WEBVTT 00:00:03.255 --> 00:00:05.840 PROFESSOR: We have a lot of fun material to cover today, 00:00:05.840 --> 00:00:07.610 so let's get started. 00:00:07.610 --> 00:00:11.400 We start with a few basic questions about-- well, 00:00:11.400 --> 00:00:13.110 first is a topology question. 00:00:13.110 --> 00:00:15.580 What is a handle? 00:00:15.580 --> 00:00:19.470 In general, a handle is really a transformation on a surface. 00:00:19.470 --> 00:00:22.190 I thought I will demonstrate in the context of glass blowing. 00:00:22.190 --> 00:00:24.860 So this is what it looks like if you're making a cup. 00:00:24.860 --> 00:00:26.800 You take some hot glass. 00:00:26.800 --> 00:00:29.470 You attach it to your surface. 00:00:29.470 --> 00:00:32.430 You cut it off your hot pipe. 00:00:32.430 --> 00:00:39.100 And then you bring it around and attach it to another point, 00:00:39.100 --> 00:00:41.100 and so that's a handle. 00:00:41.100 --> 00:00:45.490 You've probably used handles before in real life. 00:00:45.490 --> 00:00:48.790 In general, or in the mathematical setting, 00:00:48.790 --> 00:00:52.120 it's the same thing but on a 2D surface. 00:00:52.120 --> 00:00:55.700 So imagine you have some 2D surface. 00:00:55.700 --> 00:01:00.420 You take two disk-like regions on the surface 00:01:00.420 --> 00:01:08.660 and then you attach on a handle-- something like that. 00:01:08.660 --> 00:01:11.470 So it's an operation you can do to a surface. 00:01:11.470 --> 00:01:14.330 And you can keep adding handles. 00:01:14.330 --> 00:01:16.100 I don't think there's a clear way to say, 00:01:16.100 --> 00:01:18.840 oh, this is clearly a handle, except to have added it 00:01:18.840 --> 00:01:20.850 as a handle. 00:01:20.850 --> 00:01:25.230 In general, the nice theorem for two dimensional surfaces-- 00:01:25.230 --> 00:01:28.910 so here's a coffee cup being converted into a torus, 00:01:28.910 --> 00:01:30.790 because it has genus 1, meaning it 00:01:30.790 --> 00:01:33.120 has essentially one handle in it. 00:01:33.120 --> 00:01:35.410 In general, you take any orientable surface 00:01:35.410 --> 00:01:39.260 without boundary-- this is locally two dimensional-- then 00:01:39.260 --> 00:01:45.820 it will be a sphere plus some non-negative number of handles. 00:01:45.820 --> 00:01:48.970 And that's a classification theorem 00:01:48.970 --> 00:01:50.570 for orientable surfaces. 00:01:50.570 --> 00:01:52.390 It's only slightly more complicated 00:01:52.390 --> 00:01:53.700 for non-orientable surfaces. 00:01:53.700 --> 00:01:58.840 And then 3D surfaces are even harder 00:01:58.840 --> 00:02:00.410 and was only recently solved. 00:02:00.410 --> 00:02:03.950 But this is two dimensional surfaces, 00:02:03.950 --> 00:02:07.730 which is easy and clean. 00:02:07.730 --> 00:02:09.919 There is a way to compute genus and in some sense 00:02:09.919 --> 00:02:12.130 learn how many handles are there. 00:02:12.130 --> 00:02:15.000 But there isn't a unique thing of, oh, this part 00:02:15.000 --> 00:02:16.360 is clearly a handle. 00:02:16.360 --> 00:02:18.179 But when you draw it this way, it kind of 00:02:18.179 --> 00:02:19.470 becomes clear this is a handle. 00:02:19.470 --> 00:02:20.178 This is a handle. 00:02:20.178 --> 00:02:22.710 But there isn't a formal sense in which-- 00:02:22.710 --> 00:02:29.170 I don't know-- this thing is not a handle or some weird thing. 00:02:29.170 --> 00:02:32.561 I think that's all about handles. 00:02:32.561 --> 00:02:33.810 Hopefully that answers things. 00:02:33.810 --> 00:02:35.420 Of course, things get more complicated 00:02:35.420 --> 00:02:38.630 when you have a boundary, which we usually call holes. 00:02:38.630 --> 00:02:43.620 But the next question is about holes in the unfoldings. 00:02:43.620 --> 00:02:46.250 So I claim that convex polyhedron, when 00:02:46.250 --> 00:02:50.490 you unfold them, never have holes in the unfolded form. 00:02:50.490 --> 00:02:53.760 And this is to contrast this example, which is not convex. 00:02:53.760 --> 00:02:56.642 But you can unfold it by cutting these two edges. 00:02:56.642 --> 00:02:57.850 And you've got a little hole. 00:02:57.850 --> 00:02:59.620 So I showed this in lecture. 00:02:59.620 --> 00:03:01.560 And natural question is, why isn't this 00:03:01.560 --> 00:03:05.290 possible for convex polyhedron? 00:03:05.290 --> 00:03:06.900 So I thought I would prove that. 00:03:06.900 --> 00:03:09.870 And the proof uses a cool theorem, which we'll probably 00:03:09.870 --> 00:03:12.650 be seeing again, called the Gauss-Bonnet theorem. 00:03:15.270 --> 00:03:21.360 And it says that if you have some surface, which 00:03:21.360 --> 00:03:24.770 is homeomorphic to a sphere-- so I don't 00:03:24.770 --> 00:03:27.430 want any handles for this theorem. 00:03:27.430 --> 00:03:30.080 And of course, convex polyhedra our sphere-like. 00:03:30.080 --> 00:03:32.310 They don't have handles. 00:03:32.310 --> 00:03:36.770 And I take a-- let me take a better color-- 00:03:36.770 --> 00:03:40.310 I take a closed curve non-self-intersecting closed 00:03:40.310 --> 00:03:45.130 curve on that surface-- that defines 00:03:45.130 --> 00:03:49.290 an interior and an exterior. 00:03:49.290 --> 00:03:52.590 Then what the Gauss-Bonnet theorem 00:03:52.590 --> 00:03:56.850 says is that if you look at the curvature that's 00:03:56.850 --> 00:04:09.950 enclosed by the curve-- the total curvature by this closed 00:04:09.950 --> 00:04:17.140 curve on the surface-- and you add on the total turn 00:04:17.140 --> 00:04:29.400 angle along the curve-- so here, the curve is turning right. 00:04:29.400 --> 00:04:32.150 Here, it's turning left, right, left. 00:04:32.150 --> 00:04:33.440 Left is positive. 00:04:33.440 --> 00:04:34.930 Right is negative. 00:04:34.930 --> 00:04:38.757 Then these always add up to 360 degrees. 00:04:38.757 --> 00:04:40.340 We're not going to prove this theorem, 00:04:40.340 --> 00:04:42.290 but we're going to use it. 00:04:42.290 --> 00:04:45.820 So this is a nice invariant for sphere-like things. 00:04:45.820 --> 00:04:47.810 When you have handles, this number changes. 00:04:47.810 --> 00:04:50.566 I think it's 0 for a torus. 00:04:50.566 --> 00:04:53.490 Well, I won't try to guess it, because it's a little bit 00:04:53.490 --> 00:04:55.150 subtle to get right. 00:04:55.150 --> 00:04:58.420 One fun consequence of this-- let's just get warmed up-- 00:04:58.420 --> 00:05:01.640 suppose you take a sphere like object 00:05:01.640 --> 00:05:04.185 and you take a closed curve-- this 00:05:04.185 --> 00:05:07.400 is going to be pretty abstract-- in this direction, 00:05:07.400 --> 00:05:11.850 then it says, OK, the total amount of curvature 00:05:11.850 --> 00:05:16.840 in here plus the total turn angle equals 360. 00:05:16.840 --> 00:05:19.710 Now suppose that I turn the curve around. 00:05:19.710 --> 00:05:28.660 So I just reverse the direction-- like this. 00:05:28.660 --> 00:05:32.190 And so now, the interior is this stuff out here. 00:05:32.190 --> 00:05:35.950 And then we get that the total curvature outside the curve, 00:05:35.950 --> 00:05:40.120 plus the total turn angle of the blue thing, equals 360. 00:05:40.120 --> 00:05:42.410 If I add those two equations together, 00:05:42.410 --> 00:05:43.950 the total turn angle cancels. 00:05:43.950 --> 00:05:46.990 Because wherever red turns, left blue turns right. 00:05:46.990 --> 00:05:49.280 And so this term disappears. 00:05:49.280 --> 00:05:51.200 And so what we get is that the total curvature 00:05:51.200 --> 00:05:53.741 inside the curve, plus the total curvature outside the curve, 00:05:53.741 --> 00:05:55.260 equals 720. 00:05:55.260 --> 00:05:57.170 So this is a nice topological invariant 00:05:57.170 --> 00:06:01.750 of sphere-like polyhedra. 00:06:01.750 --> 00:06:04.230 You add up the total curvature everywhere-- 00:06:04.230 --> 00:06:08.150 it didn't really matter what the curve was-- you always get 720. 00:06:08.150 --> 00:06:09.670 And so this is kind of neat. 00:06:09.670 --> 00:06:11.170 For convex polyhedra, this means you 00:06:11.170 --> 00:06:13.061 have to somehow divvy up this curvature. 00:06:13.061 --> 00:06:14.560 Because all curvatures are positive. 00:06:14.560 --> 00:06:17.400 So 720 is, somehow, spread around. 00:06:17.400 --> 00:06:19.889 But you have to have exactly that much. 00:06:19.889 --> 00:06:22.430 For non-convex things, you could have some negative curvature 00:06:22.430 --> 00:06:24.570 that balances out a lot of positive curvature. 00:06:24.570 --> 00:06:26.180 So you can have a lot of both. 00:06:26.180 --> 00:06:28.490 But you have almost the same amount of each. 00:06:28.490 --> 00:06:31.262 Just you have exactly 720 excess. 00:06:31.262 --> 00:06:32.220 that's just a fun fact. 00:06:32.220 --> 00:06:34.011 We'll be using that in the future, I think. 00:06:38.370 --> 00:06:40.670 OK, that was Gauss-Bonnet. 00:06:40.670 --> 00:06:42.380 Now let's use it to prove that this 00:06:42.380 --> 00:06:46.610 can't happen for convex polyhedron. 00:06:46.610 --> 00:06:52.920 So the idea is, suppose you have an unfolding with a hole in it. 00:06:52.920 --> 00:06:54.670 So this is the surface out here. 00:06:58.100 --> 00:07:00.580 Then I'm going to take a closed curve that 00:07:00.580 --> 00:07:06.151 walks around the hole but stays inside the unfolding. 00:07:06.151 --> 00:07:08.150 And then, of course, I'm visualizing that really 00:07:08.150 --> 00:07:10.420 on the polyhedron. 00:07:10.420 --> 00:07:13.440 So I want the interior of the curve 00:07:13.440 --> 00:07:16.850 to enclose the hole-- or what becomes the hole. 00:07:16.850 --> 00:07:18.790 Now, what we learned from Gauss-Bonnet 00:07:18.790 --> 00:07:21.160 is that the curvature enclosed by this thing, 00:07:21.160 --> 00:07:24.630 plus the total turn angle of the curve, equals 360. 00:07:24.630 --> 00:07:28.670 The total turn angle of the curve is 360. 00:07:28.670 --> 00:07:31.330 It's a planar walk here. 00:07:31.330 --> 00:07:38.150 So turn angle-- total turn here is 360 degrees. 00:07:38.150 --> 00:07:40.380 Now, if we have a convex polyhedron, 00:07:40.380 --> 00:07:42.505 then this curvature has to be non-negative. 00:07:45.280 --> 00:07:46.610 Well, I mean, sorry. 00:07:46.610 --> 00:07:49.580 In any case here, this curvature better equal 0. 00:07:49.580 --> 00:07:53.320 For convex polyhedra, this is a sum of vertex curvatures. 00:07:53.320 --> 00:07:55.550 For convex polyhedra, every vertex 00:07:55.550 --> 00:07:58.160 has strictly positive curvature actually. 00:07:58.160 --> 00:08:01.690 I mean the zero curvature vertices aren't vertices. 00:08:01.690 --> 00:08:04.530 They're just points on the surface. 00:08:04.530 --> 00:08:07.230 So for convex polyhedron, if you're 00:08:07.230 --> 00:08:08.690 going to have zero total curvature, 00:08:08.690 --> 00:08:11.160 that means you actually have no curvature. 00:08:11.160 --> 00:08:13.839 So you have no vertices enclosed in here, 00:08:13.839 --> 00:08:14.880 which is a contradiction. 00:08:14.880 --> 00:08:17.640 That means there wasn't a hole to open up. 00:08:17.640 --> 00:08:20.020 Technically, you could do something weird 00:08:20.020 --> 00:08:25.350 like-- I don't know, let's take a cube. 00:08:25.350 --> 00:08:29.920 You could say, OK, I'm going to make a couple cuts here 00:08:29.920 --> 00:08:32.140 that do nothing. 00:08:32.140 --> 00:08:35.333 This is kind of a weird situation. 00:08:35.333 --> 00:08:36.966 We're going to add those cuts. 00:08:36.966 --> 00:08:39.299 And then, of course, you could draw this curve around it 00:08:39.299 --> 00:08:40.260 and say, oh yeah, look. 00:08:40.260 --> 00:08:42.851 I've got lots of zero curvature vertices inside here. 00:08:42.851 --> 00:08:45.100 In general, whenever you have zero curvature vertices, 00:08:45.100 --> 00:08:47.780 you could always just suture them back up-- uncut them-- 00:08:47.780 --> 00:08:50.040 and there'd be no difference in the unfolding. 00:08:50.040 --> 00:08:52.270 So you have to assume that you've already 00:08:52.270 --> 00:08:55.370 removed pointless cuts. 00:08:55.370 --> 00:08:57.840 Then there'll be no vertices in there. 00:08:57.840 --> 00:09:02.580 And so then, in fact, there was no hole for convex polyhedra. 00:09:02.580 --> 00:09:05.960 For non-convex polyhedra, you can have a negative curvature 00:09:05.960 --> 00:09:09.500 vertex that balances out some positive curvature vertices. 00:09:09.500 --> 00:09:12.320 And so the total curvature here equals zero. 00:09:12.320 --> 00:09:13.530 But you have three vertices. 00:09:13.530 --> 00:09:15.113 And that just can't happen for convex. 00:09:19.650 --> 00:09:23.650 Next quick question is, when we're 00:09:23.650 --> 00:09:26.980 talking about the cut locus and the ridge tree, 00:09:26.980 --> 00:09:28.654 we drew some pictures. 00:09:28.654 --> 00:09:30.320 The claim is that it was a spanning tree 00:09:30.320 --> 00:09:31.020 of the polyhedron. 00:09:31.020 --> 00:09:32.811 And in particular, the leaves of the tree-- 00:09:32.811 --> 00:09:35.060 the degree 1 vertices-- are exactly 00:09:35.060 --> 00:09:38.260 the vertices of the polyhedron. 00:09:38.260 --> 00:09:40.490 And I hadn't actually realized this, 00:09:40.490 --> 00:09:45.690 but in fact, it's not really literally true. 00:09:45.690 --> 00:09:48.380 It's kind of spiritually true. 00:09:48.380 --> 00:09:51.960 The vertices, in fact, have unique shortest paths to x. 00:09:51.960 --> 00:09:55.150 So remember, we have some point, x, on the surface. 00:09:55.150 --> 00:09:57.860 And we're looking at points, like this one, 00:09:57.860 --> 00:09:59.795 they have non-unique shortest paths to x. 00:09:59.795 --> 00:10:02.580 This is, if you grow the fire around x, 00:10:02.580 --> 00:10:04.060 where does the fire meet itself. 00:10:04.060 --> 00:10:06.250 And it will meet itself along this edge, 00:10:06.250 --> 00:10:07.750 because you could go around this way 00:10:07.750 --> 00:10:11.060 or go around this way and it's equal length path. 00:10:11.060 --> 00:10:12.559 But at the vertex, there's actually 00:10:12.559 --> 00:10:13.600 a unique way to go there. 00:10:13.600 --> 00:10:15.490 That's the black line. 00:10:15.490 --> 00:10:18.150 So technically, this point is not on the ridge tree. 00:10:18.150 --> 00:10:20.040 But all of these points are. 00:10:20.040 --> 00:10:21.890 So this is kind of like a limiting point 00:10:21.890 --> 00:10:25.006 of the ridge tree points. 00:10:25.006 --> 00:10:26.880 So we think of it as being on the ridge tree. 00:10:26.880 --> 00:10:28.520 I mean, you could think of as cut are not cut. 00:10:28.520 --> 00:10:29.710 It doesn't really matter. 00:10:29.710 --> 00:10:34.122 But you cut right next to it, so effectively the same thing. 00:10:34.122 --> 00:10:35.830 But it is a neat point-- a subtle point-- 00:10:35.830 --> 00:10:39.750 that these guys have unique shortest paths. 00:10:39.750 --> 00:10:41.030 Whereas, these do not. 00:10:41.030 --> 00:10:45.420 Still we cut all the way up to corner. 00:10:45.420 --> 00:10:49.290 All right, that is that. 00:10:49.290 --> 00:10:52.560 So now we have a bunch of newer and more exciting 00:10:52.560 --> 00:10:58.010 things-- or updates that you haven't heard of. 00:10:58.010 --> 00:11:01.030 One question that we mentioned was generalizing the star 00:11:01.030 --> 00:11:02.130 and source unfoldings. 00:11:02.130 --> 00:11:05.395 And there's a new paper about this. 00:11:05.395 --> 00:11:10.610 This is with my PhD advisor, Anna Lubiw, 00:11:10.610 --> 00:11:15.260 and this is just a warm up to get started. 00:11:15.260 --> 00:11:18.710 So here, we have a box, if we take a point x. 00:11:18.710 --> 00:11:21.830 And let's see, here we have the source unfolding from x. 00:11:21.830 --> 00:11:25.062 And here, we have the star unfolding, just for comparison. 00:11:25.062 --> 00:11:27.020 It's also kind of fun to see them side by side. 00:11:27.020 --> 00:11:28.290 They're color coded. 00:11:28.290 --> 00:11:31.870 So where you cut is the ridge tree, in this case. 00:11:31.870 --> 00:11:34.514 And you end up gluing along the ridge tree 00:11:34.514 --> 00:11:35.430 in the star unfolding. 00:11:38.090 --> 00:11:39.710 Now, we're going to generalize things 00:11:39.710 --> 00:11:42.390 a little bit in that we're going to generalize the source 00:11:42.390 --> 00:11:44.020 unfolding, specifically. 00:11:44.020 --> 00:11:46.560 So source unfolding, you have a point x. 00:11:46.560 --> 00:11:49.120 And you just sort of shoot shortest paths all 00:11:49.120 --> 00:11:51.097 from x, and that's what you keep. 00:11:51.097 --> 00:11:53.180 And you end up cutting along the ridge tree, which 00:11:53.180 --> 00:11:55.100 is the void in our diagram at this point. 00:11:55.100 --> 00:11:57.420 So I'm going to generalize that a little bit 00:11:57.420 --> 00:12:00.370 and think of this as a tiny little circle. 00:12:00.370 --> 00:12:02.290 And in general, what I'm going to do 00:12:02.290 --> 00:12:04.731 is the source unfolding outside the circle. 00:12:04.731 --> 00:12:07.230 And so when the circle is really tiny, it is just the source 00:12:07.230 --> 00:12:08.560 unfolding. 00:12:08.560 --> 00:12:11.650 But in general, I'm going to do the star unfolding 00:12:11.650 --> 00:12:14.240 inside the circle. 00:12:14.240 --> 00:12:15.180 OK. 00:12:15.180 --> 00:12:17.250 So in this case, nothing changes. 00:12:17.250 --> 00:12:19.860 But next example is going to be more general. 00:12:19.860 --> 00:12:21.770 Instead of being a single point here, 00:12:21.770 --> 00:12:23.780 I'm going to take a geodesic arc-- 00:12:23.780 --> 00:12:26.030 a straight line on the surface. 00:12:26.030 --> 00:12:28.310 So here's an example of that. 00:12:28.310 --> 00:12:31.350 We have a square-based pyramid. 00:12:31.350 --> 00:12:34.470 We drew a straight line on the surface. 00:12:34.470 --> 00:12:37.210 If you unfolded it, it would be straight. 00:12:37.210 --> 00:12:40.296 We're thinking of having a little-- 00:12:40.296 --> 00:12:43.630 I think it's called a racetrack curve in mathematics-- 00:12:43.630 --> 00:12:46.229 around that straight line. 00:12:46.229 --> 00:12:48.270 And I'm going to do star unfolding on the inside. 00:12:48.270 --> 00:12:49.940 In this case, there's no vertices on the inside, 00:12:49.940 --> 00:12:51.002 so nothing happens. 00:12:51.002 --> 00:12:52.960 And then we do source unfolding on the outside. 00:12:52.960 --> 00:12:54.334 This is actually previously known 00:12:54.334 --> 00:12:57.290 to unfold by O'Rourke and others. 00:12:57.290 --> 00:13:00.890 But we have a simpler proof, essentially. 00:13:00.890 --> 00:13:03.000 And we're going to generalize it more. 00:13:03.000 --> 00:13:05.490 But what we do is the source unfolding on the outside. 00:13:05.490 --> 00:13:07.520 So you take shortest paths-- from every point, 00:13:07.520 --> 00:13:12.300 you take its shortest path to this geodesic. 00:13:12.300 --> 00:13:15.130 That's, some of these blue lines show various shortest paths. 00:13:15.130 --> 00:13:16.810 That's what you keep. 00:13:16.810 --> 00:13:20.240 The ridge tree is if you light fire simultaneously 00:13:20.240 --> 00:13:22.630 along this entire segment, where does it burn out? 00:13:22.630 --> 00:13:24.520 And that's the purple stuff. 00:13:24.520 --> 00:13:27.740 And this is the complementary diagram, 00:13:27.740 --> 00:13:32.100 where you do the reverse, which would be the star unfolding 00:13:32.100 --> 00:13:34.230 on the outside, source unfolding on the inside. 00:13:34.230 --> 00:13:36.400 So you glue along the purple stuff 00:13:36.400 --> 00:13:39.870 instead of cutting along the purple stuff. 00:13:39.870 --> 00:13:42.390 And this, we conjecture, doesn't overlap but we don't know. 00:13:44.990 --> 00:13:46.750 All right, so fine. 00:13:46.750 --> 00:13:47.627 That looks easy. 00:13:47.627 --> 00:13:49.460 And you can prove that this doesn't overlap. 00:13:49.460 --> 00:13:51.370 Before you proved it, because you just 00:13:51.370 --> 00:13:54.630 had shortest paths emanating in all directions around x. 00:13:54.630 --> 00:13:58.600 Now, you have shortest paths emanating 00:13:58.600 --> 00:14:01.950 in 180 degrees of directions around this endpoint. 00:14:01.950 --> 00:14:03.590 Then they are all just straight. 00:14:03.590 --> 00:14:05.660 And then they rotate around 180 degrees here. 00:14:05.660 --> 00:14:07.140 And then they're all just straight. 00:14:07.140 --> 00:14:08.700 And so there can't be anything overlapping, 00:14:08.700 --> 00:14:10.300 because you're just taking a continuum 00:14:10.300 --> 00:14:12.470 of these segments of varying lengths. 00:14:12.470 --> 00:14:14.130 They don't overlap. 00:14:14.130 --> 00:14:15.540 Here, they're all parallel. 00:14:15.540 --> 00:14:16.660 Here, they sweep nicely. 00:14:16.660 --> 00:14:19.110 In general, if they always turn clockwise 00:14:19.110 --> 00:14:22.471 as you walk along the curve, you're fine. 00:14:22.471 --> 00:14:24.197 OK, here's a more general one. 00:14:24.197 --> 00:14:26.030 In general, what we can prove is that if you 00:14:26.030 --> 00:14:29.820 have a convex curve on the surface-- 00:14:29.820 --> 00:14:33.172 so it always turns to the left, I think technically. 00:14:33.172 --> 00:14:34.880 The angle on the left hand side is always 00:14:34.880 --> 00:14:37.071 less than or equal to 180 degrees. 00:14:37.071 --> 00:14:38.570 You have to be a little careful what 00:14:38.570 --> 00:14:39.370 it means to turn to the left. 00:14:39.370 --> 00:14:41.660 When you hit a vertex, you've got less than 360 total. 00:14:41.660 --> 00:14:43.120 So what does to the left mean? 00:14:43.120 --> 00:14:45.890 It just means you've got less than 180 degrees of material 00:14:45.890 --> 00:14:47.430 on your left side. 00:14:47.430 --> 00:14:49.320 So this is an example of a convex curve. 00:14:49.320 --> 00:14:50.840 It's got some circular arcs. 00:14:50.840 --> 00:14:53.442 We're no longer tracking along some-- 00:14:53.442 --> 00:14:55.400 we're no longer just doubling along some curve. 00:14:55.400 --> 00:14:57.060 Here, we enclose a vertex, which makes 00:14:57.060 --> 00:14:59.380 it a little more exciting, because now what we're 00:14:59.380 --> 00:15:02.150 going to do star unfolding on the inside, which remember 00:15:02.150 --> 00:15:05.760 was cutting along shortest paths from every vertex 00:15:05.760 --> 00:15:06.610 to your thing. 00:15:06.610 --> 00:15:08.730 In this case, your thing is no longer a point, x. 00:15:08.730 --> 00:15:10.690 It is now this convex curve. 00:15:10.690 --> 00:15:13.895 So let's say this is the shortest path here. 00:15:13.895 --> 00:15:18.777 It might be than one choice, but we're going to-- actually, no. 00:15:18.777 --> 00:15:19.860 This is the shortest path. 00:15:19.860 --> 00:15:21.050 This is a root 2 diagonal. 00:15:21.050 --> 00:15:22.896 This is length 1. 00:15:22.896 --> 00:15:24.270 So we're going to come along here 00:15:24.270 --> 00:15:26.190 and that's this dashed line. 00:15:26.190 --> 00:15:29.140 So it got opened up here, because we 00:15:29.140 --> 00:15:30.620 had too little material here. 00:15:30.620 --> 00:15:31.910 We open it up. 00:15:31.910 --> 00:15:33.410 But otherwise, it acts the same. 00:15:33.410 --> 00:15:36.050 In particular, outside this red curve, 00:15:36.050 --> 00:15:37.290 it's just source unfolding. 00:15:37.290 --> 00:15:38.907 You've got all these shortest paths. 00:15:38.907 --> 00:15:41.115 And again, what we argue-- and I'm not going to prove 00:15:41.115 --> 00:15:43.520 it here-- is that all of these shortest paths 00:15:43.520 --> 00:15:47.800 keep turning clockwise and do so exactly 360 degrees. 00:15:47.800 --> 00:15:49.950 There's some jumps when you hit these gaps. 00:15:49.950 --> 00:15:51.770 And you have to kind of jump over them. 00:15:51.770 --> 00:15:55.610 But still, you have no collision. 00:15:55.610 --> 00:15:58.240 And drawn over here is the reverse, 00:15:58.240 --> 00:16:03.616 where we would star unfold the outside of a convex curve-- 00:16:03.616 --> 00:16:05.740 because it's a convex curve, the inside and outside 00:16:05.740 --> 00:16:08.930 are different-- and source unfold on the inside. 00:16:08.930 --> 00:16:10.790 And this, we conjecture, doesn't overlap. 00:16:10.790 --> 00:16:13.390 But we don't know how to prove it. 00:16:13.390 --> 00:16:15.900 This thing is a generalization of the source unfolding. 00:16:15.900 --> 00:16:18.590 Because when this curve is super tiny, it is a source unfolding. 00:16:18.590 --> 00:16:20.090 This thing would be a generalization 00:16:20.090 --> 00:16:23.641 of the star unfolding, but we don't know whether works. 00:16:23.641 --> 00:16:25.840 I think one more example here. 00:16:25.840 --> 00:16:29.070 This is a way to generate convex curves. 00:16:29.070 --> 00:16:31.320 You start from some point, x. 00:16:31.320 --> 00:16:34.130 And you design it-- you choose a direction so 00:16:34.130 --> 00:16:36.980 that if you go straight-- so this curve goes straight 00:16:36.980 --> 00:16:38.230 everywhere except x. 00:16:38.230 --> 00:16:40.940 If you set it up right, you come back to x. 00:16:40.940 --> 00:16:42.690 So this, in particular, is a convex curve, 00:16:42.690 --> 00:16:44.440 because it's straight everywhere except x. 00:16:44.440 --> 00:16:46.290 And at x, it's convex. 00:16:46.290 --> 00:16:49.730 And so, in this case, we enclose a few vertices. 00:16:49.730 --> 00:16:53.440 On the inside, we've got V3, V7, V6. 00:16:53.440 --> 00:16:59.950 So each of those ends up getting cut here in these green lines. 00:16:59.950 --> 00:17:02.250 And so, like this one is a very tiny cut. 00:17:02.250 --> 00:17:04.160 We just cut there at V3. 00:17:04.160 --> 00:17:05.960 The other ones are little bigger. 00:17:05.960 --> 00:17:07.800 But they open up. 00:17:07.800 --> 00:17:12.359 And yeah, I guess, this is the-- well, 00:17:12.359 --> 00:17:15.520 it's a few different versions of the picture here. 00:17:15.520 --> 00:17:17.790 But again, you look at the shortest paths. 00:17:17.790 --> 00:17:19.099 Here, they're all parallel. 00:17:19.099 --> 00:17:20.300 Here, they sweep. 00:17:20.300 --> 00:17:21.670 Here, they're all parallel. 00:17:21.670 --> 00:17:24.110 Here, we jump, but it's just the same 00:17:24.110 --> 00:17:26.339 as sweeping-- it doesn't hurt you. 00:17:26.339 --> 00:17:28.300 Then they're all parallel. 00:17:28.300 --> 00:17:28.890 Then we jump. 00:17:28.890 --> 00:17:31.139 Then they're all parallel, in this particular example. 00:17:31.139 --> 00:17:33.030 But they will always proceed clockwise 00:17:33.030 --> 00:17:37.600 around the curve, which here is drawn red. 00:17:37.600 --> 00:17:39.390 Gets split up a little bit, but you 00:17:39.390 --> 00:17:41.150 can show because of that sweeping, 00:17:41.150 --> 00:17:42.640 they won't hit each other. 00:17:42.640 --> 00:17:46.290 And you have non-overlapping unfolding. 00:17:46.290 --> 00:17:51.000 So we call this sun unfolding, because of the rays 00:17:51.000 --> 00:17:52.642 that sweep around. 00:17:52.642 --> 00:17:54.100 And because we have a convex curve, 00:17:54.100 --> 00:17:56.820 it's kind of like the sun. 00:17:56.820 --> 00:18:00.150 So that's a new unfolding. 00:18:00.150 --> 00:18:02.950 The obvious open question is the reverse, 00:18:02.950 --> 00:18:04.600 when you glue around the purple sides 00:18:04.600 --> 00:18:10.585 instead of the-- It's essentially, 00:18:10.585 --> 00:18:11.960 instead of having a convex curve, 00:18:11.960 --> 00:18:14.780 you have a reflex curve, exactly the opposite. 00:18:14.780 --> 00:18:16.106 What happens, we don't know. 00:18:16.106 --> 00:18:17.230 It's a lot harder to prove. 00:18:17.230 --> 00:18:18.490 Because in particular, it's a lot harder 00:18:18.490 --> 00:18:20.531 to prove that the star unfolding doesn't overlap. 00:18:20.531 --> 00:18:22.670 And you've got to include at least that proof 00:18:22.670 --> 00:18:24.090 in any generalization of it. 00:18:26.610 --> 00:18:30.070 Next topic of unfolding kind of related 00:18:30.070 --> 00:18:32.130 we call zipper unfolding. 00:18:32.130 --> 00:18:36.960 So these are some examples of real felt models with a zipper. 00:18:36.960 --> 00:18:41.190 The goal is, I want an unfolding that has a single zipper. 00:18:41.190 --> 00:18:44.410 And you just pull the zipper, and it makes your polyhedron. 00:18:44.410 --> 00:18:46.340 So this is an example of an octahedron. 00:18:46.340 --> 00:18:48.730 This one is actually not even a polyhedron. 00:18:48.730 --> 00:18:51.650 It has two pyramidal pockets in it. 00:18:51.650 --> 00:18:54.140 And it's actually closed off in the middle. 00:18:54.140 --> 00:18:56.220 I think we have a little video here 00:18:56.220 --> 00:19:00.180 of what it looks like to open that octahedron. 00:19:00.180 --> 00:19:03.180 So what does it take to have a single zipper 00:19:03.180 --> 00:19:08.445 line, that connects everything together? 00:19:08.445 --> 00:19:09.820 Well, if you think about it, that 00:19:09.820 --> 00:19:14.160 means that the cuts that you make must follow a single path. 00:19:14.160 --> 00:19:18.380 So in general, the cuts form a tree on a convex polyhedron. 00:19:18.380 --> 00:19:21.000 We want that tree to be just a path. 00:19:21.000 --> 00:19:23.000 So it's like a Hamiltonian path. 00:19:23.000 --> 00:19:26.430 It's got to visit all the vertices in some order. 00:19:26.430 --> 00:19:28.975 And you'd like it to unfold without overlap. 00:19:28.975 --> 00:19:30.100 So is this always possible? 00:19:34.280 --> 00:19:39.780 This is a father, son, mother, son, son paper. 00:19:39.780 --> 00:19:41.790 So this is my advisor again and her two sons. 00:19:41.790 --> 00:19:44.164 Although we like to say this a paper with her three sons, 00:19:44.164 --> 00:19:47.900 because I'm her academic son-- or four, if you count 00:19:47.900 --> 00:19:48.560 Marty too. 00:19:48.560 --> 00:19:51.350 She was his advisor as well. 00:19:51.350 --> 00:19:53.800 So here are some examples of good unfolding. 00:19:53.800 --> 00:19:55.200 So we have the platonic solids. 00:19:55.200 --> 00:19:57.170 These are just typical unfoldings. 00:19:57.170 --> 00:19:59.220 These are all zipper unfoldings. 00:19:59.220 --> 00:20:02.425 So we're cutting along edges, along a Hamiltonian path 00:20:02.425 --> 00:20:03.300 that doesn't overlap. 00:20:03.300 --> 00:20:06.210 They all have this kind of nice snake like shape. 00:20:06.210 --> 00:20:07.640 And so all platonic solids can be 00:20:07.640 --> 00:20:10.210 made by zipper edge unfoldings. 00:20:10.210 --> 00:20:11.740 Next, we did Archimedean solids. 00:20:11.740 --> 00:20:13.480 This is a lot more work. 00:20:13.480 --> 00:20:17.800 But again, you get these nice S-like curves. 00:20:17.800 --> 00:20:19.224 And they're all zipper unfolding, 00:20:19.224 --> 00:20:20.890 because they're all possible and they'll 00:20:20.890 --> 00:20:27.060 have these Hamiltonian paths, cuts, and avoid overlap. 00:20:27.060 --> 00:20:28.850 But you may notice, there's one missing up 00:20:28.850 --> 00:20:30.890 there-- the great Rhombi-Cosi-Dodecahedron, 00:20:30.890 --> 00:20:32.870 my favorite Archimedean solid. 00:20:32.870 --> 00:20:37.000 And it has a rather different looking unfolding. 00:20:37.000 --> 00:20:39.430 As far as we can tell, there's no S-shaped one-- 00:20:39.430 --> 00:20:40.350 whatever that means. 00:20:40.350 --> 00:20:42.580 But you have to have a tree. 00:20:42.580 --> 00:20:45.310 These examples all had path-like-- 00:20:45.310 --> 00:20:47.470 The dual graph was roughly a path. 00:20:47.470 --> 00:20:49.970 I guess it branches a little bit here. 00:20:49.970 --> 00:20:53.810 Here, it's very tree like, I guess. 00:20:53.810 --> 00:20:57.260 So all Archimedean solids can be done. 00:20:57.260 --> 00:20:59.560 One open question, next category up, 00:20:59.560 --> 00:21:01.890 is Johnson solids, which are all polyhedron made 00:21:01.890 --> 00:21:04.570 with regular polygon faces-- convex polyhedron made 00:21:04.570 --> 00:21:06.484 by regular polygonal faces. 00:21:06.484 --> 00:21:08.150 Those we don't know, whether they always 00:21:08.150 --> 00:21:10.040 have zipper unfoldings. 00:21:10.040 --> 00:21:13.160 But we do know there are some convex polyhedra-- 00:21:13.160 --> 00:21:19.970 like this rhombic dodecahedron-- that do not have zipper 00:21:19.970 --> 00:21:21.960 unfoldings if you only cut along edges. 00:21:21.960 --> 00:21:24.710 Because this graph has no Hamiltonian path. 00:21:24.710 --> 00:21:26.674 So never mind avoiding overlap. 00:21:26.674 --> 00:21:28.840 There's some polyhedra that just aren't Hamiltonian. 00:21:28.840 --> 00:21:31.330 There's no path it visits every vertex exactly 00:21:31.330 --> 00:21:33.890 once and only follows edges. 00:21:33.890 --> 00:21:36.180 So there's nothing you could even 00:21:36.180 --> 00:21:39.100 hope to cut along and avoid overlap. 00:21:39.100 --> 00:21:41.590 So that's bad news for edge unfoldings. 00:21:41.590 --> 00:21:43.840 The big open question here is for general unfoldings 00:21:43.840 --> 00:21:46.420 if you're allowed to cut anywhere on a convex surface 00:21:46.420 --> 00:21:48.030 and do things like the sun unfolding. 00:21:48.030 --> 00:21:49.676 I mean, edge unfoldings, we don't 00:21:49.676 --> 00:21:50.800 know how to do them anyway. 00:21:50.800 --> 00:21:53.034 So if you allow general unfoldings, 00:21:53.034 --> 00:21:55.450 we've got star unfolding, source unfolding, sun unfolding, 00:21:55.450 --> 00:21:57.090 all sorts of things. 00:21:57.090 --> 00:21:59.740 But none of them are zipper unfoldings. 00:21:59.740 --> 00:22:01.520 They all cut along trees. 00:22:01.520 --> 00:22:04.500 Like star unfolding cuts along a star. 00:22:04.500 --> 00:22:06.460 The source unfolding cuts along the ridge tree, 00:22:06.460 --> 00:22:08.910 which is going to be very tree like thing. 00:22:08.910 --> 00:22:13.040 Can you always convert a tree cut into a path cut? 00:22:13.040 --> 00:22:14.520 We've tried. 00:22:14.520 --> 00:22:17.550 It seems quite challenging. 00:22:17.550 --> 00:22:18.680 So that's the open problem. 00:22:18.680 --> 00:22:22.640 Does every convex polyhedron have a general zipper 00:22:22.640 --> 00:22:26.420 unfolding, because edge unfolding is always too much 00:22:26.420 --> 00:22:28.010 to hope for. 00:22:28.010 --> 00:22:32.542 So those are some fun problems to think about. 00:22:32.542 --> 00:22:34.000 It's kind of like zipper unfolding. 00:22:36.760 --> 00:22:38.870 So the next topic-- oh, right. 00:22:38.870 --> 00:22:43.240 One more thing to show, this is a very fun talk that we gave. 00:22:43.240 --> 00:22:46.530 All five of us gave this talk at Canadian Conference 00:22:46.530 --> 00:22:48.110 on Computational Geometry. 00:22:48.110 --> 00:22:51.270 And one of the props in the talk was this cardboard box. 00:22:51.270 --> 00:22:53.844 And in the middle of the talk, it starts jiggling. 00:22:53.844 --> 00:22:56.010 And actually, initially, only four of the co-authors 00:22:56.010 --> 00:22:58.750 were giving the talk, because the fifth one 00:22:58.750 --> 00:23:01.220 was hiding inside the box. 00:23:01.220 --> 00:23:03.170 And then in the middle of the talk, 00:23:03.170 --> 00:23:05.750 he just jumps out and then starts 00:23:05.750 --> 00:23:08.180 speaking, as if nothing happened. 00:23:08.180 --> 00:23:09.720 It was a lot of fun. 00:23:09.720 --> 00:23:13.200 At this point, our son, Jonah, is in the box. 00:23:13.200 --> 00:23:14.320 He was fairly small. 00:23:14.320 --> 00:23:17.090 He's grown up a lot since, but at the time 00:23:17.090 --> 00:23:18.520 he fit nicely into this cardboard 00:23:18.520 --> 00:23:21.930 box, which was maybe this big. 00:23:21.930 --> 00:23:23.870 I think we give him a book and a flashlight 00:23:23.870 --> 00:23:26.640 and stuff to do while he was sitting there, waiting 00:23:26.640 --> 00:23:30.440 for his slides, waiting for the queue to come out. 00:23:30.440 --> 00:23:31.110 There we go. 00:23:36.610 --> 00:23:39.050 So those are unfolding of the cube or a box in general. 00:23:41.910 --> 00:23:45.851 Next topic is going back to edge unfolding. 00:23:45.851 --> 00:23:47.420 I like this comment. 00:23:47.420 --> 00:23:49.000 I thought they were pretty obvious, 00:23:49.000 --> 00:23:50.820 but now you've convinced me otherwise. 00:23:50.820 --> 00:23:54.450 And some of the evidence for edge unfolding being difficult 00:23:54.450 --> 00:23:57.720 was this polyhedron, which we proved has no edge unfolding. 00:23:57.720 --> 00:24:01.210 I didn't say it in lecture, but we call it edge on unfoldable, 00:24:01.210 --> 00:24:04.350 because they cannot be unfolded. 00:24:04.350 --> 00:24:06.700 This is actually the first example we came up with. 00:24:06.700 --> 00:24:10.092 I mention it only because it has fewer faces. 00:24:10.092 --> 00:24:12.175 We usually show this one because it's triangulated 00:24:12.175 --> 00:24:13.840 and that's kind of cooler. 00:24:13.840 --> 00:24:17.790 This one has convex faces still, so still topologically convex. 00:24:17.790 --> 00:24:20.110 I mean, if I pushed these points in, 00:24:20.110 --> 00:24:22.950 the polyhedron would be convex but has fewer faces. 00:24:22.950 --> 00:24:34.000 It has I guess six faces per hat, times 4 hats, so 24 faces. 00:24:34.000 --> 00:24:36.440 This was done in '99. 00:24:36.440 --> 00:24:38.740 It turns out at exactly the same time, 00:24:38.740 --> 00:24:40.556 there was this paper by Tarasov called, 00:24:40.556 --> 00:24:42.180 "Polyhedron with No Natural Unfolding." 00:24:42.180 --> 00:24:44.799 The paper has no figures, so I had to draw one 00:24:44.799 --> 00:24:46.090 to show you what it looks like. 00:24:46.090 --> 00:24:47.580 It's just a cube. 00:24:47.580 --> 00:24:50.100 And then at each corner of the cube, you cut off the corner 00:24:50.100 --> 00:24:53.450 and then pull the point out. 00:24:53.450 --> 00:24:56.690 And they proved, by a pretty similar argument-- I mean, 00:24:56.690 --> 00:24:58.730 essentially you treat each of these as a hat. 00:24:58.730 --> 00:25:00.350 It's kind of a very simple hat. 00:25:00.350 --> 00:25:02.730 They happen to overlap; they share edges. 00:25:02.730 --> 00:25:05.170 But again, because of these negative curvature vertices, 00:25:05.170 --> 00:25:06.930 you have to cut through the hat. 00:25:06.930 --> 00:25:08.860 And that cut has to keep going around. 00:25:08.860 --> 00:25:10.070 Eventually, it forms a cycle. 00:25:10.070 --> 00:25:15.480 So that's what Tarasov proved in the same year, '99. 00:25:15.480 --> 00:25:22.830 And then Grunbaum-- he's a famous geometer in Seattle-- 00:25:22.830 --> 00:25:24.230 came up with some more examples. 00:25:24.230 --> 00:25:27.780 So he initially wanted to make a star shaped example. 00:25:27.780 --> 00:25:30.680 Star shaped means that there's a single point, namely 00:25:30.680 --> 00:25:33.250 the center, where you can shine a light in all directions 00:25:33.250 --> 00:25:36.949 and the light reaches the entire interior of the polyhedron. 00:25:36.949 --> 00:25:38.740 He didn't know about our witch hat example. 00:25:38.740 --> 00:25:42.570 So he took the Tarasov example, made it a dodecahedron, 00:25:42.570 --> 00:25:44.500 and then it is star shaped. 00:25:44.500 --> 00:25:47.200 I think ours are already star shaped. 00:25:47.200 --> 00:25:48.130 So that's kind of fun. 00:25:48.130 --> 00:25:51.110 It looks like some scary underwater creature. 00:25:51.110 --> 00:25:53.000 And then he learned about our paper, 00:25:53.000 --> 00:25:54.541 and he said, all right, I want to get 00:25:54.541 --> 00:25:56.000 as few faces as possible. 00:25:56.000 --> 00:25:59.630 And so we had 26, was it, 24? 00:25:59.630 --> 00:26:03.580 This one has only 13 faces. 00:26:03.580 --> 00:26:06.540 And it has tetrahedral-- well, that's not actually 00:26:06.540 --> 00:26:09.030 tetrahedrally asymmetric, because this bottom spike is 00:26:09.030 --> 00:26:10.880 different from the others. 00:26:10.880 --> 00:26:13.540 But it's kind of like Tarasov's example down 00:26:13.540 --> 00:26:17.780 to its very minimal amounts, also star shaped. 00:26:17.780 --> 00:26:21.420 And his conjecture is that, if you have 12 faces or fewer, 00:26:21.420 --> 00:26:24.480 there is no un-unfoldable polyhedron. 00:26:24.480 --> 00:26:30.130 So he says polyhedra with 12 faces are un-un-unfoldable. 00:26:30.130 --> 00:26:33.259 He wanted to one up us. 00:26:33.259 --> 00:26:35.175 Open problem is to define un-un-un-unfoldable. 00:26:37.934 --> 00:26:38.850 But we're up to three. 00:26:42.580 --> 00:26:45.040 So that's some un-unfoldable polyhedra 00:26:45.040 --> 00:26:49.330 and some conjectured un-un-unfoldable polyhedra. 00:26:49.330 --> 00:26:50.940 This is fun to say. 00:26:50.940 --> 00:26:55.430 The next result, which is very recent is that-- this 00:26:55.430 --> 00:26:58.750 is with Zach Abel and myself-- that it 00:26:58.750 --> 00:27:02.330 is NP-complete to decide, given a polyhedron, 00:27:02.330 --> 00:27:05.460 is it unfoldable or un-unfoldable-- by cutting 00:27:05.460 --> 00:27:06.250 along edges. 00:27:06.250 --> 00:27:07.850 These are all cutting along edges. 00:27:07.850 --> 00:27:10.270 And we reduce from this problem, which 00:27:10.270 --> 00:27:13.620 comes from parallel computers actually-- 00:27:13.620 --> 00:27:16.500 or geometry in general, you want to pack a bunch of squares 00:27:16.500 --> 00:27:17.300 into a square. 00:27:17.300 --> 00:27:19.930 So you have squares of different sizes. 00:27:19.930 --> 00:27:22.910 And you want to pack those squares into a given square. 00:27:22.910 --> 00:27:23.640 Is it possible? 00:27:23.640 --> 00:27:24.490 Yes or no? 00:27:24.490 --> 00:27:27.030 Sometimes it is, sometimes it isn't. 00:27:27.030 --> 00:27:32.259 This proof, by many people-- Long, et al.-- 00:27:32.259 --> 00:27:34.550 is from three partition, which is a problem we've seen. 00:27:34.550 --> 00:27:35.466 I won't go through it. 00:27:35.466 --> 00:27:37.980 But essentially, it's kind of like the disk packing 00:27:37.980 --> 00:27:40.460 proof, which I showed in lecture some time ago related 00:27:40.460 --> 00:27:42.340 to tree maker. 00:27:42.340 --> 00:27:44.740 But you set up this tiny space and you end up 00:27:44.740 --> 00:27:49.800 having to partition a bunch of your squares into groups, each 00:27:49.800 --> 00:27:53.950 with the same sum groups of size 3. 00:27:53.950 --> 00:27:57.290 So starting from this problem of square packing, 00:27:57.290 --> 00:28:01.220 how do we convert it into an unfolding problem? 00:28:01.220 --> 00:28:03.430 This is a rough idea of the construction. 00:28:03.430 --> 00:28:05.780 So the big picture-- this is initially 00:28:05.780 --> 00:28:07.000 a polyhedron with boundary. 00:28:07.000 --> 00:28:08.680 Later on, we'll remove the boundary. 00:28:08.680 --> 00:28:12.482 So just imagine a square, and in the square, there's a tower. 00:28:12.482 --> 00:28:13.940 Things are not drawn to scale here. 00:28:13.940 --> 00:28:17.270 The tower is super tall. 00:28:17.270 --> 00:28:22.010 This little pipe thing is very narrow and also very long. 00:28:22.010 --> 00:28:27.130 I think even way longer than anything in this picture. 00:28:27.130 --> 00:28:30.610 OK, so now along the side of the tower-- 00:28:30.610 --> 00:28:33.010 so this is an unfolding. 00:28:33.010 --> 00:28:36.510 Like if you cut along these edges, you get a plus sign. 00:28:36.510 --> 00:28:38.370 This is the tower. 00:28:38.370 --> 00:28:40.770 Now, this grid stuff is something 00:28:40.770 --> 00:28:42.100 that I haven't shown you yet. 00:28:42.100 --> 00:28:44.870 But basically, think of it as water. 00:28:44.870 --> 00:28:45.960 It's very malleable. 00:28:45.960 --> 00:28:48.780 It can be cut open in many different ways. 00:28:48.780 --> 00:28:51.520 And essentially, you don't have to worry about it being there. 00:28:51.520 --> 00:28:54.010 It's like the glue that holds everything together. 00:28:54.010 --> 00:28:58.100 But then, there's these square faces, b1 up to bn. 00:28:58.100 --> 00:28:59.940 These are the things you need to pack. 00:28:59.940 --> 00:29:02.340 So our goal is to set things up that so, when 00:29:02.340 --> 00:29:04.550 you take this tower out, you have a square hole. 00:29:04.550 --> 00:29:07.174 That is your target square shape. 00:29:07.174 --> 00:29:08.590 And then on the side of the tower, 00:29:08.590 --> 00:29:10.090 you've got all these squares. 00:29:10.090 --> 00:29:12.580 Basically those squares have to fit into that hole. 00:29:12.580 --> 00:29:14.230 So it is square packing. 00:29:14.230 --> 00:29:15.200 That is our goal. 00:29:15.200 --> 00:29:19.560 Now, the challenge for that is to make all of this stuff 00:29:19.560 --> 00:29:22.029 get out of the way and also for these squares-- 00:29:22.029 --> 00:29:24.070 normally when you unfold, you're very constrained 00:29:24.070 --> 00:29:25.312 in how you lay out the faces. 00:29:25.312 --> 00:29:27.020 You can cut here or cut here, but there's 00:29:27.020 --> 00:29:28.680 a discrete set of choices. 00:29:28.680 --> 00:29:31.670 Our goal is to design this stuff so that these squares can just 00:29:31.670 --> 00:29:35.230 basically move willy-nilly without hitting each other. 00:29:35.230 --> 00:29:37.610 It's not easy to do, but that's the thing. 00:29:37.610 --> 00:29:40.949 This is one big face on the outside-- this kind of L shape. 00:29:40.949 --> 00:29:43.490 We didn't want to go all the way around for a couple reasons. 00:29:43.490 --> 00:29:45.930 One is we need a place for things to get out. 00:29:45.930 --> 00:29:49.840 But also, we wanted this to be topologically convex. 00:29:49.840 --> 00:29:53.870 So we didn't want to face that is a donut. 00:29:53.870 --> 00:29:56.330 OK, so what's the next part of the construction? 00:29:56.330 --> 00:29:59.280 Well, if we can arrange for these guys 00:29:59.280 --> 00:30:03.150 to move willy-nilly-- that's these very thin lines-- 00:30:03.150 --> 00:30:05.510 we can just imagine that if the squares are somehow 00:30:05.510 --> 00:30:09.780 packed, instead of having things overlapping like this, 00:30:09.780 --> 00:30:14.687 you can route all of those paths to avoid crossings. 00:30:14.687 --> 00:30:17.270 So as long as there's tiny gaps between all the squares, which 00:30:17.270 --> 00:30:20.010 doesn't turn out to change the square packing problem very 00:30:20.010 --> 00:30:24.080 much, you can do this. 00:30:24.080 --> 00:30:27.340 And then there's another issue, which is there's a lot of stuff 00:30:27.340 --> 00:30:27.840 here. 00:30:27.840 --> 00:30:30.130 You've got to put it somewhere, so you end up-- 00:30:30.130 --> 00:30:32.620 it's very, very tiny-- so in some cases, 00:30:32.620 --> 00:30:34.760 you have to do this kind of wiggling just 00:30:34.760 --> 00:30:39.510 to eat up length in certain settings. 00:30:39.510 --> 00:30:41.040 It gets complicated. 00:30:41.040 --> 00:30:43.140 So how do we do this wiggly stuff? 00:30:43.140 --> 00:30:47.980 Well, at the first level, there's 00:30:47.980 --> 00:30:50.720 a very fine grid with very small squares here. 00:30:53.370 --> 00:30:55.700 And we set it up so that we can follow everything 00:30:55.700 --> 00:30:57.150 along a single path. 00:30:57.150 --> 00:30:59.960 We can visit all of this great stuff along a single path. 00:30:59.960 --> 00:31:03.120 Occasionally, we will encounter squares, 00:31:03.120 --> 00:31:05.500 but the path length between any two squares 00:31:05.500 --> 00:31:08.640 is so big that these squares have room to kind of stretch 00:31:08.640 --> 00:31:13.010 out as far away from each other as they need to go. 00:31:13.010 --> 00:31:14.670 Now, these squares are not squares. 00:31:14.670 --> 00:31:16.540 They are actually this construction, 00:31:16.540 --> 00:31:18.440 which we call atoms. 00:31:18.440 --> 00:31:23.420 It looks like a weird set of pyramids, or outside of towers. 00:31:23.420 --> 00:31:27.280 But in fact, there's many different ways to unfold this. 00:31:27.280 --> 00:31:29.400 There's lots of different ways to cut it open. 00:31:29.400 --> 00:31:30.660 And some of them go straight. 00:31:30.660 --> 00:31:31.618 Some of them turn left. 00:31:31.618 --> 00:31:32.910 Some of them turn right. 00:31:32.910 --> 00:31:34.830 These are three of the unfoldings, I forget. 00:31:34.830 --> 00:31:37.080 There's a few dozen unfoldings that we 00:31:37.080 --> 00:31:39.700 need that all have the right parameters. 00:31:39.700 --> 00:31:41.340 This one's clearly turning left. 00:31:41.340 --> 00:31:43.677 This one's going straight in a certain sense. 00:31:43.677 --> 00:31:46.010 This is coming from a left turn and then going straight. 00:31:46.010 --> 00:31:48.060 So there's lots of combinations here, 00:31:48.060 --> 00:31:50.350 slightly different parities, and so on. 00:31:50.350 --> 00:31:54.410 But the end effect is that if you have a big grid of these 00:31:54.410 --> 00:31:58.690 and you're following them in a path, 00:31:58.690 --> 00:32:01.870 the path does some turns on the surface. 00:32:01.870 --> 00:32:03.670 But you can force it to do whatever 00:32:03.670 --> 00:32:05.320 turns you want in the unfolding. 00:32:05.320 --> 00:32:08.330 So you have complete freedom to move the squares around. 00:32:08.330 --> 00:32:10.450 As I showed you, you can avoid crossings, 00:32:10.450 --> 00:32:13.160 and it all works out. 00:32:13.160 --> 00:32:15.640 Maybe one detail, which I didn't mention, 00:32:15.640 --> 00:32:19.370 is there's going to be a ton of extra stuff. 00:32:19.370 --> 00:32:20.350 Where does it go? 00:32:25.630 --> 00:32:27.130 It's going to fill the pipe up. 00:32:30.240 --> 00:32:32.860 It's going to push these guys up somewhat. 00:32:32.860 --> 00:32:34.780 And this thing is so tall that there's 00:32:34.780 --> 00:32:39.790 room to put all the excess stuff in here. 00:32:39.790 --> 00:32:41.710 And it's also so tall that you can't just 00:32:41.710 --> 00:32:44.810 reach all the way out and put all the squares on the outside. 00:32:44.810 --> 00:32:48.729 OK, so if you make that super, super tall, none of the squares 00:32:48.729 --> 00:32:50.270 fit in here, because it's too narrow. 00:32:50.270 --> 00:32:51.686 And none of the squares can get up 00:32:51.686 --> 00:32:53.160 to the top, because it's too long. 00:32:53.160 --> 00:32:55.880 So that's how it works. 00:32:55.880 --> 00:32:58.360 Now, that was a polyhedron with boundary. 00:32:58.360 --> 00:33:00.070 Without boundary, the construction 00:33:00.070 --> 00:33:01.690 is almost the same. 00:33:01.690 --> 00:33:04.930 You just add some extra stuff on the outside 00:33:04.930 --> 00:33:08.180 to make it a regular polyhedron homeomorphic to a sphere. 00:33:08.180 --> 00:33:10.530 But in the end, you have basically the same construction 00:33:10.530 --> 00:33:11.770 of this nice square. 00:33:11.770 --> 00:33:14.490 The pipe and some other stuff, you prove basically 00:33:14.490 --> 00:33:16.490 doesn't matter. 00:33:16.490 --> 00:33:20.940 So that is NP-completeness of edge unfolding 00:33:20.940 --> 00:33:23.730 of topologically convex polyhedra, 00:33:23.730 --> 00:33:25.590 even orthogonal polyhedra, which is 00:33:25.590 --> 00:33:29.855 kind of nifty-- from last year. 00:33:32.455 --> 00:33:34.080 If you actually want to unfold things-- 00:33:34.080 --> 00:33:36.550 if you want to build things out of paper, 00:33:36.550 --> 00:33:39.430 the current best heuristic unfolder is called Pepakura. 00:33:39.430 --> 00:33:42.800 It's a free download online, probably Windows only. 00:33:42.800 --> 00:33:44.840 You can take some weird 3D model. 00:33:44.840 --> 00:33:48.330 And it uses multiple pieces, in general, but you can add tabs 00:33:48.330 --> 00:33:50.550 and it's quite practical. 00:33:50.550 --> 00:33:54.050 And cut it out, either manually or with your computer 00:33:54.050 --> 00:33:55.950 controlled sign cutter or something 00:33:55.950 --> 00:34:00.020 and then fold up your pieces. 00:34:00.020 --> 00:34:02.270 When a one piece unfolding is possible, 00:34:02.270 --> 00:34:03.480 it will typically find it. 00:34:03.480 --> 00:34:05.160 But it's not guaranteed. 00:34:05.160 --> 00:34:07.330 I don't know exactly what algorithm it uses. 00:34:07.330 --> 00:34:08.889 But some combination of brute force 00:34:08.889 --> 00:34:12.159 and just cutting into multiple pieces when it fails. 00:34:18.620 --> 00:34:23.320 Next up, we have band unfolding. 00:34:23.320 --> 00:34:25.310 So I talked very briefly about band unfolding. 00:34:25.310 --> 00:34:28.497 I thought I'd show you some pictures about it. 00:34:28.497 --> 00:34:31.080 And so remember, we're talking about volcano unfoldings, which 00:34:31.080 --> 00:34:32.830 is when you cut along all the edges 00:34:32.830 --> 00:34:35.530 from a point or it's sort of in the vertical thing. 00:34:35.530 --> 00:34:38.150 Band unfolding was when you had some side faces, 00:34:38.150 --> 00:34:42.250 you kept those intact and cut everything else away. 00:34:42.250 --> 00:34:44.150 So what do I have to show here? 00:34:44.150 --> 00:34:50.489 This is an example of a band unfolding gone wrong-- I guess. 00:34:50.489 --> 00:34:52.139 It's a little hard to see. 00:34:52.139 --> 00:34:54.170 This is, in general, a prismatoid. 00:34:54.170 --> 00:34:57.320 We had a top polygon, A, here, and then below it, 00:34:57.320 --> 00:35:00.060 this is polygon, B. We took the convex hull, which 00:35:00.060 --> 00:35:01.910 adds all these other edges. 00:35:01.910 --> 00:35:06.630 And this is an example of a bad unfolding of a primatoid. 00:35:06.630 --> 00:35:08.530 Here's an example of a bad-- this is really 00:35:08.530 --> 00:35:11.740 a band unfolding that has gone wrong. 00:35:11.740 --> 00:35:15.620 So imagine here the top polygon is this triangle. 00:35:15.620 --> 00:35:17.270 Bottom polygon is this triangle. 00:35:17.270 --> 00:35:18.005 That nicely nest. 00:35:18.005 --> 00:35:21.180 It's a very easy example, seemingly. 00:35:21.180 --> 00:35:23.590 But for whatever reason, we also added this cut. 00:35:23.590 --> 00:35:27.680 And you can actually force that if I make this not quite-- 00:35:27.680 --> 00:35:29.490 actually, maybe it's not quite a triangle. 00:35:29.490 --> 00:35:30.470 It's a quadrilateral. 00:35:30.470 --> 00:35:32.930 Yeah, and this is slightly a quadrilateral. 00:35:32.930 --> 00:35:35.200 So when you take a convex hull, you get that edge. 00:35:35.200 --> 00:35:37.900 If you cut along that edge and then cut along here 00:35:37.900 --> 00:35:41.010 and cut along here and do the band unfolding 00:35:41.010 --> 00:35:44.080 thing-- we're not even drawing the bottom polygon here-- just 00:35:44.080 --> 00:35:49.790 the band itself overlaps, which is annoying. 00:35:49.790 --> 00:35:53.070 Nonetheless-- so this is an old example-- 00:35:53.070 --> 00:35:57.776 but nonetheless, we proved that there's at least one place 00:35:57.776 --> 00:35:59.150 to cut the band-- like here would 00:35:59.150 --> 00:36:02.920 work-- that will avoid overlap. 00:36:02.920 --> 00:36:06.310 And this is some parts the proof. 00:36:06.310 --> 00:36:10.370 In general, here we're looking at the inner polygon. 00:36:10.370 --> 00:36:12.770 And we cut it somewhere. 00:36:12.770 --> 00:36:16.609 And then we argue about where that vertex goes if we open up 00:36:16.609 --> 00:36:18.150 all the angles, which is what happens 00:36:18.150 --> 00:36:20.440 when you squash it flat. 00:36:20.440 --> 00:36:23.200 And in particular, we argue this vertex 00:36:23.200 --> 00:36:26.510 must stay in the gray region, like in the picture 00:36:26.510 --> 00:36:30.850 on the right, so it can't go up here. 00:36:30.850 --> 00:36:33.470 And so in general, if you look at how these things unfold, 00:36:33.470 --> 00:36:36.280 if you're lucky, things will be convex when you open it. 00:36:36.280 --> 00:36:38.130 This is always going to be good. 00:36:38.130 --> 00:36:40.400 There's this weakly convex picture, 00:36:40.400 --> 00:36:42.880 where when you close the ends, it's 00:36:42.880 --> 00:36:45.730 not quite convex but only at one point. 00:36:48.840 --> 00:36:53.420 This is troublesome-- sometimes this works, 00:36:53.420 --> 00:36:54.510 sometimes it doesn't. 00:36:54.510 --> 00:36:56.550 If you look at this example, I believe 00:36:56.550 --> 00:36:58.850 it is in the weakly convex category. 00:36:58.850 --> 00:37:01.790 I haven't told you the other one is a spiral. 00:37:01.790 --> 00:37:06.190 Here, if you draw a 90 degree angle from this last bar, 00:37:06.190 --> 00:37:11.110 then this guy is on wrong side of that 90 degree thing. 00:37:11.110 --> 00:37:13.740 This thing can't happen. 00:37:13.740 --> 00:37:17.190 So that's comforting. 00:37:17.190 --> 00:37:19.460 And it's related to this property. 00:37:19.460 --> 00:37:21.390 But these things can cause problems 00:37:21.390 --> 00:37:23.220 like in the previous picture. 00:37:23.220 --> 00:37:25.760 And so you have to argue that there is a place 00:37:25.760 --> 00:37:29.600 to cut where you don't get this or if when you get it, 00:37:29.600 --> 00:37:30.590 it's still OK. 00:37:30.590 --> 00:37:33.440 And that's a messy argument-- a lot of case analysis 00:37:33.440 --> 00:37:35.160 I won't go through here. 00:37:35.160 --> 00:37:39.970 I think I have one picture of a nicely working example. 00:37:39.970 --> 00:37:45.300 This is, I guess, a general primatoid. 00:37:45.300 --> 00:37:48.770 But here, we have a nice cutting that works. 00:37:48.770 --> 00:37:53.510 The original question that someone here posed was, 00:37:53.510 --> 00:37:54.970 what about prismoids? 00:37:54.970 --> 00:37:57.730 So we know the band unfolds nicely. 00:37:57.730 --> 00:38:00.170 What we don't know is, can you attach the top polygon 00:38:00.170 --> 00:38:03.067 and the bottom polygons to the band and get an unfolding. 00:38:03.067 --> 00:38:05.400 We don't know, for example, whether all prismatoids have 00:38:05.400 --> 00:38:07.890 edge unfoldings, because we don't 00:38:07.890 --> 00:38:09.780 know how to place the top and bottom things. 00:38:09.780 --> 00:38:11.730 Probably possible but really hard 00:38:11.730 --> 00:38:14.820 to figure out where they would go. 00:38:14.820 --> 00:38:20.150 A simpler problem is prismoids-- which this might actually 00:38:20.150 --> 00:38:20.802 be a prismoid. 00:38:20.802 --> 00:38:22.260 If you know that all of these edges 00:38:22.260 --> 00:38:29.240 are parallel initially-- so it's a very nice situation-- then 00:38:29.240 --> 00:38:31.170 that's a more special case from prismatoids-- 00:38:31.170 --> 00:38:33.590 then maybe you could attach the interface and outerface. 00:38:33.590 --> 00:38:35.270 We don't know. 00:38:35.270 --> 00:38:37.770 We know volcano unfoldings work for prismoids. 00:38:37.770 --> 00:38:39.270 We don't know about band unfoldings. 00:38:39.270 --> 00:38:41.960 That could be an interesting open problem to work on. 00:38:41.960 --> 00:38:43.890 Maybe it's easy with prismoids. 00:38:43.890 --> 00:38:48.330 They seem pretty clean, but I don't know for sure. 00:38:48.330 --> 00:38:50.960 Oh, another fun thing, which relates to our next topic, 00:38:50.960 --> 00:38:55.090 is that if you just unfold the band part, 00:38:55.090 --> 00:38:58.720 you can do it by continuous blooming-- meaning there's 00:38:58.720 --> 00:39:01.900 a continuous motion from the folded thing 00:39:01.900 --> 00:39:04.130 to the unfolded thing, or vice versa. 00:39:04.130 --> 00:39:06.220 And the easy way to see that-- you kind of get 00:39:06.220 --> 00:39:09.860 a sense from this picture-- in general, when you squash it 00:39:09.860 --> 00:39:12.190 all the way flat, it opens. 00:39:12.190 --> 00:39:15.840 Like this initial grey diagram is a projection of the original 00:39:15.840 --> 00:39:16.510 thing. 00:39:16.510 --> 00:39:18.670 If you-- instead of opening it all the way-- 00:39:18.670 --> 00:39:20.750 if you kind of squish it from the top 00:39:20.750 --> 00:39:23.310 and just keep lowering that top face, at the end, 00:39:23.310 --> 00:39:25.080 it's fully in the plane. 00:39:25.080 --> 00:39:27.050 But in the middle, you have a motion. 00:39:27.050 --> 00:39:29.780 And by the same argument, at all times, 00:39:29.780 --> 00:39:31.030 you are non-self intersecting. 00:39:31.030 --> 00:39:33.350 So that actually gives you a continuous blooming 00:39:33.350 --> 00:39:37.385 of the band of a prismatoid, which is kind of cool. 00:39:39.890 --> 00:39:42.690 So the next topic is blooming. 00:39:42.690 --> 00:39:44.350 This is the last topic also. 00:39:44.350 --> 00:39:46.400 A bunch of people asked about continuous booming 00:39:46.400 --> 00:39:49.080 and so the obvious one to learn more about. 00:39:49.080 --> 00:39:55.530 So I have some algorithms and examples for you. 00:39:55.530 --> 00:39:57.840 This is the paper-- a bunch of authors, 00:39:57.840 --> 00:40:00.030 "Continuous Blooming of Convex Polyhedra." 00:40:00.030 --> 00:40:01.880 This problem was posed by Connolly, 00:40:01.880 --> 00:40:04.715 I believe, a bunch of years prior. 00:40:10.380 --> 00:40:13.580 It's still not known whether every unfolding continuously 00:40:13.580 --> 00:40:14.700 blooms. 00:40:14.700 --> 00:40:17.940 It could be that every unfolding-- 00:40:17.940 --> 00:40:21.020 every non-overlapping unfolding of a convex polyhedron-- 00:40:21.020 --> 00:40:22.160 continuously blooms. 00:40:22.160 --> 00:40:25.530 I would guess the answer is, it doesn't but it's plausible. 00:40:25.530 --> 00:40:27.750 That was the original question. 00:40:27.750 --> 00:40:30.000 What we found are two different ways 00:40:30.000 --> 00:40:33.060 to continuously bloom any convex polyhedron, 00:40:33.060 --> 00:40:34.784 but we choose the unfolding. 00:40:34.784 --> 00:40:36.450 So we have a couple different strategies 00:40:36.450 --> 00:40:38.920 for choosing unfoldings that do continuously bloom. 00:40:38.920 --> 00:40:41.150 Whether every unfolding continuously blooms, 00:40:41.150 --> 00:40:42.130 I don't know. 00:40:42.130 --> 00:40:44.730 There are definitely unfoldings of non-convex polyhedra that 00:40:44.730 --> 00:40:48.390 do not continuously bloom, based on knitting needles type 00:40:48.390 --> 00:40:50.880 examples, based on bad linkage stuff. 00:40:50.880 --> 00:40:53.890 But for convex polyhedra, I'm not sure. 00:40:56.930 --> 00:40:58.425 So what do we have first? 00:40:58.425 --> 00:41:02.800 First strategy actually starts from any unfolding you have. 00:41:02.800 --> 00:41:06.370 So we start here from the cross unfolding of the cube. 00:41:06.370 --> 00:41:07.920 And then it refines it. 00:41:07.920 --> 00:41:09.990 So a similar strategy to hinged dissection, 00:41:09.990 --> 00:41:14.130 although I think this was done before hinged dissection. 00:41:14.130 --> 00:41:16.670 Let's take some hinged structure, 00:41:16.670 --> 00:41:18.665 add extra cuts and extra hinges. 00:41:18.665 --> 00:41:21.040 In this case, I mean the hinges are going to be the same. 00:41:21.040 --> 00:41:22.770 It's just adding extra cuts. 00:41:22.770 --> 00:41:26.030 So we're going to cut along the red thing and also 00:41:26.030 --> 00:41:27.160 this red dashed line. 00:41:27.160 --> 00:41:31.990 The red thing is a spanning tree of the-- 00:41:31.990 --> 00:41:33.780 or it's really the dual graph. 00:41:33.780 --> 00:41:35.640 So we have, the dual graph is you 00:41:35.640 --> 00:41:37.580 have a vertex for every face. 00:41:37.580 --> 00:41:39.650 You connect them together if they share an edge. 00:41:39.650 --> 00:41:41.710 In this case, when they share an edge, 00:41:41.710 --> 00:41:43.070 we're going to cut along there. 00:41:43.070 --> 00:41:44.990 So we cut along all this red stuff. 00:41:44.990 --> 00:41:48.760 And the cool thing is, if you walk around the red structure, 00:41:48.760 --> 00:41:51.420 you get a cycle-- a Hamiltonian cycle that 00:41:51.420 --> 00:41:53.520 visits all the faces exactly once. 00:41:53.520 --> 00:41:57.100 We want to path, not a cycle, so we add one more cut. 00:41:57.100 --> 00:42:00.230 So then this blue dash thing is a path. 00:42:00.230 --> 00:42:02.720 And the claim is, any path shaped 00:42:02.720 --> 00:42:05.980 unfolding-- where the faces are connected together in a path-- 00:42:05.980 --> 00:42:07.900 can be continuously bloomed. 00:42:07.900 --> 00:42:09.410 How do you do it? 00:42:09.410 --> 00:42:09.910 Roll. 00:42:13.210 --> 00:42:15.570 So it's a little easier for me to think 00:42:15.570 --> 00:42:17.285 about unrolling rather than rolling, 00:42:17.285 --> 00:42:18.410 but they're the same thing. 00:42:18.410 --> 00:42:21.580 So imagine, you start with the cube. 00:42:21.580 --> 00:42:25.580 And then you just roll it-- you unroll one face at a time. 00:42:25.580 --> 00:42:28.490 So initially, there was a 90 degree fold angle here. 00:42:28.490 --> 00:42:31.812 Just unroll it, so that now they're in the same plane. 00:42:31.812 --> 00:42:33.770 And there was initially a 90 degree angle here. 00:42:33.770 --> 00:42:35.200 Unroll that. 00:42:35.200 --> 00:42:38.160 After three steps, I think-- this guy's 00:42:38.160 --> 00:42:42.850 been flattened, two, three-- we have this picture. 00:42:42.850 --> 00:42:44.960 In general, I would normally draw it rotated, 00:42:44.960 --> 00:42:50.140 so that the unfolded part lives in the floor of xy-plane. 00:42:50.140 --> 00:42:52.930 And the polyhedron lives in z greater than or equal to zero, 00:42:52.930 --> 00:42:55.340 so lives above that plane. 00:42:55.340 --> 00:42:58.800 What you know at all times is that the unfolded part 00:42:58.800 --> 00:43:00.910 is a subset of the unfolding. 00:43:00.910 --> 00:43:03.010 It's a subset of this picture. 00:43:03.010 --> 00:43:04.970 So it doesn't overlap itself. 00:43:04.970 --> 00:43:06.910 If you start with a non-overlapping unfolding, 00:43:06.910 --> 00:43:09.330 you do this cutting. 00:43:09.330 --> 00:43:11.390 And then you just take some subset of the faces-- 00:43:11.390 --> 00:43:13.710 a prefix of the path-- then that will 00:43:13.710 --> 00:43:15.430 be a non-overlapping thing. 00:43:15.430 --> 00:43:17.390 On the other hand, the polyhedron that remains, 00:43:17.390 --> 00:43:18.350 we haven't touched. 00:43:18.350 --> 00:43:19.940 I mean, it's rotating. 00:43:19.940 --> 00:43:22.680 You have to make sure it stays above the plane here. 00:43:22.680 --> 00:43:26.390 But it is just a subset of the faces of the cube. 00:43:26.390 --> 00:43:28.130 The cube is also not self intersecting. 00:43:28.130 --> 00:43:29.780 So any subset is not self intersecting. 00:43:29.780 --> 00:43:31.710 So this thing doesn't self intersect. 00:43:31.710 --> 00:43:34.930 This thing doesn't self intersect. 00:43:34.930 --> 00:43:39.700 Potentially, this cube will be resting on like f1 here. 00:43:39.700 --> 00:43:43.470 It's possible for this thing to unroll and just touch 00:43:43.470 --> 00:43:46.910 the plane-- xy-plane. 00:43:46.910 --> 00:43:49.790 But it won't penetrate it, just because this thing's convex 00:43:49.790 --> 00:43:51.220 and this is a plane. 00:43:51.220 --> 00:43:53.530 So you can always keep this thing above-- keep a convex 00:43:53.530 --> 00:43:56.650 shape above-- or what was a convex shape-- 00:43:56.650 --> 00:44:00.000 a partial version of a convex shape-- above a plane. 00:44:00.000 --> 00:44:02.040 But they can be potentially touching, 00:44:02.040 --> 00:44:04.900 even along two dimensional surfaces. 00:44:04.900 --> 00:44:08.720 So if you allow touching, this is the path unroll algorithm. 00:44:08.720 --> 00:44:09.990 It works fine. 00:44:09.990 --> 00:44:11.490 If you don't want to allow touching, 00:44:11.490 --> 00:44:20.970 you need a slightly better strategy, which I'll tell you. 00:44:20.970 --> 00:44:22.550 It's pretty simple. 00:44:22.550 --> 00:44:26.300 I won't argue that it works, but it's 00:44:26.300 --> 00:44:28.170 called the 2-step unfolding. 00:44:28.170 --> 00:44:30.550 This is named after the 2-step dance. 00:44:30.550 --> 00:44:33.180 So we alternate between two kinds of steps. 00:44:33.180 --> 00:44:45.430 One is unfold of edge, ei, to be almost coplanar-- sorry, 00:44:45.430 --> 00:44:57.710 this should be a face-- face, fi, 00:44:57.710 --> 00:45:03.995 to be almost coplanar with the previous one, fi minus 1. 00:45:03.995 --> 00:45:05.870 So that does correspond to unfolding an edge, 00:45:05.870 --> 00:45:08.870 but I'm going to think about numbering the faces. 00:45:08.870 --> 00:45:11.220 Almost means we unfold it to epsilon 00:45:11.220 --> 00:45:14.980 within the angle of 180 degrees. 00:45:14.980 --> 00:45:26.230 Then we finish unfolding the previous step-- fi minus 1. 00:45:26.230 --> 00:45:27.080 And then we repeat. 00:45:30.077 --> 00:45:31.035 This is a little funny. 00:45:33.830 --> 00:45:36.110 So we're worried that if we unfold all the way flat, 00:45:36.110 --> 00:45:38.900 we'll actually live in the plane and then we'll touch things. 00:45:38.900 --> 00:45:40.042 We don't want to do that. 00:45:40.042 --> 00:45:42.000 So we're going to unfold it almost all the way. 00:45:42.000 --> 00:45:44.460 But when we're done, we do want to be flat. 00:45:44.460 --> 00:45:47.729 So I don't want to just unfold everything almost all the way. 00:45:47.729 --> 00:45:49.520 That might work, but it's a little tricky-- 00:45:49.520 --> 00:45:51.840 the almost might interact. 00:45:51.840 --> 00:45:55.400 So what I'm going to do is just keep one guy almost unfolded. 00:45:55.400 --> 00:45:57.150 That will be the previous one. 00:45:57.150 --> 00:45:58.990 When I've almost unfolded the next guy, 00:45:58.990 --> 00:46:01.516 I will finish unfolding the previous guy. 00:46:01.516 --> 00:46:03.390 Then we will almost unfold the next, next guy 00:46:03.390 --> 00:46:06.390 and then finish unfolding the next guy, and so on. 00:46:06.390 --> 00:46:07.870 And this turns out to work. 00:46:07.870 --> 00:46:10.870 Essentially, you've got your fully planar part, 00:46:10.870 --> 00:46:14.180 which is everything before i minus 2. 00:46:14.180 --> 00:46:15.930 That would be completely flat. 00:46:15.930 --> 00:46:18.050 Then the next angle will be almost flat. 00:46:18.050 --> 00:46:20.260 And then the rest will be in its original state. 00:46:20.260 --> 00:46:22.610 And you can argue the appropriate definitions 00:46:22.610 --> 00:46:23.650 of almost. 00:46:23.650 --> 00:46:26.190 There will be no-- in this case, there 00:46:26.190 --> 00:46:28.300 will be no two dimensional overlap. 00:46:28.300 --> 00:46:31.890 You still can get an edge resting on another thing-- 00:46:31.890 --> 00:46:33.210 still not quite perfect. 00:46:33.210 --> 00:46:38.700 To make it completely perfect, we need a waltz. 00:46:38.700 --> 00:46:41.775 So the waltz is a three step dance. 00:46:45.960 --> 00:46:54.190 So the waltz, we have-- a little tricky-- we unfold fi 00:46:54.190 --> 00:47:02.480 to almost coplanar-- almost 180 degrees, with fi minus 1. 00:47:02.480 --> 00:47:09.270 Then we unfold fi plus 1 slightly. 00:47:12.230 --> 00:47:23.690 And then we finish unfolding fi minus 1. 00:47:23.690 --> 00:47:26.440 So there's essentially a potential interaction 00:47:26.440 --> 00:47:28.850 between fi plus 1 and fi minus 1. 00:47:28.850 --> 00:47:31.610 We do a little bit of unfolding to prevent that 00:47:31.610 --> 00:47:34.910 from being an issue here. 00:47:34.910 --> 00:47:38.690 And so you end up with this three step waltz thing. 00:47:41.790 --> 00:47:45.980 Stefan Langerman is a dancer, so he liked this terminology. 00:47:45.980 --> 00:47:48.040 I won't argue here that that works, 00:47:48.040 --> 00:47:50.510 but now actually, you avoid all touching. 00:47:50.510 --> 00:47:54.110 Last picture I wanted to show you is I 00:47:54.110 --> 00:47:55.590 said there were two ways to unfold. 00:47:55.590 --> 00:47:58.419 One was you take any unfolding, you make it Hamiltonian. 00:47:58.419 --> 00:48:00.960 And then you unroll it, using one of these three algorithms-- 00:48:00.960 --> 00:48:04.390 just regular unroll, two step, or waltz. 00:48:04.390 --> 00:48:08.660 But another strategy we know works is the source unfolding. 00:48:08.660 --> 00:48:10.710 Here, no extra cuts required. 00:48:10.710 --> 00:48:13.290 Source unfolding is like the cleanest, coolest unfolding. 00:48:13.290 --> 00:48:15.490 It turns out it just unfolds. 00:48:15.490 --> 00:48:17.080 No problem. 00:48:17.080 --> 00:48:18.200 How do we unfold it? 00:48:18.200 --> 00:48:22.020 We follow a post order traversal of the tree of faces, 00:48:22.020 --> 00:48:25.500 meaning we completely do one subtree. 00:48:25.500 --> 00:48:29.060 And now, here, we're in the middle of doing this subtree. 00:48:29.060 --> 00:48:31.920 So here, we just have a cube with x being this point. 00:48:31.920 --> 00:48:33.190 And so the reds are the cuts. 00:48:33.190 --> 00:48:34.170 Pretty simple example. 00:48:34.170 --> 00:48:36.450 You just have a star of four prongs. 00:48:36.450 --> 00:48:38.840 But here, we've completely unfolded this subtree. 00:48:38.840 --> 00:48:40.820 We've done one step of this subtree. 00:48:40.820 --> 00:48:42.800 The next thing we would do is flip it open. 00:48:42.800 --> 00:48:45.460 Then we do the next one, then we do the next one. 00:48:45.460 --> 00:48:48.550 Essentially what we argue here, in one minute, 00:48:48.550 --> 00:48:53.270 is as you unfold you can think of-- well, 00:48:53.270 --> 00:48:54.650 there's a couple things going on. 00:48:54.650 --> 00:48:57.650 One is just look at a shortest path. 00:48:57.650 --> 00:48:58.150 Right? 00:48:58.150 --> 00:49:01.900 This source unfolding is made by a star of shortest paths. 00:49:01.900 --> 00:49:07.700 So just think of each shortest path individually for now. 00:49:07.700 --> 00:49:10.800 Individually, if you look at a single shortest path, 00:49:10.800 --> 00:49:12.850 it hits some sequence of faces. 00:49:12.850 --> 00:49:17.190 Those faces form a path-like unfolding. 00:49:17.190 --> 00:49:19.360 So you just use path unroll. 00:49:19.360 --> 00:49:21.590 And that will work. 00:49:21.590 --> 00:49:24.090 The only issue is potential interactions between the paths. 00:49:24.090 --> 00:49:28.360 And that's where you have to get into the post order traversal. 00:49:28.360 --> 00:49:32.040 In general, you want this shortest path to stay shortest. 00:49:32.040 --> 00:49:34.400 What we're going to do is imagine the polyhedron 00:49:34.400 --> 00:49:37.220 in some sense as growing here. 00:49:37.220 --> 00:49:42.139 So as we unfold-- like, as we start unfolding this-- 00:49:42.139 --> 00:49:44.055 we can think of the interior of the polyhedron 00:49:44.055 --> 00:49:45.890 as just getting bigger. 00:49:45.890 --> 00:49:47.570 And in that convex polyhedron, it 00:49:47.570 --> 00:49:50.500 turns out still these paths remain shortest. 00:49:50.500 --> 00:49:52.997 And by that, all the variants to work out. 00:49:52.997 --> 00:49:54.830 And you could argue that the paths basically 00:49:54.830 --> 00:49:56.560 don't interact with each other, because 00:49:56.560 --> 00:49:57.620 of post order traversal. 00:49:57.620 --> 00:49:58.620 That was very hand wavy. 00:49:58.620 --> 00:50:01.690 The actual proof is a bit technical and more 00:50:01.690 --> 00:50:05.900 than I can do in zero minutes. 00:50:05.900 --> 00:50:09.320 That's a sketch of continuous blooming 00:50:09.320 --> 00:50:12.170 of source unfolding, which is pretty cool. 00:50:12.170 --> 00:50:14.110 Star unfolding is open. 00:50:14.110 --> 00:50:18.800 Whether all unfoldings of convex polyhedra work is open. 00:50:18.800 --> 00:50:21.042 For non-convex polyhedra, there are 00:50:21.042 --> 00:50:23.500 bunch of non-convex polyhedra, which we'll be talking about 00:50:23.500 --> 00:50:27.150 in future lectures, that we know have unfoldings. 00:50:27.150 --> 00:50:28.600 Do they have continuous blooming? 00:50:28.600 --> 00:50:29.430 We have no idea. 00:50:29.430 --> 00:50:32.610 This is the state of the art for bloomings. 00:50:32.610 --> 00:50:34.930 Still a lot of interesting open questions. 00:50:34.930 --> 00:50:38.952 Any other questions from you? 00:50:38.952 --> 00:50:41.680 All right, that's it.