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all right lecture 12 is about 10 SEC  radice like this one which you saw and  when the rigid infinitesimal rigidity  and Carpenters rule theorem all in one  quick lecture so just a couple questions  about ten surgeries well first is about  infinitesimal rigidity in general this  is a sort of extra bonus I talked about  one reason why this dot product  condition is the right thing which is  based on projection but there are other  ways to think about it so say we have an  edge VW actually I think I'll draw it W  on the left so here's VW we would think  of this point as being sea of W that's  your configuration this point is C of E  then we were talking about when if you  have a velocity vector D of W and a  velocity vector say something like this  to give V when it preserves the length  of this bar to the first order and the  condition we had was C of E minus C of W  dot product with D of V minus D of W  equals 0 so a question well so the way  we said in class this works is if you  look at the projected length of this  vector onto the segment that's  essentially how much shorter in this  case it gets shorter in this case this  is how much longer the segment gets to  the first order and that corresponds to  DV dot product with this vector because  this vector C V minus C of W is this  direction here cv minus C of W so if you  take the dot product with this vector  you get that projected length you want  those two dot products to be  equal and so you want the difference to  be zero that's one way to think about it  but in fact this quantity also has an  intuitive notion if you've done some  basic mechanics this is the relative  motion of V with respect to W so it's  like well imagine w is not moving to do  that you just subtract D of W from all  motions then W will remain stationary to  the first order solve first order  motions so then what's the motion of V  well it's going to be D of V minus D of  W because everything gets subtracted by  D of W so that is minus D of W  corresponds to drawing that vector in  the other direction and then adding that  to D of W corresponds to this kind of  pictures this is minus D of W and so  this vector is D of V minus D of W  that's the sum of those two vectors and  the claim is what we want is for these  two vectors to be perpendicular to each  other the edge and the motion and this  makes a lot of sense because we're  imagining w is not moving so it's just V  moving and we want the motion of V to be  perpendicular locally to the segment  because that's going to preserve the  length ok so that that's actually I  think an even more intuitive way to see  it and since this video this lecture  video has been online for two years  every year so I get an email saying hey  you asked what does DV minus VW mean and  yes I didn't realize it at the time but  it is just relative motion and so yeah  it's like like if you're moving along  this circle centered at w that's where  you want to be either straight up or  straight down clear some bonus intuition  it's a little harder to see the other  conditions from this perspective at  least I find it harder to see but I  guess  I shouldn't claim that are also going to  get more emails in the future but let's  see if you want it to be was the dot  product being greater or equal to zero  means it probably means that this angle  is obtuse which means that the lengths  getting longer and less equal to zero  means that the angle is getting non  obtuse and if you know the dot product  rule with the cosine of the angle you  should be able to figure that out but  it's a little bit less it's less  memorized in my head whereas dot product  equals zero everyone remembers it's  being perpendicular so this was struts  and cables ok next question was a couple  questions here one is how can you say  the tenseiga tu showed is rigid because  you can flex it this doesn't look very  rigid this guy here it looks quite  flexible and then there was another  question by someone else saying why did  you use springs to represent bars and  these are sort of the same question this  is flexible because the springs are  flexible the idea with the springs I  mean I'm just guessing why this model  was made this way springs are nice  because they do have a natural resting  length so in that sense they want to be  a given length and if you pull on them  they try to restore the original length  also if you push on them you can do it  but they try to restore their to their  resting length and right now these guys  are all in the resting length and what's  cool about this is you can kind of feel  the resistance I mean this is what  happens in any material you pull on it  it can maybe pull a little bit but it  gives a lot of force going back to its  original state these I don't know these  cables probably you can stretch that  material at very tiny amount i can feel  the force it's a little less visible  here you can actually see the force and  because it's so much smaller you can  really it's easier to feel what's going  on in here I think that's the intuition  you could of course construct these with  steel bars it's just then it wouldn't  moove be a little less exciting it's up  to you of course but this is just one  way to build models of ten securities  but it's also why it's so flexible  as these bars are not super strong but  it was kind of fun as you feel the  equilibrium stress here that things go  back to where they were originally they  kind of hold in position there because  all the stress is here balance out cool  and one more kind of question is about  sculpture ten security sculptures are  cool so i thought i'd show you a few  more examples the sort of the master  here is Kenneth's Nelson I think he  possibly invented tensei gratis and I  showed one example in lecture but these  are many more he's been doing it since  the 60s some of them are very big  because have some measurements here this  guy is 85 feet long and rests on these  three posts a little hard to see the  cables here but they are it's all struts  and cables so using these bars to be  actually there might be a bars in this  picture I think they might be rigid even  when their struts but and some more this  is one of the longest at 72 feet and  this is the tallest at 90 feet pretty  impressive this is a taller version of  the one that I showed last time or in in  lecture so I'm not sure how he designs  them he's an artist by trade what their  uses computational tools or gadgets or  I'm not sure be interesting to be  interesting to talk to him actually if  you want to see more examples of his  sculpture this is just a piece of it go  to his website on the design side our  good friend told me he wrote ah she who  did the origami organizer bunny it's  also done intense a gritty bunny and he  didn't design this by hand as she might  imagine he designed it with a new piece  of software that hasn't yet been  released called free form 10 seg ready  you start with a polyhedron and there's  a few different starting initial  constructions that  kind of set that thing up as a kind of  tense egra T and then it solves to make  it a balance to have an equilibrium  stress and on the right is the stress  these are what structural engineer would  call force polygons it's essentially the  it's like if you take this graph and you  rotate this is all in 3d by the way so a  little tricky that's rotating in this  case he's pulling on things for trying  to force vertices to come together and  the whole thing is updating and as long  as this number down here is close to  zero here it's 10 to the minus 20 it's a  pretty good approximation of 0 then this  thing is in equilibrium and it's again  solving all those constraints like we  have before so yeah right force polygons  you take every edge you rotate it 90  degrees and then sort of magic  constructing a little polygon around  each face so you scale the edge by the  stress in there and those polygons  should close up to be zero if the if you  at the vertex you satisfy equilibrium  and you can draw all those polygons  those polygons can be joined together to  make one kind of graph and if that graph  closes up then you have an equilibrium  stress so it's kind of a neat way to  visualize that the thing is rigid that  it would rather that it would hold all  of these edges at fixed length and you  have to verify that the bar structure is  rigid but this is presumably constructed  to have that property so hopefully this  offer will be released at some point it  definitely looks like a cool way to play  around with tenseiga T's and design  things question  ah i see yeah that's a good question so  can you see what happens if i like when  that one when i tweak it and it wobbles  yeah it'd be nice to see that in  simulation definitely this is this  software is based on freeform origami  designer and that has the two modes  right one where you can change the  crease pattern one where it tries to  stay it keep it fixed so i'm not sure  whether the software has the same two  modes but it certainly could that's  definitely computable you could  basically ignore the constraints for a  while pull on something then let go let  it restore the constraints without it  changing the tents a gritty like it's  doing here and you should see it wobble  in fact in that case you probably want  to use it a less good numerical method  you don't want it to stabilize as  quickly you'd like to see a jiggle for a  while that's so it should be doable  could be an interesting project to  extend the software if it hasn't been if  that hasn't been implemented yet I could  ask him I asked him last night as it I  mean plan is to release it at some point  but not quite ready for prime time yet  other questions about this so I I think  it should be fun to play with if you  want to build some tents egra tease  there's a couple descriptions on how to  do this out of various easy to find  household objects like straws or with  these well rubber bands things like that  so you can check out George heart has  two instruction web pages about them  they're linked from this slide if you go  to the lecture and yeah that was all I  had for questions any more questions  about this lecture all perfectly clear  we'll be seeing a lot more stuff about  locked linkages the Carpenters rule part  which is at the end of class in the next  two three lectures yeah  well in general a linear program is  something like you have a matrix and you  multiply it by a vector and then you  have let's say in our case we have  greater than equal to 0 so when I say 0  I mean 000 the dual of a linear program  is what you get by transposing the  matrix basically so you get you rotate  it 90 degrees or actually flip around  that diagonal so all the columns become  rose rose become columns so now you've  got some other thing here which is going  to be this big it's probably going to be  less than or equal to something that big  so this is a transpose this is why and  this is your just called see this is in  general called little B and there are  relations between these two things it's  the short version but you can kind of  see so in principle if we take this is  in our case this is basically the  rigidity matrix was this rigidity matrix  Prime in the lecture because if we want  to write greater than or equal to 0  struts are just fine cables we negate  everything in the row bars where we have  equality we need both the original  version and the negated version of that  row but if we just imagine struts for  the moment because I mean some sense  everything well can be simulated by  struts and negative struts struts and  cables then we had the number of rows  here this was essentially the number of  edges in our linkage and the number of  columns here was d times n this is the  number of degrees of freedom these are  the coordinates of all of our things and  then X here was actually our velocity  vectors and so on well actually I should  probably write that so this what we're  calling velocity vectors D this is our  it's like conflict this is different d  these are the D vectors the derivatives  so what would these things be well the  number the columns here are going to be  edges and the rows are going to be  coordinates and remember equilibrium  stress looks like this basically for  every vertex we have sum over all other  vertices let's say VW is an edge of what  is it stress of VW times C of I forget  whether c of e minus c of w or or the  reverse equals zero so this should look  good the number of these constraints is  well here it says it's the number of  vertices but in fact when you say equals  0 this is a vector sum so this has d  coordinates when you say that equals 0  that's d constraints so there are d  times n constraints and how many things  are involved in the constraints well  essentially the edges every Edge has a  term in here so that's kind of why that  looks right you have to go through the  algebra and exactly what's written here  to see that when u transpose it you do  exactly get this constraint and there's  the issue of equals 0 vs greater than  equal to 0 so that's a little bit more  subtle yeah but that's the short version  and you also have to check that the when  your strut or cable you get just a sign  constraint on this thing that's maybe a  little less obvious but at least at a  high level this looks right and their  relations about the primal and the dual  linear program about so for example the  this linear program which is  characterizing all infinitesimally  motions if you find an infinitesimally  twice that  vector or the set of vectors is also an  infinitesimal motion so the solution  space to this linear program is what's  called a convex polyhedral cone meaning  something I mean this is sort of the  beginning of a cone but it goes off to  infinity you can take any motion you  find and scale it off to infinity you  always include the origin 0 0 0 because  you can always do no motion but any  motion can be scaled up so if this thing  has a solution at all other than the 0 0  0 motion it's an unbounded LP meaning  there's you go off to infinity and when  the primal LP is unbounded you know  things about the dual which I forget but  you cannot have one being unbounded and  the other being something else and so  that guarantees that there's an  equilibrium stress that's roughly how it  works other questions

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all right lecture 12 is about 10 SEC  radice like this one which you saw and  when the rigid infinitesimal rigidity  and Carpenters rule theorem all in one  quick lecture so just a couple questions  about ten surgeries well first is about  infinitesimal rigidity in general this  is a sort of extra bonus I talked about  one reason why this dot product  condition is the right thing which is  based on projection but there are other  ways to think about it so say we have an  edge VW actually I think I'll draw it W  on the left so here's VW we would think  of this point as being sea of W that's  your configuration this point is C of E  then we were talking about when if you  have a velocity vector D of W and a  velocity vector say something like this  to give V when it preserves the length  of this bar to the first order and the  condition we had was C of E minus C of W  dot product with D of V minus D of W  equals 0 so a question well so the way  we said in class this works is if you  look at the projected length of this  vector onto the segment that's  essentially how much shorter in this  case it gets shorter in this case this  is how much longer the segment gets to  the first order and that corresponds to  DV dot product with this vector because  this vector C V minus C of W is this  direction here cv minus C of W so if you  take the dot product with this vector  you get that projected length you want  those two dot products to be  equal and so you want the difference to  be zero that's one way to think about it  but in fact this quantity also has an  intuitive notion if you've done some  basic mechanics this is the relative  motion of V with respect to W so it's  like well imagine w is not moving to do  that you just subtract D of W from all  motions then W will remain stationary to  the first order solve first order  motions so then what's the motion of V  well it's going to be D of V minus D of  W because everything gets subtracted by  D of W so that is minus D of W  corresponds to drawing that vector in  the other direction and then adding that  to D of W corresponds to this kind of  pictures this is minus D of W and so  this vector is D of V minus D of W  that's the sum of those two vectors and  the claim is what we want is for these  two vectors to be perpendicular to each  other the edge and the motion and this  makes a lot of sense because we're  imagining w is not moving so it's just V  moving and we want the motion of V to be  perpendicular locally to the segment  because that's going to preserve the  length ok so that that's actually I  think an even more intuitive way to see  it and since this video this lecture  video has been online for two years  every year so I get an email saying hey  you asked what does DV minus VW mean and  yes I didn't realize it at the time but  it is just relative motion and so yeah  it's like like if you're moving along  this circle centered at w that's where  you want to be either straight up or  straight down clear some bonus intuition  it's a little harder to see the other  conditions from this perspective at  least I find it harder to see but I  guess  I shouldn't claim that are also going to  get more emails in the future but let's  see if you want it to be was the dot  product being greater or equal to zero  means it probably means that this angle  is obtuse which means that the lengths  getting longer and less equal to zero  means that the angle is getting non  obtuse and if you know the dot product  rule with the cosine of the angle you  should be able to figure that out but  it's a little bit less it's less  memorized in my head whereas dot product  equals zero everyone remembers it's  being perpendicular so this was struts  and cables ok next question was a couple  questions here one is how can you say  the tenseiga tu showed is rigid because  you can flex it this doesn't look very  rigid this guy here it looks quite  flexible and then there was another  question by someone else saying why did  you use springs to represent bars and  these are sort of the same question this  is flexible because the springs are  flexible the idea with the springs I  mean I'm just guessing why this model  was made this way springs are nice  because they do have a natural resting  length so in that sense they want to be  a given length and if you pull on them  they try to restore the original length  also if you push on them you can do it  but they try to restore their to their  resting length and right now these guys  are all in the resting length and what's  cool about this is you can kind of feel  the resistance I mean this is what  happens in any material you pull on it  it can maybe pull a little bit but it  gives a lot of force going back to its  original state these I don't know these  cables probably you can stretch that  material at very tiny amount i can feel  the force it's a little less visible  here you can actually see the force and  because it's so much smaller you can  really it's easier to feel what's going  on in here I think that's the intuition  you could of course construct these with  steel bars it's just then it wouldn't  moove be a little less exciting it's up  to you of course but this is just one  way to build models of ten securities  but it's also why it's so flexible  as these bars are not super strong but  it was kind of fun as you feel the  equilibrium stress here that things go  back to where they were originally they  kind of hold in position there because  all the stress is here balance out cool  and one more kind of question is about  sculpture ten security sculptures are  cool so i thought i'd show you a few  more examples the sort of the master  here is Kenneth's Nelson I think he  possibly invented tensei gratis and I  showed one example in lecture but these  are many more he's been doing it since  the 60s some of them are very big  because have some measurements here this  guy is 85 feet long and rests on these  three posts a little hard to see the  cables here but they are it's all struts  and cables so using these bars to be  actually there might be a bars in this  picture I think they might be rigid even  when their struts but and some more this  is one of the longest at 72 feet and  this is the tallest at 90 feet pretty  impressive this is a taller version of  the one that I showed last time or in in  lecture so I'm not sure how he designs  them he's an artist by trade what their  uses computational tools or gadgets or  I'm not sure be interesting to be  interesting to talk to him actually if  you want to see more examples of his  sculpture this is just a piece of it go  to his website on the design side our  good friend told me he wrote ah she who  did the origami organizer bunny it's  also done intense a gritty bunny and he  didn't design this by hand as she might  imagine he designed it with a new piece  of software that hasn't yet been  released called free form 10 seg ready  you start with a polyhedron and there's  a few different starting initial  constructions that  kind of set that thing up as a kind of  tense egra T and then it solves to make  it a balance to have an equilibrium  stress and on the right is the stress  these are what structural engineer would  call force polygons it's essentially the  it's like if you take this graph and you  rotate this is all in 3d by the way so a  little tricky that's rotating in this  case he's pulling on things for trying  to force vertices to come together and  the whole thing is updating and as long  as this number down here is close to  zero here it's 10 to the minus 20 it's a  pretty good approximation of 0 then this  thing is in equilibrium and it's again  solving all those constraints like we  have before so yeah right force polygons  you take every edge you rotate it 90  degrees and then sort of magic  constructing a little polygon around  each face so you scale the edge by the  stress in there and those polygons  should close up to be zero if the if you  at the vertex you satisfy equilibrium  and you can draw all those polygons  those polygons can be joined together to  make one kind of graph and if that graph  closes up then you have an equilibrium  stress so it's kind of a neat way to  visualize that the thing is rigid that  it would rather that it would hold all  of these edges at fixed length and you  have to verify that the bar structure is  rigid but this is presumably constructed  to have that property so hopefully this  offer will be released at some point it  definitely looks like a cool way to play  around with tenseiga T's and design  things question  ah i see yeah that's a good question so  can you see what happens if i like when  that one when i tweak it and it wobbles  yeah it'd be nice to see that in  simulation definitely this is this  software is based on freeform origami  designer and that has the two modes  right one where you can change the  crease pattern one where it tries to  stay it keep it fixed so i'm not sure  whether the software has the same two  modes but it certainly could that's  definitely computable you could  basically ignore the constraints for a  while pull on something then let go let  it restore the constraints without it  changing the tents a gritty like it's  doing here and you should see it wobble  in fact in that case you probably want  to use it a less good numerical method  you don't want it to stabilize as  quickly you'd like to see a jiggle for a  while that's so it should be doable  could be an interesting project to  extend the software if it hasn't been if  that hasn't been implemented yet I could  ask him I asked him last night as it I  mean plan is to release it at some point  but not quite ready for prime time yet  other questions about this so I I think  it should be fun to play with if you  want to build some tents egra tease  there's a couple descriptions on how to  do this out of various easy to find  household objects like straws or with  these well rubber bands things like that  so you can check out George heart has  two instruction web pages about them  they're linked from this slide if you go  to the lecture and yeah that was all I  had for questions any more questions  about this lecture all perfectly clear  we'll be seeing a lot more stuff about  locked linkages the Carpenters rule part  which is at the end of class in the next  two three lectures yeah  well in general a linear program is  something like you have a matrix and you  multiply it by a vector and then you  have let's say in our case we have  greater than equal to 0 so when I say 0  I mean 000 the dual of a linear program  is what you get by transposing the  matrix basically so you get you rotate  it 90 degrees or actually flip around  that diagonal so all the columns become  rose rose become columns so now you've  got some other thing here which is going  to be this big it's probably going to be  less than or equal to something that big  so this is a transpose this is why and  this is your just called see this is in  general called little B and there are  relations between these two things it's  the short version but you can kind of  see so in principle if we take this is  in our case this is basically the  rigidity matrix was this rigidity matrix  Prime in the lecture because if we want  to write greater than or equal to 0  struts are just fine cables we negate  everything in the row bars where we have  equality we need both the original  version and the negated version of that  row but if we just imagine struts for  the moment because I mean some sense  everything well can be simulated by  struts and negative struts struts and  cables then we had the number of rows  here this was essentially the number of  edges in our linkage and the number of  columns here was d times n this is the  number of degrees of freedom these are  the coordinates of all of our things and  then X here was actually our velocity  vectors and so on well actually I should  probably write that so this what we're  calling velocity vectors D this is our  it's like conflict this is different d  these are the D vectors the derivatives  so what would these things be well the  number the columns here are going to be  edges and the rows are going to be  coordinates and remember equilibrium  stress looks like this basically for  every vertex we have sum over all other  vertices let's say VW is an edge of what  is it stress of VW times C of I forget  whether c of e minus c of w or or the  reverse equals zero so this should look  good the number of these constraints is  well here it says it's the number of  vertices but in fact when you say equals  0 this is a vector sum so this has d  coordinates when you say that equals 0  that's d constraints so there are d  times n constraints and how many things  are involved in the constraints well  essentially the edges every Edge has a  term in here so that's kind of why that  looks right you have to go through the  algebra and exactly what's written here  to see that when u transpose it you do  exactly get this constraint and there's  the issue of equals 0 vs greater than  equal to 0 so that's a little bit more  subtle yeah but that's the short version  and you also have to check that the when  your strut or cable you get just a sign  constraint on this thing that's maybe a  little less obvious but at least at a  high level this looks right and their  relations about the primal and the dual  linear program about so for example the  this linear program which is  characterizing all infinitesimally  motions if you find an infinitesimally  twice that  vector or the set of vectors is also an  infinitesimal motion so the solution  space to this linear program is what's  called a convex polyhedral cone meaning  something I mean this is sort of the  beginning of a cone but it goes off to  infinity you can take any motion you  find and scale it off to infinity you  always include the origin 0 0 0 because  you can always do no motion but any  motion can be scaled up so if this thing  has a solution at all other than the 0 0  0 motion it's an unbounded LP meaning  there's you go off to infinity and when  the primal LP is unbounded you know  things about the dual which I forget but  you cannot have one being unbounded and  the other being something else and so  that guarantees that there's an  equilibrium stress that's roughly how it  works other questions
 

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