WEBVTT 00:00:03.220 --> 00:00:05.520 PROFESSOR: Alright, lecture 12 is about tensegrities, 00:00:05.520 --> 00:00:07.810 like this one which you saw. 00:00:07.810 --> 00:00:11.360 And when they're rigid-- infinitesimal rigidity, 00:00:11.360 --> 00:00:16.560 and carpenter's rule theorem, all in one quick lecture. 00:00:16.560 --> 00:00:19.610 So, just a couple questions about tensegrities. 00:00:19.610 --> 00:00:21.530 Well first is about infinitesimal rigidity 00:00:21.530 --> 00:00:23.630 in general. 00:00:23.630 --> 00:00:27.380 This is sort of extra, bonus. 00:00:27.380 --> 00:00:31.340 I talked about one reason why this dot product condition 00:00:31.340 --> 00:00:35.060 is the right thing, which is based on projection. 00:00:35.060 --> 00:00:38.340 But there are other ways to think about it. 00:00:38.340 --> 00:00:44.520 So let's say we have an edge, vw-- 00:00:44.520 --> 00:00:46.775 actually I think I'll draw it w on the left. 00:00:49.980 --> 00:00:52.362 So here's vw. 00:00:52.362 --> 00:00:54.550 We would think of this point is being 00:00:54.550 --> 00:00:56.400 c of w, that's your configuration. 00:00:56.400 --> 00:01:00.220 This point is c of e. 00:01:00.220 --> 00:01:04.090 Then we were talking about when, if you have a velocity 00:01:04.090 --> 00:01:07.620 vector, d of w, and a velocity vector-- 00:01:14.560 --> 00:01:18.470 Say something like this, d of v, when 00:01:18.470 --> 00:01:23.790 it preserves the length of this bar to the first order. 00:01:23.790 --> 00:01:30.650 And the condition we had was c of v minus c of w dot product 00:01:30.650 --> 00:01:37.190 with d of v minus d of w equals 0. 00:01:40.380 --> 00:01:43.470 So the question-- so the way we said 00:01:43.470 --> 00:01:46.390 in class this works is if you look at the projected 00:01:46.390 --> 00:01:48.770 length of this vector onto the segment, 00:01:48.770 --> 00:01:50.720 that's essentially how much shorter, 00:01:50.720 --> 00:01:54.350 in this case it gets shorter, in this case 00:01:54.350 --> 00:01:56.610 this is how much longer the segment 00:01:56.610 --> 00:01:57.920 gets to the first order. 00:01:57.920 --> 00:02:02.450 And that corresponds to d of v dot product with this vector, 00:02:02.450 --> 00:02:05.020 because this vector, c of v minus c of w 00:02:05.020 --> 00:02:09.830 is this direction here. 00:02:09.830 --> 00:02:10.971 c of v minus c of w. 00:02:10.971 --> 00:02:12.970 So if you take the dot product with this vector, 00:02:12.970 --> 00:02:14.650 you get that projected length. 00:02:14.650 --> 00:02:17.090 You want those two dot products to be equal. 00:02:17.090 --> 00:02:19.310 And so you want the difference to be 0. 00:02:19.310 --> 00:02:21.870 That's one way to think about it. 00:02:21.870 --> 00:02:26.450 But, in fact, this quantity also has an intuitive notion, 00:02:26.450 --> 00:02:30.230 if you've done some basic mechanics. 00:02:30.230 --> 00:02:34.980 This is the relative motion of v with respect to w. 00:02:34.980 --> 00:02:41.040 So it's like-- well imagine w is not moving. 00:02:41.040 --> 00:02:44.290 To do that you just subtract d of w from all motions. 00:02:44.290 --> 00:02:47.180 Then w will remain stationary to the first order, 00:02:47.180 --> 00:02:49.310 so all first order motions. 00:02:49.310 --> 00:02:50.680 So then what's the motion of v? 00:02:50.680 --> 00:02:54.542 Well, it's going to be d of v minus d of w, 00:02:54.542 --> 00:02:57.130 because everything get's subtracted by d of w. 00:02:57.130 --> 00:03:04.320 So that it is-- minus d of w corresponds to drawing 00:03:04.320 --> 00:03:06.690 that vector in the other direction, 00:03:06.690 --> 00:03:09.680 and then adding that to d of w corresponds 00:03:09.680 --> 00:03:11.890 to this kind of picture. 00:03:11.890 --> 00:03:13.930 This is minus d of w. 00:03:13.930 --> 00:03:23.160 And so this vector is d of v minus d of w. 00:03:23.160 --> 00:03:25.690 That's the sum of those two vectors. 00:03:25.690 --> 00:03:30.060 And the claim is what we want is for these two vectors 00:03:30.060 --> 00:03:32.400 to be perpendicular to each other. 00:03:32.400 --> 00:03:35.010 The edge and the motion. 00:03:35.010 --> 00:03:38.480 And this makes a lot of sense, because we're imagining w 00:03:38.480 --> 00:03:41.540 as not moving, so it's just v moving. 00:03:41.540 --> 00:03:46.000 And we want the motion of v to be perpendicular locally, 00:03:46.000 --> 00:03:49.960 to the segment, because that's going to preserve the length. 00:03:49.960 --> 00:03:53.030 So that's actually, I think, an even more intuitive way 00:03:53.030 --> 00:03:53.790 to see it. 00:03:53.790 --> 00:03:55.820 And since this video, this lecture video, 00:03:55.820 --> 00:03:58.220 has been online for two years, every year 00:03:58.220 --> 00:04:01.800 so I get an email saying hey, you asked what is d of w 00:04:01.800 --> 00:04:02.990 minus d of w mean? 00:04:02.990 --> 00:04:05.730 And, yes, I didn't realize it at the time, 00:04:05.730 --> 00:04:07.080 but it's just relative motion. 00:04:07.080 --> 00:04:09.100 And so, yeah. 00:04:09.100 --> 00:04:12.620 It's like if you're moving along this circle, 00:04:12.620 --> 00:04:15.720 centered at w, that's where you want to be, either straight up 00:04:15.720 --> 00:04:18.120 or straight down. 00:04:18.120 --> 00:04:20.110 Clear? 00:04:20.110 --> 00:04:21.079 Some bonus intuition. 00:04:21.079 --> 00:04:23.360 It's a little harder to see the other conditions 00:04:23.360 --> 00:04:26.270 from this perspective, at least I find it harder to see. 00:04:26.270 --> 00:04:30.110 But I guess I shouldn't claim that, or else I'm 00:04:30.110 --> 00:04:33.820 going to get more emails in the future. 00:04:33.820 --> 00:04:35.870 Let's see, if you want it to be-- 00:04:35.870 --> 00:04:38.476 what does the dot product being greater or equal to 0 mean? 00:04:38.476 --> 00:04:39.850 It probably means that this angle 00:04:39.850 --> 00:04:43.956 is obtuse, which means that the lengths getting longer. 00:04:43.956 --> 00:04:45.330 And less than or equal to 0 means 00:04:45.330 --> 00:04:47.440 that the angle is getting non-obtuse. 00:04:47.440 --> 00:04:50.387 And if the dot product rule of the cosine of the angle, 00:04:50.387 --> 00:04:51.970 you should be able to figure that out. 00:04:51.970 --> 00:04:56.420 But it's a little bit less memorized in my head. 00:04:56.420 --> 00:04:58.210 Whereas dot product equals 0, everyone 00:04:58.210 --> 00:05:01.190 remembers that's being perpendicular. 00:05:01.190 --> 00:05:04.075 So this was struts and cables. 00:05:10.650 --> 00:05:13.440 Next question was, a couple questions here. 00:05:13.440 --> 00:05:17.380 One is how can you say the tensegrity you showed is rigid? 00:05:17.380 --> 00:05:18.875 Because you can flex it. 00:05:18.875 --> 00:05:22.820 This doesn't look very rigid, this guy here. 00:05:22.820 --> 00:05:24.342 It looks quite flexible. 00:05:24.342 --> 00:05:26.550 And then there was another question, by someone else, 00:05:26.550 --> 00:05:29.550 saying why did you use springs to represent bars? 00:05:29.550 --> 00:05:32.350 And these are sort of the same question. 00:05:32.350 --> 00:05:36.320 This is flexible because the springs are flexible. 00:05:36.320 --> 00:05:38.590 The idea with the springs, I mean 00:05:38.590 --> 00:05:41.520 I'm just guessing why this model is made this way, 00:05:41.520 --> 00:05:44.320 springs are nice because they do have a natural resting length. 00:05:44.320 --> 00:05:46.840 So in that sense, they want to be a given length. 00:05:46.840 --> 00:05:50.260 And if you pull on them they try to restore the original length. 00:05:50.260 --> 00:05:52.160 Also if you push on them, you can do it, 00:05:52.160 --> 00:05:54.810 but they try to restore to there resting length. 00:05:54.810 --> 00:05:57.240 And right now these guys are all on their resting length. 00:05:57.240 --> 00:05:58.614 And what's cool about this is you 00:05:58.614 --> 00:06:00.340 could kind of feel the resistance. 00:06:00.340 --> 00:06:03.230 I mean this is what happens in any material, you pull on it, 00:06:03.230 --> 00:06:05.900 it can maybe pull a little bit, but it 00:06:05.900 --> 00:06:10.580 gives a lot of force going back to its original state. 00:06:10.580 --> 00:06:13.780 These-- I don't know, these cables, probably you 00:06:13.780 --> 00:06:16.430 could stretch that material and very tiny amount. 00:06:16.430 --> 00:06:18.680 I could feel the force, it's a little less visible. 00:06:18.680 --> 00:06:20.250 Here you can actually see the force 00:06:20.250 --> 00:06:23.520 and, because it's so much smaller, 00:06:23.520 --> 00:06:25.540 it's easier to feel what's going on in here. 00:06:25.540 --> 00:06:27.010 I think that's the intuition. 00:06:27.010 --> 00:06:30.960 You could, of course, construct these with steel bars. 00:06:30.960 --> 00:06:32.473 It's just then it wouldn't move, it 00:06:32.473 --> 00:06:33.806 would be a little less exciting. 00:06:33.806 --> 00:06:35.840 So it's up to you, of course. 00:06:35.840 --> 00:06:39.810 But this is just one way to build models of tensegrities. 00:06:39.810 --> 00:06:41.460 But it's also why it's so flexible, 00:06:41.460 --> 00:06:45.640 because these bars are not super strong. 00:06:45.640 --> 00:06:48.750 But what's kind of fun is you feel the equilibrium stress 00:06:48.750 --> 00:06:53.091 here, that things go back to where they were originally. 00:06:53.091 --> 00:06:54.590 They kind of hold in position there, 00:06:54.590 --> 00:06:57.460 because all the stresses here balance out. 00:07:02.400 --> 00:07:09.760 And one more kind of question is about-- sculpture-- tensegrity 00:07:09.760 --> 00:07:12.600 sculptures are cool so I thought I'd show you a few more 00:07:12.600 --> 00:07:14.360 examples. 00:07:14.360 --> 00:07:17.110 The sort of-- the master here is Kenneth Snelson. 00:07:17.110 --> 00:07:20.410 I think he possibly invented tensegrities. 00:07:20.410 --> 00:07:25.150 And I showed one example in lecture, 00:07:25.150 --> 00:07:26.660 but these are many more. 00:07:26.660 --> 00:07:29.320 He's been doing it since the '60s. 00:07:29.320 --> 00:07:33.740 Some of them are very big, I have some measurements here. 00:07:33.740 --> 00:07:38.974 This guy is 85 feet long and rests on these three posts. 00:07:38.974 --> 00:07:40.890 And it's a little hard to see the cables here, 00:07:40.890 --> 00:07:43.730 but it's all struts and cables. 00:07:43.730 --> 00:07:48.540 So using these bars to be-- actually 00:07:48.540 --> 00:07:50.890 they might be bars in this picture. 00:07:50.890 --> 00:07:53.200 I think they might be rigid even when they're struts. 00:07:56.130 --> 00:07:58.080 And some more. 00:07:58.080 --> 00:08:00.620 This is one of the longest, at 72 feet. 00:08:00.620 --> 00:08:03.920 And this is the tallest, at 90 feet. 00:08:03.920 --> 00:08:04.820 Pretty impressive. 00:08:04.820 --> 00:08:08.580 This is a taller version of the one that I showed last time, 00:08:08.580 --> 00:08:11.120 or in lecture. 00:08:11.120 --> 00:08:16.010 So I'm not sure how he designs them, he's an artist by trade. 00:08:16.010 --> 00:08:18.370 Whether he uses computational tools, or gadgets, 00:08:18.370 --> 00:08:22.620 or I'm not sure, it would be interesting to talk 00:08:22.620 --> 00:08:23.665 to him, actually. 00:08:23.665 --> 00:08:26.630 But if you want to see more examples of this sculpture, 00:08:26.630 --> 00:08:29.420 this is just a piece of it, go to his website. 00:08:32.120 --> 00:08:35.350 On the design side, our good friend, Tomohio Tachi, 00:08:35.350 --> 00:08:38.380 who do the origami Origamizer Bunny, 00:08:38.380 --> 00:08:40.370 has also done a tensegrity bunny. 00:08:40.370 --> 00:08:43.059 And he didn't design this by hand, as you might imagine. 00:08:43.059 --> 00:08:45.940 He designed it with a new piece of software that hasn't yet 00:08:45.940 --> 00:08:48.280 been released, called Freeform Tensegrity. 00:08:48.280 --> 00:08:50.220 You start with a polyhedron, and there's 00:08:50.220 --> 00:08:54.540 a few different initial constructions that kind of 00:08:54.540 --> 00:08:57.600 set that thing up as a kind of tensegrity. 00:08:57.600 --> 00:09:01.600 And then it solves to make it balance, 00:09:01.600 --> 00:09:03.110 to have an equilibrium stress. 00:09:03.110 --> 00:09:05.340 And on the right is the stress. 00:09:05.340 --> 00:09:07.200 These are what a structural engineer 00:09:07.200 --> 00:09:09.040 would call force polygons. 00:09:09.040 --> 00:09:12.882 It's essentially the-- it's like if you take this graph 00:09:12.882 --> 00:09:15.340 and you rotate-- this is all in 3D, by the way, so a little 00:09:15.340 --> 00:09:16.330 tricky. 00:09:16.330 --> 00:09:17.450 That's rotating. 00:09:17.450 --> 00:09:19.390 In this case he's pulling on things, 00:09:19.390 --> 00:09:21.530 trying to force vertices to come together. 00:09:21.530 --> 00:09:23.150 And the whole thing is updating. 00:09:23.150 --> 00:09:25.460 And as long as this number down here is close to 0, 00:09:25.460 --> 00:09:27.510 here it's 10 to the minus 20, it's 00:09:27.510 --> 00:09:29.950 a pretty good approximation of 0, 00:09:29.950 --> 00:09:33.340 then this thing is in equilibrium. 00:09:33.340 --> 00:09:35.250 It's again, solving all those constraints 00:09:35.250 --> 00:09:36.240 like we have before. 00:09:41.870 --> 00:09:42.780 So force polygons. 00:09:42.780 --> 00:09:45.460 You take every edge, you rotate it 90 degrees, 00:09:45.460 --> 00:09:48.120 and then sort of con-- imagine constructing a little polygon 00:09:48.120 --> 00:09:50.495 around each face. 00:09:50.495 --> 00:09:53.880 So you scale the edge by the stress in there, 00:09:53.880 --> 00:09:58.250 and those polygons should close up to be 0, if, at the vertex, 00:09:58.250 --> 00:09:59.830 you satisfy equilibrium. 00:09:59.830 --> 00:10:01.520 And you can draw all those polygons. 00:10:01.520 --> 00:10:03.880 Those polygons can be joined together 00:10:03.880 --> 00:10:06.130 to make one kind of graph. 00:10:06.130 --> 00:10:08.230 And if that graph closes up, then you 00:10:08.230 --> 00:10:09.680 have an equilibrium stress. 00:10:09.680 --> 00:10:11.840 So it's kind of a neat way to visualize 00:10:11.840 --> 00:10:15.410 that the thing is rigid. 00:10:15.410 --> 00:10:18.349 Or rather that it would hold all of these edges at fixed length, 00:10:18.349 --> 00:10:20.640 and you have to verify that the bar structure is rigid, 00:10:20.640 --> 00:10:22.056 but this is presumably constructed 00:10:22.056 --> 00:10:24.020 to have that property. 00:10:24.020 --> 00:10:26.650 So hopefully this software will be released at some point. 00:10:26.650 --> 00:10:28.108 It definitely looks like a cool way 00:10:28.108 --> 00:10:30.780 to play around with tensegrities and design things. 00:10:30.780 --> 00:10:31.714 Question? 00:10:31.714 --> 00:10:33.089 AUDIENCE: Is there any way to see 00:10:33.089 --> 00:10:37.530 what happens to the equilibrium stresses [INAUDIBLE]? 00:10:37.530 --> 00:10:38.630 Uh. 00:10:38.630 --> 00:10:39.312 I see, yeah. 00:10:39.312 --> 00:10:40.270 That's a good question. 00:10:43.080 --> 00:10:45.810 So can you see what happens if I-- like when that one, when 00:10:45.810 --> 00:10:47.890 I tweak it and then wobbles. 00:10:47.890 --> 00:10:48.390 Yeah. 00:10:48.390 --> 00:10:52.080 It would be nice to see that in simulation, definitely. 00:10:52.080 --> 00:10:55.180 This software is based on free form origami designer, 00:10:55.180 --> 00:10:56.812 and that has the two modes, right? 00:10:56.812 --> 00:10:58.645 One where you can change the crease pattern, 00:10:58.645 --> 00:11:01.212 and one where it tries to stay, keep it fixed. 00:11:01.212 --> 00:11:03.420 So I'm not sure whether the software has the same two 00:11:03.420 --> 00:11:07.230 modes, but it certainly could, that's definitely computable. 00:11:07.230 --> 00:11:09.740 You could basically ignore the concerns for awhile, 00:11:09.740 --> 00:11:12.470 pull on something, then let go, let it restore the constraints 00:11:12.470 --> 00:11:13.930 without it changing the tensegrity, 00:11:13.930 --> 00:11:15.120 like it's doing here. 00:11:15.120 --> 00:11:16.620 And you should see it wobble. 00:11:16.620 --> 00:11:18.120 In fact, in that case you'd probably 00:11:18.120 --> 00:11:20.770 want to use a less good numerical method. 00:11:20.770 --> 00:11:22.890 Because you don't want it to stabilize as quickly, 00:11:22.890 --> 00:11:25.620 you'd like to see it jiggle for a while. 00:11:25.620 --> 00:11:26.920 So it should be doable. 00:11:26.920 --> 00:11:29.350 Could be an interesting project to extend the software, 00:11:29.350 --> 00:11:32.000 if it hasn't been implemented yet. 00:11:32.000 --> 00:11:33.940 I could ask him. 00:11:33.940 --> 00:11:36.770 I asked him last night, he say that the plan is to release it 00:11:36.770 --> 00:11:41.930 at some point, but not quite ready for prime time yet. 00:11:41.930 --> 00:11:45.790 Other questions about this? 00:11:45.790 --> 00:11:49.120 So I think this would be fun to play with. 00:11:49.120 --> 00:11:51.170 If you want to build some tensegrities 00:11:51.170 --> 00:11:55.900 there's a couple descriptions of how to do this out 00:11:55.900 --> 00:12:02.080 of various, easy to find, household objects like straws, 00:12:02.080 --> 00:12:08.400 or with these wove rubber bands, things like that. 00:12:08.400 --> 00:12:10.110 So you can check out-- George Hart 00:12:10.110 --> 00:12:15.420 has two construction web pages about them. 00:12:15.420 --> 00:12:20.200 They're linked from this slide, if you go to the lecture. 00:12:20.200 --> 00:12:22.130 That was all I had for questions. 00:12:22.130 --> 00:12:23.820 Any more questions about this lecture? 00:12:26.830 --> 00:12:28.580 All perfectly clear? 00:12:28.580 --> 00:12:31.182 We'll be seeing a lot more stuff about locked linkages, 00:12:31.182 --> 00:12:32.640 the carpenter's rule part, which is 00:12:32.640 --> 00:12:35.920 at the end of class in the next two-three lectures. 00:12:35.920 --> 00:12:41.520 AUDIENCE: Can you explain how infinitesimal motions-- 00:12:41.520 --> 00:12:46.090 Well, in general, a linear program 00:12:46.090 --> 00:12:48.440 is something like-- you have a matrix 00:12:48.440 --> 00:12:50.803 and you multiply it by a vector. 00:12:50.803 --> 00:12:53.386 And then you have, let's say, in our case we have greater than 00:12:53.386 --> 00:12:54.880 or equal to 0. 00:12:54.880 --> 00:12:59.547 So when I say 0 I mean 0 0 0. 00:13:03.620 --> 00:13:05.570 The dual of a linear program is what 00:13:05.570 --> 00:13:07.760 you get by transposing the matrix, basically. 00:13:07.760 --> 00:13:14.190 So you get-- you rotate it 90 degrees, 00:13:14.190 --> 00:13:18.210 or actually flip around that diagonal, so all the columns 00:13:18.210 --> 00:13:20.620 become rows, rows become columns. 00:13:20.620 --> 00:13:23.200 So now you've got some other thing here, 00:13:23.200 --> 00:13:27.115 which is going to be this big. 00:13:27.115 --> 00:13:30.630 It's probably going to be less than or equal to something 00:13:30.630 --> 00:13:31.350 that big. 00:13:33.921 --> 00:13:37.650 So this is a transpose, this is y, 00:13:37.650 --> 00:13:40.140 and this is usually called c, this is, 00:13:40.140 --> 00:13:43.240 in general called little b. 00:13:43.240 --> 00:13:46.330 And they are relations between these two things, 00:13:46.330 --> 00:13:48.870 is the short version. 00:13:48.870 --> 00:13:51.704 But you can kind of see-- so in principle, 00:13:51.704 --> 00:13:53.120 if we take-- this is, in our case, 00:13:53.120 --> 00:13:55.550 this is basically the rigidity matrix. 00:13:55.550 --> 00:13:58.360 This is rigidity matrix prime, in the lecture. 00:13:58.360 --> 00:14:00.720 Because if we want to write greater than or equal to 0, 00:14:00.720 --> 00:14:02.030 struts are just fine. 00:14:02.030 --> 00:14:05.030 Cables we negate everything in the row. 00:14:05.030 --> 00:14:08.480 Bars, where we have equality, we need both the original version 00:14:08.480 --> 00:14:10.995 and the negated version of that row. 00:14:10.995 --> 00:14:12.870 But if we just imagine struts for the moment, 00:14:12.870 --> 00:14:15.211 because I mean subsets everything-- well, 00:14:15.211 --> 00:14:17.460 can be simulated by struts and negative struts, struts 00:14:17.460 --> 00:14:19.640 and cables. 00:14:19.640 --> 00:14:21.450 Then we had the number of rows here, 00:14:21.450 --> 00:14:26.910 this was essentially the number of edges in our linkage. 00:14:26.910 --> 00:14:30.111 And the number of columns here was d times n, 00:14:30.111 --> 00:14:31.860 this was the number of degrees of freedom. 00:14:31.860 --> 00:14:34.770 These are the coordinates of all of our things. 00:14:34.770 --> 00:14:40.335 And then x here was actually our velocity vectors, and so on. 00:14:40.335 --> 00:14:42.210 Well, actually, I should probably write that. 00:14:42.210 --> 00:14:45.090 So this, what we're calling velocity vector's, d. 00:14:45.090 --> 00:14:47.537 This is our-- It's like conflict. 00:14:47.537 --> 00:14:48.370 This is different d. 00:14:48.370 --> 00:14:52.479 These are the d vectors, the derivatives. 00:14:52.479 --> 00:14:53.770 So, what would these things be? 00:14:53.770 --> 00:14:58.935 Well the columns here are going to be edges. 00:15:02.190 --> 00:15:05.460 And the rows are going to be coordinates. 00:15:10.540 --> 00:15:14.020 And remember equilibrium stress looks like this. 00:15:14.020 --> 00:15:18.060 Basically for every vertex we have 00:15:18.060 --> 00:15:21.650 sum over all other vertices, let's say 00:15:21.650 --> 00:15:34.610 v w is an edge of, what is it, stress of vw times 00:15:34.610 --> 00:15:41.020 c of-- I forget whether c of v minus c of w or the reverse, 00:15:41.020 --> 00:15:41.860 equals 0. 00:15:44.800 --> 00:15:49.510 So this should look good. 00:15:49.510 --> 00:15:52.810 The number of these constraints is-- well here 00:15:52.810 --> 00:15:54.590 it says it's the number of vertices. 00:15:54.590 --> 00:15:58.440 But in fact, when you say equals zero, this is a vector sum, 00:15:58.440 --> 00:15:59.790 so this has d coordinates. 00:15:59.790 --> 00:16:02.490 When you say that equals 0, that's d constraints. 00:16:02.490 --> 00:16:05.530 So they're d times n constraints. 00:16:05.530 --> 00:16:08.820 And how many things are involved in the constraints? 00:16:08.820 --> 00:16:13.400 Well essentially the edges, every edge has a term in here. 00:16:13.400 --> 00:16:16.954 So that's kind of why that looks right. 00:16:16.954 --> 00:16:18.370 You have to go through the algebra 00:16:18.370 --> 00:16:21.180 and exactly what's written here, to see that when you transpose 00:16:21.180 --> 00:16:23.940 it, you do exactly get this constraint. 00:16:23.940 --> 00:16:27.420 And there's the issue of equals 0, versus 00:16:27.420 --> 00:16:28.990 greater than or equal to 0. 00:16:28.990 --> 00:16:31.380 So that's a little bit more subtle. 00:16:35.200 --> 00:16:36.570 But that's the short version. 00:16:36.570 --> 00:16:40.400 And you also have to check that the-- when your strut or cable, 00:16:40.400 --> 00:16:43.450 you get just a sign constraint on this thing. 00:16:43.450 --> 00:16:44.990 That's maybe a little less obvious. 00:16:44.990 --> 00:16:47.560 But at least a high level, this looks right. 00:16:47.560 --> 00:16:50.420 And the relations about the primal 00:16:50.420 --> 00:16:52.760 and the dual linear program about-- 00:16:52.760 --> 00:16:55.140 So for example, this linear program, 00:16:55.140 --> 00:16:57.290 which is characterizing all infinitesimal motions, 00:16:57.290 --> 00:17:01.150 if you find an infinitesimal motion, twice that vector 00:17:01.150 --> 00:17:03.520 or the set of vectors is also an infinitesimal motion. 00:17:03.520 --> 00:17:05.990 So the solution space to this linear program 00:17:05.990 --> 00:17:10.050 is what's called a convex polyhedral cone. 00:17:10.050 --> 00:17:14.140 Meaning-- I mean this is sort of the beginning of a cone, 00:17:14.140 --> 00:17:15.390 but it goes off to infinity. 00:17:15.390 --> 00:17:17.240 You could take any motion you find 00:17:17.240 --> 00:17:18.520 and scale it off to infinity. 00:17:18.520 --> 00:17:21.099 You always include the origin 0, 0, 0. 00:17:21.099 --> 00:17:23.430 Because you can always do no motion. 00:17:23.430 --> 00:17:25.450 But any motion can be scaled up. 00:17:25.450 --> 00:17:29.200 So if this thing has a solution at all, other than the 0, 0, 0, 00:17:29.200 --> 00:17:34.370 motion, it's an unbounded lp, meaning you go off to infinity. 00:17:34.370 --> 00:17:37.010 And when the primal lp is unbounded 00:17:37.010 --> 00:17:42.210 you know things about the dual, which I forget. 00:17:42.210 --> 00:17:45.670 But you cannot have one being unbound and the other being 00:17:45.670 --> 00:17:46.440 something else. 00:17:46.440 --> 00:17:49.700 And so that guarantees that there's an equilibrium stress. 00:17:49.700 --> 00:17:52.900 That's roughly how it works. 00:17:52.900 --> 00:17:54.620 Are there questions?