WEBVTT 00:00:03.310 --> 00:00:07.320 PROFESSOR: All right, lecture 19 is about mostly refolding, 00:00:07.320 --> 00:00:11.330 common unfoldings of polyhedra, convex polyhedra, and also 00:00:11.330 --> 00:00:12.840 about Mozartkugel. 00:00:12.840 --> 00:00:17.390 But most questions were about the common unfolding 00:00:17.390 --> 00:00:20.280 stuff, so let me start with that. 00:00:20.280 --> 00:00:23.330 First question-- is this fractal unfolding 00:00:23.330 --> 00:00:27.150 of regular tetrahedron and the cube, 00:00:27.150 --> 00:00:33.350 which looks like this-- is it resolved? 00:00:33.350 --> 00:00:36.230 The answer is no, but it is at least published now. 00:00:36.230 --> 00:00:38.500 This is at EuroCG. 00:00:38.500 --> 00:00:40.970 One of the authors is here, Ryuhei Uehara. 00:00:40.970 --> 00:00:45.740 And it's still conjecture, although there 00:00:45.740 --> 00:00:47.465 is a way to explicitly construct this. 00:00:47.465 --> 00:00:49.820 This does actually fold into two things. 00:00:49.820 --> 00:00:53.270 It folds into a cube and a tetramonohedron, a tetrahedron 00:00:53.270 --> 00:00:55.110 with equal sides. 00:00:55.110 --> 00:00:58.270 It's not a regular tetrahedron, though, 00:00:58.270 --> 00:01:02.130 but this is actually an unfolding. 00:01:02.130 --> 00:01:06.180 And they have another one, which they haven't drawn because it 00:01:06.180 --> 00:01:12.050 has 10 to the 180 edges or so, but it is within 3 times 10 00:01:12.050 --> 00:01:16.690 to the minus 1796 of regular. 00:01:16.690 --> 00:01:21.116 So that's pretty good evidence that this is gonna work out. 00:01:21.116 --> 00:01:22.240 Let me write that down. 00:01:27.140 --> 00:01:29.820 So they have a particular iteration procedure. 00:01:29.820 --> 00:01:33.230 It's not always guaranteed to make a connected unfolding, is 00:01:33.230 --> 00:01:34.770 the troubling part. 00:01:34.770 --> 00:01:36.960 I think it will converge, but the challenge 00:01:36.960 --> 00:01:40.030 is to find an infinite sequence of examples 00:01:40.030 --> 00:01:43.950 like this one, that form a connected polygon, that 00:01:43.950 --> 00:01:46.520 would limit to the fractal. 00:01:46.520 --> 00:01:47.870 They have that instance. 00:01:47.870 --> 00:01:51.600 They have another instance which is very, very good. 00:01:51.600 --> 00:01:53.897 This is the error. 00:01:53.897 --> 00:01:55.980 But we don't yet know whether there are infinitely 00:01:55.980 --> 00:02:00.720 many examples the converge all the way to some fractal set. 00:02:00.720 --> 00:02:03.590 So that is a state of the fractal. 00:02:03.590 --> 00:02:07.220 Next we have a question about boxes. 00:02:07.220 --> 00:02:09.490 When that lecture was given, it was open 00:02:09.490 --> 00:02:11.700 whether there was a common unfolding of three boxes. 00:02:11.700 --> 00:02:14.130 That problem is now solved. 00:02:14.130 --> 00:02:15.790 There's two papers about this topic. 00:02:15.790 --> 00:02:19.070 The first one we wrote together. 00:02:19.070 --> 00:02:21.180 And this is just a fun example, not 00:02:21.180 --> 00:02:22.790 yet related to the three box problem, 00:02:22.790 --> 00:02:25.820 but you may recall there were some unfoldings of two 00:02:25.820 --> 00:02:29.530 boxes where one was 45 degrees to the other. 00:02:29.530 --> 00:02:31.010 This was the challenge of making it 00:02:31.010 --> 00:02:33.300 not 45 degrees, something else. 00:02:33.300 --> 00:02:39.430 And here it's a 3 by 1 triangle that is the axis. 00:02:39.430 --> 00:02:44.670 So you end up with root 10, root of 3 squared plus 1 squared. 00:02:44.670 --> 00:02:48.080 So that's still two boxes, but at least at a funny angle. 00:02:48.080 --> 00:02:50.590 And this was actually found computationally, 00:02:50.590 --> 00:02:53.720 with an integer linear program for this. 00:02:53.720 --> 00:02:56.360 Integer linear programming-- set up 00:02:56.360 --> 00:02:58.860 the problem of which pixels are in and which pixels are out 00:02:58.860 --> 00:03:00.510 in this representation, and could you 00:03:00.510 --> 00:03:03.719 fit-- we started with these parameters. 00:03:03.719 --> 00:03:05.260 You have to be very careful about how 00:03:05.260 --> 00:03:07.960 you choose the numbers to make the surface areas match. 00:03:07.960 --> 00:03:10.780 So we just chose that by hand, and then tried 00:03:10.780 --> 00:03:13.031 to find whether there was such a thing. 00:03:13.031 --> 00:03:15.280 You could formulate that as an integer linear program. 00:03:15.280 --> 00:03:17.330 It's not guaranteed to be solvable 00:03:17.330 --> 00:03:19.750 especially efficiently, but this one we found a solution. 00:03:19.750 --> 00:03:22.514 For fun we also turned it into a puzzle, where we were just 00:03:22.514 --> 00:03:24.430 modified the boundaries in corresponding ways, 00:03:24.430 --> 00:03:28.214 so now finding either folding is actually pretty challenging. 00:03:28.214 --> 00:03:29.630 Well, one of them's probably easy, 00:03:29.630 --> 00:03:34.910 but the off center one is challenging. 00:03:34.910 --> 00:03:38.250 But this paper also had a first kind of solution 00:03:38.250 --> 00:03:39.560 to the three box problem. 00:03:39.560 --> 00:03:44.360 So this is one net-- it's a grid unfolding, even-- 00:03:44.360 --> 00:03:49.740 and it folds into 1 by 1 by 5, 1 by 2 by 3, and 0 by 1 by 11. 00:03:49.740 --> 00:03:52.290 So this is a flat box, a doubly covered rectangle. 00:03:52.290 --> 00:03:54.516 It's also not technically a grid unfolding-- we're 00:03:54.516 --> 00:03:56.140 cutting in the middle of a pixel, here. 00:03:56.140 --> 00:04:01.310 But if you just refine by 2 and call this 0 by 2 by 22, 00:04:01.310 --> 00:04:04.530 2 by 4 by 6 and then 2 by 2 by 10, then 00:04:04.530 --> 00:04:06.939 this is a grid unfolding. 00:04:06.939 --> 00:04:09.230 I don't know if you care whether it's a grid unfolding, 00:04:09.230 --> 00:04:12.090 but you can make it one if you want. 00:04:12.090 --> 00:04:15.260 So this is the first solution, and this is actually 00:04:15.260 --> 00:04:16.435 found by computer search. 00:04:19.810 --> 00:04:22.070 This came from a long series of research 00:04:22.070 --> 00:04:23.320 in the two box problem. 00:04:23.320 --> 00:04:25.820 We started by looking at all common unfoldings of 1 00:04:25.820 --> 00:04:29.240 by 1 by 5, 1 by 2 by 3, and then saw whether any of them 00:04:29.240 --> 00:04:31.930 folded in to 0 by one by 11. 00:04:31.930 --> 00:04:34.445 So the just for fun, the way that search 00:04:34.445 --> 00:04:37.350 was done-- which is in the same paper-- 00:04:37.350 --> 00:04:41.020 is essentially to try to construct incrementally 00:04:41.020 --> 00:04:43.270 a common unfolding of those two polygons 00:04:43.270 --> 00:04:46.155 so they both have surface area 22. 00:04:46.155 --> 00:04:47.905 The idea is, you start with a single pixel 00:04:47.905 --> 00:04:51.100 and you say, oh, a single pixel can fit on both surfaces. 00:04:51.100 --> 00:04:53.722 Then you try all possible ways to add a second pixel, 00:04:53.722 --> 00:04:55.430 and make sure that at all times the thing 00:04:55.430 --> 00:04:59.306 you have is a partial unfolding of both shapes. 00:04:59.306 --> 00:05:00.680 So that way you get to throw away 00:05:00.680 --> 00:05:02.440 anything that's guaranteed to be bad, 00:05:02.440 --> 00:05:05.060 that won't fit on either shape. 00:05:05.060 --> 00:05:12.690 And so as you grow your polygon, you get more and more options. 00:05:12.690 --> 00:05:16.600 So the max we get to here is about 5 million 00:05:16.600 --> 00:05:19.850 common partial unfoldings of size 17. 00:05:19.850 --> 00:05:21.850 And for comparison, the total number 00:05:21.850 --> 00:05:26.700 of ways to build polyominals of size 17-- 00:05:26.700 --> 00:05:28.640 this is only looking at grid unfoldings-- 00:05:28.640 --> 00:05:33.450 so all possible polyominals of size 17 is 10 times that. 00:05:33.450 --> 00:05:35.620 So we're getting a lot of savings in time 00:05:35.620 --> 00:05:39.580 by throwing away things that are not common partial unfoldings. 00:05:39.580 --> 00:05:42.230 And then, conveniently, things start getting smaller. 00:05:42.230 --> 00:05:44.430 You get to throw away more and more stuff, 00:05:44.430 --> 00:05:51.640 and you end up with only 2,263 common unfoldings of these two 00:05:51.640 --> 00:05:53.640 shapes, top. 00:05:53.640 --> 00:05:57.110 And surprisingly, exactly one of them folds into a 0 00:05:57.110 --> 00:05:58.870 by 1 by 11 box. 00:05:58.870 --> 00:06:02.680 So this is unique for these three parameters. 00:06:02.680 --> 00:06:04.570 And unfortunately that's kind of the limit 00:06:04.570 --> 00:06:07.110 of how far we could easily go with a computer search. 00:06:07.110 --> 00:06:09.000 And we could maybe get a little bit bigger, 00:06:09.000 --> 00:06:10.500 but it seemed like we were not going 00:06:10.500 --> 00:06:13.430 to be able to find three boxes just by computer search 00:06:13.430 --> 00:06:16.757 if we wanted a non-flat example, because numbers are just 00:06:16.757 --> 00:06:18.340 going to have to get bigger, and we're 00:06:18.340 --> 00:06:21.570 not to be able to do this enumeration procedure. 00:06:21.570 --> 00:06:26.680 So we need to think, instead of computer search. 00:06:26.680 --> 00:06:29.590 Before we get to that, one other fun result 00:06:29.590 --> 00:06:34.130 we got from thinking is, if you allow these flat doubly covered 00:06:34.130 --> 00:06:39.220 rectangles, there's one polygon, namely a long strip, 00:06:39.220 --> 00:06:44.610 that will fold into arbitrarily many distinct rectangles. 00:06:44.610 --> 00:06:48.410 So if you take a long strip, you can do this zigzag pattern. 00:06:48.410 --> 00:06:52.610 In this case it's height 2, and you alternate covered on top, 00:06:52.610 --> 00:06:55.210 covered on the bottom, covered on top, covered on the bottom, 00:06:55.210 --> 00:06:58.020 and you turn around and do the same. 00:06:58.020 --> 00:07:01.200 And if provided your dimensions are properly divisible, 00:07:01.200 --> 00:07:04.200 this will make a doubly covered rectangle of this size. 00:07:04.200 --> 00:07:06.650 And you could also do it with-- this 00:07:06.650 --> 00:07:09.920 is height 1, height 2, height 3, and so on. 00:07:09.920 --> 00:07:14.970 In general , if you have and l by one strip you can make about 00:07:14.970 --> 00:07:18.140 l different rectangles, depending on how you choose 00:07:18.140 --> 00:07:22.040 your angle to be one of some integer thing. 00:07:22.040 --> 00:07:22.832 So that's cool. 00:07:22.832 --> 00:07:24.290 If you allow doubly covered things, 00:07:24.290 --> 00:07:26.680 then you can actually make many, many boxes. 00:07:26.680 --> 00:07:29.180 If you really want 3D boxes, that's 00:07:29.180 --> 00:07:31.090 where the next paper comes in. 00:07:31.090 --> 00:07:35.460 So next paper is about three different orthogonal boxes, 00:07:35.460 --> 00:07:37.115 all not flat. 00:07:37.115 --> 00:07:41.510 It's by Shirakawa and Uehara again. 00:07:41.510 --> 00:07:42.860 And here's one of the solutions. 00:07:42.860 --> 00:07:47.110 So this one polygon is a grid unfolding-- folds into a 7 00:07:47.110 --> 00:07:51.910 by 8 by 14, a 2 by 4 by 43, and a 2 by 13 by 16. 00:07:51.910 --> 00:07:54.500 And this is almost certainly beyond what we could ever 00:07:54.500 --> 00:07:58.430 do by computer search, by just exhaustion. 00:07:58.430 --> 00:08:02.420 This is found using human intelligence, and this grid 00:08:02.420 --> 00:08:03.670 pattern. 00:08:03.670 --> 00:08:04.930 And they have another example. 00:08:04.930 --> 00:08:09.720 This one falls into 2 by 13 by 58, 7 by 14 by 38, and 7 by 56. 00:08:09.720 --> 00:08:12.900 This is actually-- this a little more complicated, 00:08:12.900 --> 00:08:15.510 or a little bigger, I should say. 00:08:15.510 --> 00:08:18.780 They're both based on a common principle, which 00:08:18.780 --> 00:08:22.260 is to start from a solution to the two box problem 00:08:22.260 --> 00:08:25.770 and then modify it to make three boxes. 00:08:25.770 --> 00:08:30.120 So this is a common unfolding event a by b by 8a box, 00:08:30.120 --> 00:08:32.799 and an a by two a by 2a plus 3b box 00:08:32.799 --> 00:08:36.419 for, I think, for any values of a and b. 00:08:36.419 --> 00:08:40.929 And essentially the way you see that is, you either wrap it 00:08:40.929 --> 00:08:43.409 around this way, or you wrap it around this way. 00:08:43.409 --> 00:08:46.350 I think that's how the two foldings work. 00:08:46.350 --> 00:08:49.150 You have to figure out which of these lines to keep, 00:08:49.150 --> 00:08:51.470 and which ones to throw. 00:08:51.470 --> 00:08:53.020 I haven't color coded them here. 00:08:53.020 --> 00:08:57.800 But that's a general thing that's known before. 00:08:57.800 --> 00:08:59.610 I think maybe we even saw in lecture 00:08:59.610 --> 00:09:02.740 this-- there was an example where you could just repeat, 00:09:02.740 --> 00:09:06.010 and get infinitely many examples of two different boxes. 00:09:06.010 --> 00:09:09.730 Now we want to modify it to make three different boxes, 00:09:09.730 --> 00:09:12.340 or to add a third box. 00:09:12.340 --> 00:09:16.230 It's not known how to do a fourth box of nonzero volume. 00:09:16.230 --> 00:09:19.620 So the basic idea, which doesn't work, 00:09:19.620 --> 00:09:22.840 is following-- if I built some box, 00:09:22.840 --> 00:09:26.270 and I slit it in this sort of h pattern, 00:09:26.270 --> 00:09:31.010 and then I refold it to be like this, 00:09:31.010 --> 00:09:32.880 I should get a different box. 00:09:32.880 --> 00:09:34.530 That would be really cool. 00:09:34.530 --> 00:09:36.290 The problem is, if you look at what 00:09:36.290 --> 00:09:37.750 lengths have to be equal for this. 00:09:37.750 --> 00:09:42.220 So these two lengths end up being these two lengths, 00:09:42.220 --> 00:09:44.360 so those sums must be equal. 00:09:44.360 --> 00:09:47.800 And over here that same length must 00:09:47.800 --> 00:09:51.810 equal the two heights of the boxes. 00:09:51.810 --> 00:09:54.530 That implies that actually this length equals this length, 00:09:54.530 --> 00:09:57.040 and so this is actually a square. 00:09:57.040 --> 00:09:59.625 It's drawn here magically to not be squares, 00:09:59.625 --> 00:10:01.000 but in fact they must be squares, 00:10:01.000 --> 00:10:03.946 so it's good example of misleading figure. 00:10:03.946 --> 00:10:06.320 Think about it as like, oh, man, they have to be squares. 00:10:06.320 --> 00:10:08.910 That means you're converting a 2 by 1 rectangle to a 1 00:10:08.910 --> 00:10:10.112 by 2 rectangle. 00:10:10.112 --> 00:10:12.070 And that's really not going to change anything. 00:10:12.070 --> 00:10:15.310 You just rotated your box, which is frustrating. 00:10:15.310 --> 00:10:18.260 Fortunately, there's a tweak to make it work. 00:10:18.260 --> 00:10:24.680 The tweak is, you add wiggly tabs and you refold things. 00:10:24.680 --> 00:10:30.610 So this is a-- in terms of the grid, here, we've got an 8 00:10:30.610 --> 00:10:35.820 by 7 rectangle originally on this face. 00:10:35.820 --> 00:10:39.280 We do all these cuts that are in bold, 00:10:39.280 --> 00:10:42.510 and then when we refold we also add in these dash creases. 00:10:42.510 --> 00:10:44.670 That's kind of part of the magic. 00:10:44.670 --> 00:10:47.030 And we end up with a 2 by 13 rectangle. 00:10:47.030 --> 00:10:49.350 These don't even have the same surface area. 00:10:49.350 --> 00:10:53.662 Some of the material got moved into the adjacent sides. 00:10:53.662 --> 00:10:55.120 But if you stare at it long enough, 00:10:55.120 --> 00:10:59.070 this is a valid transformation of an existing box. 00:10:59.070 --> 00:11:01.700 Now, of course, the challenge is to get this trick 00:11:01.700 --> 00:11:07.029 to work with this unfolding of two boxes. 00:11:07.029 --> 00:11:08.570 And it'll be really nice if you could 00:11:08.570 --> 00:11:10.111 make it work for both the boxes, then 00:11:10.111 --> 00:11:11.450 you'd get four boxes, total. 00:11:11.450 --> 00:11:12.900 We don't know how to do that. 00:11:12.900 --> 00:11:15.370 But with this particular augmentation-- 00:11:15.370 --> 00:11:19.030 so, same picture, particular values of a and b chosen here, 00:11:19.030 --> 00:11:23.740 but these extra tabs and pockets are put in there. 00:11:23.740 --> 00:11:28.330 Then we get a common unfolding of three boxes. 00:11:28.330 --> 00:11:30.210 And you can see from the numbers here you've 00:11:30.210 --> 00:11:34.100 got the 2 by 13 and the 7 by 8 appearing, 00:11:34.100 --> 00:11:38.380 And this comes from the other folding. 00:11:38.380 --> 00:11:40.640 So, pretty cool, right? 00:11:40.640 --> 00:11:43.466 And the same gadget is being used 00:11:43.466 --> 00:11:45.090 in this the first example I showed you, 00:11:45.090 --> 00:11:47.200 which is a somewhat smaller one. 00:11:47.200 --> 00:11:50.335 Again you get 7 by 8 here, and 2 by 13. 00:11:53.360 --> 00:11:58.504 So, this is how we know how to do boxes. 00:11:58.504 --> 00:11:59.920 Now, in fact, there are infinitely 00:11:59.920 --> 00:12:02.590 many examples along these lines. 00:12:02.590 --> 00:12:04.360 You can change that grid. 00:12:04.360 --> 00:12:08.620 Parities have to be right and maintain some proportions, 00:12:08.620 --> 00:12:11.660 but there are lots of examples of taking that grid, 00:12:11.660 --> 00:12:16.670 applying it to this net, and then you get these three boxes. 00:12:16.670 --> 00:12:20.950 For any integer, k greater or equal to zero will work. 00:12:20.950 --> 00:12:27.080 And so you get a nice infinite series of examples, I think. 00:12:27.080 --> 00:12:31.212 The rough ratios are almost the same, but-- is that right? 00:12:31.212 --> 00:12:33.295 No, no, actually they're changing the proportions. 00:12:33.295 --> 00:12:40.805 You've got k, 4k, 16k, versus 4k, 4k, 7k, versus 4k, k, 4k. 00:12:40.805 --> 00:12:44.224 So-- sorry? 00:12:44.224 --> 00:12:49.841 Oh, 4k, 2k, and 24k, sorry-- and this is 32. 00:12:49.841 --> 00:12:50.340 Yeah, sorry. 00:12:50.340 --> 00:12:53.250 I need to expand on that that. 00:12:53.250 --> 00:12:55.240 So, pretty cool, and this is the state 00:12:55.240 --> 00:12:57.490 of the art of common unfoldings. 00:12:57.490 --> 00:13:01.110 Next open problem is for non-zero volume boxes. 00:13:01.110 --> 00:13:04.930 Can you prove that 100 nonzero volume boxes are impossible? 00:13:04.930 --> 00:13:07.100 Who knows. 00:13:07.100 --> 00:13:10.300 Maybe even four or five is impossible. 00:13:10.300 --> 00:13:12.540 It seems pretty tricky. 00:13:12.540 --> 00:13:16.180 OK, I have one more question, which 00:13:16.180 --> 00:13:18.160 is about the smooth unfolding stuff. 00:13:21.850 --> 00:13:25.490 I call this a limit, and there's this is issue about the area. 00:13:25.490 --> 00:13:26.640 So I want to go into this. 00:13:26.640 --> 00:13:29.620 Remember this picture-- we take a smooth prismatoid, which 00:13:29.620 --> 00:13:34.090 is two convex-- not polygons, but two convex 00:13:34.090 --> 00:13:35.330 bodies, parallel. 00:13:35.330 --> 00:13:36.550 Take the convex hull. 00:13:36.550 --> 00:13:38.830 You get the smooth surface. 00:13:38.830 --> 00:13:41.820 And then we unfold it in the volcano way 00:13:41.820 --> 00:13:44.860 of taking every rib here, every rule line, 00:13:44.860 --> 00:13:48.480 and developing it out flat. 00:13:48.480 --> 00:13:50.540 And when you do that, you take the union 00:13:50.540 --> 00:13:53.040 of the ribs and the unfolding, does not 00:13:53.040 --> 00:13:55.410 have the same area as the original surface 00:13:55.410 --> 00:13:57.960 area of the polyhedron, which seems a little weird. 00:14:01.255 --> 00:14:03.630 Sometimes we might call I might have called this a limit. 00:14:03.630 --> 00:14:06.599 It's not really a limit of the discrete case. 00:14:06.599 --> 00:14:09.140 And this is why we don't know how to solve the discrete case, 00:14:09.140 --> 00:14:13.710 and yet, we can solve the smooth case. 00:14:13.710 --> 00:14:17.330 The issue is, if you wanted to form this structure 00:14:17.330 --> 00:14:20.960 as a limit of discrete prismatoids, 00:14:20.960 --> 00:14:23.150 you want to subdivide this into a very fine polygon, 00:14:23.150 --> 00:14:25.810 subdivide this into a very fine polygon, 00:14:25.810 --> 00:14:27.710 and then try to unfold those discreetly, 00:14:27.710 --> 00:14:30.210 and then take the limit as you take finer and finer, 00:14:30.210 --> 00:14:32.660 closer and closer approximations of the smooth things 00:14:32.660 --> 00:14:35.080 as polygons. 00:14:35.080 --> 00:14:39.250 It really depends how you discretize-- maybe one way 00:14:39.250 --> 00:14:42.360 to think about that is, if I start here 00:14:42.360 --> 00:14:45.460 and I do uniform spacing from this point-- 00:14:45.460 --> 00:14:48.390 or I do it from this, point or from this point-- each of those 00:14:48.390 --> 00:14:50.810 is going to have a different-- it's going to capture 00:14:50.810 --> 00:14:53.612 a somewhat different discrete structure. 00:14:53.612 --> 00:14:55.070 I think the way to think about that 00:14:55.070 --> 00:14:57.690 is, the gaps will be spread out in different ways. 00:14:57.690 --> 00:14:59.650 If I always cut here, there will always 00:14:59.650 --> 00:15:02.500 be a gap incident to that point, whereas if I always cut here, 00:15:02.500 --> 00:15:04.583 there will always be a gap incident to this point, 00:15:04.583 --> 00:15:07.830 and not necessarily at this point. 00:15:07.830 --> 00:15:10.810 Depending on your discrete approximation, 00:15:10.810 --> 00:15:13.020 you'll get a very different unfolded diagram. 00:15:13.020 --> 00:15:14.020 You're going to get holes out here, 00:15:14.020 --> 00:15:15.645 because when you do the discrete thing, 00:15:15.645 --> 00:15:17.740 the discrete thing does preserve area. 00:15:17.740 --> 00:15:20.150 And yet, you imagine, as you take the limit somehow 00:15:20.150 --> 00:15:21.440 the area jumps at the end. 00:15:21.440 --> 00:15:24.480 That shouldn't happen for a limit, so this is not a limit. 00:15:24.480 --> 00:15:26.820 What happens is, you have many different limit points 00:15:26.820 --> 00:15:29.110 depending on your discrete representation. 00:15:29.110 --> 00:15:30.630 And this green thing is essentially 00:15:30.630 --> 00:15:35.760 the union of all those limit points. 00:15:35.760 --> 00:15:37.820 When you take the union, the area goes up. 00:15:37.820 --> 00:15:40.290 That's essentially what's going on here. 00:15:40.290 --> 00:15:41.540 That's my best interpretation. 00:15:45.750 --> 00:15:48.554 But yeah, shouldn't use the word limit. 00:15:48.554 --> 00:15:50.220 OK that's it for questions, unless there 00:15:50.220 --> 00:15:54.460 are other questions about this or anything 00:15:54.460 --> 00:15:57.250 we talked about in lecture. 00:15:57.250 --> 00:16:01.780 So the next topic I want to talk about-- this is a bonus, 00:16:01.780 --> 00:16:04.190 extra topic hasn't been covered in this class before, 00:16:04.190 --> 00:16:05.630 but I've always wanted to. 00:16:05.630 --> 00:16:07.975 I think there was a final project about it once. 00:16:07.975 --> 00:16:11.980 It's this guy, Theo Jansen, and he 00:16:11.980 --> 00:16:13.695 builds these crazy linkage structures. 00:16:13.695 --> 00:16:16.070 And the next two classes are about protein folding, which 00:16:16.070 --> 00:16:21.080 is more on the linkage folding aspect, and so 00:16:21.080 --> 00:16:24.280 thought it'd be fun to transition over. 00:16:24.280 --> 00:16:26.000 Maybe first I'll just show you some. 00:16:26.000 --> 00:16:32.620 These are two kits you can buy of his constructions. 00:16:32.620 --> 00:16:36.476 This guy-- I will blow. 00:16:36.476 --> 00:16:40.380 [BLOWS] 00:16:49.180 --> 00:16:50.450 This guy's on here. 00:16:59.190 --> 00:17:03.500 If I turn this guy, he walks-- and walks 00:17:03.500 --> 00:17:06.230 pretty elegantly if I turn this very uniformly. 00:17:06.230 --> 00:17:06.983 It's pretty cool. 00:17:06.983 --> 00:17:08.149 It can walk both directions. 00:17:13.220 --> 00:17:14.329 It's also unpowered. 00:17:14.329 --> 00:17:15.589 You blow it, and it works. 00:17:15.589 --> 00:17:16.970 Here's another example. 00:17:16.970 --> 00:17:18.261 You see just the legs. 00:17:18.261 --> 00:17:20.510 It's more clearly a linkage, although this is actually 00:17:20.510 --> 00:17:22.599 just implementing the same linkage. 00:17:22.599 --> 00:17:25.800 It's just got some thicker-- filled in some rigid triangles 00:17:25.800 --> 00:17:26.934 here. 00:17:26.934 --> 00:17:30.140 [BLOWS] 00:17:30.140 --> 00:17:33.380 This is fun, because the wind's going in one direction, 00:17:33.380 --> 00:17:35.710 and the guy's going in the opposite direction. 00:17:35.710 --> 00:17:37.650 Here it's going in the direction of the wind. 00:17:37.650 --> 00:17:39.884 Here it's going perpendicular to the wind. 00:17:39.884 --> 00:17:41.700 [BLOWS] 00:17:41.700 --> 00:17:43.523 A little bit slower, though. 00:17:48.150 --> 00:17:53.280 I'll just manually turn this guy. 00:17:53.280 --> 00:17:54.785 Is he going a little faster? 00:17:54.785 --> 00:17:58.000 So it's got really cool leg mechanism. 00:18:02.330 --> 00:18:04.060 Looks almost like an animal. 00:18:04.060 --> 00:18:08.290 Theo calls these his children, his animals. 00:18:08.290 --> 00:18:10.570 His goal is for them to be self sufficient 00:18:10.570 --> 00:18:13.440 and to live on their own in the beach. 00:18:13.440 --> 00:18:15.270 How many people have seen his TED Talk? 00:18:15.270 --> 00:18:16.057 A few people. 00:18:16.057 --> 00:18:16.890 You should watch it. 00:18:16.890 --> 00:18:19.181 I'm not going to show it here, because some people have 00:18:19.181 --> 00:18:20.820 seen it. 00:18:20.820 --> 00:18:23.300 He has this vision. 00:18:23.300 --> 00:18:24.970 He makes them all out of PVC. 00:18:24.970 --> 00:18:31.259 These are just injection molded plastic kits which you can buy. 00:18:31.259 --> 00:18:32.050 They're super cool. 00:18:32.050 --> 00:18:33.740 Let me show you the bigger ones. 00:18:33.740 --> 00:18:36.511 This is living on the beach. 00:18:36.511 --> 00:18:38.260 At this point they're not self sufficient, 00:18:38.260 --> 00:18:41.220 he has to correct them occasionally. . 00:18:41.220 --> 00:18:45.320 But this guy's just walking in the direction of the wind. 00:18:45.320 --> 00:18:50.420 I think here the wind slows down a little bit, so lose momentum, 00:18:50.420 --> 00:18:53.170 then picks up. 00:18:53.170 --> 00:18:54.600 He does this in the Netherlands. 00:18:54.600 --> 00:18:56.470 He's got this crazy beach. 00:18:56.470 --> 00:18:59.620 Here's a much bigger one. 00:18:59.620 --> 00:19:03.160 It's got a lot more wind collectors at the top. 00:19:03.160 --> 00:19:05.060 So these are all self powered. 00:19:05.060 --> 00:19:10.330 Originally he pushed them along, but now they've 00:19:10.330 --> 00:19:12.540 learned to move by themselves. 00:19:12.540 --> 00:19:16.540 And very cool legs-- I want to talk more about the leg 00:19:16.540 --> 00:19:21.050 linkage, because that's the more mathematical part. 00:19:21.050 --> 00:19:23.900 So this is his original drawing of the leg. 00:19:23.900 --> 00:19:25.580 It's a pretty simple idea. 00:19:25.580 --> 00:19:27.350 It's a pretty simple linkage. 00:19:27.350 --> 00:19:34.740 You've got a quadrilateral here, 4 bar linkage, C-K-J-B, 00:19:34.740 --> 00:19:37.090 and you've got a rigid triangle attached here. 00:19:37.090 --> 00:19:40.670 Then another quadrilateral here, and then a rigid triangle 00:19:40.670 --> 00:19:41.750 at the bottom. 00:19:41.750 --> 00:19:44.120 So it's really just two quadrilaterals 00:19:44.120 --> 00:19:45.980 connected by two rigid triangles. 00:19:45.980 --> 00:19:48.230 And so it's a one degree of freedom mechanism. 00:19:48.230 --> 00:19:50.370 As you flex this quad, this one has 00:19:50.370 --> 00:19:53.960 to adapt because that quad controls all three 00:19:53.960 --> 00:19:56.000 of these points because of the rigid triangles, 00:19:56.000 --> 00:19:58.400 and so the fourth one is determined. 00:19:58.400 --> 00:19:59.980 And then the crank is here. 00:20:02.880 --> 00:20:07.850 So this says vaste printer, which means fixed points. 00:20:07.850 --> 00:20:09.940 So you've got these two fixed points-- 00:20:09.940 --> 00:20:11.810 this is actually also fixed. 00:20:11.810 --> 00:20:14.360 This is just an x to y ratio. 00:20:14.360 --> 00:20:17.240 So these two guys are pinned, and this guy 00:20:17.240 --> 00:20:24.150 is rotating around this fixed point-- that's the actuation, 00:20:24.150 --> 00:20:28.130 and that causes this quadrilateral to fold. 00:20:28.130 --> 00:20:31.070 We are basically setting the center of rotation of this guy 00:20:31.070 --> 00:20:32.760 relative to this guy by measuring 00:20:32.760 --> 00:20:34.270 these lengths, a and l. 00:20:36.930 --> 00:20:40.170 He has these numbers, which he calls the 11 holy numbers. 00:20:40.170 --> 00:20:42.820 It's a little cause because there's actually 13 of them. 00:20:42.820 --> 00:20:45.305 There are 11 edge lengths, and then there's 00:20:45.305 --> 00:20:47.180 these two, which are not really edge lengths. 00:20:47.180 --> 00:20:49.810 It's just a measurement of how to put the center of rotation 00:20:49.810 --> 00:20:53.330 relative to this guy. 00:20:53.330 --> 00:20:55.160 And the reason why x and y matter here 00:20:55.160 --> 00:20:57.410 is because there's actually a floor at the bottom, 00:20:57.410 --> 00:21:01.650 and you have to orient yourself correctly relative to the floor 00:21:01.650 --> 00:21:04.300 so you can actually walk. 00:21:04.300 --> 00:21:09.310 So he found these numbers by genetic algorithm. 00:21:09.310 --> 00:21:12.170 In the early days there weren't-- I guess, 00:21:12.170 --> 00:21:14.590 before there were a lot of genetic algorithms around. 00:21:14.590 --> 00:21:16.349 So he tried lots of different values, 00:21:16.349 --> 00:21:17.890 simulated them on the computer, found 00:21:17.890 --> 00:21:21.420 which ones walked best, basically. 00:21:21.420 --> 00:21:23.830 Ones that were walking well in various kinds of terrain 00:21:23.830 --> 00:21:25.750 would survive to the next generation, 00:21:25.750 --> 00:21:28.260 try various random mutations, perturbing 00:21:28.260 --> 00:21:32.350 all of these numbers, and eventually finding this set 00:21:32.350 --> 00:21:35.690 to be especially good in simulation, 00:21:35.690 --> 00:21:38.930 and that's what all of the built linkages are based on. 00:21:42.160 --> 00:21:44.730 This is his computer drawing, plus some markings 00:21:44.730 --> 00:21:46.390 on top of it. 00:21:46.390 --> 00:21:48.800 If you take this linkage and go through the one 00:21:48.800 --> 00:21:51.260 degree of freedom, this point, of course, 00:21:51.260 --> 00:21:52.730 moves along a circle. 00:21:52.730 --> 00:21:55.860 Then we're just drawing lots of instances on top of each other. 00:21:55.860 --> 00:21:58.750 The foot-- this guy down here moves 00:21:58.750 --> 00:22:01.790 in this pattern, which is pretty cool. 00:22:01.790 --> 00:22:05.240 This is basically lifting your foot up off the ground, 00:22:05.240 --> 00:22:08.199 pushing it down, and then dragging along the floor. 00:22:08.199 --> 00:22:09.990 That's how you walk, if you think about it. 00:22:09.990 --> 00:22:14.540 You push and propel yourself forward. 00:22:14.540 --> 00:22:16.710 So this scraping along the floor, 00:22:16.710 --> 00:22:18.330 if you've got a high friction foot, 00:22:18.330 --> 00:22:21.100 will actually move you forward. 00:22:21.100 --> 00:22:24.162 In some sense, all of this design-- and the one way 00:22:24.162 --> 00:22:25.620 you would choose these numbers-- is 00:22:25.620 --> 00:22:27.590 to guarantee that A- you get lift, 00:22:27.590 --> 00:22:30.840 because if you have non-uniform terrain, you want to walk. 00:22:30.840 --> 00:22:33.730 He likes to call this mechanism better than the wheel. 00:22:33.730 --> 00:22:37.000 It's like every invention of something like the wheel, 00:22:37.000 --> 00:22:39.100 but even better because wheels don't really 00:22:39.100 --> 00:22:40.990 work if you have steps on the ground. 00:22:40.990 --> 00:22:42.740 Walking is superior, cause you could 00:22:42.740 --> 00:22:46.050 walk over bumps and steps and things like that. 00:22:46.050 --> 00:22:47.810 So you want a good amount of lift, 00:22:47.810 --> 00:22:51.320 and then you want a good amount, and ideally-- you 00:22:51.320 --> 00:22:52.790 can kind of see this here. 00:22:52.790 --> 00:22:55.600 These are uniformly spaced as you rotate around, 00:22:55.600 --> 00:22:57.590 and these are all-- they look uniformly 00:22:57.590 --> 00:23:00.560 spaced on the bottom, which means you're applying 00:23:00.560 --> 00:23:04.420 uniform force all along the floor contact. 00:23:04.420 --> 00:23:10.120 It's not uniform here, but it's uniform down here, mostly. 00:23:10.120 --> 00:23:12.370 It's not actually straight in the true mechanism. 00:23:12.370 --> 00:23:15.630 I think this is another drawing, two other people's 00:23:15.630 --> 00:23:17.467 drawings of the same mechanism. 00:23:17.467 --> 00:23:19.050 Here the triangles have been filled in 00:23:19.050 --> 00:23:23.900 to be-- I think this is a design for a metal version. 00:23:23.900 --> 00:23:25.860 And then this is a mathematical simulation. 00:23:25.860 --> 00:23:30.480 Again, uniformly spaced rotary cranking here. 00:23:30.480 --> 00:23:32.410 You get to see the spacing of the points here. 00:23:32.410 --> 00:23:35.970 So it's moving a little bit faster over the lift, 00:23:35.970 --> 00:23:38.350 and then pushing down and spending a fair amount of time 00:23:38.350 --> 00:23:42.940 here, and pretty uniformly applying . force. 00:23:42.940 --> 00:23:45.430 Also not completely flat. 00:23:45.430 --> 00:23:47.490 It looks like it's convex-- in fact 00:23:47.490 --> 00:23:50.180 there's a little upward dip here. 00:23:50.180 --> 00:23:53.850 But it's pretty good, pretty straight, and a really good 00:23:53.850 --> 00:23:57.160 walking mechanism, possibly the best within this family. 00:23:57.160 --> 00:24:00.770 Depends how you define best, of course. 00:24:00.770 --> 00:24:05.370 And here's a fun animation of how 00:24:05.370 --> 00:24:07.860 the parts of the linkage and adding multiple legs, which 00:24:07.860 --> 00:24:09.910 is I wanted to show it. 00:24:09.910 --> 00:24:12.110 So you start with one 4 bar mechanism. 00:24:12.110 --> 00:24:15.570 These are probably not quite the right lengths, but close. 00:24:15.570 --> 00:24:19.090 You add two rigid triangles it's a little 00:24:19.090 --> 00:24:22.190 add another quadrilateral on there you see the spitting 00:24:22.190 --> 00:24:25.940 part offset relative to that and then here you 00:24:25.940 --> 00:24:28.250 get the trajectory of the foot. 00:24:28.250 --> 00:24:31.660 And you see on a terrain how it would move you forward, 00:24:31.660 --> 00:24:34.170 especially if you have multiple legs, then one of them 00:24:34.170 --> 00:24:37.650 is always in contact so you're moving forward. 00:24:37.650 --> 00:24:40.570 And then if you want to be even cooler, 00:24:40.570 --> 00:24:45.120 you go to three dimensions, use even more legs. 00:24:45.120 --> 00:24:48.290 And you have a little crank in the center that 00:24:48.290 --> 00:24:53.229 offsets the timing of the feet so they're all sporadically 00:24:53.229 --> 00:24:54.770 moving, and that's essentially what's 00:24:54.770 --> 00:24:59.100 going on in this structure where they're nicely, 00:24:59.100 --> 00:25:01.800 evenly distributed. 00:25:01.800 --> 00:25:03.810 There's a central crank here that 00:25:03.810 --> 00:25:08.900 just offsets the timing of all these feet, 00:25:08.900 --> 00:25:11.174 and it's super cool. 00:25:11.174 --> 00:25:12.590 I mean, some ways the coolest part 00:25:12.590 --> 00:25:16.360 is how it looks like a real animal walking-- 00:25:16.360 --> 00:25:18.764 the funny timing, and the way you lift and then spend 00:25:18.764 --> 00:25:20.930 most your time with a foot on the ground, and so on. 00:25:20.930 --> 00:25:25.280 So, fun stuff. 00:25:25.280 --> 00:25:26.004 Let's see. 00:25:26.004 --> 00:25:27.420 Of course, other people have tried 00:25:27.420 --> 00:25:28.880 to improve it in various ways. 00:25:28.880 --> 00:25:32.410 There's a couple of papers proposing alternate walking 00:25:32.410 --> 00:25:34.030 linkage mechanisms. 00:25:34.030 --> 00:25:37.490 This is the Theo Jansen version, this 00:25:37.490 --> 00:25:40.750 is another version by Ghassaei. 00:25:40.750 --> 00:25:44.060 And it's actually a simpler linkage, just 00:25:44.060 --> 00:25:48.100 a 4 bar linkage over here too rigid triangles and another 4 00:25:48.100 --> 00:25:49.360 bar linkage. 00:25:49.360 --> 00:25:51.160 I guess roughly the same number of parts 00:25:51.160 --> 00:25:54.352 but, I think, one fewer edge if I counted correctly, 00:25:54.352 --> 00:25:56.060 just because of the way they're combined. 00:25:59.160 --> 00:26:02.960 And some fun things as you get-- this is the foot pattern down 00:26:02.960 --> 00:26:05.890 here, still has a nice straight part, looks pretty uniform, 00:26:05.890 --> 00:26:07.240 speeds up for the lift. 00:26:07.240 --> 00:26:12.506 Does not lift as high-- this one has a 30 percent higher step, 00:26:12.506 --> 00:26:13.630 which is considered better. 00:26:13.630 --> 00:26:15.900 You could move over rougher terrain. 00:26:15.900 --> 00:26:18.750 This one has other Jansen's has other advantages, 00:26:18.750 --> 00:26:20.807 like if you have two of them in the same plane 00:26:20.807 --> 00:26:22.390 they will not collide with each other. 00:26:22.390 --> 00:26:24.682 Two legs reflected around each other. 00:26:24.682 --> 00:26:25.890 This one, they would collide. 00:26:25.890 --> 00:26:28.982 You have to move them to separate planes. 00:26:28.982 --> 00:26:30.440 But some fun things here as you get 00:26:30.440 --> 00:26:34.170 that this foot pattern is symmetric, down the y-axis, 00:26:34.170 --> 00:26:38.030 and you get much less variation in the height, 00:26:38.030 --> 00:26:42.680 sort of more uniform, and a little less 00:26:42.680 --> 00:26:46.956 velocity variation on the stride part, the walking. 00:26:46.956 --> 00:26:48.330 And then the other thing, which I 00:26:48.330 --> 00:26:50.250 think was the main purpose of the design, 00:26:50.250 --> 00:26:52.690 was that the center of mass of the whole structure 00:26:52.690 --> 00:26:54.450 doesn't move very much. 00:26:54.450 --> 00:26:56.090 The blue dots here are the center 00:26:56.090 --> 00:26:59.960 of mass over the motion of the Jansen mechanism. 00:26:59.960 --> 00:27:05.471 And this one, the blue dots are much smaller areas, 85% less 00:27:05.471 --> 00:27:05.970 movement. 00:27:05.970 --> 00:27:09.920 I'm not sure of the exact definition of 85% there. 00:27:12.850 --> 00:27:15.400 This should be, if you were like riding on the thing, 00:27:15.400 --> 00:27:18.740 you should feel less wobble, in principle. 00:27:18.740 --> 00:27:22.110 They did build one of these out of PVC tubing as well, 00:27:22.110 --> 00:27:25.260 but I haven't seen any video so I don't know how it looks. 00:27:25.260 --> 00:27:27.830 Be kind of neat to see 00:27:27.830 --> 00:27:34.000 So this is a two fat example, apparently. 00:27:34.000 --> 00:27:39.175 Said this one was over eating, and so it tends to fall over. 00:27:39.175 --> 00:27:41.550 So it's a challenge to get these things to work and you-- 00:27:41.550 --> 00:27:44.250 I can't imagine how many hours he spends building these, 00:27:44.250 --> 00:27:47.680 and then for one of them to not live. 00:27:47.680 --> 00:27:52.759 He does things like, he'll run these experiments. 00:27:52.759 --> 00:27:54.300 Like genetic algorithms is one thing, 00:27:54.300 --> 00:27:56.070 but he'll do the real genetics thing 00:27:56.070 --> 00:27:58.820 of building lots of different variations, 00:27:58.820 --> 00:28:01.920 put them on the sand, see which one goes the fastest. 00:28:01.920 --> 00:28:03.450 The other ones he kills. 00:28:03.450 --> 00:28:05.750 The one who went was the fastest survives 00:28:05.750 --> 00:28:06.740 to the next generation. 00:28:06.740 --> 00:28:09.610 But each experiment must take hours and hours-- 00:28:09.610 --> 00:28:10.360 really impressive. 00:28:13.020 --> 00:28:16.140 So there are issues in getting these things to work, 00:28:16.140 --> 00:28:19.590 and just having the leg mechanism is one thing. 00:28:19.590 --> 00:28:21.800 These guys have a lot of very interesting mechanisms 00:28:21.800 --> 00:28:25.150 beyond the legs, in order to-- they're sensors, essentially. 00:28:25.150 --> 00:28:28.830 They're purely mechanical-- there's no electronics here 00:28:28.830 --> 00:28:30.900 that are physical, I should say. 00:28:30.900 --> 00:28:33.550 There are sensors to detect things like the wind direction. 00:28:33.550 --> 00:28:37.310 Recently we had the honor of co-exhibiting with Theo Jansen. 00:28:37.310 --> 00:28:40.850 So this is some of our curve crease paper folding in this, 00:28:40.850 --> 00:28:42.570 or some of his structures. 00:28:42.570 --> 00:28:45.510 Some of these small kits and photographs of his larger ones 00:28:45.510 --> 00:28:50.640 were at this show in Stonybrook, New York, earlier this year. 00:28:50.640 --> 00:28:52.610 It's kind of fun. 00:28:52.610 --> 00:28:56.460 And while we're on the topic of kinetic sculpture, 00:28:56.460 --> 00:28:58.685 I have to show Arthur Ganson. 00:28:58.685 --> 00:29:02.680 It's funny, Ganson and Jensen are so close. 00:29:02.680 --> 00:29:06.480 This is our local kinetic expert-- kinetic sculpture 00:29:06.480 --> 00:29:06.980 expert. 00:29:06.980 --> 00:29:10.950 These are four examples I took from the web, 00:29:10.950 --> 00:29:13.020 but all of these examples and many, many more 00:29:13.020 --> 00:29:13.900 at the MIT Museum. 00:29:13.900 --> 00:29:16.820 If you haven't seen his exhibit, you must. 00:29:16.820 --> 00:29:21.960 He hand builds lots of gears and cranks, 00:29:21.960 --> 00:29:24.330 and uses them to build crazy things. 00:29:24.330 --> 00:29:28.120 This is called the self-oiling machine. 00:29:28.120 --> 00:29:30.430 it just grabbed a bunch of oil and it' 00:29:30.430 --> 00:29:33.460 about to pour all that oil onto itself. 00:29:33.460 --> 00:29:35.935 It's really gross. 00:29:35.935 --> 00:29:40.190 There it goes, pour-- and should never get rusty. 00:29:40.190 --> 00:29:45.700 This is a very simple mechanism that's just the random and cool 00:29:45.700 --> 00:29:48.057 how the changes kind of move, wiggles around. 00:29:48.057 --> 00:29:49.640 It's a force to be planar, essentially 00:29:49.640 --> 00:29:51.360 because of that chain. 00:29:51.360 --> 00:29:54.350 This one-- you've got various handmade gears and so on, 00:29:54.350 --> 00:29:59.510 and hand cut pieces of rough paper, to make them fly. 00:29:59.510 --> 00:30:02.460 And this is another beautiful, relatively simple one. 00:30:02.460 --> 00:30:06.690 You've got a big counterweight to make this chair almost zero 00:30:06.690 --> 00:30:09.871 weight, and then there's a little cat here 00:30:09.871 --> 00:30:11.620 that's just rigid and going back and forth 00:30:11.620 --> 00:30:15.350 and it causes that chair to dance around in very cool ways. 00:30:19.367 --> 00:30:20.200 It's pretty amazing. 00:30:20.200 --> 00:30:24.310 He lives locally, and has made a huge array of sculptures, 00:30:24.310 --> 00:30:27.570 one which I don't have a video of, which is a lot of fun, 00:30:27.570 --> 00:30:30.700 is just-- there's a big crank, and then 00:30:30.700 --> 00:30:33.510 a series of like 20 to 1 gear reductions, 00:30:33.510 --> 00:30:35.440 and at the other end the turning element 00:30:35.440 --> 00:30:37.095 is in a solid block of concrete. 00:30:37.095 --> 00:30:38.970 And there's some computation, like it'll 00:30:38.970 --> 00:30:42.950 take 100,000 years before that before it turns that concrete 00:30:42.950 --> 00:30:45.760 around, which it can't do. 00:30:45.760 --> 00:30:47.585 Initially the gears are going pretty fast, 00:30:47.585 --> 00:30:48.970 and then, slower and slower, and then 00:30:48.970 --> 00:30:50.850 you just can't see them for like five iterations 00:30:50.850 --> 00:30:52.270 and then it's wedged into concrete 00:30:52.270 --> 00:31:00.050 It's a fun, practical example-- practical Anyway, lots 00:31:00.050 --> 00:31:03.840 of very cool kinetic sculpture and all somehow based 00:31:03.840 --> 00:31:08.150 around linkages although not necessarily strictly following 00:31:08.150 --> 00:31:08.890 our definition. 00:31:08.890 --> 00:31:12.680 And next week we'll be talking about protein folding.