WEBVTT 00:00:04.900 --> 00:00:08.370 PROFESSOR: Today we begin the third major part of the class. 00:00:08.370 --> 00:00:11.130 We've done paper folding, linkage folding, 00:00:11.130 --> 00:00:13.380 and now we're going to do polyhedron folding. 00:00:25.860 --> 00:00:28.220 The very last topic we did was hinged dissection, 00:00:28.220 --> 00:00:31.184 which is somewhere in the middle of all these things. 00:00:31.184 --> 00:00:32.600 But with polyhedron folding, we're 00:00:32.600 --> 00:00:34.650 thinking about a two dimensional surface 00:00:34.650 --> 00:00:40.760 in 3D, something like a cube, and we're 00:00:40.760 --> 00:00:51.290 interested in cutting along the edges of that shape 00:00:51.290 --> 00:00:55.470 or somehow cutting along the surface-- that's 00:00:55.470 --> 00:01:03.490 a good cutting-- and then unfolding into some flat shape. 00:01:10.770 --> 00:01:13.920 So this is a standard cross unfolding of the cube. 00:01:13.920 --> 00:01:18.310 This is the unfolding process, and of course the reverse 00:01:18.310 --> 00:01:25.090 is the folding process, and both of them are interesting. 00:01:25.090 --> 00:01:27.350 We're going to start thinking about unfolding. 00:01:27.350 --> 00:01:30.190 That's one of the most practical problems here. 00:01:30.190 --> 00:01:35.860 You really want to build some 3D shape out of sheet material. 00:01:35.860 --> 00:01:37.930 What shape do you cut out in order 00:01:37.930 --> 00:01:41.301 to bend it into that surface? 00:01:41.301 --> 00:01:42.550 So you start with the surface. 00:01:42.550 --> 00:01:46.110 You need to figure out where to cut so that when you unfold, 00:01:46.110 --> 00:01:48.287 you get something that has no overlap. 00:01:48.287 --> 00:01:49.870 If you want to make it from one sheet, 00:01:49.870 --> 00:01:53.410 it shouldn't have any overlap, and ideally just one piece 00:01:53.410 --> 00:01:56.970 because then you have to do less welding or edge joining 00:01:56.970 --> 00:02:01.230 to make that surface. 00:02:01.230 --> 00:02:02.996 So today and the next couple lectures 00:02:02.996 --> 00:02:05.120 will be all about unfolding, and then eventually we 00:02:05.120 --> 00:02:09.380 will turn to the reverse problem, folding problem. 00:02:09.380 --> 00:02:12.440 And there are two kinds of unfolding. 00:02:12.440 --> 00:02:15.480 The one that I just drew is an edge 00:02:15.480 --> 00:02:20.010 unfolding because it only cuts along edges. 00:02:20.010 --> 00:02:33.970 Edge unfolding only cut along edges of the surface. 00:02:33.970 --> 00:02:35.010 This property is nice. 00:02:35.010 --> 00:02:37.080 If you're building something, you 00:02:37.080 --> 00:02:40.090 don't want to have visible seems. 00:02:40.090 --> 00:02:41.850 You have to be bent at the edges, 00:02:41.850 --> 00:02:45.070 so it's not such a big deal if you're also 00:02:45.070 --> 00:02:46.967 fusing along an edge. 00:02:46.967 --> 00:02:49.050 But in general, you could imagine cutting anywhere 00:02:49.050 --> 00:02:51.340 on the surface, and this is an example 00:02:51.340 --> 00:02:53.060 where the solid red lines are cutting 00:02:53.060 --> 00:02:55.000 on what you see and the dotted red lines were 00:02:55.000 --> 00:02:57.510 cutting on the backside. 00:02:57.510 --> 00:03:02.150 And when you unfold that thing, you get this stoplight polygon, 00:03:02.150 --> 00:03:05.620 and this is what we call a general unfolding, or just 00:03:05.620 --> 00:03:08.290 unfolding. 00:03:08.290 --> 00:03:10.010 There you can cut anywhere. 00:03:10.010 --> 00:03:16.810 In both cases, we want one piece, no overlap, 00:03:16.810 --> 00:03:19.740 but you can change where you're allowed to cut. 00:03:40.230 --> 00:03:42.120 So those are the rules of the game, 00:03:42.120 --> 00:03:45.580 or two possible rules of the game. 00:03:45.580 --> 00:03:50.350 And let me tell you what's known about these kinds 00:03:50.350 --> 00:03:51.930 of unfoldings. 00:03:51.930 --> 00:03:54.060 I've mentioned it way back in lecture one. 00:04:26.829 --> 00:04:28.370 so we can think about edge unfolding, 00:04:28.370 --> 00:04:29.911 we can think about general unfolding, 00:04:29.911 --> 00:04:32.340 and we can think about them for convex polyhedra-- 00:04:32.340 --> 00:04:35.580 simple things like the cube have no dents-- 00:04:35.580 --> 00:04:40.350 or general polyhedra, non-convex polyhedra. 00:04:40.350 --> 00:04:45.760 And status is these two corners are open. 00:04:49.860 --> 00:04:52.655 This one is solved. 00:04:52.655 --> 00:04:53.960 It can always be done. 00:04:53.960 --> 00:04:57.300 You can always generally unfold a convex polyhedron, 00:04:57.300 --> 00:05:01.080 but we don't know about general polyhedra. 00:05:01.080 --> 00:05:02.810 Edge unfolding of convex polyhedra, 00:05:02.810 --> 00:05:05.319 we don't know whether it's possible, 00:05:05.319 --> 00:05:06.860 but for non-convex polyhedra, we know 00:05:06.860 --> 00:05:08.444 that's too much to hope for. 00:05:11.170 --> 00:05:14.390 Not always possible. 00:05:14.390 --> 00:05:18.400 Both of these questions could go either way, 00:05:18.400 --> 00:05:22.320 but we know in various generalizations, easy case 00:05:22.320 --> 00:05:26.450 of convex polyhedra general unfolding, we can do it. 00:05:26.450 --> 00:05:28.260 The most restrictive, hardest situation 00:05:28.260 --> 00:05:30.470 is edge unfolding non-convex polyhedra. 00:05:30.470 --> 00:05:31.680 That's too much. 00:05:31.680 --> 00:05:33.981 But each of these things, we don't know. 00:05:33.981 --> 00:05:35.980 And these are some of the biggest open questions 00:05:35.980 --> 00:05:41.330 remaining in geometric folding algorithms, this field. 00:05:41.330 --> 00:05:47.470 I'm going to talk mostly about these three things 00:05:47.470 --> 00:05:52.580 today, so this open problem and these two results. 00:05:52.580 --> 00:05:56.920 And then this topic will essentially be next lecture. 00:05:56.920 --> 00:06:00.165 Obviously it's not solved, but we still have a lot to say. 00:06:00.165 --> 00:06:01.540 There are various partial results 00:06:01.540 --> 00:06:05.640 toward solving both of those problems, 00:06:05.640 --> 00:06:08.590 I would say more on this one. 00:06:08.590 --> 00:06:10.780 This one is five centuries old. 00:06:10.780 --> 00:06:14.758 This one is one decade old or so. 00:06:26.051 --> 00:06:28.050 So I need to set the stage a little bit before I 00:06:28.050 --> 00:06:30.400 can talk about any of these things, 00:06:30.400 --> 00:06:33.680 so let me start with a little bit of terminology. 00:06:40.220 --> 00:06:43.060 We've in some sense talked about curvature before. 00:06:43.060 --> 00:06:44.810 I don't think I gave it this name, though, 00:06:44.810 --> 00:06:45.893 in the context of origami. 00:06:50.860 --> 00:06:52.990 And I use it almost subconsciously 00:06:52.990 --> 00:06:54.560 so it's good for me to define it. 00:07:00.810 --> 00:07:03.740 If you have some vertex on your polyhedron. 00:07:03.740 --> 00:07:07.070 Let's say we're looking at a corner of a cube, 00:07:07.070 --> 00:07:12.660 so this has three 90 degree angles coming together. 00:07:12.660 --> 00:07:17.200 You sum those angles, we get 270, 00:07:17.200 --> 00:07:20.110 and then you take 360 minus that sum, 00:07:20.110 --> 00:07:22.875 and that is your curvature. 00:07:22.875 --> 00:07:26.180 This is going to be 90. 00:07:26.180 --> 00:07:28.550 And what I really care about is whether the curvature 00:07:28.550 --> 00:07:31.590 is positive, 0, or negative. 00:07:39.292 --> 00:07:41.250 Positive curvature, which is what we got here-- 00:07:41.250 --> 00:07:44.720 this is plus 90-- is like a convex cone. 00:07:51.970 --> 00:07:54.360 So if you're thinking about convex polyhedra, which 00:07:54.360 --> 00:07:56.370 is one of the situations we care about, 00:07:56.370 --> 00:08:02.180 you'll always get positive or 0, maybe, curvature vertices, 00:08:02.180 --> 00:08:04.120 depending on what you're thinking about. 00:08:04.120 --> 00:08:06.250 Zero, this is like a piece of paper, 00:08:06.250 --> 00:08:10.680 so this is what you might call flat, but it's locally flat. 00:08:10.680 --> 00:08:14.200 It could be drawn flat or we know 00:08:14.200 --> 00:08:16.790 how to fold pieces of paper. 00:08:16.790 --> 00:08:18.380 You never change the curvature. 00:08:18.380 --> 00:08:19.200 The sum of the instant face angles 00:08:19.200 --> 00:08:21.730 will always stay the same, no matter how you fold this thing. 00:08:21.730 --> 00:08:22.813 Curvature is an invariant. 00:08:25.780 --> 00:08:29.020 If you just think locally and forget about the actual folding 00:08:29.020 --> 00:08:32.169 of the thing, it's kind of flat. 00:08:32.169 --> 00:08:34.360 So you can think of this as a piece of paper. 00:08:34.360 --> 00:08:35.960 We talked about these two situations 00:08:35.960 --> 00:08:38.711 in the context of Kawasaki's theorem way back when. 00:08:38.711 --> 00:08:40.169 Kawasaki's theorem applied for both 00:08:40.169 --> 00:08:42.530 of these for the negative curvature case, which 00:08:42.530 --> 00:08:45.220 is when you have lots of material all joined 00:08:45.220 --> 00:08:48.540 at a single point, something like this. 00:08:53.300 --> 00:08:58.700 Then Kawasaki's theorem changed a little bit. 00:08:58.700 --> 00:09:01.394 There were some other cases that could happen. 00:09:01.394 --> 00:09:02.810 So when you have tons of material, 00:09:02.810 --> 00:09:05.769 this can only happen in a non-convex polyhedron. 00:09:05.769 --> 00:09:06.810 That's sort of the point. 00:09:06.810 --> 00:09:08.395 If you're a convex polyhedron, you 00:09:08.395 --> 00:09:11.370 know you have to have only these two situations. 00:09:11.370 --> 00:09:12.760 Where do you get flat vertices? 00:09:12.760 --> 00:09:15.490 Well, if you think of this point as a vertex 00:09:15.490 --> 00:09:18.880 and you add up all the angles around it, well, that's 360. 00:09:18.880 --> 00:09:21.220 360 minus 360 is 0. 00:09:21.220 --> 00:09:24.600 So you have zero curvature on all the faces and on the edges. 00:09:24.600 --> 00:09:28.330 Also, if you look here, you have 180 and other 180. 00:09:28.330 --> 00:09:30.540 That adds to 360. 00:09:30.540 --> 00:09:32.530 And then at the actual vertices, that's 00:09:32.530 --> 00:09:36.340 where you have positive curvature for a convex 00:09:36.340 --> 00:09:37.820 polyhedron. 00:09:37.820 --> 00:09:39.340 So just some terminology. 00:09:39.340 --> 00:09:41.840 Get used to the idea of positive curvature, zero curvature, 00:09:41.840 --> 00:09:42.798 and negative curvature. 00:09:46.230 --> 00:09:50.680 Then we have the idea of a cutting. 00:09:50.680 --> 00:09:53.400 So cutting is just, what edges do you cut? 00:09:53.400 --> 00:09:55.740 It's the red stuff in this picture. 00:09:55.740 --> 00:09:57.860 What edges do you cut in order to unfold? 00:09:57.860 --> 00:10:02.730 So this is an unfolding, the mapping here, but the red part, 00:10:02.730 --> 00:10:03.769 I'll call it cutting. 00:10:03.769 --> 00:10:05.435 Sometimes it's also called an unfolding, 00:10:05.435 --> 00:10:06.893 but that can be a little confusing. 00:10:17.940 --> 00:10:21.130 The main point here is I want to talk about what constraints 00:10:21.130 --> 00:10:24.150 cutting must satisfy in order to be valid so we can just 00:10:24.150 --> 00:10:27.880 get a sense of what is happening here. 00:10:27.880 --> 00:10:34.930 If you look at these pictures, you can see two properties. 00:10:34.930 --> 00:10:38.320 One is that the cuttings visit all the vertices 00:10:38.320 --> 00:10:40.890 of the polyhedron. 00:10:40.890 --> 00:10:42.610 So there's this red stuff. 00:10:42.610 --> 00:10:45.740 The red stuff is connected. 00:10:45.740 --> 00:10:49.350 It's acyclic, so it's actually a tree in these pictures, 00:10:49.350 --> 00:10:51.880 and it's visiting all the corners. 00:10:51.880 --> 00:10:55.330 Even back here, it's visited on the back. 00:10:55.330 --> 00:10:56.550 Is that always true? 00:10:56.550 --> 00:10:59.660 Most of those properties are mostly true, 00:10:59.660 --> 00:11:02.150 but it depends a little bit. 00:11:02.150 --> 00:11:05.580 One thing that is sure is the cutting 00:11:05.580 --> 00:11:12.015 must span all non-zero curvature vertices. 00:11:15.630 --> 00:11:22.060 But I do have to say "non-zero curvature." "Spans" 00:11:22.060 --> 00:11:24.060 just means visits. 00:11:24.060 --> 00:11:27.330 You have to hit all the vertices of non-zero curvature 00:11:27.330 --> 00:11:29.550 because anything of non-zero curvature 00:11:29.550 --> 00:11:31.360 can't be flattened by itself. 00:11:31.360 --> 00:11:32.250 It needs to be cut. 00:11:35.590 --> 00:11:38.640 At that point, the cutting has to have degree at least one. 00:11:41.790 --> 00:11:44.090 With zero curvature vertices, even if they're not 00:11:44.090 --> 00:11:47.495 flat in three dimensions, you could flatten them 00:11:47.495 --> 00:11:49.620 because we know locally, it really is a flat thing. 00:11:49.620 --> 00:11:51.984 So there may not be any cuts at zero curvature vertices. 00:11:51.984 --> 00:11:54.150 Most of the time, we won't have to worry about them, 00:11:54.150 --> 00:11:56.700 but something to think about. 00:11:56.700 --> 00:11:58.440 Everybody else you have to span. 00:12:03.970 --> 00:12:06.020 You can say even more. 00:12:06.020 --> 00:12:09.680 If you have very negative curvature, 00:12:09.680 --> 00:12:12.260 if the curvature of a vertex is less than some integer, 00:12:12.260 --> 00:12:24.020 k times 360 with a minus sign in front, then cutting 00:12:24.020 --> 00:12:32.760 must have degree strictly more than k plus 1. 00:12:32.760 --> 00:12:34.680 So if you have negative curvature, 00:12:34.680 --> 00:12:39.264 you know you have to have at least two cuts 00:12:39.264 --> 00:12:41.180 for a negative curvature vertex because if you 00:12:41.180 --> 00:12:44.270 had negative curvature-- one of these things-- 00:12:44.270 --> 00:12:45.380 you made a single cut. 00:12:45.380 --> 00:12:47.070 Then all that material is still there. 00:12:47.070 --> 00:12:49.490 If you then flattened it, it's going to overlap itself 00:12:49.490 --> 00:12:51.365 because you're trying to flatten it into 360. 00:12:51.365 --> 00:12:54.080 There's more than 360 material there. 00:12:54.080 --> 00:12:57.590 So negative curvature already, you need two vertices. 00:12:57.590 --> 00:13:00.130 That's the plus 1 here. 00:13:00.130 --> 00:13:06.610 And as soon as you get to smaller than negative 360, 00:13:06.610 --> 00:13:09.360 then you have to have at least three cuts and so on, just 00:13:09.360 --> 00:13:11.925 to partition up into at most 360 groups. 00:13:14.700 --> 00:13:15.920 This is useful. 00:13:15.920 --> 00:13:18.170 Negative curvature is basically really tough to unfold 00:13:18.170 --> 00:13:22.320 and we'll use that to make counter examples here. 00:13:22.320 --> 00:13:22.890 What else? 00:13:35.940 --> 00:13:48.165 your polyhedron has no handles, then cutting has no cycles. 00:13:56.620 --> 00:14:00.400 A handle is something of higher genus. 00:14:00.400 --> 00:14:04.116 If this is your polyhedron, we call this a handle. 00:14:04.116 --> 00:14:05.490 You have sort of a blob down here 00:14:05.490 --> 00:14:08.730 and you have a connection from one place to the other. 00:14:08.730 --> 00:14:10.600 If you have a handle, you can have 00:14:10.600 --> 00:14:13.780 a cycle of cuts like this that does not disconnect 00:14:13.780 --> 00:14:16.550 the surface, but we basically never think 00:14:16.550 --> 00:14:18.355 about that situation. 00:14:18.355 --> 00:14:20.480 Whenever we're thinking of something of genus zero, 00:14:20.480 --> 00:14:25.739 like a convex polyhedron or a regular, non-convex polyhedron, 00:14:25.739 --> 00:14:27.780 you really shouldn't have a cycle in your cutting 00:14:27.780 --> 00:14:29.250 because if you had a cycle, you'd 00:14:29.250 --> 00:14:31.920 disconnect your surface into two parts 00:14:31.920 --> 00:14:34.400 normally, if you're like a sphere or like a disk. 00:14:37.230 --> 00:14:40.910 That's the acyclic condition and when it holds. 00:14:40.910 --> 00:14:42.060 Some other good things. 00:14:57.230 --> 00:15:08.560 If a polyhedron has no boundary and no handles, 00:15:08.560 --> 00:15:25.280 and the unfolding has no holes, then cutting 00:15:25.280 --> 00:15:28.710 is a spanning tree. 00:15:31.230 --> 00:15:34.530 Spanning tree, it's a concept we've used a few times. 00:15:34.530 --> 00:15:35.380 It's just a tree. 00:15:35.380 --> 00:15:37.600 In this case, it's got to visit all the vertices 00:15:37.600 --> 00:15:41.130 of non-zero curvature and it's a tree, so it's connected 00:15:41.130 --> 00:15:43.450 and it's acyclic. 00:15:43.450 --> 00:15:45.410 The main new thing here is that it's connected. 00:15:45.410 --> 00:15:46.270 We've already said that it should 00:15:46.270 --> 00:15:48.370 be acyclic with no handles, we've already 00:15:48.370 --> 00:15:51.030 said that it should span everybody, 00:15:51.030 --> 00:15:53.330 so it's sort of a summary theorem, 00:15:53.330 --> 00:15:55.520 but we have a whole bunch of conditions here. 00:15:55.520 --> 00:16:02.160 In particular, this will hold if your polyhedron is convex. 00:16:06.770 --> 00:16:10.450 So for convex polyhedra, you have a spanning tree, 00:16:10.450 --> 00:16:12.782 and that's what's going on in this picture even 00:16:12.782 --> 00:16:13.740 for general unfoldings. 00:16:13.740 --> 00:16:14.940 That's the interesting case. 00:16:14.940 --> 00:16:16.690 Here, it's maybe more obvious, here it's 00:16:16.690 --> 00:16:20.160 a little less obvious. 00:16:20.160 --> 00:16:25.550 And to sort of see what could go wrong here, I have an example. 00:16:25.550 --> 00:16:27.210 This is a non-convex polyhedron. 00:16:27.210 --> 00:16:29.270 There's a whole bunch of views of it 00:16:29.270 --> 00:16:33.330 up here, and in particular this top view. 00:16:33.330 --> 00:16:36.710 What we're doing is slicing along just those two 00:16:36.710 --> 00:16:39.182 edges and some other stuff around the outside, 00:16:39.182 --> 00:16:40.890 but in particular, we cut those two edges 00:16:40.890 --> 00:16:46.060 and there aren't any other cuts around there. 00:16:46.060 --> 00:16:48.610 A polyhedron is set up so that when you open it up, 00:16:48.610 --> 00:16:50.100 this works out. 00:16:50.100 --> 00:16:51.170 It's nice and flat. 00:16:51.170 --> 00:16:52.280 This is extremely rare. 00:16:52.280 --> 00:16:54.750 If you perturbed this example, it wouldn't hold. 00:16:54.750 --> 00:16:57.280 But you can set things up so that, in fact, what's 00:16:57.280 --> 00:17:00.320 happening is that the total curvature-- this vertex has 00:17:00.320 --> 00:17:03.150 positive curvature, this has negative, this has positive, 00:17:03.150 --> 00:17:06.300 and the sum of those three curvatures is zero, 00:17:06.300 --> 00:17:08.680 and that's what allows this to be flat 00:17:08.680 --> 00:17:11.550 because the total curvature in that little region is zero, 00:17:11.550 --> 00:17:13.750 and that's when things are allowed to be flat. 00:17:13.750 --> 00:17:16.990 So you can cut here and make a separate collection of cuts 00:17:16.990 --> 00:17:18.880 on the outside, but it's disconnected, 00:17:18.880 --> 00:17:20.780 so this is kind of weird. 00:17:20.780 --> 00:17:25.020 Most of the time, things will be connected, 00:17:25.020 --> 00:17:28.602 and as long as your unfolding has no holes-- 00:17:28.602 --> 00:17:30.060 and for an unfolding to have holes, 00:17:30.060 --> 00:17:31.643 you'd have to have this weird property 00:17:31.643 --> 00:17:34.270 that a bunch of curvatures sum to zero like in that picture-- 00:17:34.270 --> 00:17:36.910 as long as you don't have that, you're OK. 00:17:42.434 --> 00:17:43.600 Let's get to these theorems. 00:17:46.820 --> 00:17:52.310 Actually, one thing related to this problem, 00:17:52.310 --> 00:17:55.420 general unfolding of arbitrary polyhedra, 00:17:55.420 --> 00:17:58.120 is you can't be too general what you 00:17:58.120 --> 00:17:59.500 mean by non-convex polyhedra. 00:17:59.500 --> 00:18:01.670 So we have this example. 00:18:01.670 --> 00:18:05.330 It's the simplest nasty polyhedron there is this. 00:18:05.330 --> 00:18:08.705 It has one vertex at the top there with a big dot 00:18:08.705 --> 00:18:10.660 that has negative curvature. 00:18:10.660 --> 00:18:12.990 There's more than 360 degrees of material 00:18:12.990 --> 00:18:15.182 from all these triangles. 00:18:15.182 --> 00:18:16.640 And that's sort of all that it has. 00:18:16.640 --> 00:18:17.930 This is the polyhedron. 00:18:17.930 --> 00:18:20.860 This is what we call boundary of the polyhedron. 00:18:20.860 --> 00:18:23.360 So far, all the polyhedra I've done haven't had that, 00:18:23.360 --> 00:18:25.430 and you may have seen that I assumed 00:18:25.430 --> 00:18:29.608 that it wasn't there for that last lemma. 00:18:29.608 --> 00:18:34.660 This polyhedron can't be unfolded at all 00:18:34.660 --> 00:18:37.740 in one piece, no overlap. 00:18:37.740 --> 00:18:38.940 Why? 00:18:38.940 --> 00:18:40.309 Because it has one vertex. 00:18:40.309 --> 00:18:41.850 You could think of these as vertices, 00:18:41.850 --> 00:18:42.984 but they're kind of flat. 00:18:42.984 --> 00:18:44.650 Everything's flat except that one point. 00:18:44.650 --> 00:18:47.000 It has negative curvature. 00:18:47.000 --> 00:18:48.900 Negative curvature, we said you have 00:18:48.900 --> 00:18:50.900 to have at least two cuts coming in there. 00:18:50.900 --> 00:18:54.380 If you just made one cut, then when you flatten this thing, 00:18:54.380 --> 00:18:56.450 it'll overlap itself locally. 00:18:56.450 --> 00:18:59.940 So there's got to be at least two cuts coming in there. 00:18:59.940 --> 00:19:02.910 I don't think that even matters much. 00:19:02.910 --> 00:19:04.850 I mean, there's at least one, certainly. 00:19:04.850 --> 00:19:06.890 Where could those guys go, those cuts? 00:19:06.890 --> 00:19:08.810 They have to wander around the surface. 00:19:08.810 --> 00:19:11.220 If they just stop in the middle of nowhere, 00:19:11.220 --> 00:19:14.429 then you're sort of doing nothing because a cut that just 00:19:14.429 --> 00:19:15.970 stops in the middle of a flat vertex, 00:19:15.970 --> 00:19:18.020 well, you might as well not have done that cut, 00:19:18.020 --> 00:19:19.234 so you could erase it. 00:19:19.234 --> 00:19:20.400 So it's got to go somewhere. 00:19:20.400 --> 00:19:23.900 They could go to each other, in which case you've made a cycle 00:19:23.900 --> 00:19:26.150 and then you've disconnected your surface because this 00:19:26.150 --> 00:19:29.070 is like a disk, or they could go to the boundary. 00:19:29.070 --> 00:19:30.650 And if they both go to the boundary, 00:19:30.650 --> 00:19:33.640 again, you disconnect your surface, two pieces. 00:19:33.640 --> 00:19:37.120 So this is kind of pathetic, but you can't unfold this 00:19:37.120 --> 00:19:40.240 with one piece, no overlap. 00:19:40.240 --> 00:19:42.120 So when I say, in this picture, and I 00:19:42.120 --> 00:19:44.800 say that it's open whether non-convex polyhedra can 00:19:44.800 --> 00:19:48.020 be generally unfolded, I mean non-convex polyhedra 00:19:48.020 --> 00:19:49.770 without boundary. 00:19:49.770 --> 00:19:51.955 I'll even give you handles if you want. 00:19:51.955 --> 00:19:54.780 I'm not sure that it matters too much, 00:19:54.780 --> 00:19:56.870 but boundary seems to make a big difference. 00:19:56.870 --> 00:19:59.340 So what's wrong is this polyhedron 00:19:59.340 --> 00:20:01.210 is kind of incomplete, and as long 00:20:01.210 --> 00:20:04.380 as you close it up somehow and don't have these boundary 00:20:04.380 --> 00:20:08.560 effects, then maybe you can generally unfold everything. 00:20:08.560 --> 00:20:10.600 That's this question. 00:20:19.410 --> 00:20:21.250 That was a little bit on this. 00:20:21.250 --> 00:20:23.940 We'll come back to it more next lecture. 00:20:23.940 --> 00:20:29.360 Next, I want to do this one, general unfolding of convex 00:20:29.360 --> 00:20:29.870 polyhedra. 00:20:29.870 --> 00:20:33.110 This is really the most positive news I could give you. 00:20:36.200 --> 00:20:39.540 All this unfolding stuff, it's the one good result. 00:20:39.540 --> 00:20:42.257 We know several good results, but in terms of that table, 00:20:42.257 --> 00:20:43.340 it's our only good result. 00:20:46.020 --> 00:20:50.530 And to do that, there's a bunch of solutions to this problem. 00:20:50.530 --> 00:20:52.595 They all use the idea of shortest paths. 00:21:11.540 --> 00:21:15.210 The shortest path is a path that's shortest. 00:21:15.210 --> 00:21:19.140 So you have some surface, you have some points. 00:21:19.140 --> 00:21:20.640 You do this all the time when you're 00:21:20.640 --> 00:21:22.240 flying between two cities. 00:21:22.240 --> 00:21:25.870 You follow a shortest path on a sphere. 00:21:25.870 --> 00:21:27.620 It's not straight in three dimensions. 00:21:27.620 --> 00:21:30.890 It's the shortest thing subject to lying on that surface, what 00:21:30.890 --> 00:21:32.310 is the shortest path you can do? 00:21:32.310 --> 00:21:33.717 So you're used to it on a sphere. 00:21:33.717 --> 00:21:35.300 It's a little weirder on a polyhedron, 00:21:35.300 --> 00:21:37.520 but it's just the same idea. 00:21:37.520 --> 00:21:40.570 Take all the possible paths you could, find the shortest one. 00:21:40.570 --> 00:21:43.000 You fix the two endpoints, x and y, 00:21:43.000 --> 00:21:44.260 and you get to optimize this. 00:21:44.260 --> 00:21:46.860 Whatever is shortest is the right thing. 00:21:46.860 --> 00:21:52.660 I have some pictures of what they look like here. 00:21:52.660 --> 00:21:56.240 So these are some convex polyhedra. 00:21:56.240 --> 00:22:00.050 They're probably random points on a sphere, 00:22:00.050 --> 00:22:01.650 take the convex hull. 00:22:01.650 --> 00:22:04.810 And then we pick some vertex, or some point x in the corner 00:22:04.810 --> 00:22:06.810 there, and this is computing the shortest paths 00:22:06.810 --> 00:22:10.360 from x to every other vertex on the polyhedron. 00:22:10.360 --> 00:22:12.680 So they look very straight in this case 00:22:12.680 --> 00:22:14.430 because the polyhedron is almost a sphere. 00:22:14.430 --> 00:22:17.020 They look kind of like great circular arcs on a sphere, 00:22:17.020 --> 00:22:18.130 but they're not quite. 00:22:18.130 --> 00:22:21.210 You can look carefully at an edge here, 00:22:21.210 --> 00:22:23.470 this does bend a little bit. 00:22:23.470 --> 00:22:25.190 It's a bit subtle. 00:22:25.190 --> 00:22:28.030 It bends only a little bit because the property you want 00:22:28.030 --> 00:22:31.130 is if you look at this triangle and this triangle 00:22:31.130 --> 00:22:33.060 and you unfold them-- so right now, 00:22:33.060 --> 00:22:36.110 they have some dihedral angle between them, 00:22:36.110 --> 00:22:39.180 and if you open it out so that they're flat, 00:22:39.180 --> 00:22:41.150 then this line should be straight. 00:22:41.150 --> 00:22:44.260 Still the case that shortest paths are straight lines, 00:22:44.260 --> 00:22:45.410 but only when you unfold. 00:22:45.410 --> 00:22:46.262 That's the idea. 00:22:46.262 --> 00:22:48.220 So if you look at all the triangles are visited 00:22:48.220 --> 00:22:51.206 by a path, and you unfold them to be a straight thing-- this 00:22:51.206 --> 00:22:54.670 is called developing, you flatten it out-- then actually, 00:22:54.670 --> 00:22:56.420 the shortest path will be a straight line, 00:22:56.420 --> 00:22:58.690 so that makes it really easy to draw these things. 00:22:58.690 --> 00:23:01.250 There still might be multiple candidates, like to get here, 00:23:01.250 --> 00:23:03.610 should I go this way or the other way around, 00:23:03.610 --> 00:23:07.465 but each of those will be locally straight, 00:23:07.465 --> 00:23:09.090 and this is a property called geodesic. 00:23:13.630 --> 00:23:17.342 Shortest paths always unfold straight. 00:23:17.342 --> 00:23:19.300 And in general, anything that unfolds straight, 00:23:19.300 --> 00:23:21.700 it might not even be shortest, is a geodesic. 00:23:26.110 --> 00:23:27.590 These are like locally shortest. 00:23:27.590 --> 00:23:28.910 Locally, you can't make them any shorter, 00:23:28.910 --> 00:23:30.270 but they might have made the wrong choice. 00:23:30.270 --> 00:23:32.020 They might have gone around the wrong way. 00:23:35.400 --> 00:23:37.525 Geodesic means going straight for a long time. 00:23:37.525 --> 00:23:39.150 You could actually spiral around and do 00:23:39.150 --> 00:23:40.610 all sorts of crazy things. 00:23:40.610 --> 00:23:50.440 But shortest paths never cross themselves 00:23:50.440 --> 00:23:52.966 because if you had the shortest path that crossed itself, 00:23:52.966 --> 00:23:53.840 that wasn't shortest. 00:23:53.840 --> 00:23:56.980 You should have just gotten rid of this part 00:23:56.980 --> 00:23:59.830 and gone straight through the crossing. 00:23:59.830 --> 00:24:01.560 That's sort of trivial. 00:24:01.560 --> 00:24:03.875 And there's another good, fun property 00:24:03.875 --> 00:24:07.330 that you may not have noticed in those figures, 00:24:07.330 --> 00:24:14.865 but they never pass through positive curvature vertex. 00:24:25.360 --> 00:24:28.060 So if you look at these pictures, 00:24:28.060 --> 00:24:30.630 they might end at a positive curvature vertex 00:24:30.630 --> 00:24:32.960 because that's where we told the shortest path to go, 00:24:32.960 --> 00:24:35.050 but in the middle, they're always crossing edges. 00:24:35.050 --> 00:24:37.920 This is a convex polyhedron so everything's positive curvature 00:24:37.920 --> 00:24:38.810 or zero. 00:24:38.810 --> 00:24:41.390 In that case, the paths can really only 00:24:41.390 --> 00:24:43.800 go through zero curvature points. 00:24:43.800 --> 00:24:46.010 They might start and end wherever, 00:24:46.010 --> 00:24:49.010 but in between, they never hit a corner. 00:24:49.010 --> 00:24:49.980 Why? 00:24:49.980 --> 00:24:54.200 Because if you have a positive curvature vertex-- 00:24:54.200 --> 00:24:57.700 I'm just going to sketch this idea-- if you went up here 00:24:57.700 --> 00:25:00.550 in order to go back down somewhere like there, 00:25:00.550 --> 00:25:03.680 it's always better to shortcut a little bit 00:25:03.680 --> 00:25:05.730 and not go through the vertex. 00:25:05.730 --> 00:25:08.276 It's better to go around one way or the other. 00:25:08.276 --> 00:25:10.880 I'm just going to wave my hands at that, but it's true. 00:25:13.550 --> 00:25:14.482 So what? 00:25:14.482 --> 00:25:16.690 Shortest paths are going to be a really powerful tool 00:25:16.690 --> 00:25:17.648 for finding unfoldings. 00:25:25.530 --> 00:25:32.640 In particular, I want to define the star, that picture, 00:25:32.640 --> 00:25:35.400 of all shortest paths from one point. 00:25:50.630 --> 00:25:52.770 And do I want points here? 00:25:52.770 --> 00:25:53.960 I think I want vertices. 00:25:57.907 --> 00:25:59.240 That's the picture that we drew. 00:26:03.931 --> 00:26:06.430 The interesting thing about all the shortest paths, they all 00:26:06.430 --> 00:26:10.304 start from the same point, they never hit each other. 00:26:10.304 --> 00:26:11.970 So not only does a shortest path not hit 00:26:11.970 --> 00:26:15.530 itself, but if you take many shortest paths, or even two 00:26:15.530 --> 00:26:18.950 shortest paths from a common starting point, 00:26:18.950 --> 00:26:21.800 they can't hit each other. 00:26:21.800 --> 00:26:24.790 It could be one is a subset of another. 00:26:24.790 --> 00:26:27.859 For example, if I took the shortest path to here 00:26:27.859 --> 00:26:29.900 and then I also took the shortest path to a point 00:26:29.900 --> 00:26:32.080 just beyond it, well, one's going 00:26:32.080 --> 00:26:33.380 to be a prefix of the other. 00:26:33.380 --> 00:26:36.860 Other than that, they will never cross each other. 00:26:36.860 --> 00:26:39.930 I'm going to just assert that, not prove it here. 00:26:39.930 --> 00:26:41.650 So really, it does look like a star. 00:26:41.650 --> 00:26:43.960 In fact, you could fill in more shortest paths. 00:26:43.960 --> 00:26:46.430 You could take the shortest path to here, for example, 00:26:46.430 --> 00:26:51.210 and it'll fit in nicely, kind of bisect that angle in there. 00:26:51.210 --> 00:26:53.150 It's a very simple kind of structure. 00:26:53.150 --> 00:26:56.120 It's just from a point, you have all this stuff going out. 00:26:56.120 --> 00:27:05.650 They never hit each other except at something 00:27:05.650 --> 00:27:12.412 called the cut locus, also called the ridge tree. 00:27:18.090 --> 00:27:27.675 And these are points with non-unique shortest paths to x. 00:27:37.960 --> 00:27:39.915 So we're fixing some point, x. 00:27:45.810 --> 00:27:47.160 So here's a simpler polyhedron. 00:27:47.160 --> 00:27:51.110 It's just a square-based pyramid. 00:27:51.110 --> 00:27:54.060 We're picking x to be in the middle of this face, 00:27:54.060 --> 00:27:57.110 and drawn in these black lines are the shortest paths 00:27:57.110 --> 00:28:00.180 from x to all the vertices. 00:28:00.180 --> 00:28:02.440 Here, that's a straight line. 00:28:02.440 --> 00:28:05.620 This one, if you unfolded it, it would be straight, 00:28:05.620 --> 00:28:07.750 and there's some similar ones on the back. 00:28:07.750 --> 00:28:10.060 And I think this is the back in case you want to see. 00:28:12.730 --> 00:28:16.870 There's one back face that you can't see at all, C, behind B 00:28:16.870 --> 00:28:20.977 here, and that's what it looks like. 00:28:20.977 --> 00:28:22.310 There's no shortest paths there. 00:28:22.310 --> 00:28:23.810 Now, there's these dashed lines. 00:28:23.810 --> 00:28:25.640 That's the ridge tree. 00:28:25.640 --> 00:28:28.000 That's the cut locus. 00:28:28.000 --> 00:28:33.330 And these are points on the backside with respect to x. 00:28:33.330 --> 00:28:37.260 If you're going from x, you go around behind to C, 00:28:37.260 --> 00:28:42.330 or you could go from x around behind to D and then to C. 00:28:42.330 --> 00:28:47.040 At some point, those things meet and are still of equal length. 00:28:47.040 --> 00:28:49.970 So here, if you're trying to go to x, 00:28:49.970 --> 00:28:52.800 you could go around this way or you could go around this way. 00:28:52.800 --> 00:28:56.230 They will be the same length, and all the points 00:28:56.230 --> 00:28:58.686 with that property are the dashed lines. 00:28:58.686 --> 00:29:00.810 Now, if you're familiar with [? Vornar ?] diagrams, 00:29:00.810 --> 00:29:05.700 this is the [? Vornar ?] diagram of one point, x. 00:29:05.700 --> 00:29:10.160 Imagine you plant grass over your polyhedron, you light 00:29:10.160 --> 00:29:15.250 a fire here, it burns at uniform speed in all directions. 00:29:15.250 --> 00:29:17.710 Where the fire meets itself on the backside, 00:29:17.710 --> 00:29:19.540 that is the ridge tree. 00:29:19.540 --> 00:29:22.030 That is the [? Vornar ?] diagram. 00:29:22.030 --> 00:29:24.410 So it's kind of intuitive that it's there. 00:29:24.410 --> 00:29:26.580 Maybe less obvious is that it's a tree 00:29:26.580 --> 00:29:27.650 and it's a spanning tree. 00:29:27.650 --> 00:29:28.691 It hits all the vertices. 00:29:32.240 --> 00:29:35.390 So it's a natural cutting. 00:29:35.390 --> 00:29:38.330 That's why it's called the cut locus, 00:29:38.330 --> 00:29:39.620 and it works really well. 00:29:58.359 --> 00:29:59.900 And it's called the source unfolding. 00:30:11.690 --> 00:30:13.730 Source unfolding goes back to the mid '80s. 00:30:13.730 --> 00:30:15.530 A bunch of people discovered it for 00:30:15.530 --> 00:30:18.010 various computational geometry applications. 00:30:18.010 --> 00:30:20.630 They didn't care about unfolding at the time. 00:30:20.630 --> 00:30:25.640 And it's kind of obvious after you think about it for a while, 00:30:25.640 --> 00:30:28.280 trying to solve this general unfolding problem. 00:30:28.280 --> 00:30:36.970 Once you have this structure, cut along the cut locus, 00:30:36.970 --> 00:30:43.105 unfold the star of shortest paths from x. 00:30:46.770 --> 00:30:49.200 So let's do it for this example. 00:30:49.200 --> 00:30:50.860 It's the same example on the left. 00:30:50.860 --> 00:30:52.150 This is the star unfolding. 00:30:52.150 --> 00:30:56.890 I've just splayed out everything, so at the boundary, 00:30:56.890 --> 00:30:58.570 I'm cutting at the dashed part. 00:30:58.570 --> 00:31:00.820 That's the cut locus. 00:31:00.820 --> 00:31:02.320 Ignore these little dashed lines. 00:31:02.320 --> 00:31:06.280 Those are just edges, unfolded. 00:31:06.280 --> 00:31:10.490 And what's happening is that all the shortest paths are all 00:31:10.490 --> 00:31:12.200 here. 00:31:12.200 --> 00:31:15.490 If I look at any shortest path from x on the left diagram, 00:31:15.490 --> 00:31:20.830 I can map it to a line segment starting from x. 00:31:20.830 --> 00:31:22.100 x was a flat vertex. 00:31:22.100 --> 00:31:24.450 It has 360 degrees of material around it. 00:31:24.450 --> 00:31:26.900 I just chose it to be somewhere in the middle somewhere. 00:31:26.900 --> 00:31:28.590 Doesn't matter other than that. 00:31:28.590 --> 00:31:32.100 And now there's 360 degrees of material on the flat unfolding. 00:31:32.100 --> 00:31:33.120 That's great. 00:31:33.120 --> 00:31:35.340 And you pick any direction here, you 00:31:35.340 --> 00:31:37.620 can map it to a corresponding direction over there 00:31:37.620 --> 00:31:40.780 on the surface, see where that shortest path would 00:31:40.780 --> 00:31:43.850 go if you kept going until you were no longer shortest. 00:31:43.850 --> 00:31:47.700 When you stop being shortest, the edge of that place 00:31:47.700 --> 00:31:52.630 is the ridge tree, and that's where you cut. 00:31:52.630 --> 00:31:54.584 You stop your segment there. 00:31:54.584 --> 00:31:57.000 So it's actually pretty obvious this thing doesn't overlap 00:31:57.000 --> 00:31:58.620 because all of these shortest paths 00:31:58.620 --> 00:32:01.940 are going in different directions from x. 00:32:01.940 --> 00:32:05.450 What you have is what we call a star shaped polygon around x. 00:32:05.450 --> 00:32:08.060 Every point on the surface is visible from x 00:32:08.060 --> 00:32:12.090 because we just sort of unrolled it to be right there. 00:32:12.090 --> 00:32:14.550 That's the source unfolding. 00:32:14.550 --> 00:32:16.350 It's a little hard to see the 3D diagrams, 00:32:16.350 --> 00:32:20.585 but it's actually really easy once you draw those shortest 00:32:20.585 --> 00:32:21.085 paths. 00:32:26.680 --> 00:32:29.984 So the source unfolding is star shaped. 00:32:29.984 --> 00:32:32.150 We have another unfolding called the star unfolding, 00:32:32.150 --> 00:32:33.661 which is not star shaped. 00:32:33.661 --> 00:32:34.285 Very confusing. 00:32:37.230 --> 00:32:40.710 I mean, what you call one doesn't matter too much. 00:32:40.710 --> 00:32:44.120 This unfolding was mentioned in 1948 00:32:44.120 --> 00:32:45.740 by Alexandrov, who we'll be hearing 00:32:45.740 --> 00:32:49.270 about more in the future, but wasn't 00:32:49.270 --> 00:32:52.770 improved to non-overlap until '92. 00:32:52.770 --> 00:32:55.570 So this one is much less obvious that it doesn't overlap, 00:32:55.570 --> 00:32:57.560 and I'm not going to prove it here. 00:32:57.560 --> 00:33:00.470 But it goes back to this idea of star. 00:33:00.470 --> 00:33:02.950 Say, OK, this cut locus is nice, but let's 00:33:02.950 --> 00:33:05.050 focus on the star of shortest paths 00:33:05.050 --> 00:33:08.750 from one point to all the other vertices. 00:33:08.750 --> 00:33:14.600 Instead of keeping those paths as the things that you unroll, 00:33:14.600 --> 00:33:17.700 what if we cut along them? 00:33:17.700 --> 00:33:20.430 It's another natural thing to try and it turns out to work. 00:33:23.380 --> 00:33:27.650 So here, we cut along the star. 00:33:27.650 --> 00:33:30.930 We've already proved this result, 00:33:30.930 --> 00:33:33.040 every convex polyhedron is generally unfoldable. 00:33:33.040 --> 00:33:35.590 Just with that picture, it's very easy. 00:33:35.590 --> 00:33:37.990 But hey, it's fun to have more. 00:33:37.990 --> 00:33:39.770 And this is what the star unfolding looks 00:33:39.770 --> 00:33:41.990 like for the same example. 00:33:41.990 --> 00:33:49.075 This is a little less obvious, but if you think about it, 00:33:49.075 --> 00:33:50.450 the star, the set of all shortest 00:33:50.450 --> 00:33:53.500 paths to all the vertices, is a spanning tree. 00:33:53.500 --> 00:33:56.280 It hits all the vertices because we told it to, 00:33:56.280 --> 00:34:00.060 and it's a tree because it's just 00:34:00.060 --> 00:34:02.960 all these edges coming together at a point. 00:34:02.960 --> 00:34:05.320 So it's a natural cutting also. 00:34:05.320 --> 00:34:06.850 And magically it works. 00:34:06.850 --> 00:34:10.260 How it works is a little less obvious. 00:34:10.260 --> 00:34:14.650 You see here the ridge tree drawn on the unfolding, 00:34:14.650 --> 00:34:17.310 and you can see all the parts and map the letter E, 00:34:17.310 --> 00:34:20.409 this is the bottom square, and stays connected. 00:34:20.409 --> 00:34:22.510 It's just like you reattach everything 00:34:22.510 --> 00:34:26.510 around the ridge tree and it doesn't overlap. 00:34:30.710 --> 00:34:33.639 It's quite difficult, I think, to give intuition why this 00:34:33.639 --> 00:34:37.024 doesn't overlap, but it doesn't. 00:34:43.178 --> 00:34:44.719 I'll wave my hands at some point when 00:34:44.719 --> 00:34:47.690 we have the necessary tools to prove it, 00:34:47.690 --> 00:34:49.574 I can mention how it's done. 00:34:49.574 --> 00:34:50.699 But we don't have them yet. 00:34:50.699 --> 00:34:52.698 We'll get them in a couple of lectures, I think. 00:34:56.150 --> 00:35:02.260 These results can be extended also in various directions. 00:35:02.260 --> 00:35:04.985 Let me tell you briefly about some extensions. 00:35:10.850 --> 00:35:13.370 For a long time, these were the two ways 00:35:13.370 --> 00:35:15.380 to solve that problem and that was sort of it. 00:35:17.564 --> 00:35:19.980 It would be nice, for example, to have some general family 00:35:19.980 --> 00:35:22.521 of unfoldings that includes the star unfolding and the source 00:35:22.521 --> 00:35:24.489 unfolding and there's something in the middle, 00:35:24.489 --> 00:35:26.280 but we don't necessarily know what that is. 00:35:26.280 --> 00:35:27.292 Do you have a question? 00:35:27.292 --> 00:35:28.208 AUDIENCE: [INAUDIBLE]? 00:35:33.307 --> 00:35:35.640 PROFESSOR: How do you find the creases is your question? 00:35:40.002 --> 00:35:42.210 You find the creases because you know for every point 00:35:42.210 --> 00:35:44.370 here where it was on the surface, 00:35:44.370 --> 00:35:47.700 and if it was on an edge, then it's a crease point. 00:35:47.700 --> 00:35:48.950 That's the easy way to do it. 00:35:50.845 --> 00:35:52.720 I don't know if that's a satisfactory answer, 00:35:52.720 --> 00:35:55.061 but it can be done. 00:35:55.061 --> 00:35:57.770 AUDIENCE: Does it matter where you choose x to be? 00:35:57.770 --> 00:36:00.660 PROFESSOR: Does it matter where you choose x to be? 00:36:00.660 --> 00:36:02.560 It will change the unfolding. 00:36:02.560 --> 00:36:06.411 It will never overlap, but in that sense 00:36:06.411 --> 00:36:08.160 there's a whole family of star unfoldings, 00:36:08.160 --> 00:36:09.270 depending on where you move x. 00:36:09.270 --> 00:36:10.800 There's a whole family of source unfoldings 00:36:10.800 --> 00:36:12.049 depending on where you move x. 00:36:12.049 --> 00:36:15.740 AUDIENCE: Does it make it more efficient, any choice? 00:36:15.740 --> 00:36:18.340 PROFESSOR: I mean, some of the unfoldings 00:36:18.340 --> 00:36:21.134 might have more cuts, some might have less cuts. 00:36:21.134 --> 00:36:22.550 You're going to get more cuts when 00:36:22.550 --> 00:36:24.760 you have-- depends how you count. 00:36:24.760 --> 00:36:29.760 In some sense, there's only n cuts, n shortest paths. 00:36:29.760 --> 00:36:33.960 But a shortest path might cut over many faces 00:36:33.960 --> 00:36:37.140 or it might just cut over one face. 00:36:37.140 --> 00:36:37.947 Depends where x is. 00:36:37.947 --> 00:36:39.530 If you choose x to be in a nice place, 00:36:39.530 --> 00:36:41.450 maybe you could get away with fewer cuts 00:36:41.450 --> 00:36:46.540 and have to do less welding, but there's no theory about that. 00:36:46.540 --> 00:36:48.222 More questions. 00:36:48.222 --> 00:36:51.020 AUDIENCE: So it doesn't matter where you put x. 00:36:51.020 --> 00:36:54.966 If it works at one spot, it basically works at every spot. 00:36:54.966 --> 00:36:56.122 [INAUDIBLE]? 00:36:56.122 --> 00:36:57.580 PROFESSOR: For convex polyhedra, it 00:36:57.580 --> 00:36:59.150 works no matter where you put x. 00:36:59.150 --> 00:37:02.730 For non-convex polyhedra, some x's might work, some might not. 00:37:06.330 --> 00:37:08.874 Here, let me show you. 00:37:08.874 --> 00:37:10.040 Here's more star unfoldings. 00:37:10.040 --> 00:37:11.030 Cool. 00:37:11.030 --> 00:37:15.320 They're really crazy looking and not really star shaped. 00:37:15.320 --> 00:37:18.410 There isn't one point that can see everything. 00:37:18.410 --> 00:37:21.620 They look kind of spiky like a star, but it's quite different. 00:37:21.620 --> 00:37:23.790 These are random points on a sphere, 00:37:23.790 --> 00:37:26.736 like take 42 random points on a sphere, take the convex hull, 00:37:26.736 --> 00:37:27.860 then take a star unfolding. 00:37:30.620 --> 00:37:35.060 Here's an example with a non-convex polyhedron. 00:37:35.060 --> 00:37:39.170 So one thing we hoped for for a little while briefly 00:37:39.170 --> 00:37:41.940 was that if you only had one vertex of negative curvature, 00:37:41.940 --> 00:37:46.574 which is this one, maybe if you choose x to be right there 00:37:46.574 --> 00:37:48.240 and did the star unfolding, because that 00:37:48.240 --> 00:37:51.860 cuts x into lots of little tiny pieces, then 00:37:51.860 --> 00:37:55.710 maybe it would unfold about overlap, but it doesn't work. 00:37:55.710 --> 00:37:56.805 We were destroyed. 00:37:59.649 --> 00:38:01.190 In fact, you can show neither of them 00:38:01.190 --> 00:38:03.700 will work in general for a non-convex polyhedra, 00:38:03.700 --> 00:38:06.020 so there's no hope of solving this problem 00:38:06.020 --> 00:38:10.430 with these techniques, at least with these exact algorithms. 00:38:18.550 --> 00:38:20.500 You might get lucky, but most of the time, 00:38:20.500 --> 00:38:21.880 I think they won't work. 00:38:30.690 --> 00:38:32.410 Currently, x is a point. 00:38:32.410 --> 00:38:36.371 You can actually let x be-- I guess 00:38:36.371 --> 00:38:37.870 I'm not going to be safe-- I'm going 00:38:37.870 --> 00:38:39.750 to call it a geodesic path. 00:38:39.750 --> 00:38:43.370 There's some restrictions on when this is allowed, 00:38:43.370 --> 00:38:45.930 but the idea is you have your surface, 00:38:45.930 --> 00:38:49.067 you take some straight path on the surface, 00:38:49.067 --> 00:38:50.900 and then you take shortest paths from there. 00:38:54.080 --> 00:38:58.550 And if you just think about the source unfolding 00:38:58.550 --> 00:39:02.304 where you keep x intact and unfold from there, 00:39:02.304 --> 00:39:03.720 the picture is going to look like, 00:39:03.720 --> 00:39:04.840 well, you have this straight line 00:39:04.840 --> 00:39:06.660 when you unfold that thing, and then you 00:39:06.660 --> 00:39:10.560 have this nice star of stuff around it. 00:39:10.560 --> 00:39:14.050 And under some simple conditions, 00:39:14.050 --> 00:39:16.430 that will work without overlap. 00:39:16.430 --> 00:39:18.530 What's more impressive is that the star unfolding 00:39:18.530 --> 00:39:20.984 works from a source like that. 00:39:20.984 --> 00:39:22.400 Again, there are some restrictions 00:39:22.400 --> 00:39:24.249 on x that I'm not going to define here, 00:39:24.249 --> 00:39:25.040 but it can be done. 00:39:25.040 --> 00:39:29.120 These are two very recent papers by O'Rourke and Itoh 00:39:29.120 --> 00:39:33.209 and Vilku from the last two years. 00:39:33.209 --> 00:39:34.000 So that's exciting. 00:39:34.000 --> 00:39:37.811 We now have four general methods for unfolding convex polyhedra. 00:39:37.811 --> 00:39:39.060 Again, these are big families. 00:39:39.060 --> 00:39:40.450 You can choose any geodesic path. 00:39:40.450 --> 00:39:41.940 Maybe some are nicer than others. 00:39:45.390 --> 00:39:46.860 You can do higher dimensions. 00:39:55.275 --> 00:39:57.405 Star unfolding doesn't really make sense 00:39:57.405 --> 00:39:58.280 in higher dimensions. 00:39:58.280 --> 00:39:59.988 I don't think I've thought about it much, 00:39:59.988 --> 00:40:02.300 but source unfolding makes sense. 00:40:02.300 --> 00:40:05.420 So you have some four polytopes, a little hard to imagine. 00:40:05.420 --> 00:40:09.150 You take some point and just radiate out from there, 00:40:09.150 --> 00:40:12.440 unfold like that until you hit yourself, and then you stop, 00:40:12.440 --> 00:40:14.510 and that works in any dimension. 00:40:14.510 --> 00:40:17.240 Source unfolding works in any dimension. 00:40:17.240 --> 00:40:22.870 That's fairly recent, 2003, Miller and Pak. 00:40:22.870 --> 00:40:27.765 Another thing you can do is continuous blooming. 00:40:32.110 --> 00:40:34.810 This is an idea posed by Connolly several years 00:40:34.810 --> 00:40:39.017 back and then solved last year, I guess. 00:40:39.017 --> 00:40:40.100 It was presented in Japan. 00:40:42.660 --> 00:40:46.242 This is about folded states versus folding motions, 00:40:46.242 --> 00:40:48.450 an issue we have thought about many times in origami. 00:40:48.450 --> 00:40:51.050 It was easy for any polygonal piece of paper. 00:40:51.050 --> 00:40:54.120 We could go from unfolded to folded state 00:40:54.120 --> 00:40:56.450 by a continuous motion without self intersection. 00:40:56.450 --> 00:40:58.400 For linkages, it was the big deal. 00:40:58.400 --> 00:41:02.030 It was all about, can we get from a to b? 00:41:02.030 --> 00:41:06.590 For convex polyhedra, you can do it, continuous blooming. 00:41:06.590 --> 00:41:10.670 This is in the middle of an algorithm of continuously 00:41:10.670 --> 00:41:12.970 blooming the source unfolding. 00:41:12.970 --> 00:41:14.380 So here's our point x. 00:41:14.380 --> 00:41:16.930 This is the cube, and the source unfolding 00:41:16.930 --> 00:41:19.990 of a cube for this point x in the center 00:41:19.990 --> 00:41:23.530 of a face, these four vertical lines and then a little x 00:41:23.530 --> 00:41:26.370 on the top side. 00:41:26.370 --> 00:41:28.500 The algorithm unfolds one edge at a time 00:41:28.500 --> 00:41:30.040 but in a very specific order. 00:41:30.040 --> 00:41:34.124 So it ends up unfolding this entire house shape first, 00:41:34.124 --> 00:41:35.790 and then it's done one edge of this one, 00:41:35.790 --> 00:41:38.081 and then it's going to fold the other one down and then 00:41:38.081 --> 00:41:41.180 unfold one, two for the backside, and then one, 00:41:41.180 --> 00:41:42.670 two for the left side. 00:41:42.670 --> 00:41:44.670 And you can show that will not self intersect 00:41:44.670 --> 00:41:47.060 as long as you had a convex polyhedron. 00:41:47.060 --> 00:41:50.564 It's not obvious but it's true. 00:41:50.564 --> 00:41:52.230 There are some other results that if you 00:41:52.230 --> 00:41:57.280 have any unfolding that doesn't self intersect at the end, 00:41:57.280 --> 00:42:01.805 you can add some cuts and make it actually continuously bloom. 00:42:01.805 --> 00:42:03.365 It's kind of like hinged dissections. 00:42:03.365 --> 00:42:05.220 It may not work by itself, but you'd 00:42:05.220 --> 00:42:09.450 cut the pieces into smaller pieces and then it will work. 00:42:09.450 --> 00:42:11.750 That's pretty good news, and source unfolding just 00:42:11.750 --> 00:42:12.620 works as is. 00:42:16.070 --> 00:42:18.420 I might talk about that some future lecture. 00:42:23.610 --> 00:42:26.370 Not known, for example, whether the star unfolding continuously 00:42:26.370 --> 00:42:27.305 blooms. 00:42:27.305 --> 00:42:30.750 That sounds a little scary to me. 00:42:30.750 --> 00:42:33.490 We don't actually have an unfolding of a convex 00:42:33.490 --> 00:42:36.940 polyhedron that does not continuously bloom. 00:42:36.940 --> 00:42:40.460 I think there should be one, but that's an open problem. 00:42:40.460 --> 00:42:43.560 Is this always possible, or is there some crazy collection 00:42:43.560 --> 00:42:45.650 of cuts that you cannot escape? 00:42:49.470 --> 00:42:54.050 That's that, general unfolding of convex polyhedra. 00:42:54.050 --> 00:42:58.900 Let's turn to the-- it's not the last topic. 00:42:58.900 --> 00:43:00.710 We still want to cover both of these. 00:43:00.710 --> 00:43:05.320 Next topic is edge unfolding of convex polyhedra. 00:43:05.320 --> 00:43:07.260 Now, this problem, as I said, goes 00:43:07.260 --> 00:43:13.090 back to 1525, implicitly at least, 00:43:13.090 --> 00:43:20.445 by this guy, Albrecht Durer, who was a cool guy. 00:43:20.445 --> 00:43:22.240 This is Renaissance time. 00:43:22.240 --> 00:43:23.730 He did many different things. 00:43:23.730 --> 00:43:26.159 I guess painter is maybe his primary profession, 00:43:26.159 --> 00:43:27.700 but he did a lot of different things. 00:43:27.700 --> 00:43:32.300 He studied early perspective, all that good stuff. 00:43:32.300 --> 00:43:33.700 This is one of his famous prints. 00:43:33.700 --> 00:43:36.860 It was actually on display at the MFA last year, 00:43:36.860 --> 00:43:39.580 I got to see it, and it has a little polyhedron 00:43:39.580 --> 00:43:42.260 thrown in there. 00:43:42.260 --> 00:43:43.700 It's a nice polyhedron. 00:43:43.700 --> 00:43:44.600 How did he draw them? 00:43:44.600 --> 00:43:48.151 Well, he probably built models out of some material. 00:43:48.151 --> 00:43:49.650 And he was just generally interested 00:43:49.650 --> 00:43:51.720 in the third dimension and understanding 00:43:51.720 --> 00:43:55.640 how all these things worked, and so he made a lot of unfoldings 00:43:55.640 --> 00:43:57.080 in this book. 00:43:57.080 --> 00:43:59.290 This is the original title. 00:43:59.290 --> 00:44:00.900 Here's a translation. 00:44:00.900 --> 00:44:03.140 Titles were a lot longer in those days. 00:44:03.140 --> 00:44:04.967 The Painter's Manual is what I usually 00:44:04.967 --> 00:44:07.050 call it, "A Manual of Measurement of Lines, Areas, 00:44:07.050 --> 00:44:08.716 and Solids by Means of Compass and Ruler 00:44:08.716 --> 00:44:11.280 Assembled by Albrecht Durer for the Use of all Lovers of Art 00:44:11.280 --> 00:44:12.905 with Appropriate Illustrations Arranged 00:44:12.905 --> 00:44:16.320 to be Printed in the Year MDXXV." 00:44:16.320 --> 00:44:20.360 And I don't even know what the subtitle is, but there you go. 00:44:20.360 --> 00:44:22.210 If you read German, I'm told this 00:44:22.210 --> 00:44:26.240 is a challenge because of the old script. 00:44:26.240 --> 00:44:29.620 So unfoldings like this. 00:44:29.620 --> 00:44:32.410 He did mostly Archimedian solids in this book. 00:44:32.410 --> 00:44:34.840 This book is several hundred pages long. 00:44:34.840 --> 00:44:36.790 I have a copy. 00:44:36.790 --> 00:44:41.380 This is a cuboctahedron unfolded. 00:44:41.380 --> 00:44:45.290 He is only cutting along edges, and he liked this, I presume, 00:44:45.290 --> 00:44:47.290 for building models. 00:44:47.290 --> 00:44:48.896 Here's a fancy one, the snub cube. 00:44:48.896 --> 00:44:50.520 This one, I think, might have an error. 00:44:50.520 --> 00:44:55.380 A couple of them have very small errors, but for the most part, 00:44:55.380 --> 00:44:58.030 he was the inventor of edge unfolding. 00:44:58.030 --> 00:45:03.230 Now, he didn't ask, is this always possible, but we did. 00:45:03.230 --> 00:45:15.370 Mathematicians did in 1976, I believe, or '85. 00:45:15.370 --> 00:45:22.230 '75, G. C. Shephard, geometer. 00:45:22.230 --> 00:45:24.830 He was the first one to write it in a paper, 00:45:24.830 --> 00:45:27.390 does every convex polyhedron have an edge unfolding? 00:45:31.910 --> 00:45:36.000 So what can I tell you about this problem? 00:45:36.000 --> 00:45:37.460 It's hard. 00:45:37.460 --> 00:45:41.035 Many people have thought about it. 00:45:41.035 --> 00:45:42.660 I'm not really sure that it's possible. 00:45:48.420 --> 00:45:49.770 A lot of people do, though. 00:46:05.240 --> 00:46:08.780 It's possible for a lot of polyhedra. 00:46:08.780 --> 00:46:11.020 In fact, every polyhedron we've tried to unfold we 00:46:11.020 --> 00:46:14.100 have eventually unfolded-- convex polyhedron. 00:46:14.100 --> 00:46:17.680 Simple set of examples are the Archimedian solids. 00:46:17.680 --> 00:46:21.440 Doesn't take that much effort to find unfoldings of all of them. 00:46:21.440 --> 00:46:25.310 You can do this with exhaustive search or just playing around. 00:46:25.310 --> 00:46:28.020 A lot of people have done this over the years. 00:46:28.020 --> 00:46:30.550 This is all the unfoldings of those guys. 00:46:30.550 --> 00:46:33.370 There's a lot of heuristic software 00:46:33.370 --> 00:46:35.680 for actually doing this. 00:46:35.680 --> 00:46:39.020 I think the coolest one right now is Pepakura. 00:46:39.020 --> 00:46:41.020 If you want to build a paper model of something, 00:46:41.020 --> 00:46:43.319 you have a 3D model thrown into Pepakura 00:46:43.319 --> 00:46:45.610 and it'll probably give you a one piece unfolding, even 00:46:45.610 --> 00:46:46.690 for non-convex shapes. 00:46:46.690 --> 00:46:48.740 Even though it's not always possible, 00:46:48.740 --> 00:46:50.480 it'll do edge cuttings, it'll try 00:46:50.480 --> 00:46:53.540 to do some exhaustive search, and usually does pretty well. 00:46:53.540 --> 00:46:55.310 Sometimes it'll use multiple pieces. 00:46:55.310 --> 00:46:58.350 That's the catch. 00:46:58.350 --> 00:47:00.670 There's a link in the lecture notes. 00:47:00.670 --> 00:47:03.710 The most thorough mathematical search 00:47:03.710 --> 00:47:09.310 is by this guy, Wolfram Schlickenrieder, 00:47:09.310 --> 00:47:13.040 who wrote the equivalent of a master's thesis 00:47:13.040 --> 00:47:19.556 in Berlin 13 years ago. 00:47:19.556 --> 00:47:21.180 I just wrote to him about it yesterday. 00:47:21.180 --> 00:47:23.434 I was like, hey, this is cool stuff. 00:47:23.434 --> 00:47:24.600 Can I show it in my lecture? 00:47:24.600 --> 00:47:25.900 He was like, yeah, it lives on. 00:47:29.600 --> 00:47:33.170 He had a class of 10 different algorithms, 00:47:33.170 --> 00:47:36.750 around that many, possible algorithms 00:47:36.750 --> 00:47:38.640 for unfolding all convex polyhedra, 00:47:38.640 --> 00:47:41.110 and then he came up with a dozen different families 00:47:41.110 --> 00:47:41.750 of polyhedra. 00:47:41.750 --> 00:47:44.208 I don't have the polyhedra drawn here, just the unfoldings. 00:47:47.420 --> 00:47:51.160 They're generators of big classes of polyhedra. 00:47:51.160 --> 00:47:54.440 He applied every algorithm to every class. 00:47:54.440 --> 00:47:56.900 Ideally, you get an algorithm that works for all classes 00:47:56.900 --> 00:48:00.620 or you find a class that foils all algorithms. 00:48:00.620 --> 00:48:03.080 Sadly, he found neither. 00:48:03.080 --> 00:48:07.190 Every algorithm was foiled by some example, 00:48:07.190 --> 00:48:09.935 yet every example was foiled by some algorithm. 00:48:13.060 --> 00:48:16.960 It's really annoying, but here's some examples where they fail. 00:48:16.960 --> 00:48:20.710 They just barely fail, just a couple triangles are messed up. 00:48:20.710 --> 00:48:22.280 Little squares are messed up. 00:48:22.280 --> 00:48:24.320 I think these are called the turtle polyhedra. 00:48:24.320 --> 00:48:29.080 It's like a big, flat thing and then a dome on top. 00:48:29.080 --> 00:48:31.960 So inconclusive, I guess, is the answer. 00:48:31.960 --> 00:48:34.970 There was one algorithm that had a degree of freedom. 00:48:34.970 --> 00:48:36.830 You had to choose a direction, then 00:48:36.830 --> 00:48:39.600 you cut along all the edges that are most in that direction. 00:48:39.600 --> 00:48:41.980 And there was a conjecture on the table from this thesis 00:48:41.980 --> 00:48:44.570 that maybe for every polyhedron, there 00:48:44.570 --> 00:48:46.520 is a direction that works, but then 00:48:46.520 --> 00:48:49.950 that was destroyed four years ago. 00:48:49.950 --> 00:48:53.514 Brendan Lucier from Waterloo proved that that's not true. 00:48:53.514 --> 00:48:54.930 He found a polyhedron that doesn't 00:48:54.930 --> 00:48:56.777 work from any direction. 00:48:56.777 --> 00:48:58.860 So at the moment, we have no candidate algorithms, 00:48:58.860 --> 00:49:01.440 which makes me worry whether this could possibly be true. 00:49:04.200 --> 00:49:06.750 We have some more bad examples. 00:49:06.750 --> 00:49:11.086 Here are some annoying things, like a cube with a corner 00:49:11.086 --> 00:49:11.585 cut off. 00:49:14.370 --> 00:49:17.030 It's a very local thing, but you can mess up really easily. 00:49:17.030 --> 00:49:19.674 It's actually quite common to mess up, I would say, 00:49:19.674 --> 00:49:21.340 even though for things like Archimedian, 00:49:21.340 --> 00:49:23.470 it's much harder to mess up. 00:49:23.470 --> 00:49:28.340 Simplest polyhedron that messes up is this sliver tetrahedron. 00:49:28.340 --> 00:49:30.800 So the polyhedron is drawn at the top. 00:49:30.800 --> 00:49:39.650 It's a little hard to see, but on the backside, 00:49:39.650 --> 00:49:47.250 there's an edge like that, if you can imagine. 00:49:47.250 --> 00:49:50.050 So it's almost flat, so that's why 00:49:50.050 --> 00:49:53.330 it's drawn flat, and the four corners, 00:49:53.330 --> 00:49:55.574 like a tetrahedron should have, four faces, 00:49:55.574 --> 00:49:57.240 the front two triangles and the back two 00:49:57.240 --> 00:49:59.580 triangles with the line that I drew. 00:49:59.580 --> 00:50:03.085 And if you unfold it in a simple way, you just cut from a to b, 00:50:03.085 --> 00:50:05.240 it unfolds like this, no problem. 00:50:05.240 --> 00:50:07.340 But if you unfold the wrong way, which 00:50:07.340 --> 00:50:09.820 is to cut both of the diagonal lines, 00:50:09.820 --> 00:50:12.990 then you end up with the spears that cross each other, 00:50:12.990 --> 00:50:15.590 so you really have to be careful. 00:50:15.590 --> 00:50:21.660 In fact, if you take a random polyhedron-- 00:50:21.660 --> 00:50:25.370 I choose 80 points on a sphere, I take the convex hull. 00:50:25.370 --> 00:50:28.600 It's a very nice, round sphere-like polyhedron. 00:50:28.600 --> 00:50:30.954 And I look at all the unfoldings-- 00:50:30.954 --> 00:50:32.620 this is probably not all the unfoldings, 00:50:32.620 --> 00:50:37.750 but I randomly generate unfoldings, and I evaluate, 00:50:37.750 --> 00:50:42.870 what is the observed probability that I get overlap? 00:50:42.870 --> 00:50:46.700 And it's very close to 100%. 00:50:46.700 --> 00:50:48.650 The conjecture is that as n goes to infinity, 00:50:48.650 --> 00:50:51.560 the probability of overlap goes to 1. 00:50:51.560 --> 00:50:52.810 We don't have a proof of that. 00:50:52.810 --> 00:50:55.456 That would be nice to prove. 00:50:55.456 --> 00:50:57.580 There's good reason to believe that's easy to prove 00:50:57.580 --> 00:51:00.291 or that it's true, let's say, that there is a proof. 00:51:00.291 --> 00:51:02.040 I don't know how to actually formalize it. 00:51:05.310 --> 00:51:07.450 This is some work from the late '80s. 00:51:17.570 --> 00:51:19.380 Just because most unfoldings fail 00:51:19.380 --> 00:51:22.280 doesn't mean there isn't one unfolding that works. 00:51:22.280 --> 00:51:25.280 If I were to try to prove that there is an unfolding that 00:51:25.280 --> 00:51:28.520 works, here's my best hope, and it ties into some things 00:51:28.520 --> 00:51:31.420 that we've seen with tensegrities. 00:51:31.420 --> 00:51:35.840 If you take a really big polyhedron, which is the case 00:51:35.840 --> 00:51:40.150 you worry about, and you look at a small portion of it, 00:51:40.150 --> 00:51:43.810 that portion will be almost completely flat. 00:51:43.810 --> 00:51:48.450 Locally, this thing is mostly flat. 00:51:48.450 --> 00:51:50.350 Instead of thinking about the big polyhedron, 00:51:50.350 --> 00:51:54.350 if we at least wanted to get it to work in any patch, 00:51:54.350 --> 00:51:59.820 think about the case of an almost flat polyhedron. 00:51:59.820 --> 00:52:01.050 So it's like a little dome. 00:52:01.050 --> 00:52:02.462 It has boundary now. 00:52:02.462 --> 00:52:04.170 Makes life a little harder, but let's say 00:52:04.170 --> 00:52:05.430 it's a nice, convex boundary. 00:52:05.430 --> 00:52:07.309 This is a convex dome. 00:52:07.309 --> 00:52:08.350 It's actually polyhedral. 00:52:12.560 --> 00:52:15.170 Well, let's make it super, super flat. 00:52:15.170 --> 00:52:17.480 So we have this z-coordinate. 00:52:17.480 --> 00:52:20.864 Scale z to squash it down into the plane. 00:52:20.864 --> 00:52:22.280 What's nice about this is then you 00:52:22.280 --> 00:52:24.240 can think of your convex polyhedron 00:52:24.240 --> 00:52:28.830 just as some drawing in the plane. 00:52:28.830 --> 00:52:31.220 So here maybe is my convex polyhedron. 00:52:31.220 --> 00:52:32.740 Now, in fact, each of these vertices 00:52:32.740 --> 00:52:35.180 is lifted a tiny amount, some infinitesimal amount, 00:52:35.180 --> 00:52:38.570 but you can think of it in the plane. 00:52:38.570 --> 00:52:40.157 These are convex faces. 00:52:40.157 --> 00:52:42.490 We know this thing can be lifted to a convex polyhedron. 00:52:42.490 --> 00:52:45.840 That lifting is a positive stress 00:52:45.840 --> 00:52:48.380 on all the edges except the boundary edges. 00:52:48.380 --> 00:52:51.650 Maybe that stress gives you some useful structure. 00:52:51.650 --> 00:52:53.430 If there's any hope of this working, 00:52:53.430 --> 00:52:56.680 that structure better be useful, but I don't know how to use it. 00:52:56.680 --> 00:52:58.640 But in particular, open question. 00:52:58.640 --> 00:53:01.430 If I give you a super shallow, arbitrarily 00:53:01.430 --> 00:53:06.540 shallow, convex polyhedral dome, can it be unfolded? 00:53:06.540 --> 00:53:07.270 I don't know. 00:53:11.290 --> 00:53:13.670 Maybe we'll work on it in the problem session. 00:53:16.426 --> 00:53:17.925 Haven't thought about it for awhile. 00:53:20.890 --> 00:53:23.840 That's if you want to solve everything. 00:53:23.840 --> 00:53:27.580 What if you just want to solve some special cases? 00:53:27.580 --> 00:53:31.900 That is a special case motivated by the general case. 00:53:31.900 --> 00:53:34.920 Well, you can solve the case of a polyhedron 00:53:34.920 --> 00:53:36.130 with at most six vertices. 00:53:36.130 --> 00:53:38.950 That's as far as we've gotten. 00:53:38.950 --> 00:53:42.210 You can solve some simple examples like pyramids 00:53:42.210 --> 00:53:46.380 if you take any convex polygon, doesn't 00:53:46.380 --> 00:53:47.909 have to be irregular or anything, 00:53:47.909 --> 00:53:50.200 and then you take a point and you take the convex hull. 00:53:52.780 --> 00:53:55.360 So that's a polyhedron and it unfolds. 00:53:55.360 --> 00:53:56.580 How does it unfold? 00:53:56.580 --> 00:53:57.850 Any suggestions where to cut? 00:54:03.670 --> 00:54:05.954 How? 00:54:05.954 --> 00:54:07.430 AUDIENCE: From the top? 00:54:07.430 --> 00:54:08.870 PROFESSOR: From the top. 00:54:08.870 --> 00:54:10.500 Yeah, just cut here. 00:54:10.500 --> 00:54:12.680 Here, we're only allowed to cut along edges. 00:54:12.680 --> 00:54:17.021 This is called the volcano unfolding. 00:54:17.021 --> 00:54:18.010 What do we have? 00:54:18.010 --> 00:54:23.310 Like that, and then there's just a bunch of triangles. 00:54:23.310 --> 00:54:25.340 What's nice about the volcano unfolding 00:54:25.340 --> 00:54:31.370 is if you look at these perpendicular strips here, 00:54:31.370 --> 00:54:34.600 those triangles will fit inside those perpendicular strips, 00:54:34.600 --> 00:54:37.370 and therefore they won't intersect each other 00:54:37.370 --> 00:54:39.370 because these strips don't intersect each other. 00:54:39.370 --> 00:54:41.120 You could probably even continuously bloom 00:54:41.120 --> 00:54:41.940 this, no problem. 00:54:41.940 --> 00:54:45.997 So a very simple example, easy to do with volcanoes. 00:54:45.997 --> 00:54:47.205 That's the so-called pyramid. 00:54:51.030 --> 00:54:56.240 Prism, a little bit more interesting, but not by much. 00:54:56.240 --> 00:54:59.791 You take some polygon. 00:54:59.791 --> 00:55:02.010 Again, doesn't have to be regular, 00:55:02.010 --> 00:55:05.304 but you take two copies of it vertically offset, take 00:55:05.304 --> 00:55:05.970 the convex hull. 00:55:08.600 --> 00:55:14.510 This one you can cut in two ways. 00:55:14.510 --> 00:55:15.920 You could do a volcano-like thing 00:55:15.920 --> 00:55:20.350 where you cut all the vertical edges 00:55:20.350 --> 00:55:24.860 and then you cut all but one of the bottom edges. 00:55:24.860 --> 00:55:30.280 Then you'll have basically a volcano with little rectangles 00:55:30.280 --> 00:55:33.420 hanging off all the sides. 00:55:33.420 --> 00:55:37.654 And then there's another copy, the bottom copy of this face, 00:55:37.654 --> 00:55:39.320 and it just hangs out over here, and you 00:55:39.320 --> 00:55:41.090 could prove that that won't hit anybody 00:55:41.090 --> 00:55:43.250 in this very simple situation. 00:55:43.250 --> 00:55:46.260 There's another unfolding, though, that I mention 00:55:46.260 --> 00:55:50.080 because it's useful for other things, 00:55:50.080 --> 00:55:51.905 and it's so-called band unfolding. 00:55:51.905 --> 00:55:58.220 Band unfolding, I want to keep intact 00:55:58.220 --> 00:56:04.266 the radial band around the thing, maybe like this. 00:56:04.266 --> 00:56:07.600 What do I want to do to keep that edge? 00:56:07.600 --> 00:56:09.450 Cut, cut. 00:56:09.450 --> 00:56:11.600 So I cut on all the top edges except one, 00:56:11.600 --> 00:56:13.700 I cut on all the bottom edges except the same one, 00:56:13.700 --> 00:56:17.110 and then I also cut this vertical edge. 00:56:17.110 --> 00:56:21.560 So what I should get is I get my top face, 00:56:21.560 --> 00:56:24.020 then I get a rectangle, then I get 00:56:24.020 --> 00:56:27.020 the band that goes around the outside, 00:56:27.020 --> 00:56:32.486 and then I get another copy of my shape, the bottom side. 00:56:32.486 --> 00:56:33.985 This is what we call band unfolding. 00:56:36.890 --> 00:56:40.100 There's some nice theorems about band unfoldings working out. 00:56:44.440 --> 00:56:46.690 Let me get to more interesting polyhedra. 00:56:49.780 --> 00:56:51.650 These are some pretty simple cases, 00:56:51.650 --> 00:56:55.180 but we don't know very much on the positive side. 00:57:05.231 --> 00:57:05.730 Prismoid. 00:57:13.000 --> 00:57:20.480 Suppose you take a polygon, and you basically inset it. 00:57:20.480 --> 00:57:24.400 So I want to make a new version of the polygon 00:57:24.400 --> 00:57:32.100 where all of the angles match, so these edges are parallel 00:57:32.100 --> 00:57:32.760 to each other. 00:57:32.760 --> 00:57:36.170 I think, actually, the lengths can change. 00:57:36.170 --> 00:57:37.935 And then you take the convex hull. 00:57:40.790 --> 00:57:43.165 These unfold with the volcano unfolding. 00:57:46.024 --> 00:57:47.440 It's a little more subtle to prove 00:57:47.440 --> 00:57:50.180 but it's proved in the textbook. 00:57:54.720 --> 00:57:57.070 Edges have to stay parallel. 00:57:57.070 --> 00:57:59.576 I think that might force something. 00:57:59.576 --> 00:58:01.200 I'm not quite sure, but the constraints 00:58:01.200 --> 00:58:02.300 are that the edges have to be parallel 00:58:02.300 --> 00:58:03.633 and the angles have to be equal. 00:58:08.370 --> 00:58:09.660 Dome. 00:58:09.660 --> 00:58:13.160 Dome is actually in some ways simpler. 00:58:13.160 --> 00:58:20.720 You take some base, and then I want a whole bunch of faces 00:58:20.720 --> 00:58:22.500 that all touch the base. 00:58:22.500 --> 00:58:26.610 So in general, it's going to be like a tree of faces, something 00:58:26.610 --> 00:58:28.710 like this. 00:58:28.710 --> 00:58:31.840 Every face has to touch an edge of the base. 00:58:31.840 --> 00:58:34.405 These also unfold, and also with a volcano. 00:58:37.060 --> 00:58:40.490 This is proved also in the textbook. 00:58:40.490 --> 00:58:42.456 Both results are by Joe O'Rourke. 00:58:45.650 --> 00:58:47.380 Prismatoid. 00:58:47.380 --> 00:58:49.870 This is the coolest special case I'm 00:58:49.870 --> 00:58:53.380 going to talk about, the most interesting. 00:58:53.380 --> 00:58:59.340 I take some convex polygon, and then in a parallel offset, 00:58:59.340 --> 00:59:06.337 I take some other convex polygon-- 00:59:06.337 --> 00:59:08.212 no relation to each other except that they're 00:59:08.212 --> 00:59:09.950 in parallel planes-- and then I take 00:59:09.950 --> 00:59:15.640 the convex hull, something like that. 00:59:22.160 --> 00:59:23.595 This one, sadly, is open. 00:59:27.250 --> 00:59:29.110 Now, what we do know, and this is related 00:59:29.110 --> 00:59:32.660 to the band unfolding, if you look 00:59:32.660 --> 00:59:38.650 at the band around the sides of this prismatoid 00:59:38.650 --> 00:59:41.460 and just unfold that by itself, that 00:59:41.460 --> 00:59:45.160 will not self intersect, and that's quite nontrivial, 00:59:45.160 --> 00:59:45.820 the proof. 00:59:45.820 --> 00:59:49.040 But if you just unroll the side faces, 00:59:49.040 --> 00:59:53.041 they will wander around but they won't hit themselves. 00:59:53.041 --> 00:59:54.540 The only remaining problem is, where 00:59:54.540 --> 00:59:57.960 does the top and the bottom face go? 00:59:57.960 --> 01:00:01.920 I think I have an example where that's a little dicey. 01:00:01.920 --> 01:00:04.060 Prism. 01:00:04.060 --> 01:00:04.860 Prismoid. 01:00:04.860 --> 01:00:07.120 That gives you an answer to your offset. 01:00:07.120 --> 01:00:10.070 So they can be sheared away from each other, 01:00:10.070 --> 01:00:12.840 but this is a prismoid. 01:00:12.840 --> 01:00:17.420 All of the sides are parallel and the angles match, a and b. 01:00:17.420 --> 01:00:22.900 Here is the volcano unfolding of the prismoid, 01:00:22.900 --> 01:00:26.580 and here's showing that it's not so obvious that it always 01:00:26.580 --> 01:00:27.080 works. 01:00:27.080 --> 01:00:30.330 If you're not careful where you put the top face-- so this 01:00:30.330 --> 01:00:34.230 is a on top of b in plan-- you could 01:00:34.230 --> 01:00:36.860 attach a to any of these faces. 01:00:36.860 --> 01:00:38.250 You have to choose one that's not 01:00:38.250 --> 01:00:41.420 so inside because it's overlapping here. 01:00:41.420 --> 01:00:42.920 But if you choose the outermost one, 01:00:42.920 --> 01:00:45.890 it works and you can prove that. 01:00:45.890 --> 01:00:51.490 This is the dome. 01:00:51.490 --> 01:00:53.900 You can even have overhang and they 01:00:53.900 --> 01:00:57.190 will unfold, volcano style. 01:00:57.190 --> 01:01:00.260 You no longer have those nice perpendicular wedges 01:01:00.260 --> 01:01:01.650 to say everything is disjoined. 01:01:01.650 --> 01:01:02.719 But it still turns out. 01:01:02.719 --> 01:01:03.510 Nothing intersects. 01:01:06.380 --> 01:01:09.290 Here's the prismatoid I wanted to show you. 01:01:09.290 --> 01:01:11.820 This one's actually almost flat. 01:01:11.820 --> 01:01:16.290 So a is this triangle right on top of b, the triangle beneath. 01:01:16.290 --> 01:01:17.790 You take the convex hull and you get 01:01:17.790 --> 01:01:20.970 all those edges on the outside. 01:01:20.970 --> 01:01:23.710 And if you're not careful-- this is more volcano 01:01:23.710 --> 01:01:26.745 style unfolding-- but you can get overlap. 01:01:32.020 --> 01:01:36.910 One thing we do know, this is kind of weird. 01:01:36.910 --> 01:01:41.380 If you try to go more general, and instead of taking a convex 01:01:41.380 --> 01:01:43.970 polygon on the top and a convex polygon on the bottom, 01:01:43.970 --> 01:01:46.290 instead you take a smooth convex curve 01:01:46.290 --> 01:01:47.890 on the top and a smooth convex curve 01:01:47.890 --> 01:01:50.594 on the bottom, which is getting weird because now there's 01:01:50.594 --> 01:01:52.010 infinitely many vertices and going 01:01:52.010 --> 01:01:54.200 to be infinitely many cuts. 01:01:54.200 --> 01:01:58.440 But there is a natural notion of unfolding 01:01:58.440 --> 01:01:59.990 which sort of takes the continuum. 01:01:59.990 --> 01:02:02.815 You take all those lines that go from one side 01:02:02.815 --> 01:02:07.630 to the other, the rule lines of the convex shape, 01:02:07.630 --> 01:02:09.647 and you just unfold them. 01:02:09.647 --> 01:02:11.730 You don't actually preserve area when you do this. 01:02:11.730 --> 01:02:14.580 It's not a valid unfolding in the usual sense, 01:02:14.580 --> 01:02:16.430 but it's a natural generalization 01:02:16.430 --> 01:02:18.756 of unfolding to smooth shapes. 01:02:18.756 --> 01:02:21.380 You can prove, well, that part's going to work just fine if you 01:02:21.380 --> 01:02:23.410 just volcano it, and then you can actually 01:02:23.410 --> 01:02:26.420 find a place for the top face. 01:02:33.920 --> 01:02:35.420 Those are the special cases we know, 01:02:35.420 --> 01:02:37.780 and even some pretty simple cases that we don't know, 01:02:37.780 --> 01:02:40.480 although it's almost there. 01:02:55.359 --> 01:02:56.275 One more open problem. 01:03:21.716 --> 01:03:23.590 Just mention this because it's a fun problem. 01:03:26.430 --> 01:03:38.380 So again, I want to edge unfold all convex polyhedra, 01:03:38.380 --> 01:03:41.709 but I allow multiple pieces to make 01:03:41.709 --> 01:03:42.875 the problem a little easier. 01:03:46.140 --> 01:03:53.690 So let's say I have a polyhedron with f faces. 01:03:53.690 --> 01:03:56.740 I want to know how few pieces could I get away with? 01:03:56.740 --> 01:03:59.836 The big open problem is, can I get away with one piece? 01:03:59.836 --> 01:04:00.960 What if you make it easier? 01:04:00.960 --> 01:04:05.760 What if I just want, say, little o of f pieces? 01:04:05.760 --> 01:04:08.410 Smaller than any constant times f. 01:04:08.410 --> 01:04:09.290 This is open. 01:04:13.210 --> 01:04:15.390 What we do know is some constant times 01:04:15.390 --> 01:04:19.710 f where the constant is less than 1. 01:04:19.710 --> 01:04:23.230 Best constant I have written here is 1/2. 01:04:23.230 --> 01:04:24.900 There is a better bound, but I think 01:04:24.900 --> 01:04:27.557 it's not so easy to summarize in this form. 01:04:27.557 --> 01:04:29.140 So you can get a little less than half 01:04:29.140 --> 01:04:32.380 the faces' number of pieces, but that's pretty pathetic. 01:04:35.507 --> 01:04:37.840 It's a problem that seemed like it would be a good idea, 01:04:37.840 --> 01:04:40.450 but so far it hasn't seemed to make the problem much easier. 01:04:44.670 --> 01:04:46.810 What do I mean by pieces? 01:04:46.810 --> 01:04:49.580 Well, we're all about one piece unfolding, 01:04:49.580 --> 01:04:53.170 so now your cutting can have cycles in it 01:04:53.170 --> 01:04:55.315 and disconnect the surface into multiple parts. 01:04:55.315 --> 01:04:56.231 AUDIENCE: [INAUDIBLE]. 01:04:58.752 --> 01:05:00.460 PROFESSOR: Multiple connected components. 01:05:15.426 --> 01:05:17.050 The tricky part is to pair up the faces 01:05:17.050 --> 01:05:20.050 so that everybody has a mate. 01:05:20.050 --> 01:05:22.510 That's not always possible. 01:05:22.510 --> 01:05:24.134 AUDIENCE: [INAUDIBLE]? 01:05:24.134 --> 01:05:24.800 PROFESSOR: Yeah. 01:05:24.800 --> 01:05:28.880 You would hope that one third, but-- I mean, 01:05:28.880 --> 01:05:30.700 you can do lots of little local arguments 01:05:30.700 --> 01:05:33.799 and prove this constant, but the big question 01:05:33.799 --> 01:05:35.340 is, can you get less than a constant? 01:05:44.730 --> 01:05:50.615 So last topic for today is edge unfolding non-convex polyhedra. 01:05:56.970 --> 01:06:00.210 Sort of did that problem addressed, 01:06:00.210 --> 01:06:03.330 and now this is not always possible. 01:06:03.330 --> 01:06:06.200 So I want to give you a polyhedron where you cannot 01:06:06.200 --> 01:06:06.910 edge unfold it. 01:06:06.910 --> 01:06:10.510 This is actually pretty easy, and we did it way back in '98 01:06:10.510 --> 01:06:12.340 when we started working on folding, 01:06:12.340 --> 01:06:14.810 but it's kind of cheating. 01:06:14.810 --> 01:06:18.630 This is a box on a box, and the only edges here 01:06:18.630 --> 01:06:20.790 are the edges of the two boxes. 01:06:20.790 --> 01:06:23.460 There's no edges connecting the outside of this face 01:06:23.460 --> 01:06:24.670 to the inside of that face. 01:06:24.670 --> 01:06:28.930 That face here is a donut. 01:06:28.930 --> 01:06:29.840 It's a square donut. 01:06:33.080 --> 01:06:37.590 So if you're only cutting along edges, that face is intact. 01:06:37.590 --> 01:06:43.100 Now I ask you, where does the top box go? 01:06:43.100 --> 01:06:45.660 The top box has five square faces, not six. 01:06:45.660 --> 01:06:47.490 There's nothing on the bottom. 01:06:47.490 --> 01:06:49.920 And somehow, it has to be attached to the rest 01:06:49.920 --> 01:06:50.810 if I want one piece. 01:06:50.810 --> 01:06:54.170 That means that all five faces fit in here, 01:06:54.170 --> 01:06:56.990 but there's only room for one in terms of area 01:06:56.990 --> 01:06:59.050 so you're screwed. 01:06:59.050 --> 01:07:03.290 But this is cheating because what I really wanted to do 01:07:03.290 --> 01:07:05.920 was generalize this problem, edge unfolding 01:07:05.920 --> 01:07:08.080 of convex polyhedra. 01:07:08.080 --> 01:07:10.000 Now, I know they're not going to look convex, 01:07:10.000 --> 01:07:13.240 but that thing is really not convex. 01:07:13.240 --> 01:07:14.740 It's really not convex in the sense 01:07:14.740 --> 01:07:18.500 that you have this face that is not even topologically a disk. 01:07:18.500 --> 01:07:19.750 It has a hole in it. 01:07:19.750 --> 01:07:22.160 Convex polyhedra don't have that. 01:07:22.160 --> 01:07:25.210 So in some sense, this is a topological problem, 01:07:25.210 --> 01:07:27.500 you might say. 01:07:27.500 --> 01:07:30.800 And you might hope that if I looked at polyhedra that 01:07:30.800 --> 01:07:34.000 are topologically convex, maybe those 01:07:34.000 --> 01:07:37.020 would unfold by edge cuts. 01:07:37.020 --> 01:07:39.920 "Topologically convex" means that if you just 01:07:39.920 --> 01:07:43.570 move the vertices around but preserve the edge structure, 01:07:43.570 --> 01:07:46.509 then it becomes a convex polyhedron. 01:07:46.509 --> 01:07:48.550 In other words, I take a convex polyhedron, which 01:07:48.550 --> 01:07:50.664 we think maybe has an edge unfolding, 01:07:50.664 --> 01:07:52.330 and then I just pull the vertices around 01:07:52.330 --> 01:07:55.150 but preserve all the faces. 01:07:55.150 --> 01:07:57.070 Then can you edge unfold those? 01:07:57.070 --> 01:07:59.470 And the answer is no. 01:07:59.470 --> 01:08:03.400 I have one more example before we get there. 01:08:03.400 --> 01:08:06.160 This polyhedron, at least all the faces are disks. 01:08:06.160 --> 01:08:07.600 There's no holes. 01:08:07.600 --> 01:08:09.560 So it's a cube with little bites taken out 01:08:09.560 --> 01:08:12.320 of all the edges, same paper. 01:08:12.320 --> 01:08:13.820 And this thing also does not unfold. 01:08:13.820 --> 01:08:14.945 It's a little less obvious. 01:08:17.622 --> 01:08:19.330 What's cheating about this example is you 01:08:19.330 --> 01:08:22.090 have two faces, like this purple one and the yellow one, 01:08:22.090 --> 01:08:24.859 that share two different edges. 01:08:24.859 --> 01:08:26.890 And for convex polyhedra, two faces either 01:08:26.890 --> 01:08:29.649 share an edge or a vertex or nothing, 01:08:29.649 --> 01:08:34.300 but they can't reach around and do to joins. 01:08:34.300 --> 01:08:40.390 Again, this is not topologically convex, but this is. 01:08:40.390 --> 01:08:43.300 So this is two views of the same thing, 01:08:43.300 --> 01:08:46.640 and if I take these spikes and I just push that vertex down 01:08:46.640 --> 01:08:50.396 to be really close to this triangle, this will be convex. 01:08:50.396 --> 01:08:52.020 So it's just a convex polyhedron that I 01:08:52.020 --> 01:08:55.060 pull on four of the points. 01:08:55.060 --> 01:08:56.580 Same facial structure. 01:08:56.580 --> 01:09:00.899 This has no edge unfolding and we're going to prove that. 01:09:12.830 --> 01:09:14.990 This example is even stronger in that 01:09:14.990 --> 01:09:16.200 all the faces are triangles. 01:09:25.560 --> 01:09:30.177 So what we're going to do is take this thing. 01:09:30.177 --> 01:09:31.885 It's kind of appropriate for this season. 01:10:00.680 --> 01:10:03.690 So I've got a tetrahedral spike on top. 01:10:03.690 --> 01:10:05.830 Think of that as going out of the board. 01:10:05.830 --> 01:10:10.041 And then in the plane of the board is this triangle, 01:10:10.041 --> 01:10:11.540 and then I just add in all the edges 01:10:11.540 --> 01:10:13.480 to make it a triangulation. 01:10:13.480 --> 01:10:15.242 Adding edges is a worry because that's 01:10:15.242 --> 01:10:16.950 where we're allowed to cut, so you really 01:10:16.950 --> 01:10:18.796 have to worry about all those edges we add. 01:10:18.796 --> 01:10:21.720 If we add them in that way, I claim, 01:10:21.720 --> 01:10:24.950 you take this-- we call it the witch's hat-- 01:10:24.950 --> 01:10:29.690 then you multiply, in some sense, by a tetrahedron, 01:10:29.690 --> 01:10:32.370 meaning I take four copies of this hat, 01:10:32.370 --> 01:10:36.390 I put one on each face of the tetrahedron, 01:10:36.390 --> 01:10:39.470 and you get that example. 01:10:39.470 --> 01:10:43.050 Here's one witch's hat and they're just joined along edges 01:10:43.050 --> 01:10:45.244 to make the tetrahedron. 01:10:45.244 --> 01:10:46.910 You could do this with bigger polyhedra, 01:10:46.910 --> 01:10:49.800 too, like octahedron, anything with equilateral triangles, 01:10:49.800 --> 01:10:52.410 but tetrahedron is the smallest where it works. 01:10:52.410 --> 01:10:55.160 Just two of them glued face to face would not work, 01:10:55.160 --> 01:10:56.760 but this way does work. 01:10:56.760 --> 01:10:57.850 Is that true? 01:11:02.350 --> 01:11:05.392 Now I've got to think about two of them joined face to face. 01:11:05.392 --> 01:11:06.350 I don't think it works. 01:11:06.350 --> 01:11:07.808 Otherwise, we would have done that. 01:11:09.910 --> 01:11:12.000 Oh yes, I think I see why. 01:11:19.380 --> 01:11:21.780 Why doesn't this work? 01:11:21.780 --> 01:11:25.540 I should say it doesn't work if the spikes are really tall 01:11:25.540 --> 01:11:28.804 and the base is really flat. 01:11:28.804 --> 01:11:30.470 I'm going to define what "really" means, 01:11:30.470 --> 01:11:32.130 but we'll get there. 01:11:36.790 --> 01:11:38.890 So here's our witch's hat. 01:11:43.160 --> 01:11:47.430 When this spike is really tall, these angles, alpha, 01:11:47.430 --> 01:11:50.600 are very close to 90, a little bit under 90. 01:11:50.600 --> 01:11:54.880 This angle in the floor here-- well, this angle in the floor 01:11:54.880 --> 01:11:57.850 is about 60, probably actually is exactly 60. 01:11:57.850 --> 01:11:59.900 This is an equilateral triangle. 01:11:59.900 --> 01:12:04.720 So if this thing is very flat, this angle will be almost 300 01:12:04.720 --> 01:12:07.250 because it's 360 minus 60. 01:12:07.250 --> 01:12:15.360 It'll be a little bit less than 300 but almost 300. 01:12:15.360 --> 01:12:17.950 These matter. 01:12:17.950 --> 01:12:21.060 In particular, the total sum of angles 01:12:21.060 --> 01:12:26.210 here is 300 plus twice 90, which is big. 01:12:26.210 --> 01:12:28.170 It's bigger than 360. 01:12:28.170 --> 01:12:28.960 That's the point. 01:12:28.960 --> 01:12:33.420 In fact, 300 plus one of these angles, 90, 01:12:33.420 --> 01:12:38.260 would be almost 390, which is way above 360. 01:12:38.260 --> 01:12:40.991 So this has negative curvature, and even if you cut out 01:12:40.991 --> 01:12:42.740 one of the spike triangles, it would still 01:12:42.740 --> 01:12:44.480 have negative curvature. 01:12:44.480 --> 01:12:49.032 That's going to be bad news because let's just 01:12:49.032 --> 01:12:49.990 imagine for the moment. 01:12:49.990 --> 01:12:51.605 We know unfolding things with boundary 01:12:51.605 --> 01:12:54.960 is hard, but let's pretend for now that you wanted 01:12:54.960 --> 01:12:57.340 to unfold a hat in isolation. 01:12:57.340 --> 01:13:01.610 You wanted to unfold it into one piece without overlap. 01:13:01.610 --> 01:13:04.480 Think about what you could do. 01:13:04.480 --> 01:13:09.110 I have to cut with a spanning forest, 01:13:09.110 --> 01:13:12.260 I guess, meaning it's acyclic, it's 01:13:12.260 --> 01:13:14.030 got to hit all the vertices. 01:13:14.030 --> 01:13:16.410 There's only four vertices here. 01:13:16.410 --> 01:13:19.500 How could I hit all the vertices with a forest? 01:13:19.500 --> 01:13:22.420 You may recall from way back when that trees have leaves. 01:13:22.420 --> 01:13:25.460 Every tree has at least two leaves. 01:13:25.460 --> 01:13:28.337 Here I might have multiple trees, maybe, but unlikely, 01:13:28.337 --> 01:13:29.920 and I have to have at least to leaves. 01:13:29.920 --> 01:13:32.150 Where could those two leaves be? 01:13:32.150 --> 01:13:35.500 Could they be at any leaves or vertices at degree one? 01:13:35.500 --> 01:13:38.472 Could they be at any of these three vertices? 01:13:38.472 --> 01:13:40.180 No, because they have negative curvature. 01:13:40.180 --> 01:13:41.905 We know a vertex with negative curvature 01:13:41.905 --> 01:13:43.780 has to have at least two cuts incident to it. 01:13:43.780 --> 01:13:46.960 You can't stop at these three vertices. 01:13:46.960 --> 01:13:52.030 That only leaves one vertex and the boundary. 01:13:52.030 --> 01:13:53.510 So the only thing you can do if you 01:13:53.510 --> 01:13:56.050 want to visit all the vertices and start somewhere 01:13:56.050 --> 01:13:58.640 on the boundary and get to the x, get to the peak. 01:14:05.280 --> 01:14:06.350 That's all you can do. 01:14:06.350 --> 01:14:08.600 You couldn't have multiple connections to the boundary 01:14:08.600 --> 01:14:10.780 because then you would disconnect. 01:14:10.780 --> 01:14:12.340 Then there's really only two choices 01:14:12.340 --> 01:14:16.480 and they're reflectionally symmetric. 01:14:16.480 --> 01:14:18.310 And we're only allowed to go along edges. 01:14:18.310 --> 01:14:19.760 It's super constrained. 01:14:19.760 --> 01:14:21.870 You can go here, walk around, and go up, 01:14:21.870 --> 01:14:24.309 or you can walk around the other way and go up. 01:14:24.309 --> 01:14:26.350 Those are actually slightly different because you 01:14:26.350 --> 01:14:29.280 have two choices of which edge to follow here. 01:14:29.280 --> 01:14:33.240 They're both screwed because if you 01:14:33.240 --> 01:14:36.540 look at this point, the white dot, or the white dot 01:14:36.540 --> 01:14:40.610 up there, what remains here on the outside 01:14:40.610 --> 01:14:44.230 is almost 300 degrees of material on the base 01:14:44.230 --> 01:14:46.840 plus one of the 90 degree faces, the back one 01:14:46.840 --> 01:14:47.910 that you can't see. 01:14:47.910 --> 01:14:49.320 It's easier to see here. 01:14:49.320 --> 01:14:51.200 You have the 300 degrees on the bottom 01:14:51.200 --> 01:14:54.630 and then the white face is still attached to it, 01:14:54.630 --> 01:14:57.415 and that's almost 390. 01:14:57.415 --> 01:14:58.790 When you flatten that thing, it's 01:14:58.790 --> 01:15:01.650 going to overlap itself at that point. 01:15:01.650 --> 01:15:03.020 Bad news. 01:15:03.020 --> 01:15:05.810 So this just says, if I look at a hat in isolation, 01:15:05.810 --> 01:15:08.510 it can't unfold, but that's not what I care about. 01:15:08.510 --> 01:15:10.720 I care about four of them joined together. 01:15:17.050 --> 01:15:19.550 Suppose you had some unfolding of the whole thing, 01:15:19.550 --> 01:15:21.290 and then I look at, well, what cuts 01:15:21.290 --> 01:15:27.090 happen within the witch's hat? 01:15:27.090 --> 01:15:29.364 I know the witch's hat cannot remain in one piece. 01:15:29.364 --> 01:15:30.780 Therefore, it must be disconnected 01:15:30.780 --> 01:15:35.100 into multiple pieces, something like this. 01:15:35.100 --> 01:15:45.050 Again, the cuts have to visit all the vertices somehow, 01:15:45.050 --> 01:15:48.320 but we know from the perspective of a single hat, 01:15:48.320 --> 01:15:51.590 that hat must split into two parts. 01:15:51.590 --> 01:15:54.430 There are lots of things you could consider. 01:15:54.430 --> 01:15:57.300 Let's suppose this is possible, not even worry 01:15:57.300 --> 01:15:59.540 about these kinds of pictures. 01:15:59.540 --> 01:16:02.255 Well, I claim we have a problem. 01:16:27.521 --> 01:16:28.395 Here's a tetrahedron. 01:16:31.410 --> 01:16:34.394 These vertices are all the same point, actually. 01:16:34.394 --> 01:16:35.810 I've just unfolded the tetrahedron 01:16:35.810 --> 01:16:37.893 because it's way easier to draw in two dimensions. 01:16:41.720 --> 01:16:46.670 So if I look at the hat that is on this triangle, 01:16:46.670 --> 01:16:49.070 this hat gets disconnected into two parts. 01:16:49.070 --> 01:16:51.090 There's only three connection points 01:16:51.090 --> 01:16:52.300 to the rest of the world. 01:16:52.300 --> 01:16:54.890 So what these pictures have to look like is they 01:16:54.890 --> 01:16:59.266 connect two of the vertices by a collection of cuts. 01:16:59.266 --> 01:17:00.890 If you're going to cut into two halves, 01:17:00.890 --> 01:17:03.380 you've got to have a path across, 01:17:03.380 --> 01:17:08.150 and there's only three vertices to visit. 01:17:08.150 --> 01:17:10.026 There's some collection of cuts that go from, 01:17:10.026 --> 01:17:11.608 let's say, this vertex to this vertex. 01:17:11.608 --> 01:17:13.440 At this point, everything's symmetric. 01:17:13.440 --> 01:17:16.689 Maybe it would be more obvious if I started the center. 01:17:16.689 --> 01:17:18.230 It could be from here to there, could 01:17:18.230 --> 01:17:20.771 be from here to there or from here to there, but at least one 01:17:20.771 --> 01:17:23.880 of those things exist, and by rotational symmetry 01:17:23.880 --> 01:17:27.990 of this diagram, say it's that one. 01:17:27.990 --> 01:17:30.540 Well, what happens to this face? 01:17:30.540 --> 01:17:32.780 Could there be something like this? 01:17:32.780 --> 01:17:36.170 No, because then this would be disconnected 01:17:36.170 --> 01:17:38.480 from the rest of the world. 01:17:38.480 --> 01:17:41.720 So suddenly, this hat is constrained. 01:17:41.720 --> 01:17:43.890 Maybe it could look like this. 01:17:43.890 --> 01:17:44.640 You have a choice. 01:17:44.640 --> 01:17:46.392 It could look like that or like that, 01:17:46.392 --> 01:17:48.100 but by reflectional symmetry, same thing. 01:17:48.100 --> 01:17:49.740 So let's say this one. 01:17:49.740 --> 01:17:53.100 Remember, x is the same everywhere. 01:17:53.100 --> 01:17:56.280 Well, that means this is impossible 01:17:56.280 --> 01:17:59.420 because this edge is actually glued to this edge, 01:17:59.420 --> 01:18:01.440 and so then this thing would be disconnected 01:18:01.440 --> 01:18:03.220 from the rest of the world. 01:18:03.220 --> 01:18:05.280 So there's a couple of possibilities. 01:18:05.280 --> 01:18:08.880 It could look like this, or it could look like this. 01:18:12.417 --> 01:18:14.000 It was one of those two for this face. 01:18:14.000 --> 01:18:15.000 We've got one face left. 01:18:18.120 --> 01:18:23.190 This is impossible because this edge glues to this one, 01:18:23.190 --> 01:18:26.070 so imagine this thing being picked up and moved over here. 01:18:26.070 --> 01:18:28.615 So we have a wiggly line here and then this stuff, 01:18:28.615 --> 01:18:29.990 and so that would be disconnected 01:18:29.990 --> 01:18:33.660 from the rest of the world, so that can't happen. 01:18:33.660 --> 01:18:36.660 What about this one? 01:18:36.660 --> 01:18:38.340 Is that the harder one? 01:18:38.340 --> 01:18:41.322 I don't know. 01:18:41.322 --> 01:18:43.280 It's been a while since I've used the argument. 01:18:43.280 --> 01:18:45.200 If we have this together with this, 01:18:45.200 --> 01:18:48.010 that's clearly bad because that forms a cycle. 01:18:48.010 --> 01:18:50.290 No good. 01:18:50.290 --> 01:18:52.430 But what if I have this together with this one? 01:18:57.377 --> 01:18:58.210 It gets hard to see. 01:19:02.130 --> 01:19:03.110 It's bad. 01:19:03.110 --> 01:19:03.990 It's a cycle. 01:19:03.990 --> 01:19:08.370 It starts and ends at x, and if you fold it up right, 01:19:08.370 --> 01:19:10.470 you'll see there's really two sides to that cycle. 01:19:10.470 --> 01:19:13.550 It's actually forced, so that's bad. 01:19:13.550 --> 01:19:19.380 All right, one more choice, this one. 01:19:19.380 --> 01:19:22.394 If I do this together with this one, 01:19:22.394 --> 01:19:24.310 that's going to be bad because that's a cycle. 01:19:24.310 --> 01:19:27.360 We start and end at x. 01:19:27.360 --> 01:19:29.930 What if I do this one and that one? 01:19:29.930 --> 01:19:31.430 Well, that's also bad because here's 01:19:31.430 --> 01:19:33.880 a cycle that starts at x, ends at x. 01:19:33.880 --> 01:19:36.540 This and this form the inner of the cycle. 01:19:36.540 --> 01:19:42.595 All cases are bad, so no edge unfolding. 01:19:46.960 --> 01:19:49.910 Tragic. 01:19:49.910 --> 01:19:52.680 We can think briefly about the case-- do I have time? 01:19:52.680 --> 01:19:54.720 I have 10 seconds. 01:19:54.720 --> 01:19:56.960 About the case where there are two triangles. 01:19:56.960 --> 01:20:01.960 I guess I should draw them like this. 01:20:01.960 --> 01:20:05.276 So this point is the same as this point. 01:20:05.276 --> 01:20:07.650 So if I try to simplify this, instead of using four hats, 01:20:07.650 --> 01:20:10.220 I just use two hats, put one here, one here. 01:20:10.220 --> 01:20:16.060 Then I could do something like this, I think, 01:20:16.060 --> 01:20:20.530 and maybe that's OK if there's no cycle formed there. 01:20:20.530 --> 01:20:23.240 So we really needed the tetrahedron somehow. 01:20:23.240 --> 01:20:26.080 I think it does work for octahedron and larger also, 01:20:26.080 --> 01:20:29.610 but just two triangles is not enough. 01:20:29.610 --> 01:20:32.875 And that is unfolding. 01:20:32.875 --> 01:20:37.080 We did not always edge foldable for general polyhedra, even 01:20:37.080 --> 01:20:39.127 topologically convex polyhedra. 01:20:39.127 --> 01:20:41.210 Next time, we'll talk more about general unfolding 01:20:41.210 --> 01:20:43.440 of arbitrary polyhedra.