1
00:00:02,700 --> 00:00:05,470

alright so this lecture was about  hyperbolic paraboloids and the extent to  which they don't exist or exist here is  a regular non existing hyperbolic  paraboloid with concentric squares no  diagonals folded here and so is just a  few questions about this what does it  mean other things these are all s by and  paying i believe and they're all open so  they're all good questions we don't know  whether the the good triangulation the  alternating one works for arbitrary and  we only know up to N equals 104 the  various fixed angles that we checked we  don't have a proof technique to do  arbitrary n you could try to make do  some amount of alternation but not  somewhere in between the two extremes  and probably you get something in  between goodness but I don't know could  certainly play with that and it's very  natural to try this with larger than  squares the only trouble is the center  no longer has a single degree of freedom  so the only thing to answer there would  be how do you want to initially fold  like if you're doing a hexagon how  hexagons probably clear what you might  want to do but for general Kagan how do  you want to arrange that innermost keg  on from that you could propagate out  just like we did with the square if you  use an alternating triangulation say  probably works well but we haven't tried  it could be a cool project to do that  not that hard so those were open  problems next we have some sort of math  general math questions c1 c2 and semi  creases these are related questions so  let me do a quick visual review of these  terms I'm going to do it for  one-dimensional curves in 2d because  that's a lot easier to think about then  what we really care about which is  two-dimensional surfaces in 3d but all  the ideas carry over  so here is just a function let's say for  example this is a parabolic arc then a  straight segment this point is missing  and instead it's up here these points  are present and here we have another  parabolic arc so this function i would  call piecewise C infinity let me explain  what these things mean so we have on the  one hand c0 is a term meaning just  continuous continuous means no jumps  like this so this is not a continuous  function this so this function overall  is not even c0 c1 is a stronger  condition which means that you not only  are you continuous but you also have a  continuous and existing first derivative  so whereas here we're thinking about F  here we're thinking of F prime so not  only can you talk about the function  being continuous c0 c1 but you can talk  about individual moments in time so for  example you know this moment here is  going to be c1 it has a nice derivative  that derivative is changing continuously  here here however this is a c0 but not  see one because the tangent on the left  is different from the tangent on the  right so this is not a c1 function  because the derivative is jumping at  that point I'm not going to draw the  derivative here it's a little harder to  draw but you could draw it at the set  with the same x-axis and you'd see a  jump from one angle to another this  point is not even c0 this point is c0  but not see one this point is c1  because the tangent here is equal on the  left is equal to the tangent on the  right if I drew this properly where this  is the bottom of the parabola okay you  can ask for more than see one and we do  in plus we talk about c2 and this says  you should have a continuous second  derivative so acceleration and this this  point for example is not see two as well  the tangents meet if you take the next  derivative you see that it's suddenly  here the tangent was completely flat it  has a second derivative of zero and then  suddenly it starts increasing now you  can design functions that are even see  infinity see infinity means no matter  how many derivatives you take its  continuous and defined no jumps but if  you do a parabola for example it will  not be so well behaved I think yes  they're right got you know x y equals x  squared you take the derivative you get  2x take the derivative of that you get  12 and then the derivative of that you  get zero so the second derivative here  is to all along the curve this I guess y  prime Y double prime so in particular  here we had a second derivative 0 here  we have a second derivative of 2 alright  so that's quick crash course on  individual points being c0 c1 or c2 or C  infinities when you can go all the way  creases are points on the paper where  you're not see one that means that you  have two different tangent planes coming  together typically so discontinuities in  the first derivative that's what we call  creases semi creases our discontinuities  in the second derivative so semi crease  is where your Nazi to crease is where  you're not see one according to this  particular paper semi crease is not too  common a term but crease is very calm  this is the usual meaning of crease semi  creases maybe a little new to this paper  we're talking about non-existence of  hype ours and so for example this would  be a crease and this would be a semi  crease here the derivative changes that  looks like a crease if you had a one  dimensional piece of paper this doesn't  look like a crease it's kind of smooth  but not it's a little bit smooth at c1  but it's not see too so there's a semi  crease there and part of that paper is  worrying about semi creases because we  want to deal with see two parts because  see two functions are nice I mean  ideally we have C infinity parts but we  only need see two parts and so we  subdivided the semi creases and  basically argue most of the time so  increases don't really happen so you  don't have to worry about them but  that's what they are any questions about  smoothness these are all different kinds  of smoothness yeah all of these things  are C infinity except at these points so  every polynomial C infinity you can  differentiate it all the time eventually  get 20 Exponential's or C infinity  pretty much all functions you can think  of that are can smooth RC infinity yeah  all right so next question is about one  of the proofs which was that got to  remember the proof the proof involving  normals basically we want involving  tangent planes than the other one is  involving normals this is the polygonal  implies flat proof so this was if you  have a region of paper is bounded by  polygonal creases so they're piecewise  straight then that region of paper if it  son creased must actually be flat and  I'm just gonna I want to talk about just  one part of the proof so there was a  sequence of steps but ultimately what we  wanted was some Seng  and bf this is a boundary of your region  okay somewhere over here and we assume  that it's straight so it took a few  steps to get here but suppose we have a  straight region and locally around this  point there's some ruling okay and then  we were looking at a point Q here I'm  going to define this ruling to be RFQ  the rule line from Q and so q is an  arbitrary pointing slides along this  segment and then we're interested in a  normal vector here which we called n of  Q that's normal to this the surface at  this point and so as q slides and if Q  changes and we were claiming a few  things but that the definition of this  guy being normal is it has to be  perpendicular to normal to the surface  is perpendicular to the surface so it  has to be perpendicular to this sort of  axis of the surface and has to be  perpendicular to this axis so we know  just from definition of normal n of Q is  perpendicular to B F and n fq is  perpendicular to RFQ the rule line so  that's the definition but it wasn't  quite what we wanted we wanted to prove  for example that the derivative of the  normal funny thing is perpendicular to  bf that was one of the things we wanted  and the other one is the corresponding  perpendicular to the rule line I don't  know if I use this notation and lecture  but this just means they're  perpendicular she's like the dot product  of those two vectors is zero so how do  we go from here to here this is actually  the part that was most confusing to me I  find this one more confusing because it  uses the fact that its torso ruled but  the question was actually about this one  so unless you ask about this one now  I'll just cover this one it's a little  simpler both are in the notes so why is  this true I guess first the intuitive  reason so you've got all these different  normals here we're going to end up  concluding they're all pointing the same  but in general they're changing  direction somehow we know that at all  times the normal is perpendicular to  this segment so it's pointing away  straight away from the segments we've  got a right angle here no matter so it's  kind of spinning around the segment at  best as it moves around we claim that  the change in the normal the derivative  of the normal as you walk along this  segment must also be perpendicular to  this segment intuitively this is obvious  if you think about it enough if you were  changing in a direction that was not  perpendicular to normal to this segment  for example you're going this way or  something then while initially your  perpendicular your kind of falling over  and then you'll no longer be  perpendicular so if you change in a  first-order way that is not  perpendicular to this thing then  afterwards you will no longer be  perpendicular so that's intuitively why  this follows from this okay the form one  way to formalize that is to use Taylor  expansion so i'll just write that  quickly if you look at a point Q and I'm  going to use this notation q plus  epsilon to mean a point just a little  bit over this is Q plus Epsilon this is  Q for very small epsilon this is going  to be approximately n of Q plus epsilon  and prime of Q in Taylor expansion then  you have a pep salon square but for  small enough epsilon this is a good  approximation and so on the one hand you  have if you look at n of Q plus epsilon  and you compare it to be F you know that  these are perpendicular and what you can  write that as a product being equal to 0  so the dot product of these two vectors  should be equal to 0 that is being  perpendicular we know this is true for  all points Q so in particular is going  to be true for Q plus epsilon so that's  what we know but now we can expand this  thing and we you know get these two  terms so we get n fq dot F plus epsilon  and prime of Q  dot BF that's just distributing that sum  over this product dot products work okay  that way now this thing is also zero  because this is just another point q  dotted with BF and they should always be  perpendicular and this is the thing we  care about now if this whole sum is  equal to 0 and this term is equal to 0  then this better be equal to 0 so right  implies this thing equal to 0 and that's  what we want it that n prime of Q is  perpendicular to bf so that's the the  algebra way to see it but if you think  about it enough and you believe things  are linear to the first order so we say  then that has to be true you can get  second derivatives and things as well  but first derivative better hold in  particular any questions about that you  can do the same thing with this property  but it's messier you need to use the  tour cyl ruled property which we didn't  really go into much detail on so I think  I'll skip it last question is what does  it all mean if we're proving things are  impossible and yet here i have physical  models of them existing what's going on  what's the difference between  mathematical paper and real paper and of  course there's no real mathematical  answer to that question but there we  have a couple competing theories for  what might be happening in this model  one is that perhaps the paper is not  being isometric perhaps it's stretching  or shearing so normally we think of  paper as unstretched ball and not  sharable the only way to kind of share  is to fold it that is almost perfectly  true but of course no material is  perfect so it might be shearing or  changing the geometry the intrinsic  geometry just enough to somehow make  this possible our theory only says that  it's impossible if you're exactly  perfect so maybe if interesting to  measure how exact paper is question  it is possible to imitate this year hold  it this is not a sheer I hold it we can  add creases to my gate just to simulate  a sheer right but only by an increases  that's what the that's what we've proved  in lecture so yeah you can kind of a  cashier but only by n increases now it  doesn't look like there's extra creases  here theory number two is there are lots  of really tiny creases here that are so  small I mean here you could detect the  crease because it was a big change in  angle but if they're super super tiny  here and you know never fold these  things completely perfect it could be  lots of little tiny things maybe you  look under a microscope and you'll see  that I don't know it could be an  interesting project to try to  investigate what really happens with  paper in these kinds of models in real  life and how it differs from from  mathematics but the two theories are  maybe some stretching or maybe extra  creases we know by adding some by adding  the diagonals it folds now here you  could pretty clearly see the diagonals  are not being added but there might be  there might be effectively there might  be enough added creases that are  together so tiny you can't see them  there is not any places  semi creases are not enough the theory  says that if you have these creases and  possibly any number of semi creases you  can't fold it at all so you need to add  actual creases yeah good question all  right that's that's it for old material  now I want it to show you some cool new  material which is based on again later  than you don't give you a nice place  right the creases here do not look like  straight lines and yet we proved they  must be straight lines so something's  going on now we didn't know initially  the straight creases had to stay  straight in 3d but once we prove that we  were pretty sure this didn't exist it's  not obvious that's very Croesus stay  straight but they do so yeah it's  evidence there's a problem okay I wanted  to talk about this paper because it's  kind of related it's about how to  efficiently make pleat folding happen  and it's written by a bunch of people  John cardinel Marty our cameraman shinji  i'ma horiyoshi ito Musashi Kiyomi Stefan  langerman rue Hitler Hara who's here on  sabbatical and tacky aki no good  practice so here's the kind of setting  it's going to be familiar in that we  have a 1d piece of paper so that's a  segment and we're also going to assume  that it's uniformly creased so each of  these is a spacing of one and now  usually we think of pre assigning  mountains and valleys here and then you  can only fold mountains on the mountains  you can only fold valleys in the valleys  all right the model here is that you're  allowed to fold and unfold repeatedly on  the same segments so you can do some  layers simple folds though I think we'll  just need all layers here ok  but you can also sort of unfold any  previous folds and I'll say the unfolds  are free we don't you don't pay for the  unfolding operations because I mean it's  just a constant factor you can only  unfold things that have previously been  folded so we'll count the number of  folds you make each time you do a full  it's a mountain or Valley say through  all the layers or through some of the  layers but our goal is in the end to  have our segments creased in a  particular pattern and the pleat pattern  would be MV MV MV so we want the last  time each crease to be folded to be a  particular pattern but along the way we  could fold it in lots of it could be  this one could be mountain for a while  as long as the last time we creased this  thing its Valley we're happy so we end  up with sort of a mountain valley  strings and we want to know how many  folds how many folding operations do you  need to make in order to achieve a  particular mountain valley string and in  particular we care about strings like MV  MV MV because as we'll learn later today  when we fold these up leading takes a  lot of time it's tedious be nice if you  could do things faster now why do we  think we can do things faster because I  can take a piece of paper and imagine  this is one-dimensional and fold it in  half and then fold it in half again boom  I got how many creases two creases for  the price of one operation now I fold  again boom I get for creases for the  price of one I fold it again well I get  eight creases for the price of one so  the number of folding operations you do  here may be much much smaller than n n  is going to be the number of the creases  that the length of your strength ok so  if I fold a piece of paper like this  actually tried this and then I unfold to  see what mountain valley assignment I  got unfolding is free I get this weird  shape and I didn't do it perfectly it's  hard to do perfectly because of creep I  get a weird thing anyone know what this  thing is called I think that's pretty  much if I unfold them to 90 degrees  dragon curve yeah dragon curve looks  like this this is the Wikipedia drawing  and the more you fold in half that you  keep adding this iteration it's a nice  fractal it's kind of cool it doesn't  happens you can prove it doesn't self  intersect I mean it touches itself at  vertices but it doesn't properly cross  itself you take the limit it looks  something like this if you fill it in  very cool wikipedia calls it the I don't  want to spoil the surprise but there's  actually a book where the the sort of  section headings are iterations of this  fractal anyone know what this book is  Jurassic Park Ian Malcolm right there so  pretty cool it's called the Jurassic  Park fractal to some at least on  Wikipedia um and so that's one  particular mountain valley assignment  you can get and let me tell you what it  is it's pretty easy to figure out and  you see why it's a fractal so the first  time we make a fold let's say I always  make mountains just for simplicity you  have a little bit of choice here but not  very much okay so first I make a  mountain fold so now I've got things  folded in half now let's say I make a  mountain fold over here of course it's  already fold in half which means i get a  valley over here so i'm going to yeah so  now i make another mountain fold but I  don't maybe I should have done it the  other way anyway so it is so I make  another mountain fold here and on the  Left I get this on the right I get the  reflection of that and then I make  another mountain fold I get this on the  left and I get the reflection of that on  the right and when I reflect I'm reading  backwards and also inverting everything  v and so on so you see the kind of  fractal nature here and that's what  gives the cool curve keep repeating that  you have essentially n choices  their login folds that you make H could  be a mountain or Valley so there n  different MV strings you can get in  essentially log n folds but sadly none  of them is the one we want MV MV MV the  pleat fold so that leaves us with the  question of how many operations do you  need to get a pleat fold this is where  things get cool so yeah well why don't I  tell you some answers so if you want to  fold mm mm or MV MV please turn out to  be the same because you can do ms on the  odd positions v's on the even positions  then you can do it in log squared n  folds and you need at least log squared  n over log log n this is way way faster  than n there's an open problem between  log squared n and log squared over log  log small gap of log factor but  somewhere in between here if you want to  do let's say a random pattern then we  can prove you need a bow sorry and  divided by log in so the obvious step  the obvious way to do it is fold each  crease unfold that takes n steps so you  can always improve by a log factor and  that's the best you can do for most  strings but these strings are  sufficiently special that you can get a  huge factor a basically exponential win  instead of doing roughly n steps you're  only doing poly log steps so there's  four results here which ones would you  like to see how many people vote for log  squared n upper bound for how many  people vote for log squared over log log  lower bound completely different set of  four how many people vote for the n over  log n story upper bound for  and lower bounds okay i think the upper  bound here wins is curious they all  these similar ideas let me I'm going to  briefly sketch because I want to get to  folding stuff how you achieve n over log  n upper bound maybe then I'll do log  squared upper bound here lower bounds  are well they're lower bounds it doesn't  say anything so if you have an MV string  to get this bound arbitrary string I'm  going to consider epsilon sorry 1 minus  epsilon log in consecutive letters and  split into basically n over log n chunks  each of size roughly log in a little bit  smaller than log in why do I do that  because the number of different chunk  values is 2 to the 1 minus epsilon times  log n right each can be mountain of  valley now to to the log n is n so this  is n to the 1 minus epsilon possible  different chunk values these are chunks  why is that interesting because I have  basically n over log and chunks but  there's only n to the 1 minus epsilon  different trunk values this is much much  smaller than this I think of epsilon  being a half the square root of n this  is n over log n so there's actually most  many of the chunks have to be repeated  so this means an average chunk is  repeated what is it an over log n of  them just like n over log n times n to  the 1 minus epsilon times which is n to  the epsilon over log n right this is the  number of chunks here and we're dividing  by  the number of different ones so typical  repetition is going to be n to the  epsilon over log in time so I'll just  assume all chunks repeated this many  times it's all linear so it doesn't  matter whether some are more common some  are less common okay cool now what so if  I look at one of these chunks and all of  its repetitions I would like to fold  them all somehow more efficiently and  there's one key idea in this paper on  how to do that which is so here's my  strip maybe here's one rep instance of  the chunk here's another one they can be  kind of spaced out arbitrarily so what  I'm going to do is fold here fold here  fold here I think that's right all the  bisectors so I'll end up with one chunk  like this then another chunk like this  then another chunk something like this  then this one's very tiny okay I get my  four repetitions of the chunk all on top  of each other what I then do so this is  going to take how many repetitions is  like n to the epsilon over logins this  is about n to the epsilon over log n  steps to do all these folds now I fold  here fold here fold here fold here  simultaneously getting them all correct  that's going to take 1 minus epsilon  time basically log n steps then I unfold  and then I discover darn it half of them  were upside down because I got this one  right and this one right but these two  are flipped so that's no good so then I  recurse on the remaining half but this  turns out to be a geometric series so if  I do all four of these and then I do two  of them and then I do one of them I'll  be done and it will only cost me another  factor of two in time so it's I get log  n plus n to the epsilon over log in I  have to do this for each of the  different chunk values so i need to do n  to the 1 minus epsilon times log n plus  n to the epsilon over  again x like to because of the geometric  series and this comes out to andover  login tada magic the login is basically  coming from the trunk size being log in  and 1 minus epsilon log n is as big as  we could make it because we needed this  to be a lot smaller than n so that's how  you save a factor of log in a little  crazy but it works in theory that's the  upper bound the lower bound is actually  pretty obvious because if you think if  you have you only use k folds you're  doing there's n to the K or maybe to n  to the K different things you can do  because there's n different places you  can make a fold could be a mountain or  valley and this better be at least two  to the end because they're two to the N  different Mountain Valley patterns you  could make and you have to somehow make  them with K folds these are all the  things you can make with K folds and you  work that out and K has to be at least  about n over log n so that's why it's  optimal and because this is an  information-theoretic argument this  works in the average case as well take a  random example you need at least n over  log in most examples probability will  need and over log in so that's those  lower bounds let me briefly tell you  about this log squared and upper bound  uses the same idea but because  everything is repeated in the mmm mmm  string you don't it's a lot easier to do  this kind of folding oh sorry there's  one step I left out here so great you  make these folds you line things up you  fold these things then you unfold so  you've and recurse eventually you fix  these guys but you also destroyed this  case in this crease in this crease so  you kind of mess things up a little  you've got to do this repeatedly for  every chunk you don't want to mess up  previous chunks that you've done so you  have to go back and fix this fold unfold  fix this fold unfold but that again only  you only cost n to the epsilon over log  n so not too bad so another factor of  two here but I'm ignoring constant  factors  okay that was an over log in upper bound  let's do log squared upper bound for  mMmmm so I need to look this up it's a  little tricky we're going to use the  trick of the Dragon curve essentially  which is repeatedly fold in half and  we're going to keep doing that until we  are left with three creases which  haven't yet been folded so this is my  folded bundle there's you know many many  many layers here and then I'm going to  fold mmm on those three creases instead  of going all the way to Dragon curve  which would just be do em fold here  first I'm going to fold this unfold fold  this unfold fold this unfold so I get  this pattern when I unfold this it  actually turns out to be kind of nice  and alternating it will be something mmm  something Valley Valley Valley something  about a mountain them something Valley  Valley Valley and so on okay great half  of my things roughly our mountains I  just need to fix these valleys so I use  the same trick which is I'm going to  fold use red maybe fold in the middle of  these mountain segments I do that I line  up all the valley segments so i end up  with after folding this is going to look  like you almost need a computer or call  this out but it that we have done  implemented this it works it's going to  look like this repeat I mean with many  layers like this all the valleys are on  top of each other these question marks  are still question marks this mountain  is that mountain this mountain is that  mutton my mountain is your mountain I  don't know so now we're going to fold  mountain mountain mountain mountain on  all five of these things and these two  remain mountains and then unfold and  what i end up with is sort of twice as  good i end up with ? and then i think  seven  sorry not quite because this part also  gets messed up so i end up with Valley  Mountain ? then seven mountains then ?  then some not so pretty stuff em vvvvv  em ? and then this repeats until we get  to the end of the string and then the  end looks kind of like this so the point  is I had 3 m's in a row now i have 7 ms  in a row each time i do this i roughly  double the number of Em's in a row by  the end almost everything will be ms  after I do this log n times I have  pretty much all Em's and then I can  finish it off and each of these steps  took log n steps because I had to fold  along all these things how did I fold  along them not one at a time of course I  fold in half and fold in half and fold  in half roughly I choose the middle most  red line to fold first and about log n  steps later everything will be piled up  so I get to use this efficiency of  dragon curve things I'm messing up these  creases of course but it's okay because  I get a big chunk that all gets correct  at once half of it gets wrong half of  its correct but the half that's getting  correct gets longer and longer each step  doubling so only log squared steps  pretty sure this is optimal but the best  lower bound we can prove is log squared  over log log all right finally let's  fold some stuff I thought we would fold  some hyperbolic paraboloids and maybe  put them together into structures like  this so you have your squares if you  don't have squares bunch of squares up  here you can start folding as many as  you like we're going to put them  together I will demonstrate up here so  the first thing we want to do is fold  along the diagonals these aren't perfect  squares but it's ok so we fold along on  diagonal and the other diagonal this is  mainly so we know where the center is  but also because we need to fold a  diagonal so we kill two birds with one  stone  whereas we just saw how to kill in this  case log n birds with one stone save log  in here we're only killing two so now  you've got your diagonals near square we  want to fold it's kind of cool I color  the edges with my chalk hands now where  everything's going to be parallel to the  sides of the square so first thing we're  going to do is construct a square that's  half is big in the centered along that  point so this involves folding the  bottom edge usually do this valley-fold  fold the bottom edge to line up with  that Center and also line up the edges  now very important when you make this  fold don't fold it all the way just fold  the middle half between the two  diagonals that's really key otherwise  this will not work it's the only thing  you have to be careful of don't fall all  the way just in the middle okay that's  one quarter do that four times so we go  here line up with the center fold the  middle  once you've got the inner square you're  going to make two more there's going to  be this square at the one quarter mark  and an inner square at the three  quarters mark to make the outer square  you fold the edge to the thing that you  just folded keep keep careful note of  which crease you're folding to should be  from the previous square not from the  current Square to fold the inner square  you go all the way up to the opposite  side of the first square and you can  just look visually are these evenly  spaced if yes you did it right if not  you're in trouble probably okay if you  make a few extra creases but don't try  to make as few extra creases as possible  so then repeat that four times and  you'll get your I guess four squares if  you count the outer boundary all right  it's a little hard to see once you've  got that those squares looks like I do  one of them incorrectly  got your nice concentric squares four of  them now you're halfway done you flip it  over and do valley folds on the other  side so because you want these to be  alternating Mountain Valley Mountain  Valley and just fill in all the squares  in between those squares and you can  figure out what the reference markers  are there all the mountain folds well  actually every other mountain fold will  be one of your references don't go to  one of your new Valley folds always go  to the mountain once and just check that  you're always filling in halfway in  between two of your Mountain folds don't  forget to flip over very important and  skip every other guy otherwise you will  think refold existing crease the wrong  way this is what it looks like after one  quarter of those  alright once you've done all this crazy  precreasing you have concentric squares  alternating mountain valley then the fun  part begins this is actually literally  fun not its I guess also harder but much  more exciting than all that precreasing  so in this case you have to fold all the  creases at once and the easy way to do  that is start at the outside and make  the outermost ring fold it go around  once or twice make sure everything is  folded including that diagonal so here  I've Valley folded everything including  the diet outer diagonal and then you  want to mountain fold the next one and  get them to be against each other you're  aiming for a kind of X shape which is  drawn over here it's going to look like  that ideally just keep collapsing square  by square making sure you alternating  mountain valley as I mean it should just  fall into place pretty much but paper  likes to miss fold a little bit so you  got to fix it all here I've done two  squares a little more  it's easier and easier because the  squares get smaller and smaller we've  got three squares done so I'm like  halfway to annex fourth square centers a  little bit tricky try to get it also  alternate so now it's nice and thin it's  like repeated sync folds of a waterbomb  base and it's like an X and then this is  your opportunity to wreak Reese all your  crease is really hard just give it a  good squeeze on each of the three four  legs of the starfish and now you've got  your hyperbolic paraboloid to make it  you want to take two midpoints here of  the squares pull them apart until it's a  little bit open and then give it a twist  and then open up a little you've got  your hyperbolic paraboloid tah-dah if we  make enough of these we can assemble it  into some cool shapes how many people  have folded one at least one few people  i will show you you can fold more we're  going to assemble something might need a  bunch of people this is the algorithm we  use for converting a polyhedron into a  bunch of hyperbolic paraboloids it's in  this paper from 99 and we take each face  of the polygons of the polyhedron sorry  so if you have a cube you've got a bunch  of squares for each square we make what  we call a 4 hat which is for hyperbolic  paraboloids joined in a cycle like this  and you've got to be careful to join it  the right way but then these the tips of  the high parts that are not joined I  mean two of the one tip of each of the  hype ours comes together at the center  then these tips are going to represent  the edge of the polygon these red dots  and so these two sides are the sides of  one hype of two these two sides of the  sides of one hype are they're going to  join to an adjacent high power over here  so that's the idea yeah so I've got  already filled with a bunch of these  already so maybe what a hat looks like  so tick two of them join them along  those edges we're going to use tape or  staples to join them don't have a fancy  lock I'm afraid and join these edges  together and you get this is a 3 hat you  can put on your head whatever and that  would represent a triangle and then  we're going to join along these two  sides to an adjacent triangle or  whatever so we could make a platonic  solid i guess the simplest one would be  a tetrahedron that has six edges so it  needs 12 hype ours I was thinking we  could make this solid it's never been  made before should be cool it's the  simplest Archimedean solid terms the  number of edges the truncated  tetrahedron this requires 36 parts so  for ambitious we can go for it but I  think we're low on time so tetrahedra  might be easier bit people want to come  up and start assembling all right  finished taping that hold these this  goes here you're gonna hold this one  virtual assembly here so maybe we'll  finish this Emily next class

2
00:00:05,470 --> 00:00:08,179
alright so this lecture was about  hyperbolic paraboloids and the extent to  which they don't exist or exist here is  a regular non existing hyperbolic  paraboloid with concentric squares no  diagonals folded here and so is just a  few questions about this what does it  mean other things these are all s by and  paying i believe and they're all open so  they're all good questions we don't know  whether the the good triangulation the  alternating one works for arbitrary and  we only know up to N equals 104 the  various fixed angles that we checked we  don't have a proof technique to do  arbitrary n you could try to make do  some amount of alternation but not  somewhere in between the two extremes  and probably you get something in  between goodness but I don't know could  certainly play with that and it's very  natural to try this with larger than  squares the only trouble is the center  no longer has a single degree of freedom  so the only thing to answer there would  be how do you want to initially fold  like if you're doing a hexagon how  hexagons probably clear what you might  want to do but for general Kagan how do  you want to arrange that innermost keg  on from that you could propagate out  just like we did with the square if you  use an alternating triangulation say  probably works well but we haven't tried  it could be a cool project to do that  not that hard so those were open  problems next we have some sort of math  general math questions c1 c2 and semi  creases these are related questions so  let me do a quick visual review of these  terms I'm going to do it for  one-dimensional curves in 2d because  that's a lot easier to think about then  what we really care about which is  two-dimensional surfaces in 3d but all  the ideas carry over  so here is just a function let's say for  example this is a parabolic arc then a  straight segment this point is missing  and instead it's up here these points  are present and here we have another  parabolic arc so this function i would  call piecewise C infinity let me explain  what these things mean so we have on the  one hand c0 is a term meaning just  continuous continuous means no jumps  like this so this is not a continuous  function this so this function overall  is not even c0 c1 is a stronger  condition which means that you not only  are you continuous but you also have a  continuous and existing first derivative  so whereas here we're thinking about F  here we're thinking of F prime so not  only can you talk about the function  being continuous c0 c1 but you can talk  about individual moments in time so for  example you know this moment here is  going to be c1 it has a nice derivative  that derivative is changing continuously  here here however this is a c0 but not  see one because the tangent on the left  is different from the tangent on the  right so this is not a c1 function  because the derivative is jumping at  that point I'm not going to draw the  derivative here it's a little harder to  draw but you could draw it at the set  with the same x-axis and you'd see a  jump from one angle to another this  point is not even c0 this point is c0  but not see one this point is c1  because the tangent here is equal on the  left is equal to the tangent on the  right if I drew this properly where this  is the bottom of the parabola okay you  can ask for more than see one and we do  in plus we talk about c2 and this says  you should have a continuous second  derivative so acceleration and this this  point for example is not see two as well  the tangents meet if you take the next  derivative you see that it's suddenly  here the tangent was completely flat it  has a second derivative of zero and then  suddenly it starts increasing now you  can design functions that are even see  infinity see infinity means no matter  how many derivatives you take its  continuous and defined no jumps but if  you do a parabola for example it will  not be so well behaved I think yes  they're right got you know x y equals x  squared you take the derivative you get  2x take the derivative of that you get  12 and then the derivative of that you  get zero so the second derivative here  is to all along the curve this I guess y  prime Y double prime so in particular  here we had a second derivative 0 here  we have a second derivative of 2 alright  so that's quick crash course on  individual points being c0 c1 or c2 or C  infinities when you can go all the way  creases are points on the paper where  you're not see one that means that you  have two different tangent planes coming  together typically so discontinuities in  the first derivative that's what we call  creases semi creases our discontinuities  in the second derivative so semi crease  is where your Nazi to crease is where  you're not see one according to this  particular paper semi crease is not too  common a term but crease is very calm  this is the usual meaning of crease semi  creases maybe a little new to this paper  we're talking about non-existence of  hype ours and so for example this would  be a crease and this would be a semi  crease here the derivative changes that  looks like a crease if you had a one  dimensional piece of paper this doesn't  look like a crease it's kind of smooth  but not it's a little bit smooth at c1  but it's not see too so there's a semi  crease there and part of that paper is  worrying about semi creases because we  want to deal with see two parts because  see two functions are nice I mean  ideally we have C infinity parts but we  only need see two parts and so we  subdivided the semi creases and  basically argue most of the time so  increases don't really happen so you  don't have to worry about them but  that's what they are any questions about  smoothness these are all different kinds  of smoothness yeah all of these things  are C infinity except at these points so  every polynomial C infinity you can  differentiate it all the time eventually  get 20 Exponential's or C infinity  pretty much all functions you can think  of that are can smooth RC infinity yeah  all right so next question is about one  of the proofs which was that got to  remember the proof the proof involving  normals basically we want involving  tangent planes than the other one is  involving normals this is the polygonal  implies flat proof so this was if you  have a region of paper is bounded by  polygonal creases so they're piecewise  straight then that region of paper if it  son creased must actually be flat and  I'm just gonna I want to talk about just  one part of the proof so there was a  sequence of steps but ultimately what we  wanted was some Seng  and bf this is a boundary of your region  okay somewhere over here and we assume  that it's straight so it took a few  steps to get here but suppose we have a  straight region and locally around this  point there's some ruling okay and then  we were looking at a point Q here I'm  going to define this ruling to be RFQ  the rule line from Q and so q is an  arbitrary pointing slides along this  segment and then we're interested in a  normal vector here which we called n of  Q that's normal to this the surface at  this point and so as q slides and if Q  changes and we were claiming a few  things but that the definition of this  guy being normal is it has to be  perpendicular to normal to the surface  is perpendicular to the surface so it  has to be perpendicular to this sort of  axis of the surface and has to be  perpendicular to this axis so we know  just from definition of normal n of Q is  perpendicular to B F and n fq is  perpendicular to RFQ the rule line so  that's the definition but it wasn't  quite what we wanted we wanted to prove  for example that the derivative of the  normal funny thing is perpendicular to  bf that was one of the things we wanted  and the other one is the corresponding  perpendicular to the rule line I don't  know if I use this notation and lecture  but this just means they're  perpendicular she's like the dot product  of those two vectors is zero so how do  we go from here to here this is actually  the part that was most confusing to me I  find this one more confusing because it  uses the fact that its torso ruled but  the question was actually about this one  so unless you ask about this one now  I'll just cover this one it's a little  simpler both are in the notes so why is  this true I guess first the intuitive  reason so you've got all these different  normals here we're going to end up  concluding they're all pointing the same  but in general they're changing  direction somehow we know that at all  times the normal is perpendicular to  this segment so it's pointing away  straight away from the segments we've  got a right angle here no matter so it's  kind of spinning around the segment at  best as it moves around we claim that  the change in the normal the derivative  of the normal as you walk along this  segment must also be perpendicular to  this segment intuitively this is obvious  if you think about it enough if you were  changing in a direction that was not  perpendicular to normal to this segment  for example you're going this way or  something then while initially your  perpendicular your kind of falling over  and then you'll no longer be  perpendicular so if you change in a  first-order way that is not  perpendicular to this thing then  afterwards you will no longer be  perpendicular so that's intuitively why  this follows from this okay the form one  way to formalize that is to use Taylor  expansion so i'll just write that  quickly if you look at a point Q and I'm  going to use this notation q plus  epsilon to mean a point just a little  bit over this is Q plus Epsilon this is  Q for very small epsilon this is going  to be approximately n of Q plus epsilon  and prime of Q in Taylor expansion then  you have a pep salon square but for  small enough epsilon this is a good  approximation and so on the one hand you  have if you look at n of Q plus epsilon  and you compare it to be F you know that  these are perpendicular and what you can  write that as a product being equal to 0  so the dot product of these two vectors  should be equal to 0 that is being  perpendicular we know this is true for  all points Q so in particular is going  to be true for Q plus epsilon so that's  what we know but now we can expand this  thing and we you know get these two  terms so we get n fq dot F plus epsilon  and prime of Q  dot BF that's just distributing that sum  over this product dot products work okay  that way now this thing is also zero  because this is just another point q  dotted with BF and they should always be  perpendicular and this is the thing we  care about now if this whole sum is  equal to 0 and this term is equal to 0  then this better be equal to 0 so right  implies this thing equal to 0 and that's  what we want it that n prime of Q is  perpendicular to bf so that's the the  algebra way to see it but if you think  about it enough and you believe things  are linear to the first order so we say  then that has to be true you can get  second derivatives and things as well  but first derivative better hold in  particular any questions about that you  can do the same thing with this property  but it's messier you need to use the  tour cyl ruled property which we didn't  really go into much detail on so I think  I'll skip it last question is what does  it all mean if we're proving things are  impossible and yet here i have physical  models of them existing what's going on  what's the difference between  mathematical paper and real paper and of  course there's no real mathematical  answer to that question but there we  have a couple competing theories for  what might be happening in this model  one is that perhaps the paper is not  being isometric perhaps it's stretching  or shearing so normally we think of  paper as unstretched ball and not  sharable the only way to kind of share  is to fold it that is almost perfectly  true but of course no material is  perfect so it might be shearing or  changing the geometry the intrinsic  geometry just enough to somehow make  this possible our theory only says that  it's impossible if you're exactly  perfect so maybe if interesting to  measure how exact paper is question  it is possible to imitate this year hold  it this is not a sheer I hold it we can  add creases to my gate just to simulate  a sheer right but only by an increases  that's what the that's what we've proved  in lecture so yeah you can kind of a  cashier but only by n increases now it  doesn't look like there's extra creases  here theory number two is there are lots  of really tiny creases here that are so  small I mean here you could detect the  crease because it was a big change in  angle but if they're super super tiny  here and you know never fold these  things completely perfect it could be  lots of little tiny things maybe you  look under a microscope and you'll see  that I don't know it could be an  interesting project to try to  investigate what really happens with  paper in these kinds of models in real  life and how it differs from from  mathematics but the two theories are  maybe some stretching or maybe extra  creases we know by adding some by adding  the diagonals it folds now here you  could pretty clearly see the diagonals  are not being added but there might be  there might be effectively there might  be enough added creases that are  together so tiny you can't see them  there is not any places  semi creases are not enough the theory  says that if you have these creases and  possibly any number of semi creases you  can't fold it at all so you need to add  actual creases yeah good question all  right that's that's it for old material  now I want it to show you some cool new  material which is based on again later  than you don't give you a nice place  right the creases here do not look like  straight lines and yet we proved they  must be straight lines so something's  going on now we didn't know initially  the straight creases had to stay  straight in 3d but once we prove that we  were pretty sure this didn't exist it's  not obvious that's very Croesus stay  straight but they do so yeah it's  evidence there's a problem okay I wanted  to talk about this paper because it's  kind of related it's about how to  efficiently make pleat folding happen  and it's written by a bunch of people  John cardinel Marty our cameraman shinji  i'ma horiyoshi ito Musashi Kiyomi Stefan  langerman rue Hitler Hara who's here on  sabbatical and tacky aki no good  practice so here's the kind of setting  it's going to be familiar in that we  have a 1d piece of paper so that's a  segment and we're also going to assume  that it's uniformly creased so each of  these is a spacing of one and now  usually we think of pre assigning  mountains and valleys here and then you  can only fold mountains on the mountains  you can only fold valleys in the valleys  all right the model here is that you're  allowed to fold and unfold repeatedly on  the same segments so you can do some  layers simple folds though I think we'll  just need all layers here ok  but you can also sort of unfold any  previous folds and I'll say the unfolds  are free we don't you don't pay for the  unfolding operations because I mean it's  just a constant factor you can only  unfold things that have previously been  folded so we'll count the number of  folds you make each time you do a full  it's a mountain or Valley say through  all the layers or through some of the  layers but our goal is in the end to  have our segments creased in a  particular pattern and the pleat pattern  would be MV MV MV so we want the last  time each crease to be folded to be a  particular pattern but along the way we  could fold it in lots of it could be  this one could be mountain for a while  as long as the last time we creased this  thing its Valley we're happy so we end  up with sort of a mountain valley  strings and we want to know how many  folds how many folding operations do you  need to make in order to achieve a  particular mountain valley string and in  particular we care about strings like MV  MV MV because as we'll learn later today  when we fold these up leading takes a  lot of time it's tedious be nice if you  could do things faster now why do we  think we can do things faster because I  can take a piece of paper and imagine  this is one-dimensional and fold it in  half and then fold it in half again boom  I got how many creases two creases for  the price of one operation now I fold  again boom I get for creases for the  price of one I fold it again well I get  eight creases for the price of one so  the number of folding operations you do  here may be much much smaller than n n  is going to be the number of the creases  that the length of your strength ok so  if I fold a piece of paper like this  actually tried this and then I unfold to  see what mountain valley assignment I  got unfolding is free I get this weird  shape and I didn't do it perfectly it's  hard to do perfectly because of creep I  get a weird thing anyone know what this  thing is called I think that's pretty  much if I unfold them to 90 degrees  dragon curve yeah dragon curve looks  like this this is the Wikipedia drawing  and the more you fold in half that you  keep adding this iteration it's a nice  fractal it's kind of cool it doesn't  happens you can prove it doesn't self  intersect I mean it touches itself at  vertices but it doesn't properly cross  itself you take the limit it looks  something like this if you fill it in  very cool wikipedia calls it the I don't  want to spoil the surprise but there's  actually a book where the the sort of  section headings are iterations of this  fractal anyone know what this book is  Jurassic Park Ian Malcolm right there so  pretty cool it's called the Jurassic  Park fractal to some at least on  Wikipedia um and so that's one  particular mountain valley assignment  you can get and let me tell you what it  is it's pretty easy to figure out and  you see why it's a fractal so the first  time we make a fold let's say I always  make mountains just for simplicity you  have a little bit of choice here but not  very much okay so first I make a  mountain fold so now I've got things  folded in half now let's say I make a  mountain fold over here of course it's  already fold in half which means i get a  valley over here so i'm going to yeah so  now i make another mountain fold but I  don't maybe I should have done it the  other way anyway so it is so I make  another mountain fold here and on the  Left I get this on the right I get the  reflection of that and then I make  another mountain fold I get this on the  left and I get the reflection of that on  the right and when I reflect I'm reading  backwards and also inverting everything  v and so on so you see the kind of  fractal nature here and that's what  gives the cool curve keep repeating that  you have essentially n choices  their login folds that you make H could  be a mountain or Valley so there n  different MV strings you can get in  essentially log n folds but sadly none  of them is the one we want MV MV MV the  pleat fold so that leaves us with the  question of how many operations do you  need to get a pleat fold this is where  things get cool so yeah well why don't I  tell you some answers so if you want to  fold mm mm or MV MV please turn out to  be the same because you can do ms on the  odd positions v's on the even positions  then you can do it in log squared n  folds and you need at least log squared  n over log log n this is way way faster  than n there's an open problem between  log squared n and log squared over log  log small gap of log factor but  somewhere in between here if you want to  do let's say a random pattern then we  can prove you need a bow sorry and  divided by log in so the obvious step  the obvious way to do it is fold each  crease unfold that takes n steps so you  can always improve by a log factor and  that's the best you can do for most  strings but these strings are  sufficiently special that you can get a  huge factor a basically exponential win  instead of doing roughly n steps you're  only doing poly log steps so there's  four results here which ones would you  like to see how many people vote for log  squared n upper bound for how many  people vote for log squared over log log  lower bound completely different set of  four how many people vote for the n over  log n story upper bound for  and lower bounds okay i think the upper  bound here wins is curious they all  these similar ideas let me I'm going to  briefly sketch because I want to get to  folding stuff how you achieve n over log  n upper bound maybe then I'll do log  squared upper bound here lower bounds  are well they're lower bounds it doesn't  say anything so if you have an MV string  to get this bound arbitrary string I'm  going to consider epsilon sorry 1 minus  epsilon log in consecutive letters and  split into basically n over log n chunks  each of size roughly log in a little bit  smaller than log in why do I do that  because the number of different chunk  values is 2 to the 1 minus epsilon times  log n right each can be mountain of  valley now to to the log n is n so this  is n to the 1 minus epsilon possible  different chunk values these are chunks  why is that interesting because I have  basically n over log and chunks but  there's only n to the 1 minus epsilon  different trunk values this is much much  smaller than this I think of epsilon  being a half the square root of n this  is n over log n so there's actually most  many of the chunks have to be repeated  so this means an average chunk is  repeated what is it an over log n of  them just like n over log n times n to  the 1 minus epsilon times which is n to  the epsilon over log n right this is the  number of chunks here and we're dividing  by  the number of different ones so typical  repetition is going to be n to the  epsilon over log in time so I'll just  assume all chunks repeated this many  times it's all linear so it doesn't  matter whether some are more common some  are less common okay cool now what so if  I look at one of these chunks and all of  its repetitions I would like to fold  them all somehow more efficiently and  there's one key idea in this paper on  how to do that which is so here's my  strip maybe here's one rep instance of  the chunk here's another one they can be  kind of spaced out arbitrarily so what  I'm going to do is fold here fold here  fold here I think that's right all the  bisectors so I'll end up with one chunk  like this then another chunk like this  then another chunk something like this  then this one's very tiny okay I get my  four repetitions of the chunk all on top  of each other what I then do so this is  going to take how many repetitions is  like n to the epsilon over logins this  is about n to the epsilon over log n  steps to do all these folds now I fold  here fold here fold here fold here  simultaneously getting them all correct  that's going to take 1 minus epsilon  time basically log n steps then I unfold  and then I discover darn it half of them  were upside down because I got this one  right and this one right but these two  are flipped so that's no good so then I  recurse on the remaining half but this  turns out to be a geometric series so if  I do all four of these and then I do two  of them and then I do one of them I'll  be done and it will only cost me another  factor of two in time so it's I get log  n plus n to the epsilon over log in I  have to do this for each of the  different chunk values so i need to do n  to the 1 minus epsilon times log n plus  n to the epsilon over  again x like to because of the geometric  series and this comes out to andover  login tada magic the login is basically  coming from the trunk size being log in  and 1 minus epsilon log n is as big as  we could make it because we needed this  to be a lot smaller than n so that's how  you save a factor of log in a little  crazy but it works in theory that's the  upper bound the lower bound is actually  pretty obvious because if you think if  you have you only use k folds you're  doing there's n to the K or maybe to n  to the K different things you can do  because there's n different places you  can make a fold could be a mountain or  valley and this better be at least two  to the end because they're two to the N  different Mountain Valley patterns you  could make and you have to somehow make  them with K folds these are all the  things you can make with K folds and you  work that out and K has to be at least  about n over log n so that's why it's  optimal and because this is an  information-theoretic argument this  works in the average case as well take a  random example you need at least n over  log in most examples probability will  need and over log in so that's those  lower bounds let me briefly tell you  about this log squared and upper bound  uses the same idea but because  everything is repeated in the mmm mmm  string you don't it's a lot easier to do  this kind of folding oh sorry there's  one step I left out here so great you  make these folds you line things up you  fold these things then you unfold so  you've and recurse eventually you fix  these guys but you also destroyed this  case in this crease in this crease so  you kind of mess things up a little  you've got to do this repeatedly for  every chunk you don't want to mess up  previous chunks that you've done so you  have to go back and fix this fold unfold  fix this fold unfold but that again only  you only cost n to the epsilon over log  n so not too bad so another factor of  two here but I'm ignoring constant  factors  okay that was an over log in upper bound  let's do log squared upper bound for  mMmmm so I need to look this up it's a  little tricky we're going to use the  trick of the Dragon curve essentially  which is repeatedly fold in half and  we're going to keep doing that until we  are left with three creases which  haven't yet been folded so this is my  folded bundle there's you know many many  many layers here and then I'm going to  fold mmm on those three creases instead  of going all the way to Dragon curve  which would just be do em fold here  first I'm going to fold this unfold fold  this unfold fold this unfold so I get  this pattern when I unfold this it  actually turns out to be kind of nice  and alternating it will be something mmm  something Valley Valley Valley something  about a mountain them something Valley  Valley Valley and so on okay great half  of my things roughly our mountains I  just need to fix these valleys so I use  the same trick which is I'm going to  fold use red maybe fold in the middle of  these mountain segments I do that I line  up all the valley segments so i end up  with after folding this is going to look  like you almost need a computer or call  this out but it that we have done  implemented this it works it's going to  look like this repeat I mean with many  layers like this all the valleys are on  top of each other these question marks  are still question marks this mountain  is that mountain this mountain is that  mutton my mountain is your mountain I  don't know so now we're going to fold  mountain mountain mountain mountain on  all five of these things and these two  remain mountains and then unfold and  what i end up with is sort of twice as  good i end up with ? and then i think  seven  sorry not quite because this part also  gets messed up so i end up with Valley  Mountain ? then seven mountains then ?  then some not so pretty stuff em vvvvv  em ? and then this repeats until we get  to the end of the string and then the  end looks kind of like this so the point  is I had 3 m's in a row now i have 7 ms  in a row each time i do this i roughly  double the number of Em's in a row by  the end almost everything will be ms  after I do this log n times I have  pretty much all Em's and then I can  finish it off and each of these steps  took log n steps because I had to fold  along all these things how did I fold  along them not one at a time of course I  fold in half and fold in half and fold  in half roughly I choose the middle most  red line to fold first and about log n  steps later everything will be piled up  so I get to use this efficiency of  dragon curve things I'm messing up these  creases of course but it's okay because  I get a big chunk that all gets correct  at once half of it gets wrong half of  its correct but the half that's getting  correct gets longer and longer each step  doubling so only log squared steps  pretty sure this is optimal but the best  lower bound we can prove is log squared  over log log all right finally let's  fold some stuff I thought we would fold  some hyperbolic paraboloids and maybe  put them together into structures like  this so you have your squares if you  don't have squares bunch of squares up  here you can start folding as many as  you like we're going to put them  together I will demonstrate up here so  the first thing we want to do is fold  along the diagonals these aren't perfect  squares but it's ok so we fold along on  diagonal and the other diagonal this is  mainly so we know where the center is  but also because we need to fold a  diagonal so we kill two birds with one  stone  whereas we just saw how to kill in this  case log n birds with one stone save log  in here we're only killing two so now  you've got your diagonals near square we  want to fold it's kind of cool I color  the edges with my chalk hands now where  everything's going to be parallel to the  sides of the square so first thing we're  going to do is construct a square that's  half is big in the centered along that  point so this involves folding the  bottom edge usually do this valley-fold  fold the bottom edge to line up with  that Center and also line up the edges  now very important when you make this  fold don't fold it all the way just fold  the middle half between the two  diagonals that's really key otherwise  this will not work it's the only thing  you have to be careful of don't fall all  the way just in the middle okay that's  one quarter do that four times so we go  here line up with the center fold the  middle  once you've got the inner square you're  going to make two more there's going to  be this square at the one quarter mark  and an inner square at the three  quarters mark to make the outer square  you fold the edge to the thing that you  just folded keep keep careful note of  which crease you're folding to should be  from the previous square not from the  current Square to fold the inner square  you go all the way up to the opposite  side of the first square and you can  just look visually are these evenly  spaced if yes you did it right if not  you're in trouble probably okay if you  make a few extra creases but don't try  to make as few extra creases as possible  so then repeat that four times and  you'll get your I guess four squares if  you count the outer boundary all right  it's a little hard to see once you've  got that those squares looks like I do  one of them incorrectly  got your nice concentric squares four of  them now you're halfway done you flip it  over and do valley folds on the other  side so because you want these to be  alternating Mountain Valley Mountain  Valley and just fill in all the squares  in between those squares and you can  figure out what the reference markers  are there all the mountain folds well  actually every other mountain fold will  be one of your references don't go to  one of your new Valley folds always go  to the mountain once and just check that  you're always filling in halfway in  between two of your Mountain folds don't  forget to flip over very important and  skip every other guy otherwise you will  think refold existing crease the wrong  way this is what it looks like after one  quarter of those  alright once you've done all this crazy  precreasing you have concentric squares  alternating mountain valley then the fun  part begins this is actually literally  fun not its I guess also harder but much  more exciting than all that precreasing  so in this case you have to fold all the  creases at once and the easy way to do  that is start at the outside and make  the outermost ring fold it go around  once or twice make sure everything is  folded including that diagonal so here  I've Valley folded everything including  the diet outer diagonal and then you  want to mountain fold the next one and  get them to be against each other you're  aiming for a kind of X shape which is  drawn over here it's going to look like  that ideally just keep collapsing square  by square making sure you alternating  mountain valley as I mean it should just  fall into place pretty much but paper  likes to miss fold a little bit so you  got to fix it all here I've done two  squares a little more  it's easier and easier because the  squares get smaller and smaller we've  got three squares done so I'm like  halfway to annex fourth square centers a  little bit tricky try to get it also  alternate so now it's nice and thin it's  like repeated sync folds of a waterbomb  base and it's like an X and then this is  your opportunity to wreak Reese all your  crease is really hard just give it a  good squeeze on each of the three four  legs of the starfish and now you've got  your hyperbolic paraboloid to make it  you want to take two midpoints here of  the squares pull them apart until it's a  little bit open and then give it a twist  and then open up a little you've got  your hyperbolic paraboloid tah-dah if we  make enough of these we can assemble it  into some cool shapes how many people  have folded one at least one few people  i will show you you can fold more we're  going to assemble something might need a  bunch of people this is the algorithm we  use for converting a polyhedron into a  bunch of hyperbolic paraboloids it's in  this paper from 99 and we take each face  of the polygons of the polyhedron sorry  so if you have a cube you've got a bunch  of squares for each square we make what  we call a 4 hat which is for hyperbolic  paraboloids joined in a cycle like this  and you've got to be careful to join it  the right way but then these the tips of  the high parts that are not joined I  mean two of the one tip of each of the  hype ours comes together at the center  then these tips are going to represent  the edge of the polygon these red dots  and so these two sides are the sides of  one hype of two these two sides of the  sides of one hype are they're going to  join to an adjacent high power over here  so that's the idea yeah so I've got  already filled with a bunch of these  already so maybe what a hat looks like  so tick two of them join them along  those edges we're going to use tape or  staples to join them don't have a fancy  lock I'm afraid and join these edges  together and you get this is a 3 hat you  can put on your head whatever and that  would represent a triangle and then  we're going to join along these two  sides to an adjacent triangle or  whatever so we could make a platonic  solid i guess the simplest one would be  a tetrahedron that has six edges so it  needs 12 hype ours I was thinking we  could make this solid it's never been  made before should be cool it's the  simplest Archimedean solid terms the  number of edges the truncated  tetrahedron this requires 36 parts so  for ambitious we can go for it but I  think we're low on time so tetrahedra  might be easier bit people want to come  up and start assembling all right  finished taping that hold these this  goes here you're gonna hold this one  virtual assembly here so maybe we'll  finish this Emily next class
 

3
00:00:08,179 --> 00:00:11,270

 

4
00:00:11,270 --> 00:00:13,610

 

5
00:00:13,610 --> 00:00:17,930

 

6
00:00:17,930 --> 00:00:21,620

 

7
00:00:21,620 --> 00:00:24,710

 

8
00:00:24,710 --> 00:00:28,310

 

9
00:00:28,310 --> 00:00:31,450

 

10
00:00:31,450 --> 00:00:35,720

 

11
00:00:35,720 --> 00:00:38,240

 

12
00:00:38,240 --> 00:00:40,130

 

13
00:00:40,130 --> 00:00:43,130

 

14
00:00:43,130 --> 00:00:45,800

 

15
00:00:45,800 --> 00:00:46,970

 

16
00:00:46,970 --> 00:00:50,030

 

17
00:00:50,030 --> 00:00:52,000

 

18
00:00:52,000 --> 00:00:53,840

 

19
00:00:53,840 --> 00:00:55,910

 

20
00:00:55,910 --> 00:00:59,330

 

21
00:00:59,330 --> 00:01:01,790

 

22
00:01:01,790 --> 00:01:03,860

 

23
00:01:03,860 --> 00:01:06,889

 

24
00:01:06,889 --> 00:01:08,330

 

25
00:01:08,330 --> 00:01:11,480

 

26
00:01:11,480 --> 00:01:13,580

 

27
00:01:13,580 --> 00:01:15,770

 

28
00:01:15,770 --> 00:01:17,569

 

29
00:01:17,569 --> 00:01:21,349

 

30
00:01:21,349 --> 00:01:23,270

 

31
00:01:23,270 --> 00:01:25,270

 

32
00:01:25,270 --> 00:01:27,949

 

33
00:01:27,949 --> 00:01:29,469

 

34
00:01:29,469 --> 00:01:31,730

 

35
00:01:31,730 --> 00:01:33,559

 

36
00:01:33,559 --> 00:01:37,999

 

37
00:01:37,999 --> 00:01:44,330

 

38
00:01:44,330 --> 00:01:49,910

 

39
00:01:49,910 --> 00:01:52,760

 

40
00:01:52,760 --> 00:01:56,269

 

41
00:01:56,269 --> 00:02:01,510

 

42
00:02:01,510 --> 00:02:03,949

 

43
00:02:03,949 --> 00:02:05,899

 

44
00:02:05,899 --> 00:02:07,580

 

45
00:02:07,580 --> 00:02:10,219

 

46
00:02:10,219 --> 00:02:13,970

 

47
00:02:13,970 --> 00:02:30,380

 

48
00:02:30,380 --> 00:02:32,479

 

49
00:02:32,479 --> 00:02:34,430

 

50
00:02:34,430 --> 00:02:36,920

 

51
00:02:36,920 --> 00:02:38,270

 

52
00:02:38,270 --> 00:02:42,860

 

53
00:02:42,860 --> 00:02:50,740

 

54
00:02:50,740 --> 00:02:53,839

 

55
00:02:53,839 --> 00:02:58,099

 

56
00:02:58,099 --> 00:03:03,559

 

57
00:03:03,559 --> 00:03:05,479

 

58
00:03:05,479 --> 00:03:08,990

 

59
00:03:08,990 --> 00:03:10,910

 

60
00:03:10,910 --> 00:03:13,400

 

61
00:03:13,400 --> 00:03:15,530

 

62
00:03:15,530 --> 00:03:25,000

 

63
00:03:25,000 --> 00:03:27,349

 

64
00:03:27,349 --> 00:03:31,729

 

65
00:03:31,729 --> 00:03:33,620

 

66
00:03:33,620 --> 00:03:35,569

 

67
00:03:35,569 --> 00:03:37,909

 

68
00:03:37,909 --> 00:03:40,970

 

69
00:03:40,970 --> 00:03:43,940

 

70
00:03:43,940 --> 00:03:46,300

 

71
00:03:46,300 --> 00:03:52,520

 

72
00:03:52,520 --> 00:03:57,319

 

73
00:03:57,319 --> 00:03:58,909

 

74
00:03:58,909 --> 00:04:01,610

 

75
00:04:01,610 --> 00:04:03,619

 

76
00:04:03,619 --> 00:04:05,119

 

77
00:04:05,119 --> 00:04:06,259

 

78
00:04:06,259 --> 00:04:07,699

 

79
00:04:07,699 --> 00:04:10,069

 

80
00:04:10,069 --> 00:04:13,580

 

81
00:04:13,580 --> 00:04:19,370

 

82
00:04:19,370 --> 00:04:26,090

 

83
00:04:26,090 --> 00:04:29,720

 

84
00:04:29,720 --> 00:04:31,310

 

85
00:04:31,310 --> 00:04:33,410

 

86
00:04:33,410 --> 00:04:36,440

 

87
00:04:36,440 --> 00:04:38,240

 

88
00:04:38,240 --> 00:04:41,960

 

89
00:04:41,960 --> 00:04:44,870

 

90
00:04:44,870 --> 00:04:49,610

 

91
00:04:49,610 --> 00:04:53,660

 

92
00:04:53,660 --> 00:04:55,220

 

93
00:04:55,220 --> 00:04:57,260

 

94
00:04:57,260 --> 00:04:59,570

 

95
00:04:59,570 --> 00:05:01,520

 

96
00:05:01,520 --> 00:05:03,710

 

97
00:05:03,710 --> 00:05:05,900

 

98
00:05:05,900 --> 00:05:07,670

 

99
00:05:07,670 --> 00:05:09,110

 

100
00:05:09,110 --> 00:05:13,220

 

101
00:05:13,220 --> 00:05:15,850

 

102
00:05:15,850 --> 00:05:20,320

 

103
00:05:20,320 --> 00:05:24,680

 

104
00:05:24,680 --> 00:05:26,660

 

105
00:05:26,660 --> 00:05:30,880

 

106
00:05:30,880 --> 00:05:34,490

 

107
00:05:34,490 --> 00:05:39,170

 

108
00:05:39,170 --> 00:05:44,840

 

109
00:05:44,840 --> 00:05:47,660

 

110
00:05:47,660 --> 00:05:49,280

 

111
00:05:49,280 --> 00:05:52,190

 

112
00:05:52,190 --> 00:05:54,500

 

113
00:05:54,500 --> 00:05:59,690

 

114
00:05:59,690 --> 00:06:02,740

 

115
00:06:02,740 --> 00:06:06,470

 

116
00:06:06,470 --> 00:06:09,020

 

117
00:06:09,020 --> 00:06:11,810

 

118
00:06:11,810 --> 00:06:15,500

 

119
00:06:15,500 --> 00:06:16,790

 

120
00:06:16,790 --> 00:06:20,780

 

121
00:06:20,780 --> 00:06:29,200

 

122
00:06:29,200 --> 00:06:35,390

 

123
00:06:35,390 --> 00:06:38,120

 

124
00:06:38,120 --> 00:06:40,730

 

125
00:06:40,730 --> 00:06:42,740

 

126
00:06:42,740 --> 00:06:44,570

 

127
00:06:44,570 --> 00:06:46,760

 

128
00:06:46,760 --> 00:06:48,320

 

129
00:06:48,320 --> 00:06:52,580

 

130
00:06:52,580 --> 00:06:56,990

 

131
00:06:56,990 --> 00:07:02,930

 

132
00:07:02,930 --> 00:07:04,400

 

133
00:07:04,400 --> 00:07:06,560

 

134
00:07:06,560 --> 00:07:08,180

 

135
00:07:08,180 --> 00:07:10,730

 

136
00:07:10,730 --> 00:07:12,920

 

137
00:07:12,920 --> 00:07:15,440

 

138
00:07:15,440 --> 00:07:17,690

 

139
00:07:17,690 --> 00:07:19,640

 

140
00:07:19,640 --> 00:07:22,909

 

141
00:07:22,909 --> 00:07:24,590

 

142
00:07:24,590 --> 00:07:26,780

 

143
00:07:26,780 --> 00:07:28,159

 

144
00:07:28,159 --> 00:07:29,570

 

145
00:07:29,570 --> 00:07:31,310

 

146
00:07:31,310 --> 00:07:32,390

 

147
00:07:32,390 --> 00:07:35,390

 

148
00:07:35,390 --> 00:07:37,219

 

149
00:07:37,219 --> 00:07:44,060

 

150
00:07:44,060 --> 00:07:49,760

 

151
00:07:49,760 --> 00:07:51,650

 

152
00:07:51,650 --> 00:07:53,719

 

153
00:07:53,719 --> 00:07:57,409

 

154
00:07:57,409 --> 00:07:59,120

 

155
00:07:59,120 --> 00:08:07,060

 

156
00:08:07,060 --> 00:08:12,110

 

157
00:08:12,110 --> 00:08:17,659

 

158
00:08:17,659 --> 00:08:18,890

 

159
00:08:18,890 --> 00:08:21,110

 

160
00:08:21,110 --> 00:08:22,370

 

161
00:08:22,370 --> 00:08:24,380

 

162
00:08:24,380 --> 00:08:28,159

 

163
00:08:28,159 --> 00:08:30,440

 

164
00:08:30,440 --> 00:08:33,200

 

165
00:08:33,200 --> 00:08:37,190

 

166
00:08:37,190 --> 00:08:39,850

 

167
00:08:39,850 --> 00:08:42,440

 

168
00:08:42,440 --> 00:08:44,840

 

169
00:08:44,840 --> 00:08:46,790

 

170
00:08:46,790 --> 00:08:48,170

 

171
00:08:48,170 --> 00:08:51,760

 

172
00:08:51,760 --> 00:08:54,920

 

173
00:08:54,920 --> 00:08:59,180

 

174
00:08:59,180 --> 00:09:01,040

 

175
00:09:01,040 --> 00:09:03,470

 

176
00:09:03,470 --> 00:09:07,220

 

177
00:09:07,220 --> 00:09:11,269

 

178
00:09:11,269 --> 00:09:15,579

 

179
00:09:15,579 --> 00:09:20,449

 

180
00:09:20,449 --> 00:09:21,860

 

181
00:09:21,860 --> 00:09:24,860

 

182
00:09:24,860 --> 00:09:26,840

 

183
00:09:26,840 --> 00:09:31,370

 

184
00:09:31,370 --> 00:09:33,740

 

185
00:09:33,740 --> 00:09:36,680

 

186
00:09:36,680 --> 00:09:40,250

 

187
00:09:40,250 --> 00:09:42,440

 

188
00:09:42,440 --> 00:09:45,320

 

189
00:09:45,320 --> 00:09:46,850

 

190
00:09:46,850 --> 00:09:48,410

 

191
00:09:48,410 --> 00:09:49,820

 

192
00:09:49,820 --> 00:09:52,130

 

193
00:09:52,130 --> 00:09:54,650

 

194
00:09:54,650 --> 00:09:59,800

 

195
00:09:59,800 --> 00:10:06,980

 

196
00:10:06,980 --> 00:10:08,329

 

197
00:10:08,329 --> 00:10:10,579

 

198
00:10:10,579 --> 00:10:14,060

 

199
00:10:14,060 --> 00:10:17,650

 

200
00:10:17,650 --> 00:10:20,300

 

201
00:10:20,300 --> 00:10:23,710

 

202
00:10:23,710 --> 00:10:26,480

 

203
00:10:26,480 --> 00:10:27,769

 

204
00:10:27,769 --> 00:10:28,760

 

205
00:10:28,760 --> 00:10:30,800

 

206
00:10:30,800 --> 00:10:35,569

 

207
00:10:35,569 --> 00:10:37,130

 

208
00:10:37,130 --> 00:10:38,930

 

209
00:10:38,930 --> 00:10:41,810

 

210
00:10:41,810 --> 00:10:44,600

 

211
00:10:44,600 --> 00:10:45,829

 

212
00:10:45,829 --> 00:10:47,449

 

213
00:10:47,449 --> 00:10:49,069

 

214
00:10:49,069 --> 00:10:51,980

 

215
00:10:51,980 --> 00:10:55,850

 

216
00:10:55,850 --> 00:10:58,340

 

217
00:10:58,340 --> 00:11:00,440

 

218
00:11:00,440 --> 00:11:01,870

 

219
00:11:01,870 --> 00:11:04,630

 

220
00:11:04,630 --> 00:11:08,140

 

221
00:11:08,140 --> 00:11:10,690

 

222
00:11:10,690 --> 00:11:13,230

 

223
00:11:13,230 --> 00:11:15,280

 

224
00:11:15,280 --> 00:11:18,670

 

225
00:11:18,670 --> 00:11:20,050

 

226
00:11:20,050 --> 00:11:25,000

 

227
00:11:25,000 --> 00:11:27,370

 

228
00:11:27,370 --> 00:11:28,840

 

229
00:11:28,840 --> 00:11:31,840

 

230
00:11:31,840 --> 00:11:35,590

 

231
00:11:35,590 --> 00:11:38,410

 

232
00:11:38,410 --> 00:11:40,390

 

233
00:11:40,390 --> 00:11:42,340

 

234
00:11:42,340 --> 00:11:43,900

 

235
00:11:43,900 --> 00:11:46,600

 

236
00:11:46,600 --> 00:11:48,370

 

237
00:11:48,370 --> 00:11:49,600

 

238
00:11:49,600 --> 00:11:51,520

 

239
00:11:51,520 --> 00:11:52,450

 

240
00:11:52,450 --> 00:11:54,190

 

241
00:11:54,190 --> 00:11:55,570

 

242
00:11:55,570 --> 00:11:58,210

 

243
00:11:58,210 --> 00:12:03,430

 

244
00:12:03,430 --> 00:12:05,080

 

245
00:12:05,080 --> 00:12:07,030

 

246
00:12:07,030 --> 00:12:10,600

 

247
00:12:10,600 --> 00:12:11,980

 

248
00:12:11,980 --> 00:12:14,260

 

249
00:12:14,260 --> 00:12:17,080

 

250
00:12:17,080 --> 00:12:22,060

 

251
00:12:22,060 --> 00:12:25,120

 

252
00:12:25,120 --> 00:12:31,060

 

253
00:12:31,060 --> 00:12:32,980

 

254
00:12:32,980 --> 00:12:34,270

 

255
00:12:34,270 --> 00:12:39,490

 

256
00:12:39,490 --> 00:12:47,560

 

257
00:12:47,560 --> 00:12:49,300

 

258
00:12:49,300 --> 00:12:51,280

 

259
00:12:51,280 --> 00:12:53,680

 

260
00:12:53,680 --> 00:12:55,360

 

261
00:12:55,360 --> 00:12:57,550

 

262
00:12:57,550 --> 00:12:58,840

 

263
00:12:58,840 --> 00:13:00,490

 

264
00:13:00,490 --> 00:13:02,560

 

265
00:13:02,560 --> 00:13:04,540

 

266
00:13:04,540 --> 00:13:07,210

 

267
00:13:07,210 --> 00:13:13,930

 

268
00:13:13,930 --> 00:13:15,700

 

269
00:13:15,700 --> 00:13:20,200

 

270
00:13:20,200 --> 00:13:22,660

 

271
00:13:22,660 --> 00:13:25,920

 

272
00:13:25,920 --> 00:13:28,270

 

273
00:13:28,270 --> 00:13:29,980

 

274
00:13:29,980 --> 00:13:31,780

 

275
00:13:31,780 --> 00:13:34,930

 

276
00:13:34,930 --> 00:13:36,640

 

277
00:13:36,640 --> 00:13:39,190

 

278
00:13:39,190 --> 00:13:42,100

 

279
00:13:42,100 --> 00:13:43,870

 

280
00:13:43,870 --> 00:13:46,540

 

281
00:13:46,540 --> 00:13:48,370

 

282
00:13:48,370 --> 00:13:50,110

 

283
00:13:50,110 --> 00:13:52,270

 

284
00:13:52,270 --> 00:13:56,350

 

285
00:13:56,350 --> 00:13:57,640

 

286
00:13:57,640 --> 00:13:59,890

 

287
00:13:59,890 --> 00:14:04,810

 

288
00:14:04,810 --> 00:14:06,340

 

289
00:14:06,340 --> 00:14:07,900

 

290
00:14:07,900 --> 00:14:10,060

 

291
00:14:10,060 --> 00:14:12,580

 

292
00:14:12,580 --> 00:14:19,270

 

293
00:14:19,270 --> 00:14:21,700

 

294
00:14:21,700 --> 00:14:23,260

 

295
00:14:23,260 --> 00:14:26,800

 

296
00:14:26,800 --> 00:14:27,460

 

297
00:14:27,460 --> 00:14:31,210

 

298
00:14:31,210 --> 00:14:32,950

 

299
00:14:32,950 --> 00:14:35,110

 

300
00:14:35,110 --> 00:14:37,000

 

301
00:14:37,000 --> 00:14:39,660

 

302
00:14:39,660 --> 00:14:43,600

 

303
00:14:43,600 --> 00:14:45,850

 

304
00:14:45,850 --> 00:14:48,610

 

305
00:14:48,610 --> 00:14:53,470

 

306
00:14:53,470 --> 00:14:55,720

 

307
00:14:55,720 --> 00:15:00,010

 

308
00:15:00,010 --> 00:15:01,510

 

309
00:15:01,510 --> 00:15:03,640

 

310
00:15:03,640 --> 00:15:06,880

 

311
00:15:06,880 --> 00:15:08,980

 

312
00:15:08,980 --> 00:15:11,950

 

313
00:15:11,950 --> 00:15:13,180

 

314
00:15:13,180 --> 00:15:15,670

 

315
00:15:15,670 --> 00:15:18,879

 

316
00:15:18,879 --> 00:15:24,129

 

317
00:15:24,129 --> 00:15:31,059

 

318
00:15:31,059 --> 00:15:33,789

 

319
00:15:33,789 --> 00:15:36,220

 

320
00:15:36,220 --> 00:15:37,689

 

321
00:15:37,689 --> 00:15:40,599

 

322
00:15:40,599 --> 00:15:42,280

 

323
00:15:42,280 --> 00:15:43,629

 

324
00:15:43,629 --> 00:15:45,879

 

325
00:15:45,879 --> 00:15:47,829

 

326
00:15:47,829 --> 00:15:50,470

 

327
00:15:50,470 --> 00:15:51,849

 

328
00:15:51,849 --> 00:15:54,429

 

329
00:15:54,429 --> 00:15:55,960

 

330
00:15:55,960 --> 00:15:58,449

 

331
00:15:58,449 --> 00:16:00,609

 

332
00:16:00,609 --> 00:16:01,929

 

333
00:16:01,929 --> 00:16:03,729

 

334
00:16:03,729 --> 00:16:04,720

 

335
00:16:04,720 --> 00:16:06,729

 

336
00:16:06,729 --> 00:16:09,729

 

337
00:16:09,729 --> 00:16:12,460

 

338
00:16:12,460 --> 00:16:14,549

 

339
00:16:14,549 --> 00:16:17,650

 

340
00:16:17,650 --> 00:16:19,720

 

341
00:16:19,720 --> 00:16:21,909

 

342
00:16:21,909 --> 00:16:23,739

 

343
00:16:23,739 --> 00:16:25,569

 

344
00:16:25,569 --> 00:16:28,150

 

345
00:16:28,150 --> 00:16:30,999

 

346
00:16:30,999 --> 00:16:33,030

 

347
00:16:33,030 --> 00:16:35,840

 

348
00:16:35,840 --> 00:16:39,600

 

349
00:16:39,600 --> 00:16:42,180

 

350
00:16:42,180 --> 00:16:44,190

 

351
00:16:44,190 --> 00:16:47,040

 

352
00:16:47,040 --> 00:16:51,660

 

353
00:16:51,660 --> 00:16:56,520

 

354
00:16:56,520 --> 00:16:59,280

 

355
00:16:59,280 --> 00:17:04,980

 

356
00:17:04,980 --> 00:17:07,820

 

357
00:17:07,820 --> 00:17:10,050

 

358
00:17:10,050 --> 00:17:11,490

 

359
00:17:11,490 --> 00:17:13,170

 

360
00:17:13,170 --> 00:17:16,110

 

361
00:17:16,110 --> 00:17:17,850

 

362
00:17:17,850 --> 00:17:19,980

 

363
00:17:19,980 --> 00:17:23,190

 

364
00:17:23,190 --> 00:17:24,840

 

365
00:17:24,840 --> 00:17:28,020

 

366
00:17:28,020 --> 00:17:32,250

 

367
00:17:32,250 --> 00:17:33,510

 

368
00:17:33,510 --> 00:17:34,800

 

369
00:17:34,800 --> 00:17:37,580

 

370
00:17:37,580 --> 00:17:40,020

 

371
00:17:40,020 --> 00:17:43,500

 

372
00:17:43,500 --> 00:17:48,210

 

373
00:17:48,210 --> 00:17:51,000

 

374
00:17:51,000 --> 00:17:54,540

 

375
00:17:54,540 --> 00:17:59,370

 

376
00:17:59,370 --> 00:18:02,130

 

377
00:18:02,130 --> 00:18:05,790

 

378
00:18:05,790 --> 00:18:08,430

 

379
00:18:08,430 --> 00:18:10,470

 

380
00:18:10,470 --> 00:18:15,960

 

381
00:18:15,960 --> 00:18:17,670

 

382
00:18:17,670 --> 00:18:19,380

 

383
00:18:19,380 --> 00:18:20,790

 

384
00:18:20,790 --> 00:18:22,230

 

385
00:18:22,230 --> 00:18:24,660

 

386
00:18:24,660 --> 00:18:27,150

 

387
00:18:27,150 --> 00:18:30,390

 

388
00:18:30,390 --> 00:18:34,140

 

389
00:18:34,140 --> 00:18:39,900

 

390
00:18:39,900 --> 00:18:44,280

 

391
00:18:44,280 --> 00:18:51,870

 

392
00:18:51,870 --> 00:18:55,380

 

393
00:18:55,380 --> 00:18:57,180

 

394
00:18:57,180 --> 00:18:58,680

 

395
00:18:58,680 --> 00:19:00,090

 

396
00:19:00,090 --> 00:19:01,590

 

397
00:19:01,590 --> 00:19:03,750

 

398
00:19:03,750 --> 00:19:05,370

 

399
00:19:05,370 --> 00:19:06,630

 

400
00:19:06,630 --> 00:19:10,050

 

401
00:19:10,050 --> 00:19:11,790

 

402
00:19:11,790 --> 00:19:13,800

 

403
00:19:13,800 --> 00:19:19,170

 

404
00:19:19,170 --> 00:19:21,210

 

405
00:19:21,210 --> 00:19:23,400

 

406
00:19:23,400 --> 00:19:26,640

 

407
00:19:26,640 --> 00:19:28,110

 

408
00:19:28,110 --> 00:19:30,030

 

409
00:19:30,030 --> 00:19:33,840

 

410
00:19:33,840 --> 00:19:35,430

 

411
00:19:35,430 --> 00:19:39,890

 

412
00:19:39,890 --> 00:19:43,050

 

413
00:19:43,050 --> 00:19:45,270

 

414
00:19:45,270 --> 00:19:53,190

 

415
00:19:53,190 --> 00:19:54,990

 

416
00:19:54,990 --> 00:19:58,530

 

417
00:19:58,530 --> 00:20:01,080

 

418
00:20:01,080 --> 00:20:03,300

 

419
00:20:03,300 --> 00:20:06,360

 

420
00:20:06,360 --> 00:20:08,160

 

421
00:20:08,160 --> 00:20:11,700

 

422
00:20:11,700 --> 00:20:13,470

 

423
00:20:13,470 --> 00:20:16,890

 

424
00:20:16,890 --> 00:20:20,100

 

425
00:20:20,100 --> 00:20:22,440

 

426
00:20:22,440 --> 00:20:25,080

 

427
00:20:25,080 --> 00:20:27,630

 

428
00:20:27,630 --> 00:20:29,820

 

429
00:20:29,820 --> 00:20:32,250

 

430
00:20:32,250 --> 00:20:35,160

 

431
00:20:35,160 --> 00:20:38,880

 

432
00:20:38,880 --> 00:20:42,300

 

433
00:20:42,300 --> 00:20:44,210

 

434
00:20:44,210 --> 00:20:47,340

 

435
00:20:47,340 --> 00:20:49,020

 

436
00:20:49,020 --> 00:20:53,139

 

437
00:20:53,139 --> 00:20:56,469

 

438
00:20:56,469 --> 00:20:58,779

 

439
00:20:58,779 --> 00:21:00,399

 

440
00:21:00,399 --> 00:21:04,029

 

441
00:21:04,029 --> 00:21:06,839

 

442
00:21:06,839 --> 00:21:10,839

 

443
00:21:10,839 --> 00:21:13,499

 

444
00:21:13,499 --> 00:21:16,959

 

445
00:21:16,959 --> 00:21:18,549

 

446
00:21:18,549 --> 00:21:21,759

 

447
00:21:21,759 --> 00:21:23,200

 

448
00:21:23,200 --> 00:21:24,879

 

449
00:21:24,879 --> 00:21:27,430

 

450
00:21:27,430 --> 00:21:29,769

 

451
00:21:29,769 --> 00:21:31,289

 

452
00:21:31,289 --> 00:21:35,109

 

453
00:21:35,109 --> 00:21:36,430

 

454
00:21:36,430 --> 00:21:38,950

 

455
00:21:38,950 --> 00:21:41,409

 

456
00:21:41,409 --> 00:21:43,259

 

457
00:21:43,259 --> 00:21:48,549

 

458
00:21:48,549 --> 00:21:50,560

 

459
00:21:50,560 --> 00:21:52,180

 

460
00:21:52,180 --> 00:21:55,989

 

461
00:21:55,989 --> 00:21:57,310

 

462
00:21:57,310 --> 00:22:00,609

 

463
00:22:00,609 --> 00:22:04,930

 

464
00:22:04,930 --> 00:22:07,149

 

465
00:22:07,149 --> 00:22:08,379

 

466
00:22:08,379 --> 00:22:10,450

 

467
00:22:10,450 --> 00:22:11,739

 

468
00:22:11,739 --> 00:22:15,369

 

469
00:22:15,369 --> 00:22:16,690

 

470
00:22:16,690 --> 00:22:18,339

 

471
00:22:18,339 --> 00:22:20,379

 

472
00:22:20,379 --> 00:22:21,789

 

473
00:22:21,789 --> 00:22:26,499

 

474
00:22:26,499 --> 00:22:30,129

 

475
00:22:30,129 --> 00:22:31,869

 

476
00:22:31,869 --> 00:22:34,719

 

477
00:22:34,719 --> 00:22:37,690

 

478
00:22:37,690 --> 00:22:39,849

 

479
00:22:39,849 --> 00:22:42,519

 

480
00:22:42,519 --> 00:22:44,709

 

481
00:22:44,709 --> 00:22:46,719

 

482
00:22:46,719 --> 00:22:48,249

 

483
00:22:48,249 --> 00:22:54,989

 

484
00:22:54,989 --> 00:22:58,060

 

485
00:22:58,060 --> 00:23:00,249

 

486
00:23:00,249 --> 00:23:03,899

 

487
00:23:03,899 --> 00:23:06,820

 

488
00:23:06,820 --> 00:23:08,980

 

489
00:23:08,980 --> 00:23:10,509

 

490
00:23:10,509 --> 00:23:16,200

 

491
00:23:16,200 --> 00:23:21,159

 

492
00:23:21,159 --> 00:23:23,680

 

493
00:23:23,680 --> 00:23:27,070

 

494
00:23:27,070 --> 00:23:28,779

 

495
00:23:28,779 --> 00:23:36,940

 

496
00:23:36,940 --> 00:23:44,620

 

497
00:23:44,620 --> 00:23:46,779

 

498
00:23:46,779 --> 00:23:52,720

 

499
00:23:52,720 --> 00:23:54,700

 

500
00:23:54,700 --> 00:23:56,769

 

501
00:23:56,769 --> 00:24:00,690

 

502
00:24:00,690 --> 00:24:04,870

 

503
00:24:04,870 --> 00:24:09,039

 

504
00:24:09,039 --> 00:24:11,590

 

505
00:24:11,590 --> 00:24:13,330

 

506
00:24:13,330 --> 00:24:16,389

 

507
00:24:16,389 --> 00:24:19,750

 

508
00:24:19,750 --> 00:24:27,430

 

509
00:24:27,430 --> 00:24:34,950

 

510
00:24:34,950 --> 00:24:40,269

 

511
00:24:40,269 --> 00:24:41,680

 

512
00:24:41,680 --> 00:24:44,289

 

513
00:24:44,289 --> 00:24:46,149

 

514
00:24:46,149 --> 00:24:48,100

 

515
00:24:48,100 --> 00:24:50,080

 

516
00:24:50,080 --> 00:24:52,930

 

517
00:24:52,930 --> 00:24:54,820

 

518
00:24:54,820 --> 00:24:56,950

 

519
00:24:56,950 --> 00:24:59,860

 

520
00:24:59,860 --> 00:25:02,649

 

521
00:25:02,649 --> 00:25:05,529

 

522
00:25:05,529 --> 00:25:10,000

 

523
00:25:10,000 --> 00:25:12,070

 

524
00:25:12,070 --> 00:25:14,830

 

525
00:25:14,830 --> 00:25:17,409

 

526
00:25:17,409 --> 00:25:19,950

 

527
00:25:19,950 --> 00:25:24,150

 

528
00:25:24,150 --> 00:25:26,490

 

529
00:25:26,490 --> 00:25:30,120

 

530
00:25:30,120 --> 00:25:31,410

 

531
00:25:31,410 --> 00:25:35,790

 

532
00:25:35,790 --> 00:25:37,500

 

533
00:25:37,500 --> 00:25:39,630

 

534
00:25:39,630 --> 00:25:46,350

 

535
00:25:46,350 --> 00:25:50,090

 

536
00:25:50,090 --> 00:25:53,520

 

537
00:25:53,520 --> 00:25:59,670

 

538
00:25:59,670 --> 00:26:03,930

 

539
00:26:03,930 --> 00:26:09,690

 

540
00:26:09,690 --> 00:26:12,720

 

541
00:26:12,720 --> 00:26:16,260

 

542
00:26:16,260 --> 00:26:19,050

 

543
00:26:19,050 --> 00:26:23,040

 

544
00:26:23,040 --> 00:26:24,780

 

545
00:26:24,780 --> 00:26:29,160

 

546
00:26:29,160 --> 00:26:33,660

 

547
00:26:33,660 --> 00:26:43,130

 

548
00:26:43,130 --> 00:26:45,900

 

549
00:26:45,900 --> 00:26:48,720

 

550
00:26:48,720 --> 00:26:50,070

 

551
00:26:50,070 --> 00:26:52,080

 

552
00:26:52,080 --> 00:26:54,480

 

553
00:26:54,480 --> 00:26:56,430

 

554
00:26:56,430 --> 00:26:59,280

 

555
00:26:59,280 --> 00:27:01,140

 

556
00:27:01,140 --> 00:27:07,460

 

557
00:27:07,460 --> 00:27:15,270

 

558
00:27:15,270 --> 00:27:20,520

 

559
00:27:20,520 --> 00:27:24,750

 

560
00:27:24,750 --> 00:27:30,450

 

561
00:27:30,450 --> 00:27:32,610

 

562
00:27:32,610 --> 00:27:32,620

 

563
00:27:32,620 --> 00:27:34,170


564
00:27:34,170 --> 00:27:36,880

 

565
00:27:36,880 --> 00:27:38,050

 

566
00:27:38,050 --> 00:27:39,370

 

567
00:27:39,370 --> 00:27:40,720

 

568
00:27:40,720 --> 00:27:42,940

 

569
00:27:42,940 --> 00:27:44,860

 

570
00:27:44,860 --> 00:27:56,200

 

571
00:27:56,200 --> 00:28:01,330

 

572
00:28:01,330 --> 00:28:03,190

 

573
00:28:03,190 --> 00:28:05,610

 

574
00:28:05,610 --> 00:28:08,710

 

575
00:28:08,710 --> 00:28:15,520

 

576
00:28:15,520 --> 00:28:19,210

 

577
00:28:19,210 --> 00:28:20,980

 

578
00:28:20,980 --> 00:28:23,590

 

579
00:28:23,590 --> 00:28:25,390

 

580
00:28:25,390 --> 00:28:29,260

 

581
00:28:29,260 --> 00:28:31,510

 

582
00:28:31,510 --> 00:28:34,110

 

583
00:28:34,110 --> 00:28:38,280

 

584
00:28:38,280 --> 00:28:41,860

 

585
00:28:41,860 --> 00:28:44,260

 

586
00:28:44,260 --> 00:28:47,590

 

587
00:28:47,590 --> 00:28:49,900

 

588
00:28:49,900 --> 00:28:51,520

 

589
00:28:51,520 --> 00:28:53,350

 

590
00:28:53,350 --> 00:28:57,970

 

591
00:28:57,970 --> 00:28:59,620

 

592
00:28:59,620 --> 00:29:02,200

 

593
00:29:02,200 --> 00:29:04,750

 

594
00:29:04,750 --> 00:29:08,740

 

595
00:29:08,740 --> 00:29:11,590

 

596
00:29:11,590 --> 00:29:14,350

 

597
00:29:14,350 --> 00:29:16,270

 

598
00:29:16,270 --> 00:29:19,240

 

599
00:29:19,240 --> 00:29:22,360

 

600
00:29:22,360 --> 00:29:23,860

 

601
00:29:23,860 --> 00:29:25,570

 

602
00:29:25,570 --> 00:29:27,070

 

603
00:29:27,070 --> 00:29:28,990

 

604
00:29:28,990 --> 00:29:33,280

 

605
00:29:33,280 --> 00:29:36,130

 

606
00:29:36,130 --> 00:29:37,390

 

607
00:29:37,390 --> 00:29:41,350

 

608
00:29:41,350 --> 00:29:46,030

 

609
00:29:46,030 --> 00:29:47,500

 

610
00:29:47,500 --> 00:29:51,100

 

611
00:29:51,100 --> 00:29:54,370

 

612
00:29:54,370 --> 00:30:01,540

 

613
00:30:01,540 --> 00:30:03,160

 

614
00:30:03,160 --> 00:30:05,740

 

615
00:30:05,740 --> 00:30:07,180

 

616
00:30:07,180 --> 00:30:11,650

 

617
00:30:11,650 --> 00:30:13,210

 

618
00:30:13,210 --> 00:30:17,320

 

619
00:30:17,320 --> 00:30:19,090

 

620
00:30:19,090 --> 00:30:22,120

 

621
00:30:22,120 --> 00:30:25,150

 

622
00:30:25,150 --> 00:30:28,870

 

623
00:30:28,870 --> 00:30:30,790

 

624
00:30:30,790 --> 00:30:32,200

 

625
00:30:32,200 --> 00:30:33,580

 

626
00:30:33,580 --> 00:30:36,880

 

627
00:30:36,880 --> 00:30:38,380

 

628
00:30:38,380 --> 00:30:40,510

 

629
00:30:40,510 --> 00:30:42,250

 

630
00:30:42,250 --> 00:30:43,660

 

631
00:30:43,660 --> 00:30:45,790

 

632
00:30:45,790 --> 00:30:47,620

 

633
00:30:47,620 --> 00:30:50,590

 

634
00:30:50,590 --> 00:30:52,300

 

635
00:30:52,300 --> 00:30:53,980

 

636
00:30:53,980 --> 00:30:55,840

 

637
00:30:55,840 --> 00:30:58,210

 

638
00:30:58,210 --> 00:31:01,750

 

639
00:31:01,750 --> 00:31:04,300

 

640
00:31:04,300 --> 00:31:07,810

 

641
00:31:07,810 --> 00:31:09,940

 

642
00:31:09,940 --> 00:31:13,050

 

643
00:31:13,050 --> 00:31:15,790

 

644
00:31:15,790 --> 00:31:18,850

 

645
00:31:18,850 --> 00:31:21,100

 

646
00:31:21,100 --> 00:31:23,920

 

647
00:31:23,920 --> 00:31:25,660

 

648
00:31:25,660 --> 00:31:29,260

 

649
00:31:29,260 --> 00:31:31,720

 

650
00:31:31,720 --> 00:31:34,240

 

651
00:31:34,240 --> 00:31:36,010

 

652
00:31:36,010 --> 00:31:37,210

 

653
00:31:37,210 --> 00:31:38,980

 

654
00:31:38,980 --> 00:31:40,300

 

655
00:31:40,300 --> 00:31:42,100

 

656
00:31:42,100 --> 00:31:43,840

 

657
00:31:43,840 --> 00:31:46,360

 

658
00:31:46,360 --> 00:31:48,820

 

659
00:31:48,820 --> 00:31:51,730

 

660
00:31:51,730 --> 00:31:55,120

 

661
00:31:55,120 --> 00:31:55,130

 

662
00:31:55,130 --> 00:31:57,210


663
00:31:57,210 --> 00:32:00,810

 

664
00:32:00,810 --> 00:32:04,789

 

665
00:32:04,789 --> 00:32:11,789

 

666
00:32:11,789 --> 00:32:14,640

 

667
00:32:14,640 --> 00:32:16,500

 

668
00:32:16,500 --> 00:32:18,649

 

669
00:32:18,649 --> 00:32:21,990

 

670
00:32:21,990 --> 00:32:25,649

 

671
00:32:25,649 --> 00:32:27,960

 

672
00:32:27,960 --> 00:32:29,669

 

673
00:32:29,669 --> 00:32:31,919

 

674
00:32:31,919 --> 00:32:35,909

 

675
00:32:35,909 --> 00:32:37,320

 

676
00:32:37,320 --> 00:32:39,720

 

677
00:32:39,720 --> 00:32:42,120

 

678
00:32:42,120 --> 00:32:44,970

 

679
00:32:44,970 --> 00:32:48,000

 

680
00:32:48,000 --> 00:32:49,409

 

681
00:32:49,409 --> 00:32:54,169

 

682
00:32:54,169 --> 00:32:57,480

 

683
00:32:57,480 --> 00:33:00,390

 

684
00:33:00,390 --> 00:33:04,770

 

685
00:33:04,770 --> 00:33:07,549

 

686
00:33:07,549 --> 00:33:13,020

 

687
00:33:13,020 --> 00:33:16,440

 

688
00:33:16,440 --> 00:33:21,419

 

689
00:33:21,419 --> 00:33:25,500

 

690
00:33:25,500 --> 00:33:28,200

 

691
00:33:28,200 --> 00:33:32,640

 

692
00:33:32,640 --> 00:33:35,430

 

693
00:33:35,430 --> 00:33:36,720

 

694
00:33:36,720 --> 00:33:40,140

 

695
00:33:40,140 --> 00:33:42,299

 

696
00:33:42,299 --> 00:33:43,799

 

697
00:33:43,799 --> 00:33:45,810

 

698
00:33:45,810 --> 00:33:47,549

 

699
00:33:47,549 --> 00:33:49,080

 

700
00:33:49,080 --> 00:33:51,090

 

701
00:33:51,090 --> 00:33:53,460

 

702
00:33:53,460 --> 00:33:54,810

 

703
00:33:54,810 --> 00:34:00,029

 

704
00:34:00,029 --> 00:34:02,640

 

705
00:34:02,640 --> 00:34:04,590

 

706
00:34:04,590 --> 00:34:08,899

 

707
00:34:08,899 --> 00:34:08,909

 

708
00:34:08,909 --> 00:34:10,139


709
00:34:10,139 --> 00:34:12,480

 

710
00:34:12,480 --> 00:34:14,210

 

711
00:34:14,210 --> 00:34:23,089

 

712
00:34:23,089 --> 00:34:28,740

 

713
00:34:28,740 --> 00:34:33,240

 

714
00:34:33,240 --> 00:34:34,529

 

715
00:34:34,529 --> 00:34:37,289

 

716
00:34:37,289 --> 00:34:39,750

 

717
00:34:39,750 --> 00:34:41,129

 

718
00:34:41,129 --> 00:34:42,779

 

719
00:34:42,779 --> 00:34:44,309

 

720
00:34:44,309 --> 00:34:48,480

 

721
00:34:48,480 --> 00:34:49,919

 

722
00:34:49,919 --> 00:34:52,619

 

723
00:34:52,619 --> 00:34:55,349

 

724
00:34:55,349 --> 00:34:57,059

 

725
00:34:57,059 --> 00:34:58,799

 

726
00:34:58,799 --> 00:35:00,450

 

727
00:35:00,450 --> 00:35:02,519

 

728
00:35:02,519 --> 00:35:05,339

 

729
00:35:05,339 --> 00:35:07,260

 

730
00:35:07,260 --> 00:35:09,120

 

731
00:35:09,120 --> 00:35:10,799

 

732
00:35:10,799 --> 00:35:13,589

 

733
00:35:13,589 --> 00:35:16,140

 

734
00:35:16,140 --> 00:35:18,240

 

735
00:35:18,240 --> 00:35:20,130

 

736
00:35:20,130 --> 00:35:21,720

 

737
00:35:21,720 --> 00:35:24,089

 

738
00:35:24,089 --> 00:35:26,490

 

739
00:35:26,490 --> 00:35:28,589

 

740
00:35:28,589 --> 00:35:33,480

 

741
00:35:33,480 --> 00:35:35,519

 

742
00:35:35,519 --> 00:35:38,010

 

743
00:35:38,010 --> 00:35:41,190

 

744
00:35:41,190 --> 00:35:45,990

 

745
00:35:45,990 --> 00:35:47,549

 

746
00:35:47,549 --> 00:35:52,019

 

747
00:35:52,019 --> 00:35:53,760

 

748
00:35:53,760 --> 00:36:00,210

 

749
00:36:00,210 --> 00:36:02,460

 

750
00:36:02,460 --> 00:36:05,339

 

751
00:36:05,339 --> 00:36:10,529

 

752
00:36:10,529 --> 00:36:15,269

 

753
00:36:15,269 --> 00:36:16,470

 

754
00:36:16,470 --> 00:36:17,849

 

755
00:36:17,849 --> 00:36:19,529

 

756
00:36:19,529 --> 00:36:19,539

 

757
00:36:19,539 --> 00:36:24,470


758
00:36:24,470 --> 00:36:27,539

 

759
00:36:27,539 --> 00:36:31,140

 

760
00:36:31,140 --> 00:36:33,420

 

761
00:36:33,420 --> 00:36:36,240

 

762
00:36:36,240 --> 00:36:37,950

 

763
00:36:37,950 --> 00:36:40,980

 

764
00:36:40,980 --> 00:36:42,599

 

765
00:36:42,599 --> 00:36:44,490

 

766
00:36:44,490 --> 00:36:46,230

 

767
00:36:46,230 --> 00:36:48,960

 

768
00:36:48,960 --> 00:36:52,019

 

769
00:36:52,019 --> 00:36:53,940

 

770
00:36:53,940 --> 00:36:55,829

 

771
00:36:55,829 --> 00:36:57,980

 

772
00:36:57,980 --> 00:37:00,809

 

773
00:37:00,809 --> 00:37:03,480

 

774
00:37:03,480 --> 00:37:06,359

 

775
00:37:06,359 --> 00:37:08,250

 

776
00:37:08,250 --> 00:37:10,200

 

777
00:37:10,200 --> 00:37:11,880

 

778
00:37:11,880 --> 00:37:14,309

 

779
00:37:14,309 --> 00:37:20,670

 

780
00:37:20,670 --> 00:37:25,740

 

781
00:37:25,740 --> 00:37:25,750

 

782
00:37:25,750 --> 00:37:36,570


783
00:37:36,570 --> 00:37:39,180

 

784
00:37:39,180 --> 00:37:41,790

 

785
00:37:41,790 --> 00:37:43,440

 

786
00:37:43,440 --> 00:37:46,500

 

787
00:37:46,500 --> 00:37:48,810

 

788
00:37:48,810 --> 00:37:51,570

 

789
00:37:51,570 --> 00:37:55,590

 

790
00:37:55,590 --> 00:37:58,410

 

791
00:37:58,410 --> 00:38:01,290

 

792
00:38:01,290 --> 00:38:03,660

 

793
00:38:03,660 --> 00:38:06,090

 

794
00:38:06,090 --> 00:38:09,720

 

795
00:38:09,720 --> 00:38:11,100

 

796
00:38:11,100 --> 00:38:13,340

 

797
00:38:13,340 --> 00:38:17,460

 

798
00:38:17,460 --> 00:38:19,410

 

799
00:38:19,410 --> 00:38:23,300

 

800
00:38:23,300 --> 00:38:26,340

 

801
00:38:26,340 --> 00:38:29,880

 

802
00:38:29,880 --> 00:38:52,800

 

803
00:38:52,800 --> 00:38:54,960

 

804
00:38:54,960 --> 00:38:57,390

 

805
00:38:57,390 --> 00:39:02,800

 

806
00:39:02,800 --> 00:39:06,340

 

807
00:39:06,340 --> 00:39:09,970

 

808
00:39:09,970 --> 00:39:12,940

 

809
00:39:12,940 --> 00:39:14,320

 

810
00:39:14,320 --> 00:39:15,550

 

811
00:39:15,550 --> 00:39:17,980

 

812
00:39:17,980 --> 00:39:20,980

 

813
00:39:20,980 --> 00:39:22,330

 

814
00:39:22,330 --> 00:39:26,020

 

815
00:39:26,020 --> 00:39:27,520

 

816
00:39:27,520 --> 00:39:30,310

 

817
00:39:30,310 --> 00:39:32,350

 

818
00:39:32,350 --> 00:39:36,310

 

819
00:39:36,310 --> 00:39:37,540

 

820
00:39:37,540 --> 00:39:40,750

 

821
00:39:40,750 --> 00:39:51,100

 

822
00:39:51,100 --> 00:39:53,080

 

823
00:39:53,080 --> 00:39:57,340

 

824
00:39:57,340 --> 00:39:59,770

 

825
00:39:59,770 --> 00:40:49,810

 

826
00:40:49,810 --> 00:40:52,370

 

827
00:40:52,370 --> 00:40:54,410

 

828
00:40:54,410 --> 00:40:56,510

 

829
00:40:56,510 --> 00:40:58,460

 

830
00:40:58,460 --> 00:41:04,520

 

831
00:41:04,520 --> 00:41:05,960

 

832
00:41:05,960 --> 00:41:08,030

 

833
00:41:08,030 --> 00:41:10,310

 

834
00:41:10,310 --> 00:41:12,650

 

835
00:41:12,650 --> 00:41:19,760

 

836
00:41:19,760 --> 00:41:21,770

 

837
00:41:21,770 --> 00:41:24,710

 

838
00:41:24,710 --> 00:41:26,510

 

839
00:41:26,510 --> 00:41:28,640

 

840
00:41:28,640 --> 00:41:30,110

 

841
00:41:30,110 --> 00:41:32,360

 

842
00:41:32,360 --> 00:41:34,850

 

843
00:41:34,850 --> 00:41:37,640

 

844
00:41:37,640 --> 00:41:41,330

 

845
00:41:41,330 --> 00:41:44,750

 

846
00:41:44,750 --> 00:41:46,310

 

847
00:41:46,310 --> 00:41:51,080

 

848
00:41:51,080 --> 00:41:53,750

 

849
00:41:53,750 --> 00:41:56,000

 

850
00:41:56,000 --> 00:42:02,230

 

851
00:42:02,230 --> 00:42:04,120

 

852
00:42:04,120 --> 00:42:06,609

 

853
00:42:06,609 --> 00:42:08,440

 

854
00:42:08,440 --> 00:42:15,820

 

855
00:42:15,820 --> 00:42:18,280

 

856
00:42:18,280 --> 00:42:20,620

 

857
00:42:20,620 --> 00:42:23,050

 

858
00:42:23,050 --> 00:42:27,400

 

859
00:42:27,400 --> 00:42:29,020

 

860
00:42:29,020 --> 00:42:31,450

 

861
00:42:31,450 --> 00:42:34,240

 

862
00:42:34,240 --> 00:42:39,340

 

863
00:42:39,340 --> 00:42:42,640

 

864
00:42:42,640 --> 00:42:45,460

 

865
00:42:45,460 --> 00:42:49,720

 

866
00:42:49,720 --> 00:42:51,520

 

867
00:42:51,520 --> 00:42:55,870

 

868
00:42:55,870 --> 00:43:01,359

 

869
00:43:01,359 --> 00:43:03,220

 

870
00:43:03,220 --> 00:43:08,349

 

871
00:43:08,349 --> 00:43:14,080

 

872
00:43:14,080 --> 00:43:20,440

 

873
00:43:20,440 --> 00:43:21,820

 

874
00:43:21,820 --> 00:43:24,940

 

875
00:43:24,940 --> 00:43:27,730

 

876
00:43:27,730 --> 00:43:31,300

 

877
00:43:31,300 --> 00:43:34,690

 

878
00:43:34,690 --> 00:43:37,270

 

879
00:43:37,270 --> 00:43:38,560

 

880
00:43:38,560 --> 00:43:41,380

 

881
00:43:41,380 --> 00:43:43,930

 

882
00:43:43,930 --> 00:43:47,349

 

883
00:43:47,349 --> 00:43:49,510

 

884
00:43:49,510 --> 00:43:52,540

 

885
00:43:52,540 --> 00:43:54,550

 

886
00:43:54,550 --> 00:43:58,150

 

887
00:43:58,150 --> 00:43:59,560

 

888
00:43:59,560 --> 00:44:01,420

 

889
00:44:01,420 --> 00:44:03,580

 

890
00:44:03,580 --> 00:44:08,380

 

891
00:44:08,380 --> 00:44:11,859

 

892
00:44:11,859 --> 00:44:13,840

 

893
00:44:13,840 --> 00:44:16,360

 

894
00:44:16,360 --> 00:44:23,320

 

895
00:44:23,320 --> 00:44:25,190

 

896
00:44:25,190 --> 00:44:34,870

 

897
00:44:34,870 --> 00:44:39,800

 

898
00:44:39,800 --> 00:44:42,320

 

899
00:44:42,320 --> 00:44:44,360

 

900
00:44:44,360 --> 00:44:48,140

 

901
00:44:48,140 --> 00:44:55,250

 

902
00:44:55,250 --> 00:44:58,150

 

903
00:44:58,150 --> 00:45:01,100

 

904
00:45:01,100 --> 00:45:02,300

 

905
00:45:02,300 --> 00:45:04,670

 

906
00:45:04,670 --> 00:45:12,140

 

907
00:45:12,140 --> 00:45:13,610

 

908
00:45:13,610 --> 00:45:17,270

 

909
00:45:17,270 --> 00:45:21,260

 

910
00:45:21,260 --> 00:45:23,030

 

911
00:45:23,030 --> 00:45:24,530

 

912
00:45:24,530 --> 00:45:26,180

 

913
00:45:26,180 --> 00:45:27,290

 

914
00:45:27,290 --> 00:45:33,140

 

915
00:45:33,140 --> 00:45:36,050

 

916
00:45:36,050 --> 00:45:38,840

 

917
00:45:38,840 --> 00:45:42,860

 

918
00:45:42,860 --> 00:45:45,740

 

919
00:45:45,740 --> 00:45:50,360

 

920
00:45:50,360 --> 00:45:54,580

 

921
00:45:54,580 --> 00:45:58,010

 

922
00:45:58,010 --> 00:46:01,190