WEBVTT 00:00:03.179 --> 00:00:05.470 PROFESSOR: So we are honored to have Professor Tomohiro 00:00:05.470 --> 00:00:09.490 Tachi here today visiting from the University of Tokyo. 00:00:09.490 --> 00:00:13.370 We first met in SIGGRAPH 2006, which 00:00:13.370 --> 00:00:16.810 was in Boston, big graphics conference, 00:00:16.810 --> 00:00:18.640 when he was a PhD student. 00:00:18.640 --> 00:00:21.800 And Tomohiro has very quickly become a star 00:00:21.800 --> 00:00:24.290 in the area of computational origami, 00:00:24.290 --> 00:00:27.950 and it's really exciting to see in particular his perspective 00:00:27.950 --> 00:00:30.030 coming from an architecture background 00:00:30.030 --> 00:00:31.631 and how that influences his work. 00:00:31.631 --> 00:00:33.630 He's going to talk about lots of exciting things 00:00:33.630 --> 00:00:37.190 in the context of architectural origami for our grand finale 00:00:37.190 --> 00:00:37.690 lecture. 00:00:37.690 --> 00:00:39.962 So please welcome Tomohiro. 00:00:39.962 --> 00:00:42.560 [APPLAUSE] 00:00:42.560 --> 00:00:45.570 PROFESSOR: Thank you very much, Eric. 00:00:45.570 --> 00:00:49.540 So I'm Tomohiro Tachi from the University of Tokyo, 00:00:49.540 --> 00:00:53.110 and I'm going to talk about architecture origami, which 00:00:53.110 --> 00:00:55.750 is architectural form design systems based 00:00:55.750 --> 00:00:58.010 in computational origami. 00:00:58.010 --> 00:01:01.650 So first, introduction. 00:01:01.650 --> 00:01:06.960 I have been doing origami as a hobby, as well as my research. 00:01:06.960 --> 00:01:13.140 Well, it was before my research started that I began folding. 00:01:13.140 --> 00:01:18.980 These are examples of folding using a traditional way 00:01:18.980 --> 00:01:26.480 of box pleating or a 3D curved surface. 00:01:26.480 --> 00:01:30.730 This is pure origami, and there is also 00:01:30.730 --> 00:01:33.860 applied origami, which is applying 00:01:33.860 --> 00:01:36.400 origami for engineering purposes. 00:01:36.400 --> 00:01:40.930 For example, we can use sheet folding 00:01:40.930 --> 00:01:48.590 for manufacturing a 3D surface or use it 00:01:48.590 --> 00:01:52.220 for increasing the structure's stiffness. 00:01:52.220 --> 00:01:58.050 These are examples of a formed 3D surface 00:01:58.050 --> 00:02:01.600 from a sheet of metal. 00:02:01.600 --> 00:02:05.850 We can use a dynamic property of origami 00:02:05.850 --> 00:02:09.940 for deployable structure, such as this 00:02:09.940 --> 00:02:16.910 is folding of solar panels used for spatial structures. 00:02:16.910 --> 00:02:20.940 A good thing about origami is that unlike the truss structure 00:02:20.940 --> 00:02:24.280 or scissors structure, you can form basically 00:02:24.280 --> 00:02:26.730 a continuous surface, and that continuity 00:02:26.730 --> 00:02:32.980 is preserved all the way of the transformation. 00:02:32.980 --> 00:02:37.060 So this makes origami potentially 00:02:37.060 --> 00:02:39.880 useful for adaptive environment, which 00:02:39.880 --> 00:02:44.600 includes context customized design or personal design 00:02:44.600 --> 00:02:49.380 so that you can make some kind of origami structures 00:02:49.380 --> 00:02:54.850 that fit customized to your body or customised to the design 00:02:54.850 --> 00:02:56.300 context. 00:02:56.300 --> 00:03:01.490 And you can, of course, do fabrication oriented design 00:03:01.490 --> 00:03:07.020 so that you can efficiently use the material 00:03:07.020 --> 00:03:11.370 to build the 3D structure that you want. 00:03:11.370 --> 00:03:15.990 What I'm proposing with the architectural origami, 00:03:15.990 --> 00:03:18.960 it's different from origami architecture. 00:03:18.960 --> 00:03:23.670 I have been using the word "origami architecture," which 00:03:23.670 --> 00:03:27.895 might mean the application of origami to design. 00:03:32.980 --> 00:03:37.160 So this is the direction of application. 00:03:37.160 --> 00:03:40.350 We start from some kind of shape or pattern 00:03:40.350 --> 00:03:45.810 and see how it is folded or see the behavior 00:03:45.810 --> 00:03:52.070 by simulating, or directly applying in physical world. 00:03:52.070 --> 00:03:59.460 But that can result in very restricted design 00:03:59.460 --> 00:04:02.120 because origami is very constrained, 00:04:02.120 --> 00:04:04.950 and that would just result in just 00:04:04.950 --> 00:04:08.820 a copy of some known origami pattern to design. 00:04:08.820 --> 00:04:14.860 Or sometimes, there are lots of origami inspired designs 00:04:14.860 --> 00:04:19.709 that don't use any of the origami properties that 00:04:19.709 --> 00:04:22.860 might be useful for engineering purpose. 00:04:22.860 --> 00:04:25.960 So what I am proposing with the word 00:04:25.960 --> 00:04:30.400 "architectural origami" is that origami theory for design 00:04:30.400 --> 00:04:33.800 and a design system that uses that. 00:04:33.800 --> 00:04:37.880 So the idea is in this direction. 00:04:37.880 --> 00:04:41.820 We want to get some kind of designed property, 00:04:41.820 --> 00:04:48.910 like how it behaves in 3D or in time, also, 00:04:48.910 --> 00:04:59.360 and we want to extract the characteristics of origami 00:04:59.360 --> 00:05:05.630 and obtain the solution from that required condition, 00:05:05.630 --> 00:05:07.730 and also design contexts that are 00:05:07.730 --> 00:05:12.600 given by the design purpose. 00:05:12.600 --> 00:05:13.740 This is the outline. 00:05:13.740 --> 00:05:16.750 First is Origamizer. 00:05:16.750 --> 00:05:21.490 I'm going to talk about basically the software I've 00:05:21.490 --> 00:05:24.630 been developing, the systems. 00:05:24.630 --> 00:05:25.380 One is Origamizer. 00:05:46.400 --> 00:05:53.370 Some of you have attended Eric's class, 00:05:53.370 --> 00:05:57.680 so the theoretic side of the Origamizer altorithm, 00:05:57.680 --> 00:06:03.710 but I'm going to talk about the software itself 00:06:03.710 --> 00:06:06.780 and the algorithm that efficiently works. 00:06:06.780 --> 00:06:09.740 So first, Origamizer. 00:06:09.740 --> 00:06:14.780 This is the software is available on my web page. 00:06:14.780 --> 00:06:17.255 It's freely available. 00:06:21.780 --> 00:06:23.210 It's about origami design. 00:06:23.210 --> 00:06:27.870 This is my oldest work as research. 00:06:27.870 --> 00:06:36.130 We had a tree method or circle river packing method 00:06:36.130 --> 00:06:39.190 developed by Meguro and Lang, which 00:06:39.190 --> 00:06:46.590 is using stick figure for representing the model 00:06:46.590 --> 00:06:52.590 and designed by packing circles and rivers. 00:06:52.590 --> 00:06:57.320 I think Jason has talked in the class. 00:06:57.320 --> 00:07:02.290 So this is existing method for origami design, 00:07:02.290 --> 00:07:05.520 and what I wanted to do is make 3D 00:07:05.520 --> 00:07:08.470 instead of this kind of 1D figure. 00:07:08.470 --> 00:07:12.760 So this is what we can get from the circle river packing 00:07:12.760 --> 00:07:17.650 method, and with Origamizer or Freeform Origami, 00:07:17.650 --> 00:07:22.382 we can get this kind of 3D form. 00:07:22.382 --> 00:07:24.340 You can say that what you see is what you fold. 00:07:30.900 --> 00:07:38.540 In this case, what you see is not what you fold, actually. 00:07:38.540 --> 00:07:43.060 So I manually designed something like that. 00:07:43.060 --> 00:07:45.760 It's a laptop PC. 00:07:45.760 --> 00:07:48.090 Well, I think it's the most complicated one. 00:07:48.090 --> 00:07:53.450 So for example, you can see there is an RGB output here 00:07:53.450 --> 00:07:57.710 and USB outputs here, and this is the touch pad, 00:07:57.710 --> 00:08:00.950 and this is keyboard. 00:08:00.950 --> 00:08:02.930 Good thing about that is the function key 00:08:02.930 --> 00:08:07.940 is a little bit narrower than the normal key here. 00:08:07.940 --> 00:08:11.760 Anyway, this is what I've been working on. 00:08:11.760 --> 00:08:16.810 Well, I think it was my hobby, but I 00:08:16.810 --> 00:08:22.590 was trying to make some kind of box pleating to make anything. 00:08:22.590 --> 00:08:26.040 And also, this is more three dimensional figure. 00:08:29.900 --> 00:08:37.000 The body is represented basically by blocks of cubes. 00:08:37.000 --> 00:08:45.270 And I did more purely geometric things with concave vertex. 00:08:45.270 --> 00:08:49.520 This was very important for me. 00:08:49.520 --> 00:08:54.150 It's very easy to make a convex polyhedron with paper. 00:08:54.150 --> 00:08:58.130 Just wrapping paper will result in folding. 00:08:58.130 --> 00:09:04.130 But it's very hard to make concave vertex. 00:09:04.130 --> 00:09:12.710 When I finished this model, I thought everything is possible, 00:09:12.710 --> 00:09:18.470 so I did a software to do this for me. 00:09:21.410 --> 00:09:25.070 The program is to realize arbitrarily given polyhedra 00:09:25.070 --> 00:09:29.040 surface with a developable surface by folding. 00:09:31.610 --> 00:09:33.700 The geometric constraints is, of course, 00:09:33.700 --> 00:09:37.215 developability so that it can be folded from a sheet of paper. 00:09:40.180 --> 00:09:42.650 In this case, we forget about continuous motion 00:09:42.650 --> 00:09:45.980 from a sheet of paper to the resulting state. 00:09:45.980 --> 00:09:51.640 We believe that physically it exists, so it works. 00:09:54.640 --> 00:09:59.550 It can be applied in engineering sense for fabrication 00:09:59.550 --> 00:10:02.630 by folding and bending. 00:10:02.630 --> 00:10:06.630 We put arbitrary polyhedron and get the crease pattern, 00:10:06.630 --> 00:10:09.640 and by only folding this part, you 00:10:09.640 --> 00:10:15.130 get the folded state that's the same as this arbitrarily given 00:10:15.130 --> 00:10:17.730 polyhedron. 00:10:17.730 --> 00:10:20.640 The idea is to use tuck. 00:10:20.640 --> 00:10:32.150 This is a given polyhedron, and this grey area is our tucks. 00:10:32.150 --> 00:10:35.860 By folding a tuck, it's hidden. 00:10:35.860 --> 00:10:40.160 It's flat folded and it's hidden behind a polyhedron's surface, 00:10:40.160 --> 00:10:43.960 like in this movie. 00:10:43.960 --> 00:10:46.560 When it's in developed state, it forms 00:10:46.560 --> 00:10:51.470 a plane with a surface polyhedron. 00:10:51.470 --> 00:10:54.760 This is good because we can make a negative curvature 00:10:54.760 --> 00:11:00.270 vertex, a concave or negative curvature vertex. 00:11:07.730 --> 00:11:15.470 The basic algorithm is to start from making the problem 00:11:15.470 --> 00:11:22.070 into laying out the surface polygons onto a plane. 00:11:22.070 --> 00:11:27.660 By properly aligning, we can get the edge tucking molecule that 00:11:27.660 --> 00:11:32.180 folds the edge to edge and vertex tucking molecule which 00:11:32.180 --> 00:11:39.060 folds vertices to vertex, and we can pack them and tesselate 00:11:39.060 --> 00:11:43.750 the surface with these elements. 00:11:43.750 --> 00:11:47.692 We can parameterize this configuration, 00:11:47.692 --> 00:11:53.640 and we can solve it by solving non-linear equation 00:11:53.640 --> 00:11:55.930 and inequality conditions. 00:11:55.930 --> 00:11:57.750 That's the basic idea. 00:11:57.750 --> 00:12:01.790 What we have is geometric constraints, 00:12:01.790 --> 00:12:06.540 which are represented by equations. 00:12:06.540 --> 00:12:09.130 So these are the equations, where 00:12:09.130 --> 00:12:14.140 this one represents the total sum, sums up 2 pi, 00:12:14.140 --> 00:12:24.010 and this shows that this forms a closed loop for each vertex. 00:12:24.010 --> 00:12:29.090 We assign these conditions for each vertex, 00:12:29.090 --> 00:12:34.770 and we solve that by two step linear mapping, which 00:12:34.770 --> 00:12:41.530 is basically solving this equation and this equation. 00:12:41.530 --> 00:12:46.040 Good thing about that is that it becomes linear, 00:12:46.040 --> 00:12:49.480 which means that you can pre-compute 00:12:49.480 --> 00:12:56.000 and you can go around the solution space. 00:12:56.000 --> 00:13:00.990 I will show you later the design system. 00:13:12.130 --> 00:13:17.790 I don't go into details, but there are several inequality 00:13:17.790 --> 00:13:22.745 conditions, which are for making crease pattern 00:13:22.745 --> 00:13:30.430 and for making it to fit into 3D configuration of the polyhedron 00:13:30.430 --> 00:13:36.150 and tuck property, which is assumed 00:13:36.150 --> 00:13:42.820 place of the tuck that's hidden. 00:13:42.820 --> 00:13:43.790 So this is a system. 00:13:46.980 --> 00:13:51.280 You put the input here in this window, 00:13:51.280 --> 00:13:58.500 and you have the layout of the surface polygons, 00:13:58.500 --> 00:14:00.530 and then you get the crease pattern like that. 00:14:04.560 --> 00:14:07.580 It automatically generates the crease pattern, 00:14:07.580 --> 00:14:14.080 and here you see it's edited in real time. 00:14:14.080 --> 00:14:15.340 By the way, this is real time. 00:14:21.920 --> 00:14:26.130 The linear equations are represented 00:14:26.130 --> 00:14:34.790 by linear equations, so you can pre-compute the linear matrix 00:14:34.790 --> 00:14:40.135 and then you can calculate the configuration interactively. 00:14:42.720 --> 00:14:46.515 You can also do some kind of boundary editing. 00:14:46.515 --> 00:14:47.901 Do you have questions? 00:14:47.901 --> 00:14:48.484 AUDIENCE: Yes. 00:14:48.484 --> 00:14:51.890 Does this maximize the amout of paper on the outside? 00:14:51.890 --> 00:14:56.910 PROFESSOR: It's about maximizing the size 00:14:56.910 --> 00:15:00.240 of the model with respect to the paper size. 00:15:00.240 --> 00:15:00.770 It's not. 00:15:04.670 --> 00:15:06.890 Yes, you can, of course, implement 00:15:06.890 --> 00:15:09.280 some kind of optimization onto the program, 00:15:09.280 --> 00:15:17.510 but I keep it more free so that the software user can search 00:15:17.510 --> 00:15:21.855 within the solution space that satisfies these origamizing 00:15:21.855 --> 00:15:22.355 conditions. 00:15:25.510 --> 00:15:30.730 So this is a series of results that are folded. 00:15:34.870 --> 00:15:38.000 This is how to fold an origami bunny. 00:15:38.000 --> 00:15:41.620 First thing you do is to get crease pattern 00:15:41.620 --> 00:15:44.670 using Origamizer, and then fold along 00:15:44.670 --> 00:15:50.410 the given crease pattern like this. 00:15:55.460 --> 00:15:57.880 Interesting thing about that is that it's 00:15:57.880 --> 00:16:00.460 changing the lighting conditions. 00:16:00.460 --> 00:16:02.630 It starts from natural light and it 00:16:02.630 --> 00:16:05.640 becomes into artificial light. 00:16:05.640 --> 00:16:07.885 It takes about 10 hours or something. 00:16:10.740 --> 00:16:11.940 And it's done. 00:16:11.940 --> 00:16:16.935 And I have this one. 00:16:16.935 --> 00:16:20.400 [APPLAUSE] 00:16:25.360 --> 00:16:29.440 There are two existing models, not one, 00:16:29.440 --> 00:16:31.745 so it's very easy to reproduce. 00:16:40.060 --> 00:16:44.380 One thing that I'm doing with Eric 00:16:44.380 --> 00:16:48.370 is to prove that anything is possible. 00:16:48.370 --> 00:16:53.280 The software is a little bit approximation, 00:16:53.280 --> 00:16:56.650 or it doesn't work in some cases, 00:16:56.650 --> 00:17:02.520 so we are doing really make it possible. 00:17:02.520 --> 00:17:04.900 This is the part of the Origamizer, 00:17:04.900 --> 00:17:08.109 so if you have questions for Origamizer. 00:17:08.109 --> 00:17:09.183 Yes? 00:17:09.183 --> 00:17:12.850 AUDIENCE: So for the input of the model 00:17:12.850 --> 00:17:16.894 that you can use, does it take [INAUDIBLE], the format? 00:17:16.894 --> 00:17:19.023 AUDIENCE: It's polygons. 00:17:19.023 --> 00:17:20.931 AUDIENCE: What's the file format that you 00:17:20.931 --> 00:17:23.859 have to import in in order to-- 00:17:23.859 --> 00:17:27.300 PROFESSOR: Format of 3D input. 00:17:27.300 --> 00:17:34.770 It's basically OBJ file that this particular software 00:17:34.770 --> 00:17:37.360 accepts, but basically, it's OK if it's 00:17:37.360 --> 00:17:42.900 a polyhedral surface with actually any topology 00:17:42.900 --> 00:17:48.110 because you can basically assign the boundary of paper onto it 00:17:48.110 --> 00:17:51.420 so that it always becomes a disk. 00:17:51.420 --> 00:17:53.814 Yes? 00:17:53.814 --> 00:17:57.742 AUDIENCE: You mentioned that earlier on in the process, 00:17:57.742 --> 00:18:01.179 you need to properly lay out the polygons onto the plane. 00:18:01.179 --> 00:18:03.143 What constitutes proper layout? 00:18:08.460 --> 00:18:16.460 PROFESSOR: Your question is, what is a proper layout? 00:18:16.460 --> 00:18:22.610 This is basically given by these conditions. 00:18:28.700 --> 00:18:36.230 In an intuitive way, if you lay out polygons, 00:18:36.230 --> 00:18:42.290 I want to fold this line onto this line with, actually, 00:18:42.290 --> 00:18:46.590 a single line, so you have to keep 00:18:46.590 --> 00:18:50.220 this symmetric against some line. 00:18:50.220 --> 00:18:52.825 So that gives a constraint. 00:18:58.530 --> 00:19:04.000 This is the basic equations that you have to solve, 00:19:04.000 --> 00:19:08.360 and there are several others, such as the boundary 00:19:08.360 --> 00:19:12.170 should be convex because we don't want to end up 00:19:12.170 --> 00:19:18.770 in some kind of very complex boundary. 00:19:18.770 --> 00:19:22.110 If it's foldable form a convex paper, 00:19:22.110 --> 00:19:24.740 then it's foldable from a square, 00:19:24.740 --> 00:19:27.910 so that's the convexity of paper. 00:19:27.910 --> 00:19:32.195 No intersection, of course, between polygons. 00:19:35.240 --> 00:19:38.840 It's a little bit difficult, but when you place crease lines, 00:19:38.840 --> 00:19:42.900 then it might hit adjacent crease lines. 00:19:42.900 --> 00:19:47.680 That's pretty bad, so you have to avoid that. 00:19:47.680 --> 00:19:51.890 Also, you have to fit to the 3D surface. 00:19:51.890 --> 00:19:55.970 Sometimes a tuck that's coming inside 00:19:55.970 --> 00:20:05.000 is bad, so you have to adjust that, 00:20:05.000 --> 00:20:08.346 and in order to do that, you need a condition. 00:20:08.346 --> 00:20:10.542 AUDIENCE: Did you find that you were moving around 00:20:10.542 --> 00:20:13.714 where the edges of the [INAUDIBLE] model, 00:20:13.714 --> 00:20:18.106 or can you just start anywhere and randomly come up 00:20:18.106 --> 00:20:19.082 with a solution? 00:20:22.822 --> 00:20:24.030 PROFESSOR: It's how to solve? 00:20:24.030 --> 00:20:24.655 AUDIENCE: Yeah. 00:20:24.655 --> 00:20:30.790 Does it get a lot harder as the polygons increase? 00:20:30.790 --> 00:20:36.770 PROFESSOR: It's not a step by step approach to laying out 00:20:36.770 --> 00:20:38.700 one piece at a time, but it's more 00:20:38.700 --> 00:20:44.700 like solving all the equations all at the same time using 00:20:44.700 --> 00:20:47.700 big vector equation. 00:20:47.700 --> 00:20:50.154 AUDIENCE: Maybe what he means is, is it numerically well 00:20:50.154 --> 00:20:52.970 conditioned? 00:20:52.970 --> 00:20:54.320 PROFESSOR: No. 00:20:54.320 --> 00:20:58.396 That's why we need extra work for proving. 00:20:58.396 --> 00:21:00.020 AUDIENCE: That it's always [INAUDIBLE]. 00:21:02.900 --> 00:21:04.950 PROFESSOR: That's a question. 00:21:04.950 --> 00:21:11.900 Well, it's not always. 00:21:11.900 --> 00:21:13.960 There are sometimes that produces 00:21:13.960 --> 00:21:20.310 some kind of over constrained part and free part. 00:21:20.310 --> 00:21:23.280 AUDIENCE: Also, are the tucks under mechanical tension? 00:21:23.280 --> 00:21:27.000 Does it want to stay tucked or does it want to come apart? 00:21:29.770 --> 00:21:31.530 PROFESSOR: Physical property of the tuck. 00:21:31.530 --> 00:21:33.884 AUDIENCE: Yes. 00:21:33.884 --> 00:21:35.300 PROFESSOR: Because of the crimping 00:21:35.300 --> 00:21:44.160 that's done to each of these vertices to fit the curvature, 00:21:44.160 --> 00:21:47.230 it stays in this form. 00:21:47.230 --> 00:21:50.740 You don't have any glues or crimps here. 00:21:50.740 --> 00:21:57.791 This is actually a good property of this design method. 00:21:57.791 --> 00:21:58.290 Yes? 00:22:02.406 --> 00:22:03.780 AUDIENCE: To approximate a curve, 00:22:03.780 --> 00:22:09.850 you're breaking it down into smaller straight edge segments. 00:22:09.850 --> 00:22:12.932 Is that useful or plannable as to what the lower bound is 00:22:12.932 --> 00:22:19.610 to how granular you make the curve approximation? 00:22:19.610 --> 00:22:24.310 PROFESSOR: About the condition for the input. 00:22:28.330 --> 00:22:32.530 In this system, I don't give any kind 00:22:32.530 --> 00:22:39.440 of numerical upper boundary for the regularity 00:22:39.440 --> 00:22:44.220 of triangles or something. 00:22:44.220 --> 00:22:47.820 But I think it's good if we can do for anything, 00:22:47.820 --> 00:22:53.010 so that's why I'm working with Eric on proving. 00:23:00.276 --> 00:23:04.590 AUDIENCE: Is there a Mac version of the program? 00:23:04.590 --> 00:23:05.755 PROFESSOR: Mac version? 00:23:05.755 --> 00:23:11.970 No, it's still Windows version only. 00:23:11.970 --> 00:23:21.295 It's written with cross platform library, so sometime in future. 00:23:28.340 --> 00:23:33.840 So let's move on to the next topic and software, which 00:23:33.840 --> 00:23:34.920 is Freeform Origami. 00:23:38.470 --> 00:23:42.960 So the objective of Freeform Origami 00:23:42.960 --> 00:23:52.270 is that we don't want to spend too much time to make 00:23:52.270 --> 00:23:56.960 this kind of 3D form, and also, we 00:23:56.960 --> 00:24:02.810 want some kind of conditions, not only developability. 00:24:02.810 --> 00:24:07.840 We want some condition that most origami has. 00:24:10.560 --> 00:24:13.380 This is flat foldability so that it 00:24:13.380 --> 00:24:17.820 folds flat, and also transformability or elastic 00:24:17.820 --> 00:24:18.790 properties. 00:24:18.790 --> 00:24:23.400 We want to use these good properties from origami 00:24:23.400 --> 00:24:28.840 while we want to make some kind of freeform surface. 00:24:28.840 --> 00:24:34.030 So the approach here is start from existing origami models 00:24:34.030 --> 00:24:39.970 and then modify that to a different shape 00:24:39.970 --> 00:24:47.000 while keeping the origami conditions 00:24:47.000 --> 00:24:51.520 with direct, straightforward user interface. 00:24:51.520 --> 00:25:03.970 Here, I used triangular mesh for representing the origami model, 00:25:03.970 --> 00:25:08.490 and we represent the origami models 00:25:08.490 --> 00:25:14.070 by the vertex coordinates of the model. 00:25:14.070 --> 00:25:19.730 These variables are constrained by developability and flat 00:25:19.730 --> 00:25:27.720 foldability This is a very direct way of implementing 00:25:27.720 --> 00:25:30.765 origami deformation. 00:25:34.520 --> 00:25:37.060 First thing we want to keep is developability, 00:25:37.060 --> 00:25:40.315 which is that you can fold from a sheet of paper. 00:25:43.190 --> 00:25:45.770 In engineering sense, it means that it 00:25:45.770 --> 00:25:48.840 can be manufactured from a sheet material, which 00:25:48.840 --> 00:25:51.860 is very nice, by folding and bending only. 00:25:55.930 --> 00:25:59.660 The condition is globally represented 00:25:59.660 --> 00:26:03.800 that there exists some isometric mapping to a plane, 00:26:03.800 --> 00:26:07.250 but if the surface is a topological disk, 00:26:07.250 --> 00:26:14.590 it can be represented by a local condition of every point. 00:26:14.590 --> 00:26:20.610 So every point on the surface, the Gauss curvature is zero. 00:26:20.610 --> 00:26:26.980 Actually, we are thinking of a non-smooth surface, 00:26:26.980 --> 00:26:32.110 but we are thinking of a piecewise linear surface, which 00:26:32.110 --> 00:26:35.590 allows variation. 00:26:35.590 --> 00:26:39.610 Smooth developer surface is only allowed to be in these forms, 00:26:39.610 --> 00:26:44.280 but we have lots of different form variations 00:26:44.280 --> 00:26:45.140 if we allow creases. 00:26:51.520 --> 00:26:55.380 Even in this case, we have to think about Gauss area, 00:26:55.380 --> 00:26:57.980 and it's very easy to define Gauss area 00:26:57.980 --> 00:27:08.810 for a C2 version of surface by multiplying curvatures, 00:27:08.810 --> 00:27:16.280 but for polyhedra case, we use instead Gauss area, which 00:27:16.280 --> 00:27:23.290 is represented by this, which is 2 pi minus sum of angles 00:27:23.290 --> 00:27:25.520 around the vertex. 00:27:25.520 --> 00:27:28.950 It's a very simple way to express 00:27:28.950 --> 00:27:32.070 that it's folded from a plane. 00:27:32.070 --> 00:27:35.320 This sums up to 2 pi. 00:27:35.320 --> 00:27:39.790 This is what you can use for developability. 00:27:39.790 --> 00:27:43.420 We want to also have flat foldability for the surface. 00:27:46.940 --> 00:27:54.570 This is applicable for compactly packaging a surface 00:27:54.570 --> 00:27:57.800 from 3D to 2D and from 2D to 3D. 00:28:00.680 --> 00:28:04.580 It's also represented by isometric condition, 00:28:04.580 --> 00:28:06.270 like in developability. 00:28:06.270 --> 00:28:10.270 And also, we have a little bit of layering condition, which 00:28:10.270 --> 00:28:18.190 is actually NP complete, which is hard, 00:28:18.190 --> 00:28:24.110 but for practical purposes, we can avoid that. 00:28:26.930 --> 00:28:29.200 The first one is isometry. 00:28:29.200 --> 00:28:32.190 That can be similarly represented 00:28:32.190 --> 00:28:40.510 by angle condition, which is called Kawasaki's theorem, so 00:28:40.510 --> 00:28:46.240 that the alternating sum of each vertex is zero. 00:28:46.240 --> 00:28:50.600 And the layer ordering is NP complete, 00:28:50.600 --> 00:28:57.660 but we can use, for example, sufficient condition given 00:28:57.660 --> 00:29:01.940 by Kawasaki or empirical condition, 00:29:01.940 --> 00:29:09.040 which basically given for each local adjacent angle, 00:29:09.040 --> 00:29:11.500 so it's very easy to implement. 00:29:11.500 --> 00:29:15.350 And it works, so we forget about NP complete part. 00:29:19.030 --> 00:29:21.740 And also, we can give several constraints. 00:29:21.740 --> 00:29:23.750 For example, this fold doesn't want 00:29:23.750 --> 00:29:28.190 to fold so that it forms a planar surface, 00:29:28.190 --> 00:29:36.300 or you can fix the point to a point in 3D space, 00:29:36.300 --> 00:29:41.600 or we can make some edge to be rigid. 00:29:44.270 --> 00:29:49.440 So we have these coordinates, and these are the variables 00:29:49.440 --> 00:29:52.825 to represent the configuration, and be assign developability, 00:29:52.825 --> 00:29:54.720 flat foldability, or other constraints. 00:29:58.260 --> 00:30:03.000 This forms an under-determined system, 00:30:03.000 --> 00:30:07.820 which means that it gives you a multi-dimensional solution 00:30:07.820 --> 00:30:08.950 space. 00:30:08.950 --> 00:30:11.530 Within the solution space, we move. 00:30:11.530 --> 00:30:16.520 We transform the surface so that you 00:30:16.520 --> 00:30:21.190 will get the always valid solution. 00:30:21.190 --> 00:30:24.760 That's the method we are applying. 00:30:24.760 --> 00:30:30.920 In order to solve that, we can use 00:30:30.920 --> 00:30:36.360 the Jacobian of the constraints and calculate it numerically. 00:30:39.130 --> 00:30:47.210 So for each step, it's a given assumed transformation mold, 00:30:47.210 --> 00:30:51.790 and this gives you a valid transformation mold 00:30:51.790 --> 00:30:55.515 by using the generalized inverse or pseudo inverse. 00:30:59.850 --> 00:31:03.420 You can implement software like this. 00:31:09.640 --> 00:31:13.990 The top is what happens with Freeform Origami. 00:31:13.990 --> 00:31:17.130 This is a crease pattern of the paper 00:31:17.130 --> 00:31:21.620 and this is the folded pattern when it's X-rayed. 00:31:21.620 --> 00:31:24.510 You can see that if you drag this point up, 00:31:24.510 --> 00:31:27.660 then all the crease pattern and the flat folded pattern 00:31:27.660 --> 00:31:30.640 changes at the same time. 00:31:30.640 --> 00:31:31.880 This is for comparison. 00:31:31.880 --> 00:31:37.210 This is kind of a simulation. 00:31:37.210 --> 00:31:44.200 In this case, the model cannot change the crease pattern, 00:31:44.200 --> 00:31:48.635 which makes a less flexible motion for origami. 00:31:57.540 --> 00:32:03.541 So we want to do some mesh modification 00:32:03.541 --> 00:32:09.290 because if you change the crease pattern, then 00:32:09.290 --> 00:32:14.920 it will end up being degenerate triangles, 00:32:14.920 --> 00:32:19.860 so that we need to do some kind of edge collapse operation. 00:32:19.860 --> 00:32:25.000 However, in order to do that, we have several conditions. 00:32:25.000 --> 00:32:27.380 We have to keep Maekawa's theorem 00:32:27.380 --> 00:32:31.350 if the surface is flat foldable, and we 00:32:31.350 --> 00:32:34.860 use that for doing edge collapsing 00:32:34.860 --> 00:32:37.180 and mesh modification. 00:32:37.180 --> 00:32:42.360 You see these patterns change, and these 00:32:42.360 --> 00:32:48.350 are getting very degenerate triangles 00:32:48.350 --> 00:32:53.190 that are removed by this operation. 00:32:53.190 --> 00:32:55.540 This is interesting because you will end up 00:32:55.540 --> 00:32:56.740 in different patterns. 00:33:00.070 --> 00:33:05.950 This part is similar to diamond pattern or Yoshimura pattern, 00:33:05.950 --> 00:33:12.480 and this part is kept the same as Miura-Ori pattern. 00:33:16.340 --> 00:33:21.350 Mesh modification enables to transform 00:33:21.350 --> 00:33:23.730 to produce more variations for pattern. 00:33:26.270 --> 00:33:31.690 From now, I will show some examples of Freeform Origami. 00:33:31.690 --> 00:33:35.740 First one is starting from Miura-Ori. 00:33:35.740 --> 00:33:38.570 I think it's well known. 00:33:38.570 --> 00:33:45.860 This is actually very old, you can see from napkin folds. 00:33:45.860 --> 00:33:49.870 For paper models, I think you can see pictures from Bauhaus. 00:33:52.990 --> 00:33:56.170 Anyway, this is called Miura-Ori, 00:33:56.170 --> 00:34:00.650 and it gives you expansive motion, 00:34:00.650 --> 00:34:04.810 and also it can be compactly packaged 00:34:04.810 --> 00:34:08.850 and it's developable, of course. 00:34:08.850 --> 00:34:15.929 So this is a variation of that, and this is another example. 00:34:24.190 --> 00:34:26.565 This model is called Melting Ice Cream. 00:34:30.429 --> 00:34:37.745 The idea is that using a variation of Miura-Ori, 00:34:37.745 --> 00:34:47.719 it forms a 3D surface that is irregular and asymmetric, 00:34:47.719 --> 00:34:53.179 while it can be folded flat so that you can roll it up, 00:34:53.179 --> 00:34:56.159 carry out. 00:34:56.159 --> 00:34:57.510 This is a model. 00:35:04.050 --> 00:35:06.320 This is another example. 00:35:06.320 --> 00:35:09.720 Here, I wanted to start from regular Miura-Ori 00:35:09.720 --> 00:35:16.740 and then to transform into a more free form surface. 00:35:19.910 --> 00:35:23.620 You can also generalize Ron Resch pattern. 00:35:23.620 --> 00:35:28.220 This is one of Ron Resch's designed triangular 00:35:28.220 --> 00:35:30.820 tessellations. 00:35:30.820 --> 00:35:35.890 Good thing about this is that it forms 00:35:35.890 --> 00:35:38.790 a kind of composite surface when its folded. 00:35:38.790 --> 00:35:43.980 So it's composed with top surface and tucks 00:35:43.980 --> 00:35:49.440 inside, just like this bunny, but it's more flexible 00:35:49.440 --> 00:35:54.870 and you can fold from a sheet to the 3D form. 00:35:54.870 --> 00:35:59.990 In order to design this, we assigned the condition 00:35:59.990 --> 00:36:07.870 that in 3D state, these three vertices form one vertex. 00:36:07.870 --> 00:36:11.470 That's extra constraints to enable generalization 00:36:11.470 --> 00:36:12.790 of Ron Resch pattern. 00:36:12.790 --> 00:36:15.810 So for example, you can get this kind 00:36:15.810 --> 00:36:24.890 of asymmetric polyhedral surface with this generalized pattern, 00:36:24.890 --> 00:36:28.710 or this form with this pattern. 00:36:28.710 --> 00:36:32.350 Each triangle has different shapes. 00:36:32.350 --> 00:36:36.730 A little bit of differentiated shape. 00:36:36.730 --> 00:36:41.920 Or you can just apply to some regular triangular mesh 00:36:41.920 --> 00:36:53.320 and then get some nice crumpled paper like that. 00:36:53.320 --> 00:36:59.390 I think this is a good way for using this software. 00:36:59.390 --> 00:37:01.310 You can design crumpled paper. 00:37:04.010 --> 00:37:06.748 It's very hard to crumple paper in real life. 00:37:10.010 --> 00:37:12.600 If you feel it's very difficult, then you 00:37:12.600 --> 00:37:15.280 can use this software to crumple up paper. 00:37:18.250 --> 00:37:23.030 This is a generalized version of Yoshimura pattern or diamond 00:37:23.030 --> 00:37:29.560 pattern that you can use for a kind of shell-like structure. 00:37:29.560 --> 00:37:35.075 This is another pattern called waterbomb pattern. 00:37:47.890 --> 00:37:54.910 It's supposed to be playing the puffer fish 00:37:54.910 --> 00:37:56.986 motion by [INAUDIBLE]. 00:37:59.650 --> 00:38:06.020 Anyway, this type of pattern is called waterbomb pattern. 00:38:06.020 --> 00:38:13.320 I think the oldest known is by Shuzo Fujimoto. 00:38:13.320 --> 00:38:15.995 It's kind of a flexible pattern. 00:38:19.750 --> 00:38:21.620 Because it's flat foldable, it can 00:38:21.620 --> 00:38:25.260 be used for deployable structure, 00:38:25.260 --> 00:38:30.260 and because of the elastic property 00:38:30.260 --> 00:38:34.230 that the folding gives, you can use for textured material 00:38:34.230 --> 00:38:38.820 or cloth folding. 00:38:38.820 --> 00:38:42.930 This is an example of deformation. 00:38:42.930 --> 00:38:45.890 This is the normal waterbomb pattern. 00:38:48.760 --> 00:38:52.830 This is based on triangles. 00:38:52.830 --> 00:38:59.120 This one forms more planar surface when it's in 3D, 00:38:59.120 --> 00:39:00.800 and this is some generalization. 00:39:04.810 --> 00:39:06.940 This is the folded form. 00:39:06.940 --> 00:39:10.850 I think it's a little bit too small to show. 00:39:15.540 --> 00:39:22.660 This is the pattern, and you fold it 00:39:22.660 --> 00:39:26.550 into a kind of a butterfly shape, 00:39:26.550 --> 00:39:31.660 and it forms a hyperbolic paraboloidal shape, 00:39:31.660 --> 00:39:36.480 but it's a subtle shape. 00:39:36.480 --> 00:39:39.930 This is another way to make a [INAUDIBLE] in paper, 00:39:39.930 --> 00:39:41.710 and this exists. 00:39:49.260 --> 00:39:52.756 This was Freeform Origami. 00:39:52.756 --> 00:39:55.410 Do you have any questions about Freeform Origami? 00:39:59.971 --> 00:40:02.554 AUDIENCE: Why can't you solve directly for the final shape 00:40:02.554 --> 00:40:06.380 that you want? 00:40:06.380 --> 00:40:09.790 PROFESSOR: The question is, why do I not 00:40:09.790 --> 00:40:17.910 search for the 3D shape instead of gradually changing? 00:40:17.910 --> 00:40:27.710 This is because the constraints are non-linear, 00:40:27.710 --> 00:40:33.460 and that makes it possible that the solution doesn't exist. 00:40:33.460 --> 00:40:37.440 If you do it continuously, then it 00:40:37.440 --> 00:40:40.290 is always true that you get the right answer. 00:40:53.450 --> 00:40:54.680 This is a demonstration. 00:40:59.030 --> 00:41:03.660 This is good because basically, you 00:41:03.660 --> 00:41:08.160 can interact with the geometry of the origami 00:41:08.160 --> 00:41:11.155 to find some new results. 00:41:18.010 --> 00:41:24.020 It's more similar to when you design with paper, like when 00:41:24.020 --> 00:41:28.430 you interact with real material, but this 00:41:28.430 --> 00:41:36.540 gives more flexible change of material than simulation. 00:41:36.540 --> 00:41:41.790 I wanted to make that kind of interaction possible, 00:41:41.790 --> 00:41:48.620 so this moves like that, and that changes the pattern. 00:42:03.560 --> 00:42:09.060 That's the reason I do it in this way, 00:42:09.060 --> 00:42:13.420 solve the equation by deformation. 00:42:17.730 --> 00:42:18.450 Any questions? 00:42:21.342 --> 00:42:22.842 AUDIENCE: I'm just curious, how long 00:42:22.842 --> 00:42:28.220 did it take you to develop both this and the Origamizer? 00:42:28.220 --> 00:42:32.290 PROFESSOR: The time for developing the software. 00:42:32.290 --> 00:42:37.940 Well, it's always continuing. 00:42:37.940 --> 00:42:39.800 I'm always containing developing, 00:42:39.800 --> 00:42:41.490 so it's very hard to say. 00:42:45.610 --> 00:42:47.560 I don't know. 00:42:47.560 --> 00:42:51.550 I think basically it starts to work in half a year, 00:42:51.550 --> 00:42:54.620 and then polish a lot of times. 00:42:59.900 --> 00:43:02.467 AUDIENCE: Did you want to show the demo one? 00:43:02.467 --> 00:43:03.050 PROFESSOR: No. 00:43:03.050 --> 00:43:04.050 Not yet. 00:43:13.350 --> 00:43:18.290 So then we talk about the rigid origami. 00:43:20.852 --> 00:43:21.776 Question? 00:43:21.776 --> 00:43:24.548 AUDIENCE: I might not understand it correctly, 00:43:24.548 --> 00:43:28.435 but in either of these programs, or is there a program that 00:43:28.435 --> 00:43:32.510 exists that allows the user to design dynamically 00:43:32.510 --> 00:43:35.790 like this curved crease origami? 00:43:38.690 --> 00:43:44.320 PROFESSOR: So curved crease origami is quite hard. 00:43:44.320 --> 00:43:53.130 I have tried with this software, and it works for some cases, 00:43:53.130 --> 00:43:56.970 and it doesn't work for most of the cases 00:43:56.970 --> 00:44:02.030 because there are too much degrees of freedom. 00:44:02.030 --> 00:44:14.470 If you simulate curve folding by very thin quadrilateral strips, 00:44:14.470 --> 00:44:19.230 so that's a future work so that the software 00:44:19.230 --> 00:44:25.010 can work with curve folding. 00:44:25.010 --> 00:44:28.840 If it's a coarse approximation, then it works. 00:44:36.410 --> 00:44:38.835 So rigid origami. 00:44:45.090 --> 00:44:50.905 This is the closest part of computational origami 00:44:50.905 --> 00:44:51.730 to architecture. 00:44:55.530 --> 00:45:00.140 So rigid origami is plates and hinges model 00:45:00.140 --> 00:45:04.740 for origami like this. 00:45:04.740 --> 00:45:14.990 So each rigid panel is connected to adjacent panel 00:45:14.990 --> 00:45:19.670 with a rotational hinge, one axis hinge. 00:45:22.570 --> 00:45:28.450 The panels do not deform, and it produces a synchronized motion. 00:45:28.450 --> 00:45:34.390 So that's a model when you want to apply 00:45:34.390 --> 00:45:39.980 origami's dynamic features to some designs. 00:45:43.960 --> 00:45:54.370 And this is a comparison, and this illustrates the energy 00:45:54.370 --> 00:45:58.860 that's caused by the distortion of facets. 00:45:58.860 --> 00:46:03.060 For example, this model falls to this state, 00:46:03.060 --> 00:46:10.750 but there is no path that gives you zero energy deformation. 00:46:10.750 --> 00:46:15.500 But this is rigid foldable, which 00:46:15.500 --> 00:46:18.855 keeps each of the faces not deforming. 00:46:22.230 --> 00:46:26.520 And it's useful for a self deployable structure 00:46:26.520 --> 00:46:30.760 or architectural structure that is very large. 00:46:30.760 --> 00:46:34.990 You can substitute panels with thick panels. 00:46:34.990 --> 00:46:37.170 I will talk in the last part. 00:46:40.990 --> 00:46:41.910 So like this. 00:46:49.062 --> 00:46:49.770 So rigid origami. 00:47:05.610 --> 00:47:13.300 Here, we want to apply rigid origami to design purposes. 00:47:13.300 --> 00:47:17.200 In order to do that, we want to think about first 00:47:17.200 --> 00:47:21.570 to also generalize rigid foldability 00:47:21.570 --> 00:47:30.560 to different shapes, and also to generalize into cylinders 00:47:30.560 --> 00:47:39.295 and other topology, like compound structures. 00:47:42.860 --> 00:47:48.505 Before that, I would like to show example of rigid folding. 00:47:52.850 --> 00:47:57.382 This is the simulation of folding. 00:47:57.382 --> 00:48:00.340 AUDIENCE: Could you play the lower left again? 00:48:00.340 --> 00:48:05.280 PROFESSOR: So this is a triangulated model 00:48:05.280 --> 00:48:12.760 of a tesselation designed by Ray Schamp. 00:48:12.760 --> 00:48:14.550 This is a waterbomb tessellation. 00:48:19.900 --> 00:48:23.910 So first thing we want to do is to simulate folding 00:48:23.910 --> 00:48:31.840 to understand the geometry, the kinematics of rigid origami. 00:48:36.470 --> 00:48:39.900 This can be represented. 00:48:39.900 --> 00:48:43.410 First easy representation is a truss model, 00:48:43.410 --> 00:48:50.610 which you basically use vertex coordinates as the variables 00:48:50.610 --> 00:48:52.930 and then constrain them with rigid bars that 00:48:52.930 --> 00:48:57.050 are connected to vertices. 00:48:57.050 --> 00:48:59.500 And this one is different representation 00:48:59.500 --> 00:49:01.690 that uses folding angle. 00:49:01.690 --> 00:49:05.780 This is a more direct way, and I have a software 00:49:05.780 --> 00:49:09.630 called Rigid Origami Simulator which basically 00:49:09.630 --> 00:49:15.700 is based on this constraint and representation. 00:49:15.700 --> 00:49:20.210 The configuration is represented by folding angles, 00:49:20.210 --> 00:49:25.270 and then the folding angles are constrained at each vertex 00:49:25.270 --> 00:49:31.470 by this equation, which is a three by three matrix 00:49:31.470 --> 00:49:34.780 form, which actually gives nine equations, 00:49:34.780 --> 00:49:39.475 but only three of them are independent. 00:49:43.070 --> 00:49:48.490 We have three equations for each vertex. 00:49:48.490 --> 00:49:52.760 That will give also under-determined system 00:49:52.760 --> 00:50:00.040 so that you can simulate folding using also generalized inverse 00:50:00.040 --> 00:50:02.720 so that you can search within the solution space 00:50:02.720 --> 00:50:13.290 by giving some pieces kind of a force that's asserted to each 00:50:13.290 --> 00:50:16.270 of the edges, and then this results 00:50:16.270 --> 00:50:19.320 in the deformation that is valid. 00:50:24.610 --> 00:50:26.570 In generic case, because there are 00:50:26.570 --> 00:50:30.190 three constraints for each vertex 00:50:30.190 --> 00:50:39.340 and one variable for each edge, we have this degree of freedom. 00:50:43.960 --> 00:50:47.480 First way to design rigid origami is 00:50:47.480 --> 00:50:57.960 to use this information from Euler's polyhedral formula, 00:50:57.960 --> 00:51:02.750 we can say that any triangular mesh-- well, not any. 00:51:02.750 --> 00:51:09.960 It's a generic triangular mesh-- produces a degree of freedom, 00:51:09.960 --> 00:51:15.330 which is number of edges on the boundary minus 3. 00:51:15.330 --> 00:51:18.810 So for example, in this [? hyper ?] triangular model, 00:51:18.810 --> 00:51:24.120 this gives one degree of freedom because the boundary 00:51:24.120 --> 00:51:27.840 edges are four. 00:51:27.840 --> 00:51:31.490 And we can generalize it to cases 00:51:31.490 --> 00:51:33.550 where there are holes also. 00:51:38.000 --> 00:51:41.670 With this idea, you can make hexagonal tripod shell, which 00:51:41.670 --> 00:51:48.510 is using six edged, hexagonal boundary, 00:51:48.510 --> 00:51:53.110 so that the degree of freedom is three, 00:51:53.110 --> 00:51:58.870 and then we pin joint three of the vertices, which 00:51:58.870 --> 00:51:59.830 gives nine constraints. 00:52:02.690 --> 00:52:08.960 Because of the rigid body degree of freedom, which is six, 00:52:08.960 --> 00:52:14.630 we have nine degrees of freedom of transformation, 00:52:14.630 --> 00:52:17.600 and then that's our constraint by nine, 00:52:17.600 --> 00:52:23.610 so it produces static structure. 00:52:23.610 --> 00:52:26.710 By changing the position of these points, 00:52:26.710 --> 00:52:30.410 you can get the different forms. 00:52:30.410 --> 00:52:36.670 For example, this is the shape, triangular form. 00:52:36.670 --> 00:52:45.430 The position of these pods changes the overall shape, 00:52:45.430 --> 00:52:49.805 and it's static when it's fixed, it's all pin joint. 00:52:59.360 --> 00:53:00.300 This is an example. 00:53:03.090 --> 00:53:11.820 If you change the position of the leg, 00:53:11.820 --> 00:53:14.500 then you get the different shapes, 00:53:14.500 --> 00:53:17.430 3D configuration like this. 00:53:20.410 --> 00:53:30.540 This is one way to build rigid origami structures. 00:53:30.540 --> 00:53:45.940 This is quite obvious result. 00:53:45.940 --> 00:53:49.830 Theoretically, it's more interesting to think about 00:53:49.830 --> 00:53:53.010 the one that you cannot imagine the transformation, 00:53:53.010 --> 00:53:57.010 for example, quadrilateral mesh. 00:53:57.010 --> 00:54:04.900 So for quadrilateral mesh, each vertex 00:54:04.900 --> 00:54:10.560 has also three constraints so that this itself 00:54:10.560 --> 00:54:14.330 forms a one DOF structure. 00:54:14.330 --> 00:54:21.920 Think about when you form a mesh with this, like Miura-Ori. 00:54:21.920 --> 00:54:31.700 Basically, if you define this folding angle, 00:54:31.700 --> 00:54:34.200 then all the folding angles here are defined. 00:54:37.930 --> 00:54:42.010 If you have another degree for vertex, 00:54:42.010 --> 00:54:47.610 then everything is here defined, and you 00:54:47.610 --> 00:54:52.420 want to have another vertex here that 00:54:52.420 --> 00:54:55.130 is defined by this folding angle. 00:54:55.130 --> 00:55:00.809 Then you will have a problem here because this folding angle 00:55:00.809 --> 00:55:02.225 and this folding angle contradict. 00:55:05.530 --> 00:55:14.360 In general, it contradicts so that you cannot make an array 00:55:14.360 --> 00:55:20.040 of folding by quadrilateral mesh. 00:55:20.040 --> 00:55:22.980 However, as you can see from this figure, 00:55:22.980 --> 00:55:26.430 there exist examples like Miura-Ori 00:55:26.430 --> 00:55:29.710 that give one degree of freedom motion. 00:55:29.710 --> 00:55:36.380 This is actually a very interesting thing happening. 00:55:36.380 --> 00:55:40.700 And also, this is great because all the constraints 00:55:40.700 --> 00:55:42.170 are redundant. 00:55:42.170 --> 00:55:47.250 You can remove this part, so you can put a hole in the center, 00:55:47.250 --> 00:55:52.490 but still, it results in the same motion. 00:55:52.490 --> 00:55:57.900 You can design more freely with this kind 00:55:57.900 --> 00:56:01.700 of robust, redundant constraint structure. 00:56:04.970 --> 00:56:13.760 What I want to do is to generalize this to a freeform. 00:56:13.760 --> 00:56:16.510 We start from Miura-Ori and also start 00:56:16.510 --> 00:56:20.480 from this pattern, which is not a developable surface, 00:56:20.480 --> 00:56:24.100 but also gives one degree of freedom motion 00:56:24.100 --> 00:56:31.230 with redundant constraints in its quadrilateral mesh design. 00:56:31.230 --> 00:56:36.345 These can be generalized to rigid, non-symmetric forms. 00:56:44.060 --> 00:56:46.290 Miura-Ori is something like that, 00:56:46.290 --> 00:56:57.110 and we can extract the property of rigid folding 00:56:57.110 --> 00:56:58.590 by looking at these vertex. 00:57:03.520 --> 00:57:06.550 If it's a flat foldable vertex, it's 00:57:06.550 --> 00:57:13.140 a degree four vertex with flat foldability, 00:57:13.140 --> 00:57:16.620 and this gives a folding motion where 00:57:16.620 --> 00:57:20.940 the opposite folding angle are the same 00:57:20.940 --> 00:57:24.110 and these are the same, and also, 00:57:24.110 --> 00:57:29.870 that the folding angle here and here are related. 00:57:29.870 --> 00:57:33.300 If you look at the tangent of half of the angle, 00:57:33.300 --> 00:57:37.660 then it's linearly related. 00:57:37.660 --> 00:57:42.690 This is very useful because if you find one configuration that 00:57:42.690 --> 00:57:48.710 works, then you will have continuous folding that works. 00:57:51.800 --> 00:57:56.330 In this way, you can make sufficient condition 00:57:56.330 --> 00:58:02.450 for making a quadrilateral mesh rigid foldable. 00:58:02.450 --> 00:58:10.360 This is easy to get because it's only the finite folding motion. 00:58:10.360 --> 00:58:15.070 So the continuous folding motion from a sheet 00:58:15.070 --> 00:58:19.270 to the folded state is guaranteed 00:58:19.270 --> 00:58:26.065 by one 3D configuration that satisfies these conditions. 00:58:29.210 --> 00:58:31.690 In order to get one configuration that 00:58:31.690 --> 00:58:36.640 satisfies this configuration, we can use the Freeform Origami, 00:58:36.640 --> 00:58:42.650 so it's very easy for us to do that. 00:58:42.650 --> 00:58:46.230 This is an example of rigid origami 00:58:46.230 --> 00:58:53.000 that's using redundant constraints. 00:58:53.000 --> 00:58:56.760 It produces one degree of freedom motion 00:58:56.760 --> 00:59:00.410 and it's very light because this part and this part 00:59:00.410 --> 00:59:04.080 counterbalance the gravity. 00:59:08.520 --> 00:59:09.590 This is another example. 00:59:16.430 --> 00:59:20.975 Quadrilateral mesh can be a rigid folding motion. 00:59:32.680 --> 00:59:36.340 You can apply to something like curve folding. 00:59:36.340 --> 00:59:39.200 It's rigid foldable curve folding. 00:59:39.200 --> 00:59:43.300 This is kind of contradictory, but we 00:59:43.300 --> 00:59:47.760 can rationalize curve folding into quadrilateral mesh. 00:59:47.760 --> 00:59:54.130 Then you can produce one degree of freedom motion like that. 00:59:54.130 --> 00:59:59.620 Another example with thickness is here. 00:59:59.620 --> 01:00:06.160 It's interesting that it's one degree of freedom motion 01:00:06.160 --> 01:00:11.260 so that every motion is coded in the pattern here. 01:00:14.670 --> 01:00:18.150 Another example is using Eggbox pattern. 01:00:22.340 --> 01:00:24.740 This actually has the same property 01:00:24.740 --> 01:00:31.200 as the Miura-Ori vertex or flat foldable degree for vertex, 01:00:31.200 --> 01:00:36.540 so that you can use the same idea 01:00:36.540 --> 01:00:40.110 to produce rigid foldable variations. 01:00:42.750 --> 01:00:46.960 You can actually combine with origami structure 01:00:46.960 --> 01:00:53.740 if you define this mountain fold, valley fold, 01:00:53.740 --> 01:00:56.340 and also add complementary mountain fold 01:00:56.340 --> 01:01:05.320 and complementary valley fold, defined here. 01:01:05.320 --> 01:01:12.070 If you use that kind of extension of the origami, 01:01:12.070 --> 01:01:20.950 then you will get design variations 01:01:20.950 --> 01:01:26.740 that combine origami. 01:01:26.740 --> 01:01:29.010 This part is origami vertex but this is not. 01:01:31.580 --> 01:01:36.660 This can have positive Gaussian curvature 01:01:36.660 --> 01:01:40.230 while this part is zero Gaussian curvature. 01:01:40.230 --> 01:01:41.880 The interesting point is that you 01:01:41.880 --> 01:01:45.250 can develop the flat folded state 01:01:45.250 --> 01:01:50.460 or actually, in this case, it's two flat folded states because 01:01:50.460 --> 01:01:57.190 in this case, it's not totally developed, 01:01:57.190 --> 01:02:02.510 but complementary fold lines are folded. 01:02:02.510 --> 01:02:05.850 So that's why I call it bidirectionally flat foldable 01:02:05.850 --> 01:02:10.669 planar quadrilateral mesh, and that can be rigidity foldable. 01:02:10.669 --> 01:02:11.460 This is an example. 01:02:28.180 --> 01:02:31.480 This was about a disk, and we can 01:02:31.480 --> 01:02:34.380 develop into a cylindrical structure. 01:02:34.380 --> 01:02:43.460 And cylinder is not trivial because, for example, this 01:02:43.460 --> 01:02:45.680 is known origami pattern, but this 01:02:45.680 --> 01:02:47.380 doesn't transform to this pattern 01:02:47.380 --> 01:02:53.725 because this is has a different number or edges. 01:02:53.725 --> 01:02:58.790 You cannot produce this kind of motion in rigid folding 01:02:58.790 --> 01:03:01.550 mechanism. 01:03:01.550 --> 01:03:07.940 You can see what's the problem by looking at this kind of disk 01:03:07.940 --> 01:03:10.700 surface that is rigidly foldable, 01:03:10.700 --> 01:03:15.220 and it forms a cylinder at one state 01:03:15.220 --> 01:03:20.740 but it doesn't produce in a continuous motion that 01:03:20.740 --> 01:03:24.540 fix this loop. 01:03:24.540 --> 01:03:26.380 But there exists. 01:03:26.380 --> 01:03:35.220 So the idea is to mirror reflect one part of Miura-Ori vertex. 01:03:35.220 --> 01:03:41.200 Actually, you can see from the origami model by Thoki Yenn. 01:03:41.200 --> 01:03:43.060 It's called Flip Flop. 01:03:43.060 --> 01:03:44.435 It's a very interesting model. 01:03:47.770 --> 01:03:56.740 You can develop more generalized origami cylinders like these. 01:03:56.740 --> 01:04:01.690 I think this is rigid foldable cylinders. 01:04:01.690 --> 01:04:05.700 Since it's using quadrilateral panels, 01:04:05.700 --> 01:04:09.500 it similarly produces one degree of freedom motion. 01:04:09.500 --> 01:04:13.250 And you can see, for example, like that. 01:04:13.250 --> 01:04:14.600 This is the motion produced. 01:04:17.410 --> 01:04:22.900 Any part produces the whole motion. 01:04:28.960 --> 01:04:33.180 And also, you can make some kind of design variations. 01:04:33.180 --> 01:04:37.185 This is using grasshopper on rhinoceros. 01:04:40.320 --> 01:04:47.010 This is the given shape, the section, and from the section, 01:04:47.010 --> 01:04:52.410 you can make a rigid foldable cylinder and also 01:04:52.410 --> 01:04:54.200 composite structures like this. 01:04:59.000 --> 01:05:01.150 So we can make composite structures 01:05:01.150 --> 01:05:06.765 like this, so it's great that it fills the space. 01:05:10.480 --> 01:05:14.390 It tessellates while preserving the one 01:05:14.390 --> 01:05:15.640 degree of freedom motion. 01:05:20.020 --> 01:05:25.050 This is an implementation with thick panels. 01:05:25.050 --> 01:05:29.950 Think I have got the cylindrical tessellation model. 01:05:32.530 --> 01:05:40.180 This is a tessellated model, which moves like that. 01:05:40.180 --> 01:05:46.573 It's kind of a volume that flattens into two states, 01:05:46.573 --> 01:05:49.490 and it's good that it's rigidly foldable 01:05:49.490 --> 01:05:51.600 and it's one degree of freedom mechanism. 01:05:59.480 --> 01:06:00.800 I want to generalize more. 01:06:05.990 --> 01:06:08.800 This is symmetric. 01:06:08.800 --> 01:06:13.740 There exists one axis that repeats the unit structure. 01:06:13.740 --> 01:06:19.950 I want to make it more general. 01:06:19.950 --> 01:06:22.870 I like the asymmetry than symmetry. 01:06:22.870 --> 01:06:28.720 I think you do also. 01:06:28.720 --> 01:06:34.130 But you have to think about the conditions around a hole, 01:06:34.130 --> 01:06:39.990 and actually, this is a difficult condition 01:06:39.990 --> 01:06:45.760 to be solved then the disk version. 01:06:45.760 --> 01:06:53.620 However, you can do that by fixing the first loop, 01:06:53.620 --> 01:06:56.800 then you know that this part is rigid foldable, 01:06:56.800 --> 01:07:01.600 so you can continue deformation from there. 01:07:01.600 --> 01:07:04.550 This is kind of sufficient condition, 01:07:04.550 --> 01:07:09.220 but useful for designing something like this. 01:07:09.220 --> 01:07:13.080 It's more free form. 01:07:13.080 --> 01:07:17.360 For example, this is also one degree 01:07:17.360 --> 01:07:20.430 of freedom rigid foldable cylinder. 01:07:24.930 --> 01:07:27.600 This is an example folded. 01:07:37.210 --> 01:07:38.220 It folds like that. 01:07:46.740 --> 01:07:49.010 This is also an example. 01:07:49.010 --> 01:07:54.290 This forms a torus when it's unfolded. 01:07:58.170 --> 01:08:01.155 Or this type of structure. 01:08:04.449 --> 01:08:05.490 So it's also cylindrical. 01:08:08.250 --> 01:08:13.220 And for kind of architectural image. 01:08:15.840 --> 01:08:20.830 This is a rigid origami for cylinder. 01:08:20.830 --> 01:08:31.520 And how to implement rigid origami to real model, 01:08:31.520 --> 01:08:37.510 because in architecture design or any other designs, 01:08:37.510 --> 01:08:45.779 we want some kind of finite thickness model. 01:08:45.779 --> 01:08:48.720 There's no ideal origami. 01:08:48.720 --> 01:08:53.770 Well, probably Origamido can produce. 01:08:53.770 --> 01:09:00.819 Anyway, we want some thick panels and rotating hinges 01:09:00.819 --> 01:09:02.410 to produce rigid origami. 01:09:06.180 --> 01:09:09.250 This is the typical way to do that. 01:09:09.250 --> 01:09:13.500 So you put the hinge on the valley side. 01:09:13.500 --> 01:09:22.800 The main problem of this one is that each vertex, there 01:09:22.800 --> 01:09:27.700 are several fold lines that are joining at the vertex, 01:09:27.700 --> 01:09:33.600 but this will no longer be concurrent. 01:09:33.600 --> 01:09:38.040 But rigid origami mechanism assumes that the fold lines are 01:09:38.040 --> 01:09:46.870 concurrent, which means that it includes translation 01:09:46.870 --> 01:09:49.910 constraints and it increases the number 01:09:49.910 --> 01:09:52.140 of constraints for each vertex. 01:09:52.140 --> 01:09:56.070 So previously, it was three constraints there, 01:09:56.070 --> 01:09:59.610 but it becomes six constraints because it includes 01:09:59.610 --> 01:10:03.210 transformation, which is very bad, 01:10:03.210 --> 01:10:08.080 but there is asymmetric vertex that allows that. 01:10:08.080 --> 01:10:11.170 So using asymmetry, you can make something like that. 01:10:11.170 --> 01:10:19.070 Also, there is a way that you can slide the hinge like that, 01:10:19.070 --> 01:10:24.950 but there is a problem in the case like this. 01:10:24.950 --> 01:10:32.500 The problem of sliding is in global accumulation of errors. 01:10:32.500 --> 01:10:37.780 So what I do is instead of shifting hinge, 01:10:37.780 --> 01:10:45.100 you can trim the volume of the vertex valley 01:10:45.100 --> 01:10:47.380 side of the model. 01:10:50.040 --> 01:10:53.590 This is actually very easy. 01:10:53.590 --> 01:10:58.660 You assume that this folds to pi minus delta, 01:10:58.660 --> 01:11:04.500 and you can just remove this part 01:11:04.500 --> 01:11:09.010 according to this maximum folding angle. 01:11:09.010 --> 01:11:18.590 And you can define this point by offsetting the edges, which 01:11:18.590 --> 01:11:24.800 is basically calculating with a straight skeleton. 01:11:24.800 --> 01:11:29.830 This also can be applied to constant thickness 01:11:29.830 --> 01:11:35.710 panels, which is more easy to be manufactured, 01:11:35.710 --> 01:11:38.240 only with three axis milling machine. 01:11:42.420 --> 01:11:48.610 This is a thick panel, thick rigid origami 01:11:48.610 --> 01:11:49.560 using this method. 01:11:52.300 --> 01:11:57.490 This is an example using a constant thickness model 01:11:57.490 --> 01:12:02.970 where no error is accumulated, like this model, 01:12:02.970 --> 01:12:05.600 a slideable hinge. 01:12:05.600 --> 01:12:09.690 This is implemented by using grasshopper again. 01:12:09.690 --> 01:12:14.670 Grasshopper is a good tool to implement something 01:12:14.670 --> 01:12:22.250 that does not require iterative optimization calculation, 01:12:22.250 --> 01:12:31.840 but it's very good for forward calculation of geometry. 01:12:31.840 --> 01:12:34.220 That calculates the pattern, and then you 01:12:34.220 --> 01:12:37.690 can lay out the pattern. 01:12:37.690 --> 01:12:43.040 This is a cloth, and we put the thick panels 01:12:43.040 --> 01:12:48.050 on both sides of the cloth to make a rigid foldable structure 01:12:48.050 --> 01:12:49.830 like this. 01:12:49.830 --> 01:12:52.120 In the center, there is a cloth that 01:12:52.120 --> 01:12:55.355 represents the ideal origami. 01:12:58.690 --> 01:12:59.975 This is what happens. 01:13:05.260 --> 01:13:11.090 I would like to give an example of rigid origami 01:13:11.090 --> 01:13:13.750 design for architecture. 01:13:13.750 --> 01:13:22.260 So this example is that we are given existing building 01:13:22.260 --> 01:13:27.880 openings, and we want to connect these openings temporarily. 01:13:31.890 --> 01:13:38.070 I want that to be compactly folded to fit to this facade, 01:13:38.070 --> 01:13:43.150 but because this is quite large like that, 01:13:43.150 --> 01:13:45.770 you cannot make from a flexible material. 01:13:45.770 --> 01:13:54.620 That will be a problem if you make it as a structure. 01:13:54.620 --> 01:13:57.240 So in order to solve that kind of problem, 01:13:57.240 --> 01:14:04.190 we can use freeform origami to design to fit the condition. 01:14:04.190 --> 01:14:12.650 So these red lines are the openings of two buildings that 01:14:12.650 --> 01:14:23.410 are actually not parallel, and the sizes are different. 01:14:23.410 --> 01:14:28.130 But still, you can connect it on freefrom origami, 01:14:28.130 --> 01:14:32.600 and also, you can keep the boundary to fit on the ground, 01:14:32.600 --> 01:14:37.220 and then you can get these form variations 01:14:37.220 --> 01:14:45.275 and you can choose one of them and calculate the paneling 01:14:45.275 --> 01:14:45.775 patterns. 01:14:48.690 --> 01:14:54.450 This is a rendering representation, 01:14:54.450 --> 01:14:57.270 but basically, you can make this kind of structure 01:14:57.270 --> 01:15:02.825 that can connect to buildings. 01:15:06.910 --> 01:15:08.762 Thank you very much. 01:15:08.762 --> 01:15:17.610 [APPLAUSE] 01:15:17.610 --> 01:15:27.680 This was more hand craft thing. 01:15:27.680 --> 01:15:34.310 I did this with Dukes there, and we 01:15:34.310 --> 01:15:41.560 wanted to test the metal folding, 01:15:41.560 --> 01:15:46.030 whether curved folding is useful for manufacturing things. 01:15:46.030 --> 01:15:51.050 I think that's true, but this model 01:15:51.050 --> 01:15:57.815 required a lot of hammering and it wasn't so easy. 01:16:01.450 --> 01:16:04.930 I think you can have a way to make 01:16:04.930 --> 01:16:08.460 it more easily constructed. 01:16:08.460 --> 01:16:11.735 So this is a table just by folding 01:16:11.735 --> 01:16:15.110 a sheet of metal or chair. 01:16:19.070 --> 01:16:20.500 This one? 01:16:20.500 --> 01:16:26.375 So this one is a combination of four papers. 01:16:31.490 --> 01:16:35.440 In this case, I did not restrict to be 01:16:35.440 --> 01:16:37.360 foldable from one sheet of paper. 01:16:41.190 --> 01:16:46.260 Well, for design, I do not do that. 01:16:46.260 --> 01:16:48.550 You can only find it by simulating 01:16:48.550 --> 01:16:54.890 and see the collision. 01:16:54.890 --> 01:16:59.160 Basically, there will be no local collision 01:16:59.160 --> 01:17:03.320 for this quadrilateral mesh, but there 01:17:03.320 --> 01:17:07.730 might be some global collisions where this and this part 01:17:07.730 --> 01:17:09.768 are colliding. 01:17:09.768 --> 01:17:13.254 AUDIENCE: For example, for Ron Resch pattern. 01:17:13.254 --> 01:17:16.830 PROFESSOR: Ron Resch pattern. 01:17:16.830 --> 01:17:19.340 I only do a local collision test, 01:17:19.340 --> 01:17:25.540 which is between facets that are adjacent, 01:17:25.540 --> 01:17:31.020 which still works for that kind of good model. 01:17:43.160 --> 01:17:47.850 I think I will show all of the things here. 01:17:47.850 --> 01:17:48.380 Like that. 01:17:56.439 --> 01:17:57.730 PROFESSOR: Any other questions? 01:18:03.230 --> 01:18:03.730 All right. 01:18:03.730 --> 01:18:04.030 Thanks again. 01:18:04.030 --> 01:18:05.321 PROFESSOR: Thank you very much. 01:18:05.321 --> 01:18:08.280 [APPLAUSE]