1 00:00:02,575 --> 00:00:03,450 PROFESSOR: All right. 2 00:00:03,450 --> 00:00:06,830 Lecture 17 was about folding polygons into polyhedra, 3 00:00:06,830 --> 00:00:10,520 and today we will do it with real pieces of paper. 4 00:00:10,520 --> 00:00:12,697 But before we get there, I want to talk 5 00:00:12,697 --> 00:00:14,030 about some things we could make. 6 00:00:16,860 --> 00:00:21,320 In lecture, I demonstrated this perimeter having technique 7 00:00:21,320 --> 00:00:24,360 where you take any convex polygon 8 00:00:24,360 --> 00:00:26,580 and you pick any point on the perimeter. 9 00:00:26,580 --> 00:00:30,040 Then you measure out the perimeter halfway on each side. 10 00:00:30,040 --> 00:00:31,890 You get the antipodal point, y. 11 00:00:31,890 --> 00:00:34,940 And then just glue-- locally, you 12 00:00:34,940 --> 00:00:39,210 glue everything around x and around y. 13 00:00:39,210 --> 00:00:40,420 So I mean, you just start. 14 00:00:40,420 --> 00:00:42,920 We call this zipping where you don't glue any extra material 15 00:00:42,920 --> 00:00:45,490 in here, you just start zipping these guys together. 16 00:00:45,490 --> 00:00:48,590 I guess these circles or when you happen to hit this vertex, 17 00:00:48,590 --> 00:00:51,842 well, then you just keep going-- zip, zip, zip, zip, zip. 18 00:00:54,920 --> 00:00:57,600 So that's one thing we could make with convex polygons. 19 00:00:57,600 --> 00:01:01,430 Slightly more general is called a pita form. 20 00:01:05,450 --> 00:01:08,810 This is defined in the textbook. 21 00:01:08,810 --> 00:01:12,700 And the idea is instead of a convex polygon, 22 00:01:12,700 --> 00:01:14,760 you could take any convex body. 23 00:01:14,760 --> 00:01:18,760 It could have some corners or it could be smooth. 24 00:01:18,760 --> 00:01:20,980 So this is what we call a convex, 25 00:01:20,980 --> 00:01:27,480 or what I'm going to call today a convex 2D body, meaning 26 00:01:27,480 --> 00:01:29,255 it can be smooth in addition to polygonal. 27 00:01:31,880 --> 00:01:34,170 And then you pick any point on the boundary 28 00:01:34,170 --> 00:01:37,470 and you measure out the perimeter halfway, x and y, 29 00:01:37,470 --> 00:01:42,040 and you glue in exactly the same way. 30 00:01:45,730 --> 00:01:50,970 Now, it does not follow from Alexandrov's theorem 31 00:01:50,970 --> 00:01:54,540 that this will make a convex thing, 32 00:01:54,540 --> 00:01:57,220 but it follows from a slightly more general form 33 00:01:57,220 --> 00:02:01,080 of Alexandrov's theorem called Alexandrov-Pogorelov theorem. 34 00:02:10,380 --> 00:02:17,040 So Pogorelov was a student of Alexandrov 35 00:02:17,040 --> 00:02:21,740 and he edited some things that Alexandrov wrote, 36 00:02:21,740 --> 00:02:25,960 but then also Pogorelov wrote his own papers 37 00:02:25,960 --> 00:02:27,540 and proved some stronger versions 38 00:02:27,540 --> 00:02:31,040 of Alexandrov's theorem that hold for smooth bodies, not 39 00:02:31,040 --> 00:02:33,220 just polygonal folding. 40 00:02:33,220 --> 00:02:34,760 So Alexandrov's theorem, you recall, 41 00:02:34,760 --> 00:02:38,940 if you have a convex polyhedral metric homeomorphic 42 00:02:38,940 --> 00:02:45,200 to a sphere, then it's realized by a unique convex polyhedron. 43 00:02:45,200 --> 00:02:50,940 So the Pogorelov's extension is that if you have everything 44 00:02:50,940 --> 00:02:53,520 except the polyhedral part-- so if you 45 00:02:53,520 --> 00:03:03,880 have a convex metric homeomorphic to a sphere-- so 46 00:03:03,880 --> 00:03:08,444 topologically it's a sphere, but we omit the polyhedral part. 47 00:03:08,444 --> 00:03:10,735 So polyhedral was that you have finitely many vertices. 48 00:03:13,620 --> 00:03:17,850 In some sense, I have infinitely many vertices, 49 00:03:17,850 --> 00:03:19,920 infinitesimal curvature at every point here. 50 00:03:22,550 --> 00:03:39,410 Then it's realized by a unique convex 3D body. 51 00:03:39,410 --> 00:03:43,793 So same concept of a convex object. 52 00:03:46,430 --> 00:03:52,153 Well, I should say this is really the surface of. 53 00:03:55,540 --> 00:03:58,150 So a convex body, I should probably define that. 54 00:03:58,150 --> 00:04:00,390 Convex body is one where you take any two points 55 00:04:00,390 --> 00:04:01,890 and you draw a straight line and you 56 00:04:01,890 --> 00:04:05,050 stay within the body for the entire straight line. 57 00:04:05,050 --> 00:04:06,290 So that's convex. 58 00:04:06,290 --> 00:04:08,704 You take any two points, like this one and this one, 59 00:04:08,704 --> 00:04:10,620 draw a line between them, and you stay inside. 60 00:04:14,980 --> 00:04:17,660 The body is the piece of paper we're gluing up. 61 00:04:17,660 --> 00:04:21,480 Here we have a 3D body, but then we 62 00:04:21,480 --> 00:04:23,580 just look at the surface of it, the boundary. 63 00:04:23,580 --> 00:04:27,950 And that's going to be the surface we make. 64 00:04:27,950 --> 00:04:29,704 Technically, that surface is not convex. 65 00:04:29,704 --> 00:04:32,120 It's the interior that's convex because you can draw lines 66 00:04:32,120 --> 00:04:33,411 through it and you stay inside. 67 00:04:36,200 --> 00:04:39,140 This is exactly the same thing, but instead of polyhedron here, 68 00:04:39,140 --> 00:04:42,680 we have body because we didn't have finitely many vertices. 69 00:04:42,680 --> 00:04:44,660 And I'm not going to prove this theorem. 70 00:04:44,660 --> 00:04:48,390 But existence is basically the same proof. 71 00:04:48,390 --> 00:04:49,460 You just take a limit. 72 00:04:49,460 --> 00:04:54,020 So you can polygonalize with lots of little edges 73 00:04:54,020 --> 00:04:58,340 here, add little points, and take the limit 74 00:04:58,340 --> 00:05:02,150 as that approximation gets very, very close to the actual curve. 75 00:05:02,150 --> 00:05:03,770 At all times you have a polyhedron, 76 00:05:03,770 --> 00:05:07,780 it's easy to show you will converge to a convex body. 77 00:05:07,780 --> 00:05:09,850 The uniqueness is the harder part, I believe. 78 00:05:09,850 --> 00:05:14,020 I haven't read the proof, but it was more challenging. 79 00:05:14,020 --> 00:05:17,210 I think that took 20 years or so to settle uniqueness. 80 00:05:17,210 --> 00:05:19,020 Alexandrov's like 1950. 81 00:05:19,020 --> 00:05:23,750 This theorem I think is 1973 the final version. 82 00:05:23,750 --> 00:05:29,230 But the version without unique was proved back in 1950. 83 00:05:29,230 --> 00:05:30,820 So that's something else we can do. 84 00:05:30,820 --> 00:05:32,410 Instead of cutting out a polygon, 85 00:05:32,410 --> 00:05:36,585 we can cut out a nice smooth thing or you can add some kinks 86 00:05:36,585 --> 00:05:37,820 and be smooth elsewhere. 87 00:05:37,820 --> 00:05:41,540 As long as it's convex, then you're 88 00:05:41,540 --> 00:05:44,370 guaranteed that-- we're only gluing together 89 00:05:44,370 --> 00:05:45,535 two convex points. 90 00:05:45,535 --> 00:05:48,620 It could be flat or it could be strictly convex, 91 00:05:48,620 --> 00:05:51,210 which means we'll have at most 360 degrees of material at any 92 00:05:51,210 --> 00:05:54,780 point because it's always doing two at most 180's. 93 00:05:54,780 --> 00:05:57,820 And so we will have a convex metric, 94 00:05:57,820 --> 00:06:01,100 meaning always at most 360 of material everywhere. 95 00:06:01,100 --> 00:06:04,820 So then this theorem applies so we'll get a unique thing. 96 00:06:04,820 --> 00:06:09,380 Another thing we could make is called a D-form. 97 00:06:13,710 --> 00:06:17,350 And I have some examples of D-forms here. 98 00:06:17,350 --> 00:06:21,494 So the idea with the D-form is you take two convex 2D bodies-- 99 00:06:21,494 --> 00:06:23,660 so here they have some straight parts and some curve 100 00:06:23,660 --> 00:06:27,500 parts-- of the same perimeter. 101 00:06:27,500 --> 00:06:30,470 I pick one point from one body and I pick another point 102 00:06:30,470 --> 00:06:31,750 from the other body. 103 00:06:31,750 --> 00:06:33,290 I attach them there. 104 00:06:33,290 --> 00:06:35,230 And then I zip. 105 00:06:35,230 --> 00:06:38,014 So it would be zip, zip, zip, zip, zip. 106 00:06:41,201 --> 00:06:42,950 Wherever my fingers were at the same time, 107 00:06:42,950 --> 00:06:45,120 those are points that get glued together. 108 00:06:45,120 --> 00:06:47,750 And this is with the convex body you get. 109 00:06:47,750 --> 00:06:51,260 Again, Alexandrov-Pogorelov applies because we're still 110 00:06:51,260 --> 00:06:54,844 only gluing two convex points together at any point. 111 00:06:54,844 --> 00:06:56,760 As long as these guys have matching perimeter, 112 00:06:56,760 --> 00:06:57,700 we'll be OK. 113 00:06:57,700 --> 00:07:01,060 And so we will get a unique convex 3D body. 114 00:07:01,060 --> 00:07:06,370 In this case, we get this fun kind of shape. 115 00:07:06,370 --> 00:07:08,390 So this is what I suggest you make. 116 00:07:08,390 --> 00:07:11,350 D-forms tend to look a little bit cooler somehow 117 00:07:11,350 --> 00:07:13,910 because they have two polygons. 118 00:07:13,910 --> 00:07:17,680 And the one that's particularly easy to build 119 00:07:17,680 --> 00:07:20,534 is two copies of the same shape, but then you 120 00:07:20,534 --> 00:07:22,200 don't want to glue corresponding points. 121 00:07:22,200 --> 00:07:24,445 Like if I glued this point to that point, 122 00:07:24,445 --> 00:07:26,320 I would just get a flat, doubly covered thing 123 00:07:26,320 --> 00:07:27,334 and that's boring. 124 00:07:27,334 --> 00:07:28,750 But you just pick some other point 125 00:07:28,750 --> 00:07:31,920 and glue it to non-matching points, 126 00:07:31,920 --> 00:07:36,170 then it acts like you have two very different convex 127 00:07:36,170 --> 00:07:37,770 bodies which is kind of cool. 128 00:07:37,770 --> 00:07:40,390 I have some images of D-forms here. 129 00:07:40,390 --> 00:07:43,780 There's actually a whole book on D-forms by John Sharp. 130 00:07:43,780 --> 00:07:46,160 They're invented by this guy Tony Wills who's an artist. 131 00:07:46,160 --> 00:07:47,720 At the top, the quote is, "there is 132 00:07:47,720 --> 00:07:49,940 no such thing as an ugly D-form." 133 00:07:49,940 --> 00:07:53,400 So we are guaranteed success here. 134 00:07:53,400 --> 00:07:55,650 And also it says, "they surprise, confuse, 135 00:07:55,650 --> 00:07:56,580 can be addictive. 136 00:07:56,580 --> 00:07:57,830 They're an intellectual virus. 137 00:07:57,830 --> 00:07:58,950 Beware." 138 00:07:58,950 --> 00:08:00,510 So you've been warned. 139 00:08:03,210 --> 00:08:05,420 This is Tony Wills, the inventor. 140 00:08:05,420 --> 00:08:08,300 Now, from the artistic perspective, 141 00:08:08,300 --> 00:08:10,470 you don't have to start from convex polygons. 142 00:08:10,470 --> 00:08:12,690 We're going to start from convex bodies 143 00:08:12,690 --> 00:08:14,230 because they're guaranteed to work. 144 00:08:14,230 --> 00:08:19,210 If you're careful to satisfy Alexandrov-Pogorelov, 145 00:08:19,210 --> 00:08:21,072 you can have a convex metric even though you 146 00:08:21,072 --> 00:08:24,940 start from nonconvex bodies, although in this case, 147 00:08:24,940 --> 00:08:27,050 he's actually getting nonconvex shapes as well. 148 00:08:27,050 --> 00:08:29,285 More typical D-form is something like this. 149 00:08:29,285 --> 00:08:33,600 This is made from two ellipses and he builds them out 150 00:08:33,600 --> 00:08:35,299 of metal. 151 00:08:35,299 --> 00:08:40,520 This is a simulation by this guy Kenneth Brakke 152 00:08:40,520 --> 00:08:43,530 who has this software called Surface Evolver. 153 00:08:43,530 --> 00:08:46,230 And it's usually used for computing minimal services, 154 00:08:46,230 --> 00:08:47,800 soap bundles and things like that. 155 00:08:47,800 --> 00:08:50,270 But here it's computing what D-forms look like, 156 00:08:50,270 --> 00:08:51,750 approximately. 157 00:08:51,750 --> 00:08:55,160 And so this is an example of taking two ellipses 158 00:08:55,160 --> 00:08:58,090 and instead of-- if you glue at matching points, 159 00:08:58,090 --> 00:09:00,770 they would be flat-- but then you change the rotation. 160 00:09:00,770 --> 00:09:04,080 So here, they're rotated by 22.5 degrees, rotated by 45 degrees, 161 00:09:04,080 --> 00:09:06,830 rotated by 90 degrees, so you get this nice continuum 162 00:09:06,830 --> 00:09:09,780 of different things that you could zip together. 163 00:09:12,540 --> 00:09:15,750 So I propose we make some. 164 00:09:15,750 --> 00:09:18,180 And so you have your scissors and tape. 165 00:09:20,742 --> 00:09:25,470 If you don't have two sheets of paper, get some from the front. 166 00:09:25,470 --> 00:09:28,112 And so everyone have scissors, tape, everything? 167 00:09:28,112 --> 00:09:29,570 We're going to have to share a bit. 168 00:09:33,310 --> 00:09:37,830 So what we want to do is make two identical convex shapes. 169 00:09:37,830 --> 00:09:39,460 So you have some here. 170 00:09:39,460 --> 00:09:41,270 Two rectangles are valid convex shapes. 171 00:09:41,270 --> 00:09:43,000 You could cut not all. 172 00:09:43,000 --> 00:09:45,990 Everyone should do something different. 173 00:09:45,990 --> 00:09:48,800 I'm just going to make sure they're nicely aligned. 174 00:09:48,800 --> 00:09:51,190 Hold on tight because you don't want them to let go, 175 00:09:51,190 --> 00:09:52,110 and then just cut. 176 00:09:54,630 --> 00:09:56,420 Always turn left with your cutting. 177 00:10:00,000 --> 00:10:00,750 You can be smooth. 178 00:10:00,750 --> 00:10:02,708 You can have some corners, what you want to do. 179 00:10:06,090 --> 00:10:07,115 Just always turn left. 180 00:10:13,080 --> 00:10:18,702 And once you have your two shapes, you pull them apart. 181 00:10:18,702 --> 00:10:20,660 Make sure you do not glue corresponding points. 182 00:10:20,660 --> 00:10:27,330 Pick some other points and attach them together with tape. 183 00:10:27,330 --> 00:10:30,000 Here's where it gets a little bit challenging. 184 00:10:30,000 --> 00:10:32,590 With smooth curves, it's little harder to tape them together, 185 00:10:32,590 --> 00:10:35,030 so you're probably going to have to use a bunch of points 186 00:10:35,030 --> 00:10:44,955 of tape, so to speak, not literal points. 187 00:10:44,955 --> 00:10:46,516 And I just tape, tape, tape. 188 00:11:25,296 --> 00:11:26,796 Before you get all the way, you want 189 00:11:26,796 --> 00:11:29,295 to make sure you can still pop it up and be convex. 190 00:11:29,295 --> 00:11:30,190 I think we'll be able to do that by [INAUDIBLE] 191 00:11:30,190 --> 00:11:31,582 inside the limit. 192 00:11:37,038 --> 00:11:38,030 Give me some scissors. 193 00:11:48,942 --> 00:11:50,450 All right. 194 00:11:50,450 --> 00:11:52,440 There's my D-form. 195 00:11:52,440 --> 00:11:55,114 Mostly convex, kind of fun. 196 00:12:04,740 --> 00:12:07,200 So a funny thing you'll notice about D-forms, 197 00:12:07,200 --> 00:12:09,980 if you made yours smooth-- so if you don't 198 00:12:09,980 --> 00:12:12,760 have any kinks on the outer boundary-- then 199 00:12:12,760 --> 00:12:17,640 the surface looks smooth, which is kind of nice. 200 00:12:17,640 --> 00:12:20,900 The other thing is there's this noticeable seam where 201 00:12:20,900 --> 00:12:22,381 you taped things. 202 00:12:22,381 --> 00:12:23,880 There, you're not smooth, obviously. 203 00:12:23,880 --> 00:12:27,530 You've got a crease along the seam. 204 00:12:27,530 --> 00:12:31,640 But also it looks like all that the surface is doing 205 00:12:31,640 --> 00:12:33,790 is kind of taking the convex hull, 206 00:12:33,790 --> 00:12:37,339 the envelope of that seam. 207 00:12:37,339 --> 00:12:38,880 So there's two properties we observe. 208 00:12:38,880 --> 00:12:41,850 One is that it looks smooth, except at the seam. 209 00:12:41,850 --> 00:12:44,160 And the other is that the hull surface 210 00:12:44,160 --> 00:12:48,040 is the convex hull of the 3D seam. 211 00:12:48,040 --> 00:12:51,910 And I want to prove both of those things. 212 00:12:51,910 --> 00:12:55,200 Those are both true facts. 213 00:12:55,200 --> 00:12:58,500 But first I need to define some things. 214 00:12:58,500 --> 00:13:01,300 So we're going to prove that D-forms 215 00:13:01,300 --> 00:13:04,780 are smooth and other good things. 216 00:13:04,780 --> 00:13:06,710 This is a paper that was originally 217 00:13:06,710 --> 00:13:12,485 a class project in this class, I think in 2007. 218 00:13:12,485 --> 00:13:13,640 Sounds about right. 219 00:13:13,640 --> 00:13:16,955 Submitted during class, it looks like, or just after semester 220 00:13:16,955 --> 00:13:19,030 with Greg Price. 221 00:13:19,030 --> 00:13:21,560 D-form have no spurious creases. 222 00:13:21,560 --> 00:13:23,080 So what's a D-form? 223 00:13:23,080 --> 00:13:25,340 You take two, for us, it's going to be 224 00:13:25,340 --> 00:13:28,900 two convex shapes of equal perimeter. 225 00:13:28,900 --> 00:13:31,810 You glue two points, two corresponding points P and Q 226 00:13:31,810 --> 00:13:36,341 together, and then you zip from there. 227 00:13:36,341 --> 00:13:38,590 We can actually talk about something even more general 228 00:13:38,590 --> 00:13:40,230 which we'll call a seam form. 229 00:13:44,720 --> 00:13:49,890 A seam form, you can have multiple convex shapes. 230 00:13:54,040 --> 00:13:56,130 It's a little harder to imagine, but you 231 00:13:56,130 --> 00:13:58,820 can join multiple points together. 232 00:13:58,820 --> 00:14:03,150 Maybe you join all three of those corners together. 233 00:14:03,150 --> 00:14:05,650 As long as you have at most 360 of material everywhere, 234 00:14:05,650 --> 00:14:06,780 it'll be fine. 235 00:14:06,780 --> 00:14:09,594 Maybe do some zipping. 236 00:14:09,594 --> 00:14:12,010 I'm not going to try to figure out exactly how this gluing 237 00:14:12,010 --> 00:14:13,843 pattern works, but you find a gluing pattern 238 00:14:13,843 --> 00:14:15,540 where you don't violate anything. 239 00:14:15,540 --> 00:14:17,451 So it looks like if I do this one, 240 00:14:17,451 --> 00:14:19,200 I better have these guys also a little bit 241 00:14:19,200 --> 00:14:25,522 sharp so that they don't add up to too much angle there. 242 00:14:25,522 --> 00:14:27,480 So these three points would also join together. 243 00:14:27,480 --> 00:14:32,410 This would be like three petals in like a seed pod 244 00:14:32,410 --> 00:14:36,510 or something, like those leaves that wrap around little fruits 245 00:14:36,510 --> 00:14:38,880 or something. 246 00:14:38,880 --> 00:14:41,860 I won't try to draw it but-- I guess I will try to draw it. 247 00:14:46,160 --> 00:14:47,190 Something like this. 248 00:14:50,530 --> 00:14:51,980 So that's a thing. 249 00:14:51,980 --> 00:14:54,190 This could even fold not exactly a sphere, 250 00:14:54,190 --> 00:14:55,590 but something like a sphere. 251 00:14:55,590 --> 00:14:58,594 So seam forms, you could have any number of convex shapes 252 00:14:58,594 --> 00:15:00,260 and glue them together however you want, 253 00:15:00,260 --> 00:15:03,020 provided at all points you satisfy Alexandrov-Pogorelov, 254 00:15:03,020 --> 00:15:06,450 so you only have at most 360 degrees of material 255 00:15:06,450 --> 00:15:09,310 and you are topologically a sphere. 256 00:15:09,310 --> 00:15:10,482 Then you have a seam form. 257 00:15:10,482 --> 00:15:12,690 So most of the theorems I'm going to talk about apply 258 00:15:12,690 --> 00:15:16,120 to general seam forms, but we're interested in D-forms and pita 259 00:15:16,120 --> 00:15:19,110 forms in particular. 260 00:15:19,110 --> 00:15:25,200 So let me tell you some theorems about seam forms. 261 00:15:25,200 --> 00:15:35,560 Theorem 1 is that a seam form equals 262 00:15:35,560 --> 00:15:45,330 convex hull of its seams. 263 00:15:45,330 --> 00:15:47,330 So the seams in general are these parts 264 00:15:47,330 --> 00:15:50,740 where you did gluing, wherever you taped stuff. 265 00:15:50,740 --> 00:15:53,550 The boundaries of the convex polygons, those are the seams. 266 00:15:53,550 --> 00:15:55,607 They map to some curves in 3D. 267 00:15:55,607 --> 00:15:57,190 If you take the convex hull, the claim 268 00:15:57,190 --> 00:16:00,129 is you get the entire seam form. 269 00:16:00,129 --> 00:16:01,670 That's actually pretty easy to prove. 270 00:16:07,999 --> 00:16:09,540 Then the second claim is there aren't 271 00:16:09,540 --> 00:16:13,010 many creases other than the seams of course. 272 00:16:13,010 --> 00:16:14,885 The seams are usually going to be creased. 273 00:16:18,600 --> 00:16:22,305 And these creases are going to be line segments. 274 00:16:25,956 --> 00:16:28,080 And not just any line segments, but their endpoints 275 00:16:28,080 --> 00:16:28,871 are pretty special. 276 00:16:59,120 --> 00:17:02,070 So the claim is that the creases have 277 00:17:02,070 --> 00:17:07,530 to be line segments basically connecting strict vertices. 278 00:17:07,530 --> 00:17:09,930 What do I mean by strict vertex? 279 00:17:09,930 --> 00:17:13,829 Something like these three points which come together, 280 00:17:13,829 --> 00:17:17,380 provided the total angle here is strictly less than 360. 281 00:17:17,380 --> 00:17:18,760 I call that a strict vertex. 282 00:17:18,760 --> 00:17:24,540 Whereas these points connecting from here to here, 283 00:17:24,540 --> 00:17:26,290 those don't count as strict vertices. 284 00:17:26,290 --> 00:17:29,530 They do have curvature, but only infinitesimal curvature. 285 00:17:29,530 --> 00:17:32,610 Because this is essentially 180 degrees of material, 286 00:17:32,610 --> 00:17:35,100 just slightly less because it's curved. 287 00:17:35,100 --> 00:17:38,050 So only where I have kinks can I potentially 288 00:17:38,050 --> 00:17:40,770 have strictly less than 180. 289 00:17:40,770 --> 00:17:45,580 And if I join them together to be strictly less than 360, 290 00:17:45,580 --> 00:17:47,060 that's a strict vertex. 291 00:17:47,060 --> 00:17:50,254 So potentially, I have a seam coming from here, also 292 00:17:50,254 --> 00:17:51,170 at the other endpoint. 293 00:17:51,170 --> 00:17:56,264 So in this picture, I might have a crease like this, 294 00:17:56,264 --> 00:17:57,680 I might have a crease like this, I 295 00:17:57,680 --> 00:18:01,990 might have a crease like this, potentially. 296 00:18:01,990 --> 00:18:05,460 But in something like a D-form, there are no strict vertices. 297 00:18:05,460 --> 00:18:12,190 If these are smooth-- so I'll assume here these are smooth. 298 00:18:12,190 --> 00:18:14,370 We could call it a smooth D-form if you like. 299 00:18:14,370 --> 00:18:18,690 Then there's no vertices, no strict vertices, 300 00:18:18,690 --> 00:18:22,630 only these sort of barely vertices. 301 00:18:22,630 --> 00:18:25,670 So there could be no creases. 302 00:18:25,670 --> 00:18:27,310 There's one other situation which 303 00:18:27,310 --> 00:18:32,059 is creases could be tangent to seams. 304 00:18:32,059 --> 00:18:33,850 This should seem impossible because the way 305 00:18:33,850 --> 00:18:35,630 I've defined things it is impossible. 306 00:18:35,630 --> 00:18:38,380 You're tangent to a seam, that would be you're like this. 307 00:18:38,380 --> 00:18:40,810 How am I supposed to have a crease inside that's 308 00:18:40,810 --> 00:18:42,065 tangent to the seam? 309 00:18:42,065 --> 00:18:43,640 The answer is you can't. 310 00:18:43,640 --> 00:18:48,930 If these regions are convex, you can't have them. 311 00:18:48,930 --> 00:18:51,170 This statement is actually about a more general form 312 00:18:51,170 --> 00:18:56,090 of seam forms where you can have a nonconvex piece. 313 00:18:56,090 --> 00:18:59,980 So in general seam form, you have a bunch of flat pieces 314 00:18:59,980 --> 00:19:02,130 and you somehow search for them together so 315 00:19:02,130 --> 00:19:03,820 that you satisfy Alexandrov-Pogorelov. 316 00:19:06,590 --> 00:19:09,803 And now you could have a tangent. 317 00:19:09,803 --> 00:19:13,191 I could, for example, have a crease that emanates from there 318 00:19:13,191 --> 00:19:14,940 or have a crease that emanates from there. 319 00:19:14,940 --> 00:19:16,630 But right now, it has nowhere to stop. 320 00:19:16,630 --> 00:19:19,060 The only way for these creases to actually exist 321 00:19:19,060 --> 00:19:23,860 is if there's a kink here, then this 322 00:19:23,860 --> 00:19:25,410 could go from there to there. 323 00:19:25,410 --> 00:19:28,947 So this is starting at a vertex and ending tangent. 324 00:19:28,947 --> 00:19:31,280 You could, of course, also start tangent and end tangent 325 00:19:31,280 --> 00:19:33,280 if there's another bend. 326 00:19:33,280 --> 00:19:35,810 But the claim is, all crease look 327 00:19:35,810 --> 00:19:38,250 like that, which means if you carefully 328 00:19:38,250 --> 00:19:40,380 designed your thing or vaguely carefully 329 00:19:40,380 --> 00:19:42,460 such as a smooth D-form, then you're 330 00:19:42,460 --> 00:19:45,020 guaranteed there are no creases which is what we observe. 331 00:19:45,020 --> 00:19:49,885 So this is justifying our intuition from these examples. 332 00:19:52,470 --> 00:19:56,280 One other case is the pita form. 333 00:19:56,280 --> 00:19:59,430 So pita form does actually make vertices. 334 00:19:59,430 --> 00:20:01,720 This point x is going to be a strict vertex because it 335 00:20:01,720 --> 00:20:04,350 has less than 180 of material, so definitely 336 00:20:04,350 --> 00:20:06,050 strictly less than 360. 337 00:20:06,050 --> 00:20:07,460 Same with y. 338 00:20:07,460 --> 00:20:11,090 So pita form is potentially going 339 00:20:11,090 --> 00:20:14,135 to have a seam from x to y, but that's it. 340 00:20:14,135 --> 00:20:15,800 Sorry, a crease from x to y. 341 00:20:15,800 --> 00:20:18,290 It has at most one crease. 342 00:20:18,290 --> 00:20:20,390 I think most examples we've made do 343 00:20:20,390 --> 00:20:22,800 have such a crease from x to y. 344 00:20:28,400 --> 00:20:36,860 Let's prove these theorems, or at least sketch the proofs. 345 00:20:36,860 --> 00:20:38,410 Don't want to get too technical here. 346 00:20:45,040 --> 00:20:50,050 So to prove the first part that the seam form is 347 00:20:50,050 --> 00:20:54,680 convex hull of its seams, it's helpful to have a tool here 348 00:20:54,680 --> 00:20:57,794 which is a theorem by Minkowski. 349 00:20:57,794 --> 00:20:59,210 We'll call it Minkowski's theorem, 350 00:20:59,210 --> 00:21:00,209 although he has a bunch. 351 00:21:06,310 --> 00:21:07,060 Hermann Minkowski. 352 00:21:14,390 --> 00:21:16,660 It relates convex things to convex hulls. 353 00:21:35,020 --> 00:21:36,980 It says, any convex body is the convex hull 354 00:21:36,980 --> 00:21:38,440 of its extreme points. 355 00:21:38,440 --> 00:21:40,460 What are extreme points? 356 00:21:40,460 --> 00:21:43,610 Extreme points are points on the surface where 357 00:21:43,610 --> 00:21:46,982 you can touch just that point with a tangent plane. 358 00:21:46,982 --> 00:21:48,440 So if you think of polyhedra, these 359 00:21:48,440 --> 00:21:50,087 are vertices of the polyhedra. 360 00:21:50,087 --> 00:21:51,920 But I want to handle things that are smooth, 361 00:21:51,920 --> 00:21:54,790 so they may might not really have any vertices. 362 00:21:54,790 --> 00:21:57,430 In general, I have some convex body. 363 00:21:57,430 --> 00:22:00,770 If I can draw-- imagine this is in two dimensions-- 364 00:22:00,770 --> 00:22:03,450 if I can draw a tangent plane that just 365 00:22:03,450 --> 00:22:06,095 touches at a single point, then I call that point extreme. 366 00:22:10,570 --> 00:22:13,420 Let's look at an example here. 367 00:22:13,420 --> 00:22:16,350 Here I believe every seam point is 368 00:22:16,350 --> 00:22:18,200 going to be an extreme point because I 369 00:22:18,200 --> 00:22:19,200 can put a tangent plane. 370 00:22:19,200 --> 00:22:21,910 It just touches at that point. 371 00:22:21,910 --> 00:22:23,160 What am I distinguishing from? 372 00:22:23,160 --> 00:22:27,390 Well, for example, this point, this surface is developable. 373 00:22:27,390 --> 00:22:28,317 We know it's ruled. 374 00:22:28,317 --> 00:22:29,650 So there's a straight line here. 375 00:22:29,650 --> 00:22:34,030 You can see it in the shadow, in the silhouette. 376 00:22:34,030 --> 00:22:35,510 If I said, is this point extreme? 377 00:22:35,510 --> 00:22:37,750 I try to put a tangent plane on, the only way 378 00:22:37,750 --> 00:22:42,100 to get a tangent plane is to include this entire line. 379 00:22:42,100 --> 00:22:44,760 So it's impossible to just hit this point 380 00:22:44,760 --> 00:22:46,460 or any point along that line. 381 00:22:46,460 --> 00:22:48,570 In fact, every interior point, I claim, 382 00:22:48,570 --> 00:22:50,300 you cannot just hit that point. 383 00:22:50,300 --> 00:22:52,920 You've got to hit an entire line. 384 00:22:52,920 --> 00:22:54,590 Whereas at the seam, I can do it. 385 00:22:54,590 --> 00:22:59,360 I can angle between this and this. 386 00:22:59,360 --> 00:23:01,240 There's a tangent plane, because this 387 00:23:01,240 --> 00:23:03,070 is curving, that only hits at one point. 388 00:23:03,070 --> 00:23:04,630 If I had a straight segment, then 389 00:23:04,630 --> 00:23:06,859 the endpoints of the straight segment of the seam 390 00:23:06,859 --> 00:23:09,400 would be extreme, but the points in the middle of the segment 391 00:23:09,400 --> 00:23:11,050 would not be extreme. 392 00:23:11,050 --> 00:23:12,850 So that's the meaning of extreme. 393 00:23:12,850 --> 00:23:15,120 And we're going to use this because I 394 00:23:15,120 --> 00:23:19,445 claim that these interior points can never be extreme. 395 00:23:25,430 --> 00:23:27,640 It can only be the seam points that are extreme. 396 00:23:27,640 --> 00:23:31,650 And therefore, we are the convex hull of the seam points 397 00:23:31,650 --> 00:23:35,340 because we are the convex hull of the extreme points. 398 00:23:35,340 --> 00:23:38,740 How do we argue that? 399 00:23:38,740 --> 00:23:54,870 So I claim an extreme point can't be locally flat 400 00:23:54,870 --> 00:24:03,120 and at the same time be convex, because convex in 3D, that's 401 00:24:03,120 --> 00:24:03,800 what we need. 402 00:24:03,800 --> 00:24:08,130 If you had a point and it just meets a tangent plane 403 00:24:08,130 --> 00:24:10,780 at that single point, that means locally 404 00:24:10,780 --> 00:24:13,376 you're kind of going down from that plane, 405 00:24:13,376 --> 00:24:15,000 if you think of that plane as vertical. 406 00:24:20,160 --> 00:24:22,700 So this is a situation. 407 00:24:22,700 --> 00:24:24,326 I want to prove this. 408 00:24:24,326 --> 00:24:26,450 To prove it, I need to introduce another tool which 409 00:24:26,450 --> 00:24:28,033 is a generally good thing for you guys 410 00:24:28,033 --> 00:24:29,960 to know about, so kind of an excuse 411 00:24:29,960 --> 00:24:35,900 to tell you about the Gauss sphere, which I don't think we 412 00:24:35,900 --> 00:24:39,350 actually cover in lectures, but might come up again. 413 00:24:39,350 --> 00:24:40,900 Gauss sphere is a simple idea. 414 00:24:40,900 --> 00:24:43,770 For every tangent plane-- let's just think 415 00:24:43,770 --> 00:24:46,070 of this single point. 416 00:24:46,070 --> 00:24:48,400 I want to look at the-- it's called the tangent space. 417 00:24:48,400 --> 00:24:51,050 Look at all the tangent planes that touch this point. 418 00:24:51,050 --> 00:24:53,680 Because this particular tangent plan 419 00:24:53,680 --> 00:24:57,710 I drew only touches that one point, it has some wiggle room. 420 00:24:57,710 --> 00:25:01,210 I can pick any direction and just wiggle 421 00:25:01,210 --> 00:25:03,132 and rotate the plane around this point 422 00:25:03,132 --> 00:25:04,840 and it won't immediately hit the surface. 423 00:25:04,840 --> 00:25:07,340 I've got a little bit of time before it hits the surface. 424 00:25:07,340 --> 00:25:12,270 So there's a two-dimensional space of tangent planes 425 00:25:12,270 --> 00:25:15,334 that touch just this point. 426 00:25:15,334 --> 00:25:16,750 Because there's at least one, I've 427 00:25:16,750 --> 00:25:18,070 got to have some wiggle room. 428 00:25:18,070 --> 00:25:19,840 So I want to draw that space. 429 00:25:19,840 --> 00:25:23,050 And a clean way to draw the space is for every such plane 430 00:25:23,050 --> 00:25:27,830 to draw a normal vector perpendicular to the plane, 431 00:25:27,830 --> 00:25:30,600 take the direction of that normal vector, 432 00:25:30,600 --> 00:25:32,650 and draw it on a sphere. 433 00:25:32,650 --> 00:25:33,670 It's a unit sphere. 434 00:25:33,670 --> 00:25:34,565 So this is a sphere. 435 00:25:37,049 --> 00:25:39,340 This one looks pretty vertical so that would correspond 436 00:25:39,340 --> 00:25:41,501 to the north pole of the sphere. 437 00:25:41,501 --> 00:25:43,500 So that direction becomes a point on the sphere. 438 00:25:43,500 --> 00:25:47,030 Think of it as this vector from the center of the sphere 439 00:25:47,030 --> 00:25:48,950 to the north pole. 440 00:25:48,950 --> 00:25:50,760 But we'll just draw it as a point. 441 00:25:50,760 --> 00:25:52,770 And because I've got a two-dimensional space 442 00:25:52,770 --> 00:25:55,550 of maneuverability or rotatability of this plane, 443 00:25:55,550 --> 00:25:58,490 I'm going to get some region-- which maybe I 444 00:25:58,490 --> 00:26:01,620 should draw like this-- of the sphere. 445 00:26:01,620 --> 00:26:08,270 Those are all the possible normal vectors 446 00:26:08,270 --> 00:26:12,550 of tangent planes at that point. 447 00:26:12,550 --> 00:26:15,600 So this is called the Gauss map where you map all these normals 448 00:26:15,600 --> 00:26:20,080 to the Gauss sphere, which Gauss sphere is just a unit sphere. 449 00:26:20,080 --> 00:26:22,010 Fun fact. 450 00:26:22,010 --> 00:26:29,870 The area of that thing, which is the map of all 451 00:26:29,870 --> 00:26:34,687 the tangent normal directions, equals the curvature 452 00:26:34,687 --> 00:26:35,270 at that point. 453 00:26:38,470 --> 00:26:40,060 You may recall at some point curvature 454 00:26:40,060 --> 00:26:42,180 is actually Gaussian curvature, so that's 455 00:26:42,180 --> 00:26:44,710 why Gauss is all over the place here. 456 00:26:44,710 --> 00:26:47,710 Now because I claim this is a two-dimensional space, 457 00:26:47,710 --> 00:26:49,410 this thing will have positive area. 458 00:26:49,410 --> 00:26:51,890 Therefore, at this point, that vertex 459 00:26:51,890 --> 00:26:55,210 has positive curvature on the surface. 460 00:26:55,210 --> 00:27:00,170 And yet, it was supposed to be in the middle of one 461 00:27:00,170 --> 00:27:02,150 of these flat shapes. 462 00:27:02,150 --> 00:27:03,890 That's supposed to have zero curvature. 463 00:27:03,890 --> 00:27:05,500 Contradiction. 464 00:27:05,500 --> 00:27:08,720 Or stated more positively, if I have 465 00:27:08,720 --> 00:27:10,332 a point that is an extreme point, 466 00:27:10,332 --> 00:27:12,040 it's only touched by one of these planes, 467 00:27:12,040 --> 00:27:13,124 I have this flexibility. 468 00:27:13,124 --> 00:27:14,040 Therefore, I get area. 469 00:27:14,040 --> 00:27:18,880 Therefore, it is not locally flat, 470 00:27:18,880 --> 00:27:21,330 so it must be a seam point. 471 00:27:21,330 --> 00:27:24,180 It could be one like this where I'm barely non-flat, 472 00:27:24,180 --> 00:27:25,040 but I am non-flat. 473 00:27:25,040 --> 00:27:27,290 There's kind of infinitesimal curvature here. 474 00:27:27,290 --> 00:27:30,900 Or it could be one of these points where I'm very non-flat, 475 00:27:30,900 --> 00:27:35,890 or I guess X and Y here are other examples of here 476 00:27:35,890 --> 00:27:36,730 I'm very not flat. 477 00:27:36,730 --> 00:27:39,790 I've only got 180 degrees of curvature roughly. 478 00:27:39,790 --> 00:27:42,080 So it's got to be one of those if you're 479 00:27:42,080 --> 00:27:43,360 going to be an extreme point. 480 00:27:43,360 --> 00:27:46,900 Therefore, all the extreme points are seam points. 481 00:27:46,900 --> 00:27:51,120 Therefore, convex hull of the extreme points 482 00:27:51,120 --> 00:27:55,520 is the convex hull of the seams, and that is your seam form. 483 00:27:55,520 --> 00:27:57,350 So that's part one. 484 00:27:57,350 --> 00:27:59,470 It's pretty easy. 485 00:27:59,470 --> 00:28:01,390 Let me tell you about part two. 486 00:28:01,390 --> 00:28:03,230 Part two builds on part one. 487 00:28:03,230 --> 00:28:06,780 It's one reason why we care about. 488 00:28:06,780 --> 00:28:10,870 So part two is that there are no spurious creases. 489 00:28:10,870 --> 00:28:12,772 Unless you have strict vertices, then you 490 00:28:12,772 --> 00:28:13,730 could connect those up. 491 00:28:17,280 --> 00:28:27,940 So let's look at a locally flat crease point. 492 00:28:31,280 --> 00:28:33,840 Actually, at this point, I should probably mention. 493 00:28:33,840 --> 00:28:39,660 This paper, I think of as a forerunner to the "How Paper 494 00:28:39,660 --> 00:28:41,260 Folds Between Creases" paper which 495 00:28:41,260 --> 00:28:44,500 is that hyperbolic parabolas don't exist. 496 00:28:44,500 --> 00:28:47,110 You may recall there were theorems about what creases 497 00:28:47,110 --> 00:28:49,020 look like, what ruled surfaces look like, 498 00:28:49,020 --> 00:28:52,214 and so on between creases. 499 00:28:52,214 --> 00:28:54,630 In that setting, we've proved things like straight creases 500 00:28:54,630 --> 00:28:57,806 stay straight and all these good things. 501 00:28:57,806 --> 00:28:59,180 Here, it's a little bit different 502 00:28:59,180 --> 00:29:01,190 because we have kind of two levels of creases. 503 00:29:01,190 --> 00:29:03,570 There's the seam which is special. 504 00:29:03,570 --> 00:29:06,180 And then we're imagining hypothetical creases 505 00:29:06,180 --> 00:29:06,960 between the seam. 506 00:29:06,960 --> 00:29:08,620 And could you have a curved crease? 507 00:29:08,620 --> 00:29:10,670 Claim is no. 508 00:29:10,670 --> 00:29:13,250 Claim is all the creases in between the seams 509 00:29:13,250 --> 00:29:15,236 have to be straight. 510 00:29:15,236 --> 00:29:16,860 And in fact, they can't look like this. 511 00:29:16,860 --> 00:29:18,700 They have to be at corners somehow. 512 00:29:18,700 --> 00:29:20,200 So that's what we're going to prove. 513 00:29:20,200 --> 00:29:21,820 But a lot of the same techniques, 514 00:29:21,820 --> 00:29:23,650 most of the same definitions. 515 00:29:23,650 --> 00:29:25,155 This paper was kind of a warm up, 516 00:29:25,155 --> 00:29:26,930 I feel like, for the nonexistence 517 00:29:26,930 --> 00:29:28,605 of hyperbolic parabolas. 518 00:29:28,605 --> 00:29:31,570 Although they use a lot of the same tools. 519 00:29:31,570 --> 00:29:33,270 They have two of the same authors, 520 00:29:33,270 --> 00:29:35,547 but there's no theorem that really 521 00:29:35,547 --> 00:29:36,630 is shared between the two. 522 00:29:36,630 --> 00:29:38,960 They're not identical. 523 00:29:38,960 --> 00:29:40,894 So we're looking inside a flat region. 524 00:29:40,894 --> 00:29:42,810 We're imagining, let's look at a crease point. 525 00:29:42,810 --> 00:29:46,440 It lies on some crease locally. 526 00:29:46,440 --> 00:29:48,130 And it's a fold crease. 527 00:29:48,130 --> 00:29:53,430 So I'm going to wave my hands a little bit. 528 00:29:53,430 --> 00:29:54,709 But imagine a folded crease. 529 00:29:54,709 --> 00:29:55,500 Maybe it's a curve. 530 00:29:55,500 --> 00:29:57,680 We don't know. 531 00:29:57,680 --> 00:30:00,090 It's folded by some angle, some nonzero angle. 532 00:30:00,090 --> 00:30:01,610 Otherwise, it's not a crease. 533 00:30:01,610 --> 00:30:06,750 So locally, it looks like two kind of planes, at least 534 00:30:06,750 --> 00:30:09,040 to the first order they're going to be planes. 535 00:30:09,040 --> 00:30:13,087 Maybe something like this. 536 00:30:13,087 --> 00:30:14,670 I'm going to draw the creases straight 537 00:30:14,670 --> 00:30:17,270 because to the first order, it is straight. 538 00:30:17,270 --> 00:30:20,770 But it might be slightly curved. 539 00:30:20,770 --> 00:30:23,230 So we're looking at a point here. 540 00:30:23,230 --> 00:30:26,290 And my point is, it's bent by some angle. 541 00:30:26,290 --> 00:30:28,130 So I'm going to draw the two tangent planes. 542 00:30:28,130 --> 00:30:30,200 There's a tangent plane on the right. 543 00:30:30,200 --> 00:30:32,190 Whatever the surface is doing on the right, 544 00:30:32,190 --> 00:30:33,390 there's a tangent plane over there. 545 00:30:33,390 --> 00:30:35,250 Whatever the surface is doing on the left of the crease, 546 00:30:35,250 --> 00:30:36,090 there's a tangent plane there. 547 00:30:36,090 --> 00:30:38,480 They're not the same plane because we are creased. 548 00:30:38,480 --> 00:30:39,560 Whatever the crease angle is, that's 549 00:30:39,560 --> 00:30:41,018 the angle between these two planes. 550 00:30:43,950 --> 00:30:46,460 I want to look at this tangent space again a little bit. 551 00:30:46,460 --> 00:30:48,410 You can think of it, the tangent space 552 00:30:48,410 --> 00:30:51,700 at the least it has this tangent plane. 553 00:30:51,700 --> 00:30:53,880 For this point, you have at least this tangent plane 554 00:30:53,880 --> 00:30:55,540 and at least this other tangent plane. 555 00:30:55,540 --> 00:31:00,180 And so in particular, you have to have the sweep between them. 556 00:31:00,180 --> 00:31:02,780 So you get at least a one-dimensional set there. 557 00:31:02,780 --> 00:31:04,420 Now, this can happen. 558 00:31:04,420 --> 00:31:06,240 So we're not going to get a contradiction. 559 00:31:06,240 --> 00:31:07,740 We're not going to suddenly discover 560 00:31:07,740 --> 00:31:10,330 there's a two-dimensional area and get that it wasn't flat. 561 00:31:10,330 --> 00:31:12,950 But at least we've got a one-dimensional arc. 562 00:31:12,950 --> 00:31:17,560 On the Gaussian sphere here, we've got something like this 563 00:31:17,560 --> 00:31:18,060 so far. 564 00:31:18,060 --> 00:31:21,050 This is one extreme tangent plane on the left. 565 00:31:21,050 --> 00:31:25,850 This is extreme tangent plane on the right. 566 00:31:25,850 --> 00:31:28,430 These two points correspond to those two. 567 00:31:28,430 --> 00:31:30,860 And then we've got the arc of-- you 568 00:31:30,860 --> 00:31:33,770 can sweep the tangent plane around and still 569 00:31:33,770 --> 00:31:37,740 stay outside the surface, 570 00:31:37,740 --> 00:31:38,790 Well, that's interesting. 571 00:31:38,790 --> 00:31:43,660 It's interesting because all of those tangent planes 572 00:31:43,660 --> 00:31:51,350 share a line, which is going to be this line that I drew here. 573 00:31:51,350 --> 00:31:53,610 I mean, you look at these two planes, 574 00:31:53,610 --> 00:31:55,350 there's a line that they share. 575 00:31:55,350 --> 00:31:58,260 And if you rotate the plane through that line, 576 00:31:58,260 --> 00:31:59,760 you'll still pass through this point 577 00:31:59,760 --> 00:32:01,345 and you won't hit the surface. 578 00:32:01,345 --> 00:32:03,770 So in fact, all those tangent planes share this line. 579 00:32:03,770 --> 00:32:05,436 I think we actually just need these two. 580 00:32:05,436 --> 00:32:07,420 So if you don't see that, don't worry. 581 00:32:07,420 --> 00:32:08,860 There are these two planes. 582 00:32:08,860 --> 00:32:10,190 There's a line there. 583 00:32:10,190 --> 00:32:12,660 They're tangent, which means that line 584 00:32:12,660 --> 00:32:17,180 is on or outside the surface because tangent planes don't 585 00:32:17,180 --> 00:32:17,940 hit the surface. 586 00:32:17,940 --> 00:32:19,640 I mean, they touch the surface barely, 587 00:32:19,640 --> 00:32:20,765 but they don't go interior. 588 00:32:32,010 --> 00:32:37,900 Well, I claim that in fact the surface must touch this line. 589 00:32:42,283 --> 00:32:54,750 I claim the surface, at least locally, 590 00:32:54,750 --> 00:32:59,060 has got to follow along that intersection 591 00:32:59,060 --> 00:33:08,520 line of the left and right tangent planes. 592 00:33:18,020 --> 00:33:20,010 Why? 593 00:33:20,010 --> 00:33:23,890 It comes down to this Gaussian sphere thing. 594 00:33:23,890 --> 00:33:26,801 We know that the surface lies inside these two planes. 595 00:33:26,801 --> 00:33:27,800 There's this wedge here. 596 00:33:27,800 --> 00:33:29,100 The surface must be below. 597 00:33:29,100 --> 00:33:32,150 We also know the surface comes and touches it at this point. 598 00:33:32,150 --> 00:33:37,220 Could it, from here, kind of dip down away from this line? 599 00:33:37,220 --> 00:33:38,430 I claim no. 600 00:33:38,430 --> 00:33:43,980 If it dipped down, then there'd be a third tangent plane. 601 00:33:43,980 --> 00:33:47,560 The tangent plane could follow and dip down as well. 602 00:33:47,560 --> 00:33:49,710 And then we've got not just a one-dimensional arc, 603 00:33:49,710 --> 00:33:52,730 but we're actually going to get a whole triangle in the Gauss 604 00:33:52,730 --> 00:33:53,540 sphere. 605 00:33:53,540 --> 00:33:56,040 Once you have a whole triangle, that means you weren't flat. 606 00:33:56,040 --> 00:33:57,562 You have positive curvature. 607 00:33:57,562 --> 00:33:59,520 So we're looking at a locally flat crease point 608 00:33:59,520 --> 00:34:02,120 somewhere in the middle of one of these polygons. 609 00:34:02,120 --> 00:34:05,724 So in fact, you can't afford to dip down because then you'd 610 00:34:05,724 --> 00:34:07,640 have at least a little bit of curvature there. 611 00:34:07,640 --> 00:34:11,197 Because you have zero curvature, you've got to stay straight. 612 00:34:11,197 --> 00:34:12,780 So that means that the surface locally 613 00:34:12,780 --> 00:34:18,860 has to exist along the segment in both directions 614 00:34:18,860 --> 00:34:20,030 until something happens. 615 00:34:20,030 --> 00:34:22,489 Now, we assumed that our point was 616 00:34:22,489 --> 00:34:24,420 locally flat and a crease point. 617 00:34:27,130 --> 00:34:30,670 So the surface must continue in this direction 618 00:34:30,670 --> 00:34:33,820 until we reach a point that is either not locally flat 619 00:34:33,820 --> 00:34:36,290 or not a crease point. 620 00:34:36,290 --> 00:34:39,659 I claim it's got to remain a crease point. 621 00:34:39,659 --> 00:34:41,909 Because look, you're following along this intersection 622 00:34:41,909 --> 00:34:43,284 line of these two tangent planes. 623 00:34:43,284 --> 00:34:44,490 These remain tangent planes. 624 00:34:44,490 --> 00:34:46,489 These are not only tangent planes at this point. 625 00:34:46,489 --> 00:34:48,197 They're also tangent planes at this point 626 00:34:48,197 --> 00:34:50,280 and potentially any point along this segment. 627 00:34:50,280 --> 00:34:53,710 As long as the surface is here, they are tangent planes. 628 00:34:53,710 --> 00:34:55,670 Therefore, you are creased here. 629 00:34:55,670 --> 00:34:58,820 I mean, you can't be smooth if you're butting up 630 00:34:58,820 --> 00:35:01,500 against these two tangent planes. 631 00:35:01,500 --> 00:35:06,180 So crease must be preserved, so the only issue is locally flat. 632 00:35:06,180 --> 00:35:10,090 It could be at some point you become not locally flat. 633 00:35:10,090 --> 00:35:11,540 And that indeed happens. 634 00:35:11,540 --> 00:35:14,120 That's when you hit the seam. 635 00:35:14,120 --> 00:35:17,010 When you hit the seam, then you're no longer locally flat. 636 00:35:17,010 --> 00:35:18,580 So this proves the creases must be 637 00:35:18,580 --> 00:35:23,660 line segments between two points on the seam. 638 00:35:23,660 --> 00:35:28,440 There's a little bit more to prove which is it either hits 639 00:35:28,440 --> 00:35:32,610 a strict vertex or it's tangent to seams. 640 00:35:32,610 --> 00:35:34,810 And this basically follows the same argument 641 00:35:34,810 --> 00:35:39,620 that if you hit something other than a strict vertex or a seam, 642 00:35:39,620 --> 00:35:42,200 you would have positive curvature where 643 00:35:42,200 --> 00:35:43,992 you shouldn't, basically. 644 00:35:43,992 --> 00:35:46,331 So I'll just leave it at that. 645 00:35:46,331 --> 00:35:47,830 I think that's enough for the proof. 646 00:35:47,830 --> 00:35:51,430 You get the idea why creases can't exist. 647 00:35:54,298 --> 00:35:55,760 Any questions about that? 648 00:36:01,680 --> 00:36:06,040 I have few more questions. 649 00:36:06,040 --> 00:36:13,160 One question is, can we see more examples of rolling belts? 650 00:36:13,160 --> 00:36:18,700 And in some sense, this example is 651 00:36:18,700 --> 00:36:20,470 an example of a rolling belt. 652 00:36:20,470 --> 00:36:24,340 So D-forms, you take two, say, ellipses. 653 00:36:24,340 --> 00:36:27,010 And the seam that connects them is a rolling belt. 654 00:36:27,010 --> 00:36:29,950 And this is an illustration of the rolling action. 655 00:36:29,950 --> 00:36:34,216 As you change, which point glues to which point? 656 00:36:34,216 --> 00:36:35,340 Let's say I fix this point. 657 00:36:35,340 --> 00:36:37,420 I can glue to this one or to this one or to this one 658 00:36:37,420 --> 00:36:38,045 or to this one. 659 00:36:38,045 --> 00:36:39,370 And then I zip around. 660 00:36:39,370 --> 00:36:42,190 That's the rolling of the belt. 661 00:36:42,190 --> 00:36:44,835 More generally, I'll draw a picture. 662 00:36:49,700 --> 00:36:53,950 More generally, a rolling belt from the viewpoint 663 00:36:53,950 --> 00:36:57,330 of a gluing tree is that you have basically it's 664 00:36:57,330 --> 00:37:03,910 like a conveyor belt like at the airport 665 00:37:03,910 --> 00:37:05,240 or on a tank or something. 666 00:37:05,240 --> 00:37:08,310 You've got treads here and it's going to roll. 667 00:37:08,310 --> 00:37:10,630 So if I mark a point, let's say I mark this point. 668 00:37:10,630 --> 00:37:13,300 And I roll it around this way a little bit, 669 00:37:13,300 --> 00:37:15,890 this point maybe stays stationary 670 00:37:15,890 --> 00:37:19,390 and I end up with something like this. 671 00:37:19,390 --> 00:37:22,230 And what that means is whatever this guy glued to over here 672 00:37:22,230 --> 00:37:23,490 is now over here. 673 00:37:23,490 --> 00:37:25,960 So we're changing the gluing. 674 00:37:25,960 --> 00:37:30,880 This guy's now getting glued to some point like this 675 00:37:30,880 --> 00:37:32,520 because it kind of rolls around. 676 00:37:32,520 --> 00:37:33,980 So in general, you're rolling this, 677 00:37:33,980 --> 00:37:37,600 but then you're always just gluing across this path. 678 00:37:37,600 --> 00:37:41,640 For this to work, every angle out here must be less than 679 00:37:41,640 --> 00:37:44,170 or equal to 180, like the material 680 00:37:44,170 --> 00:37:45,910 that's out on the outside. 681 00:37:45,910 --> 00:37:48,000 Remember, the gluing tree has the outside 682 00:37:48,000 --> 00:37:49,226 of the polygon in here. 683 00:37:49,226 --> 00:37:50,600 The inside of polygon's out here. 684 00:37:50,600 --> 00:37:53,270 So all the material's on the outside here. 685 00:37:53,270 --> 00:37:55,950 All of these things must be less than or equal to 180. 686 00:37:55,950 --> 00:37:57,885 If you can find a path in your gluing tree 687 00:37:57,885 --> 00:38:00,592 where you've got less than 180 material all around you, 688 00:38:00,592 --> 00:38:01,800 then you can do this rolling. 689 00:38:01,800 --> 00:38:04,470 And no matter how you roll, and then just glue across, 690 00:38:04,470 --> 00:38:07,620 you will still have a valid gluing, a valid Alexandrov 691 00:38:07,620 --> 00:38:08,480 gluing. 692 00:38:08,480 --> 00:38:11,270 So that's what rolling belts look like in general. 693 00:38:11,270 --> 00:38:15,890 We see them all over the place with these D-forms. 694 00:38:15,890 --> 00:38:17,390 The simple example I showed in class 695 00:38:17,390 --> 00:38:22,480 was when you take a rectangle, you glue into a cylinder. 696 00:38:22,480 --> 00:38:24,800 And now up here, I've got a rolling belt. 697 00:38:24,800 --> 00:38:27,680 Because if you look at the sides here, 698 00:38:27,680 --> 00:38:30,680 I've got actually exactly 180 degrees of material 699 00:38:30,680 --> 00:38:32,620 all around the belt. 700 00:38:32,620 --> 00:38:34,530 And so I could collapse it like this 701 00:38:34,530 --> 00:38:38,406 or I could collapse it-- let me tape it. 702 00:38:38,406 --> 00:38:39,780 A little hard to hold everything. 703 00:38:47,307 --> 00:38:49,640 If you look at this belt, I could collapse it like this. 704 00:38:49,640 --> 00:38:52,560 Or I could roll it a little bit and classify like that. 705 00:38:52,560 --> 00:38:54,990 Or I could roll it a little bit, collapse it like that. 706 00:38:54,990 --> 00:38:56,360 Roll it a little more. 707 00:38:56,360 --> 00:38:57,420 Collapse it like that. 708 00:38:57,420 --> 00:38:59,586 All of these are valid gluing because they're always 709 00:38:59,586 --> 00:39:01,126 gluing 180 to 180. 710 00:39:01,126 --> 00:39:03,190 And that's what rolling belts are. 711 00:39:13,170 --> 00:39:15,660 Does the broken applet work now? 712 00:39:15,660 --> 00:39:17,420 I don't know why it wasn't working. 713 00:39:17,420 --> 00:39:20,007 Probably some conflict with suspension 714 00:39:20,007 --> 00:39:21,590 or just the Java installation was bad. 715 00:39:21,590 --> 00:39:22,215 But here it is. 716 00:39:22,215 --> 00:39:24,250 It's freely available online. 717 00:39:24,250 --> 00:39:27,370 You draw your graph. 718 00:39:27,370 --> 00:39:29,550 It's a little hard to just use. 719 00:39:29,550 --> 00:39:31,290 It's hard to draw an exciting example. 720 00:39:31,290 --> 00:39:33,780 Here I'll a draw a tetrahedron because that's simple. 721 00:39:33,780 --> 00:39:36,900 In this case, it's abstract, so all of these edges 722 00:39:36,900 --> 00:39:37,670 are unit lengths. 723 00:39:37,670 --> 00:39:38,836 It's kind of like TreeMaker. 724 00:39:38,836 --> 00:39:40,510 You separately specify the lengths. 725 00:39:40,510 --> 00:39:42,100 Then you say Compute. 726 00:39:42,100 --> 00:39:47,230 And it will find, in this case, it's a tetrahedron, 727 00:39:47,230 --> 00:39:48,310 an equilateral. 728 00:39:48,310 --> 00:39:50,000 A regular tetrahedron is a little hard 729 00:39:50,000 --> 00:39:53,570 to see because we've got either a weird field of view 730 00:39:53,570 --> 00:39:55,974 or a lot of perspective here. 731 00:39:55,974 --> 00:39:57,390 But this is a regular tetrahedron. 732 00:39:57,390 --> 00:40:01,586 You can verify that by say making some of these lengths 733 00:40:01,586 --> 00:40:02,085 longer. 734 00:40:02,085 --> 00:40:05,810 I'll make three of them five. 735 00:40:05,810 --> 00:40:10,010 So then this is like a pyramid. 736 00:40:10,010 --> 00:40:11,760 Of course, we're just essentially applying 737 00:40:11,760 --> 00:40:15,460 Cauchy's rigidity theorem here and say, oh, this uniquely 738 00:40:15,460 --> 00:40:19,390 assembles into this spiky tetrahedron. 739 00:40:19,390 --> 00:40:22,480 You could, in principle, drawn an across here 740 00:40:22,480 --> 00:40:23,889 with the appropriate gluing. 741 00:40:23,889 --> 00:40:25,680 It's a little tricky to draw in because you 742 00:40:25,680 --> 00:40:27,270 have to also triangulate. 743 00:40:27,270 --> 00:40:30,010 It'd be hard to do from scratch to compute gluings. 744 00:40:30,010 --> 00:40:32,210 But in theory, especially if you use software 745 00:40:32,210 --> 00:40:34,001 to compute the gluings and compute shortest 746 00:40:34,001 --> 00:40:36,820 path in a glued polygon-- which we'll 747 00:40:36,820 --> 00:40:38,680 be getting more to next class. 748 00:40:38,680 --> 00:40:42,400 Next lecture is about algorithms for finding gluings. 749 00:40:42,400 --> 00:40:44,010 Then computing shortest paths. 750 00:40:44,010 --> 00:40:47,460 Then you could use that as input to this software which 751 00:40:47,460 --> 00:40:51,390 will then compute what the 3D polyhedron is. 752 00:40:51,390 --> 00:40:54,440 So that's the Alexandrov implementation. 753 00:40:54,440 --> 00:40:57,410 This is by, I think, a student of Bobenko and Izmestiev. 754 00:41:02,000 --> 00:41:04,440 One quick question about-- running out 755 00:41:04,440 --> 00:41:07,460 of time-- about the nonconvex case. 756 00:41:07,460 --> 00:41:10,350 So we're doing all this work about making convex surfaces. 757 00:41:10,350 --> 00:41:13,350 I said nonconvex case is trivial. 758 00:41:13,350 --> 00:41:15,690 In a sense, the decision problem is trivial. 759 00:41:15,690 --> 00:41:17,980 It's not an easy theorem to prove. 760 00:41:17,980 --> 00:41:22,040 And aha, taped it up. 761 00:41:22,040 --> 00:41:26,430 It follows from what's called the Burago-Zalgaller theorem. 762 00:41:26,430 --> 00:41:30,320 And it says that if you tap any polyhedral metric-- so that 763 00:41:30,320 --> 00:41:34,190 means you take a polygon, you glue it however you want. 764 00:41:34,190 --> 00:41:35,409 You could even add boundary. 765 00:41:35,409 --> 00:41:37,950 Pick any polyhedron metrics, so you take any polygon and glue 766 00:41:37,950 --> 00:41:39,280 it however you want. 767 00:41:39,280 --> 00:41:40,750 It could have handles. 768 00:41:40,750 --> 00:41:42,390 It could be a doughnut. 769 00:41:42,390 --> 00:41:43,820 It could be whatever. 770 00:41:43,820 --> 00:41:49,290 Then it has an isometric polyhedral realization in 3D. 771 00:41:49,290 --> 00:41:51,614 So there is a way-- so we're doing it abstractly. 772 00:41:51,614 --> 00:41:52,780 We're gluing stuff together. 773 00:41:52,780 --> 00:41:53,905 It's kind of topologically. 774 00:41:53,905 --> 00:41:55,290 We don't know what we're doing. 775 00:41:55,290 --> 00:41:57,190 But then there is a way to actually embed it 776 00:41:57,190 --> 00:42:00,790 as a polyhedral surface in 3D without stretching 777 00:42:00,790 --> 00:42:02,120 any of the lengths. 778 00:42:02,120 --> 00:42:05,120 So it's a folding of the piece of paper. 779 00:42:05,120 --> 00:42:07,330 It could have creases in it, will have actually 780 00:42:07,330 --> 00:42:09,310 tons of creases in it. 781 00:42:09,310 --> 00:42:12,622 And it has exactly the right-- it connects 782 00:42:12,622 --> 00:42:14,580 all the things you want it to connect together. 783 00:42:14,580 --> 00:42:17,640 It's kind of crazy. 784 00:42:17,640 --> 00:42:21,800 Furthermore, it could be self-crossing. 785 00:42:21,800 --> 00:42:26,020 But if your surface is either orientable-- 786 00:42:26,020 --> 00:42:32,840 so it doesn't have any cross caps or Mobius strips in it-- 787 00:42:32,840 --> 00:42:36,070 or it has boundary, then there's even 788 00:42:36,070 --> 00:42:38,250 a way to do without crossings. 789 00:42:38,250 --> 00:42:40,640 So that's basically all the cases we care about. 790 00:42:40,640 --> 00:42:44,630 You could glue this together into a sphere or into a disk 791 00:42:44,630 --> 00:42:48,789 or into a torus or two-handled torus or whatever. 792 00:42:48,789 --> 00:42:50,330 No matter what you do, there is a way 793 00:42:50,330 --> 00:42:53,930 to embed it in 3D as a presumably nonconvex very 794 00:42:53,930 --> 00:42:56,440 creased surface. 795 00:42:56,440 --> 00:42:58,830 This is a hard to prove. 796 00:42:58,830 --> 00:43:00,570 If you're familiar with Nash's embedding 797 00:43:00,570 --> 00:43:02,650 theorem about embedding Riemannian surfaces, 798 00:43:02,650 --> 00:43:05,510 it uses a bunch of techniques from there called spiraling 799 00:43:05,510 --> 00:43:07,250 perturbations, which I don't know. 800 00:43:07,250 --> 00:43:10,620 And one description I read calls the resulting surfaces 801 00:43:10,620 --> 00:43:14,730 "strongly corrugated," which I think means tons of creases. 802 00:43:14,730 --> 00:43:16,280 It is finitely many creases. 803 00:43:16,280 --> 00:43:17,220 That's the theorem. 804 00:43:17,220 --> 00:43:18,262 There's no bound on them. 805 00:43:18,262 --> 00:43:19,845 As far as I know, there's no algorithm 806 00:43:19,845 --> 00:43:20,860 to compute this thing. 807 00:43:20,860 --> 00:43:22,734 I've never seen a picture of them, otherwise, 808 00:43:22,734 --> 00:43:23,750 I'd show it to you. 809 00:43:23,750 --> 00:43:27,270 So lots of open problems in making this real. 810 00:43:27,270 --> 00:43:29,810 But at least the decision problem is kind of boring. 811 00:43:29,810 --> 00:43:30,920 You do any gluing. 812 00:43:30,920 --> 00:43:33,044 It will make a nonconvex thing. 813 00:43:33,044 --> 00:43:34,960 To find that thing is still pretty interesting 814 00:43:34,960 --> 00:43:36,430 question though. 815 00:43:36,430 --> 00:43:38,280 That's it.