38,419 --> 00:33:40,999

 

683
00:33:40,999 --> 00:33:42,409

 

684
00:33:42,409 --> 00:33:43,789

 

685
00:33:43,789 --> 00:33:47,299

 

686
00:33:47,299 --> 00:33:48,289

 

687
00:33:48,289 --> 00:33:50,180

 

688
00:33:50,180 --> 00:33:51,889

 

689
00:33:51,889 --> 00:33:55,100

 

690
00:33:55,100 --> 00:33:57,860

 

691
00:33:57,860 --> 00:34:00,530

 

692
00:34:00,530 --> 00:34:04,970

 

693
00:34:04,970 --> 00:34:08,869

 

694
00:34:08,869 --> 00:34:15,379

 

695
00:34:15,379 --> 00:34:17,570

 

696
00:34:17,570 --> 00:34:19,960

 

697
00:34:19,960 --> 00:34:23,540

 

698
00:34:23,540 --> 00:34:25,159

 

699
00:34:25,159 --> 00:34:27,409

 

700
00:34:27,409 --> 00:34:29,119

 

701
00:34:29,119 --> 00:34:30,889

 

702
00:34:30,889 --> 00:34:32,840

 

703
00:34:32,840 --> 00:34:35,060

 

704
00:34:35,060 --> 00:34:38,510

 

705
00:34:38,510 --> 00:34:40,460

 

706
00:34:40,460 --> 00:34:45,340

 

707
00:34:45,340 --> 00:34:47,690

 

708
00:34:47,690 --> 00:34:49,310

 

709
00:34:49,310 --> 00:34:51,440

 

710
00:34:51,440 --> 00:34:52,879

 

711
00:34:52,879 --> 00:34:54,409

 

712
00:34:54,409 --> 00:34:56,240

 

713
00:34:56,240 --> 00:34:58,550

 

714
00:34:58,550 --> 00:35:00,710

 

715
00:35:00,710 --> 00:35:05,450

 

716
00:35:05,450 --> 00:35:07,430

 

717
00:35:07,430 --> 00:35:10,400

 

718
00:35:10,400 --> 00:35:11,840

 

719
00:35:11,840 --> 00:35:12,920

 

720
00:35:12,920 --> 00:35:14,120

 

721
00:35:14,120 --> 00:35:17,570

 

722
00:35:17,570 --> 00:35:19,760

 

723
00:35:19,760 --> 00:35:25,310

 

724
00:35:25,310 --> 00:35:27,349

 

725
00:35:27,349 --> 00:35:31,550

 

726
00:35:31,550 --> 00:35:35,099

 

727
00:35:35,099 --> 00:35:41,109

 

728
00:35:41,109 --> 00:35:45,310

 

729
00:35:45,310 --> 00:35:47,109

 

730
00:35:47,109 --> 00:35:49,900

 

731
00:35:49,900 --> 00:35:55,839

 

732
00:35:55,839 --> 00:35:58,660

 

733
00:35:58,660 --> 00:36:01,450

 

734
00:36:01,450 --> 00:36:05,370

 

735
00:36:05,370 --> 00:36:08,020

 

736
00:36:08,020 --> 00:36:11,370

 

737
00:36:11,370 --> 00:36:16,030

 

738
00:36:16,030 --> 00:36:18,040

 

739
00:36:18,040 --> 00:36:20,319

 

740
00:36:20,319 --> 00:36:23,520

 

741
00:36:23,520 --> 00:36:26,260

 

742
00:36:26,260 --> 00:36:28,120

 

743
00:36:28,120 --> 00:36:29,680

 

744
00:36:29,680 --> 00:36:31,569

 

745
00:36:31,569 --> 00:36:34,089

 

746
00:36:34,089 --> 00:36:36,339

 

747
00:36:36,339 --> 00:36:38,920

 

748
00:36:38,920 --> 00:36:41,650

 

749
00:36:41,650 --> 00:36:44,349

 

750
00:36:44,349 --> 00:36:54,270

 

751
00:36:54,270 --> 00:36:58,540

 

752
00:36:58,540 --> 00:37:00,520

 

753
00:37:00,520 --> 00:37:03,960

 

754
00:37:03,960 --> 00:37:09,400

 

755
00:37:09,400 --> 00:37:14,500

 

756
00:37:14,500 --> 00:37:16,030

 

757
00:37:16,030 --> 00:37:17,829

 

758
00:37:17,829 --> 00:37:24,880

 

759
00:37:24,880 --> 00:37:26,650

 

760
00:37:26,650 --> 00:37:29,589

 

761
00:37:29,589 --> 00:37:32,920

 

762
00:37:32,920 --> 00:37:34,900

 

763
00:37:34,900 --> 00:37:36,910

 

764
00:37:36,910 --> 00:37:39,520

 

765
00:37:39,520 --> 00:37:42,910

 

766
00:37:42,910 --> 00:37:45,510

 

767
00:37:45,510 --> 00:37:47,950

 

768
00:37:47,950 --> 00:37:52,300

 

769
00:37:52,300 --> 00:38:11,079

 

770
00:38:11,079 --> 00:38:15,310

 

771
00:38:15,310 --> 00:38:17,520

 

772
00:38:17,520 --> 00:38:21,280

 

773
00:38:21,280 --> 00:38:23,290

 

774
00:38:23,290 --> 00:38:25,960

 

775
00:38:25,960 --> 00:38:28,060

 

776
00:38:28,060 --> 00:38:29,800

 

777
00:38:29,800 --> 00:38:31,510

 

778
00:38:31,510 --> 00:38:34,510

 

779
00:38:34,510 --> 00:38:37,089

 

780
00:38:37,089 --> 00:38:38,920

 

781
00:38:38,920 --> 00:38:40,300

 

782
00:38:40,300 --> 00:38:43,150

 

783
00:38:43,150 --> 00:38:47,079

 

784
00:38:47,079 --> 00:38:48,640

 

785
00:38:48,640 --> 00:38:51,609

 

786
00:38:51,609 --> 00:38:54,420

 

787
00:38:54,420 --> 00:38:56,589

 

788
00:38:56,589 --> 00:38:58,270

 

789
00:38:58,270 --> 00:39:02,170

 

790
00:39:02,170 --> 00:39:08,730

 

791
00:39:08,730 --> 00:39:10,870

 

792
00:39:10,870 --> 00:39:13,089

 

793
00:39:13,089 --> 00:39:14,740

 

794
00:39:14,740 --> 00:39:17,650

 

795
00:39:17,650 --> 00:39:19,240

 

796
00:39:19,240 --> 00:39:20,710

 

797
00:39:20,710 --> 00:39:22,780

 

798
00:39:22,780 --> 00:39:24,400

 

799
00:39:24,400 --> 00:39:26,020

 

800
00:39:26,020 --> 00:39:29,380

 

801
00:39:29,380 --> 00:39:31,300

 

802
00:39:31,300 --> 00:39:33,670

 

803
00:39:33,670 --> 00:39:35,849

 

804
00:39:35,849 --> 00:39:38,079

 

805
00:39:38,079 --> 00:39:39,040

 

806
00:39:39,040 --> 00:39:42,280

 

807
00:39:42,280 --> 00:39:46,330

 

808
00:39:46,330 --> 00:39:51,220

 

809
00:39:51,220 --> 00:39:53,750

 

810
00:39:53,750 --> 00:39:55,760

 

811
00:39:55,760 --> 00:39:57,410

 

812
00:39:57,410 --> 00:39:59,240

 

813
00:39:59,240 --> 00:40:04,720

 

814
00:40:04,720 --> 00:40:06,770

 

815
00:40:06,770 --> 00:40:10,040

 

816
00:40:10,040 --> 00:40:12,800

 

817
00:40:12,800 --> 00:40:14,359

 

818
00:40:14,359 --> 00:40:16,100

 

819
00:40:16,100 --> 00:40:18,980

 

820
00:40:18,980 --> 00:40:21,859

 

821
00:40:21,859 --> 00:40:23,660

 

822
00:40:23,660 --> 00:40:26,540

 

823
00:40:26,540 --> 00:40:28,609

 

824
00:40:28,609 --> 00:40:30,500

 

825
00:40:30,500 --> 00:40:33,620

 

826
00:40:33,620 --> 00:40:35,000

 

827
00:40:35,000 --> 00:40:37,490

 

828
00:40:37,490 --> 00:40:40,510

 

829
00:40:40,510 --> 00:40:43,760

 

830
00:40:43,760 --> 00:40:45,320

 

831
00:40:45,320 --> 00:40:48,770

 

832
00:40:48,770 --> 00:40:53,150

 

833
00:40:53,150 --> 00:40:54,380

 

834
00:40:54,380 --> 00:40:57,200

 

835
00:40:57,200 --> 00:41:00,050

 

836
00:41:00,050 --> 00:41:03,520

 

837
00:41:03,520 --> 00:41:08,450

 

838
00:41:08,450 --> 00:41:17,920

 

839
00:41:17,920 --> 00:41:22,370

 

840
00:41:22,370 --> 00:41:25,390

 

841
00:41:25,390 --> 00:41:29,270

 

842
00:41:29,270 --> 00:41:31,640

 

843
00:41:31,640 --> 00:41:34,340

 

844
00:41:34,340 --> 00:41:35,690

 

845
00:41:35,690 --> 00:41:37,610

 

846
00:41:37,610 --> 00:41:39,920

 

847
00:41:39,920 --> 00:41:41,810

 

848
00:41:41,810 --> 00:41:46,640

 

849
00:41:46,640 --> 00:41:49,250

 

850
00:41:49,250 --> 00:41:51,980

 

851
00:41:51,980 --> 00:41:53,270

 

852
00:41:53,270 --> 00:41:55,070

 

853
00:41:55,070 --> 00:41:59,840

 

854
00:41:59,840 --> 00:42:02,450

 

855
00:42:02,450 --> 00:42:05,840

 

856
00:42:05,840 --> 00:42:06,800

 

857
00:42:06,800 --> 00:42:08,090

 

858
00:42:08,090 --> 00:42:10,160

 

859
00:42:10,160 --> 00:42:12,410

 

860
00:42:12,410 --> 00:42:15,020

 

861
00:42:15,020 --> 00:42:18,080

 

862
00:42:18,080 --> 00:42:21,470

 

863
00:42:21,470 --> 00:42:23,870

 

864
00:42:23,870 --> 00:42:26,630

 

865
00:42:26,630 --> 00:42:29,080

 

866
00:42:29,080 --> 00:42:31,340

 

867
00:42:31,340 --> 00:42:33,860

 

868
00:42:33,860 --> 00:42:37,100

 

869
00:42:37,100 --> 00:42:39,410

 

870
00:42:39,410 --> 00:42:41,600

 

871
00:42:41,600 --> 00:42:43,460

 

872
00:42:43,460 --> 00:42:45,800

 

873
00:42:45,800 --> 00:42:47,930

 

874
00:42:47,930 --> 00:42:49,940

 

875
00:42:49,940 --> 00:42:51,620

 

876
00:42:51,620 --> 00:42:52,880

 

877
00:42:52,880 --> 00:42:55,340

 

878
00:42:55,340 --> 00:42:57,170

 

879
00:42:57,170 --> 00:42:58,430

 

880
00:42:58,430 --> 00:43:01,010

 

881
00:43:01,010 --> 00:43:03,200

 

882
00:43:03,200 --> 00:43:05,210

 

883
00:43:05,210 --> 00:43:07,580

 

884
00:43:07,580 --> 00:43:09,230

 

885
00:43:09,230 --> 00:43:10,970

 

886
00:43:10,970 --> 00:43:12,830

 

887
00:43:12,830 --> 00:43:14,270

 

888
00:43:14,270 --> 00:43:15,800

 

889
00:43:15,800 --> 00:43:18,380

 

890
00:43:18,380 --> 00:43:21,590

 

891
00:43:21,590 --> 00:43:23,270

 

892
00:43:23,270 --> 00:43:25,430

 

893
00:43:25,430 --> 00:43:27,500

 

894
00:43:27,500 --> 00:43:29,540

 

895
00:43:29,540 --> 00:43:30,990

 

896
00:43:30,990 --> 00:43:32,820

 

897
00:43:32,820 --> 00:43:35,340

 

898
00:43:35,340 --> 00:43:38,010

 

899
00:43:38,010 --> 00:43:40,140

 

900
00:43:40,140 --> 00:43:42,870

 

901
00:43:42,870 --> 00:43:44,670

 

902
00:43:44,670 --> 00:43:47,160

 

903
00:43:47,160 --> 00:43:48,330

 

904
00:43:48,330 --> 00:43:50,190

 

905
00:43:50,190 --> 00:43:54,060

 

906
00:43:54,060 --> 00:43:56,670

 

907
00:43:56,670 --> 00:43:58,380

 

908
00:43:58,380 --> 00:43:59,760

 

909
00:43:59,760 --> 00:44:02,280

 

910
00:44:02,280 --> 00:44:05,310

 

911
00:44:05,310 --> 00:44:06,780

 

912
00:44:06,780 --> 00:44:13,980

 

913
00:44:13,980 --> 00:44:16,590

 

914
00:44:16,590 --> 00:44:19,110

 

915
00:44:19,110 --> 00:44:21,060

 

916
00:44:21,060 --> 00:44:23,600

 

917
00:44:23,600 --> 00:44:26,310

 

918
00:44:26,310 --> 00:44:28,770

 

919
00:44:28,770 --> 00:44:31,680

 

920
00:44:31,680 --> 00:44:33,360

 

921
00:44:33,360 --> 00:44:34,920

 

922
00:44:34,920 --> 00:44:39,090

 

923
00:44:39,090 --> 00:44:41,310

 

924
00:44:41,310 --> 00:44:43,260

 

925
00:44:43,260 --> 00:44:45,690

 

926
00:44:45,690 --> 00:44:47,730

 

927
00:44:47,730 --> 00:44:49,680

 

928
00:44:49,680 --> 00:44:51,870

 

929
00:44:51,870 --> 00:44:54,330

 

930
00:44:54,330 --> 00:44:55,470

 

931
00:44:55,470 --> 00:44:57,840

 

932
00:44:57,840 --> 00:44:59,700

 

933
00:44:59,700 --> 00:45:01,200

 

934
00:45:01,200 --> 00:45:02,580

 

935
00:45:02,580 --> 00:45:04,320

 

936
00:45:04,320 --> 00:45:06,570

 

937
00:45:06,570 --> 00:45:07,800

 

938
00:45:07,800 --> 00:45:14,160

 

939
00:45:14,160 --> 00:45:15,600

 

940
00:45:15,600 --> 00:45:20,690

 

941
00:45:20,690 --> 00:45:20,700

 

942
00:45:20,700 --> 00:45:25,450


943
00:45:25,450 --> 00:45:32,029

 

944
00:45:32,029 --> 00:45:35,599

 

945
00:45:35,599 --> 00:45:39,170

 

946
00:45:39,170 --> 00:45:40,730

 

947
00:45:40,730 --> 00:45:46,880

 

948
00:45:46,880 --> 00:45:49,069

 

949
00:45:49,069 --> 00:45:51,739

 

950
00:45:51,739 --> 00:45:53,749

 

951
00:45:53,749 --> 00:45:56,509

 

952
00:45:56,509 --> 00:45:59,900

 

953
00:45:59,900 --> 00:46:03,470

 

954
00:46:03,470 --> 00:46:06,739

 

955
00:46:06,739 --> 00:46:09,980

 

956
00:46:09,980 --> 00:46:11,660

 

957
00:46:11,660 --> 00:46:14,089

 

958
00:46:14,089 --> 00:46:15,739

 

959
00:46:15,739 --> 00:46:17,599

 

960
00:46:17,599 --> 00:46:18,759

 

961
00:46:18,759 --> 00:46:20,390

 

962
00:46:20,390 --> 00:46:23,150

 

963
00:46:23,150 --> 00:46:27,170

 

964
00:46:27,170 --> 00:46:31,160

 

965
00:46:31,160 --> 00:46:33,650

 

966
00:46:33,650 --> 00:46:37,640

 

967
00:46:37,640 --> 00:46:39,620

 

968
00:46:39,620 --> 00:46:42,319

 

969
00:46:42,319 --> 00:46:43,999

 

970
00:46:43,999 --> 00:46:46,190

 

971
00:46:46,190 --> 00:46:47,660

 

972
00:46:47,660 --> 00:46:51,829

 

973
00:46:51,829 --> 00:46:54,380

 

974
00:46:54,380 --> 00:46:55,999

 

975
00:46:55,999 --> 00:46:59,960

 

976
00:46:59,960 --> 00:47:02,809

 

977
00:47:02,809 --> 00:47:06,109

 

978
00:47:06,109 --> 00:47:08,749

 

979
00:47:08,749 --> 00:47:10,460

 

980
00:47:10,460 --> 00:47:12,950

 

981
00:47:12,950 --> 00:47:14,359

 

982
00:47:14,359 --> 00:47:19,549

 

983
00:47:19,549 --> 00:47:22,249

 

984
00:47:22,249 --> 00:47:24,109

 

985
00:47:24,109 --> 00:47:26,150

 

986
00:47:26,150 --> 00:47:29,150

 

987
00:47:29,150 --> 00:47:32,599

 

988
00:47:32,599 --> 00:47:34,069

 

989
00:47:34,069 --> 00:47:35,539

 

990
00:47:35,539 --> 00:47:37,849

 

991
00:47:37,849 --> 00:47:38,790

 

992
00:47:38,790 --> 00:47:40,980

 

993
00:47:40,980 --> 00:47:42,480

 

994
00:47:42,480 --> 00:47:45,990

 

995
00:47:45,990 --> 00:47:47,580

 

996
00:47:47,580 --> 00:47:50,040

 

997
00:47:50,040 --> 00:47:54,120

 

998
00:47:54,120 --> 00:47:56,970

 

999
00:47:56,970 --> 00:48:02,130

 

1000
00:48:02,130 --> 00:48:03,570

 

1001
00:48:03,570 --> 00:48:05,490

 

1002
00:48:05,490 --> 00:48:08,700

 

1003
00:48:08,700 --> 00:48:10,530

 

1004
00:48:10,530 --> 00:48:12,720

 

1005
00:48:12,720 --> 00:48:14,910

 

1006
00:48:14,910 --> 00:48:17,700

 

1007
00:48:17,700 --> 00:48:20,760

 

1008
00:48:20,760 --> 00:48:23,970

 

1009
00:48:23,970 --> 00:48:27,360

 

1010
00:48:27,360 --> 00:48:30,450

 

1011
00:48:30,450 --> 00:48:31,920

 

1012
00:48:31,920 --> 00:48:33,480

 

1013
00:48:33,480 --> 00:48:40,380

 

1014
00:48:40,380 --> 00:48:42,270

 

1015
00:48:42,270 --> 00:48:44,310

 

1016
00:48:44,310 --> 00:48:46,050

 

1017
00:48:46,050 --> 00:48:50,430

 

1018
00:48:50,430 --> 00:48:52,650

 

1019
00:48:52,650 --> 00:48:54,750

 

1020
00:48:54,750 --> 00:48:58,020

 

1021
00:48:58,020 --> 00:49:00,240

 

1022
00:49:00,240 --> 00:49:02,790

 

1023
00:49:02,790 --> 00:49:05,570

 

1024
00:49:05,570 --> 00:49:09,600

 

1025
00:49:09,600 --> 00:49:13,170

 

1026
00:49:13,170 --> 00:49:16,980

 

1027
00:49:16,980 --> 00:49:19,950

 

1028
00:49:19,950 --> 00:49:21,210

 

1029
00:49:21,210 --> 00:49:23,400

 

1030
00:49:23,400 --> 00:49:25,440

 

1031
00:49:25,440 --> 00:49:27,600

 

1032
00:49:27,600 --> 00:49:29,610

 

1033
00:49:29,610 --> 00:49:33,870

 

1034
00:49:33,870 --> 00:49:36,840

 

1035
00:49:36,840 --> 00:49:38,430

 

1036
00:49:38,430 --> 00:49:41,130

 

1037
00:49:41,130 --> 00:49:43,020

 

1038
00:49:43,020 --> 00:49:45,390

 

1039
00:49:45,390 --> 00:49:48,510

 

1040
00:49:48,510 --> 00:49:51,450

 

1041
00:49:51,450 --> 00:49:52,290

 

1042
00:49:52,290 --> 00:49:53,340

 

1043
00:49:53,340 --> 00:49:54,960

 

1044
00:49:54,960 --> 00:49:56,970

 

1045
00:49:56,970 --> 00:49:59,070

 

1046
00:49:59,070 --> 00:50:02,460

 

1047
00:50:02,460 --> 00:50:04,860

 

1048
00:50:04,860 --> 00:50:07,350

 

1049
00:50:07,350 --> 00:50:09,750

 

1050
00:50:09,750 --> 00:50:12,360

 

1051
00:50:12,360 --> 00:50:15,180

 

1052
00:50:15,180 --> 00:50:16,620

 

1053
00:50:16,620 --> 00:50:18,540

 

1054
00:50:18,540 --> 00:50:20,540

 

1055
00:50:20,540 --> 00:50:23,850

 

1056
00:50:23,850 --> 00:50:26,220

 

1057
00:50:26,220 --> 00:50:28,560

 

1058
00:50:28,560 --> 00:50:30,020

 

1059
00:50:30,020 --> 00:50:33,090

 

1060
00:50:33,090 --> 00:50:36,030

 

1061
00:50:36,030 --> 00:50:38,160

 

1062
00:50:38,160 --> 00:50:39,570

 

1063
00:50:39,570 --> 00:50:41,460

 

1064
00:50:41,460 --> 00:50:43,170

 

1065
00:50:43,170 --> 00:50:49,130

 

1066
00:50:49,130 --> 00:50:57,120

 

1067
00:50:57,120 --> 00:51:00,030

 

1068
00:51:00,030 --> 00:51:01,800

 

1069
00:51:01,800 --> 00:51:03,930

 

1070
00:51:03,930 --> 00:51:06,060

 

1071
00:51:06,060 --> 00:51:09,810

 

1072
00:51:09,810 --> 00:51:11,490

 

1073
00:51:11,490 --> 00:51:14,160

 

1074
00:51:14,160 --> 00:51:15,990

 

1075
00:51:15,990 --> 00:51:18,630

 

1076
00:51:18,630 --> 00:51:20,370

 

1077
00:51:20,370 --> 00:51:25,710

 

1078
00:51:25,710 --> 00:51:28,680