1 00:00:02,785 --> 00:00:03,660 PROFESSOR: All right. 2 00:00:03,660 --> 00:00:07,770 Welcome back to 6849. 3 00:00:07,770 --> 00:00:12,530 You may recall or have heard that this film crew from Japan 4 00:00:12,530 --> 00:00:13,380 is here. 5 00:00:13,380 --> 00:00:15,390 They're from a program called [? Gachan, ?] 6 00:00:15,390 --> 00:00:17,147 which is about showing all the cool things 7 00:00:17,147 --> 00:00:18,980 that happen in universities around the world 8 00:00:18,980 --> 00:00:20,860 to students in Japan. 9 00:00:20,860 --> 00:00:22,500 So don't mind them, but you're going 10 00:00:22,500 --> 00:00:25,616 to have to sign a release form at the end. 11 00:00:25,616 --> 00:00:27,740 I apparently don't need you to sign a release form. 12 00:00:27,740 --> 00:00:28,380 It just works. 13 00:00:33,444 --> 00:00:34,860 We're talking about origami today. 14 00:00:34,860 --> 00:00:38,850 We began a series of lectures on computational origami. 15 00:00:38,850 --> 00:00:42,330 We're going to do some design, some actual folding, 16 00:00:42,330 --> 00:00:47,080 and some foldability algorithm stuff. 17 00:00:47,080 --> 00:00:54,230 Does everyone have one of these strips and a square of paper? 18 00:00:54,230 --> 00:00:55,010 You don't. 19 00:00:55,010 --> 00:00:56,950 Who has the master? 20 00:00:56,950 --> 00:01:00,380 Why don't you grab a few? 21 00:01:00,380 --> 00:01:02,050 Some of these. 22 00:01:02,050 --> 00:01:03,180 Here's the package. 23 00:01:06,780 --> 00:01:09,001 Lots of stuff. 24 00:01:09,001 --> 00:01:11,140 The paper comes from the [? Gachan ?] team, 25 00:01:11,140 --> 00:01:12,650 so thanks to them. 26 00:01:12,650 --> 00:01:17,870 This is stuff you use in your adding machine, I guess. 27 00:01:17,870 --> 00:01:20,630 That's where it comes from. 28 00:01:20,630 --> 00:01:21,220 Yes. 29 00:01:21,220 --> 00:01:22,095 You need one of each. 30 00:01:22,095 --> 00:01:22,805 Go for it. 31 00:01:22,805 --> 00:01:24,110 Open it. 32 00:01:24,110 --> 00:01:25,790 Let's get started. 33 00:01:25,790 --> 00:01:26,995 Where is my chalk? 34 00:01:30,580 --> 00:01:37,160 So just to warm you up, I want to do some folding 35 00:01:37,160 --> 00:01:38,810 and give you some terminology. 36 00:01:38,810 --> 00:01:41,280 I know some of you have done origami before, 37 00:01:41,280 --> 00:01:44,920 but a lot of you haven't, so bear with me. 38 00:01:44,920 --> 00:01:47,752 I will try to do this quickly. 39 00:01:47,752 --> 00:01:49,460 And even for those who have done origami, 40 00:01:49,460 --> 00:01:51,540 you may not have done origami math before. 41 00:01:51,540 --> 00:01:52,810 That's why you're here. 42 00:01:52,810 --> 00:01:55,940 So things like a piece of paper, which 43 00:01:55,940 --> 00:01:58,000 you might take for granted, I'm going 44 00:01:58,000 --> 00:02:03,860 to define here to be a 2D polygon. 45 00:02:03,860 --> 00:02:05,880 There's actually many interesting definitions 46 00:02:05,880 --> 00:02:08,130 of a piece of paper, but for today at least, 47 00:02:08,130 --> 00:02:10,110 most of the time, we'll think about polygon. 48 00:02:10,110 --> 00:02:12,026 Usually, you think of a square piece of paper, 49 00:02:12,026 --> 00:02:15,440 but "polygon" would be the geometer's generalization 50 00:02:15,440 --> 00:02:17,000 of that. 51 00:02:17,000 --> 00:02:20,955 Also, the polygon has distinguished sides. 52 00:02:24,820 --> 00:02:29,970 So distinguished top and bottom side. 53 00:02:29,970 --> 00:02:36,550 In practice, with standard kami paper from Japan, 54 00:02:36,550 --> 00:02:38,707 you have the colored side and the white side, 55 00:02:38,707 --> 00:02:40,040 so that's one clear distinction. 56 00:02:40,040 --> 00:02:41,700 Maybe we call the colored side the top. 57 00:02:41,700 --> 00:02:44,080 Doesn't really matter, just so we can tell things apart, 58 00:02:44,080 --> 00:02:47,720 and in particular tell mountains from valleys. 59 00:02:47,720 --> 00:02:49,120 That's our motivation. 60 00:02:49,120 --> 00:02:55,730 So a crease in a piece of paper, this I'm 61 00:02:55,730 --> 00:03:00,630 going to think abstractly as just a line segment or some one 62 00:03:00,630 --> 00:03:06,325 dimensional curve drawn on the piece of paper. 63 00:03:13,220 --> 00:03:19,760 And then we have crease pattern. 64 00:03:26,130 --> 00:03:31,180 This is just a bunch of creases, so a bunch of line segments 65 00:03:31,180 --> 00:03:32,860 drawn on the piece of paper. 66 00:03:32,860 --> 00:03:35,620 If you are graph theoretically inclined, 67 00:03:35,620 --> 00:03:37,380 I would think of this as a planar 68 00:03:37,380 --> 00:03:42,635 graph drawn on the paper. 69 00:03:45,589 --> 00:03:47,380 If you're not graph theoretically inclined, 70 00:03:47,380 --> 00:03:49,340 just ignore that definition. 71 00:03:49,340 --> 00:03:52,900 They're the same thing. 72 00:03:52,900 --> 00:03:55,130 That's a crease pattern. 73 00:03:55,130 --> 00:04:00,090 So for example, the thing we're going to be folding 74 00:04:00,090 --> 00:04:03,270 is this very simple pinwheel, and this 75 00:04:03,270 --> 00:04:05,760 is an example of a crease pattern that 76 00:04:05,760 --> 00:04:07,640 folds into that pinwheel. 77 00:04:07,640 --> 00:04:10,537 So here, the lines are distinguished. 78 00:04:10,537 --> 00:04:13,120 There's two different kinds, the red lines and the blue lines, 79 00:04:13,120 --> 00:04:14,561 but ignore that for now. 80 00:04:14,561 --> 00:04:16,019 That's a bunch of creases which you 81 00:04:16,019 --> 00:04:17,399 could put in and make that thing. 82 00:04:20,769 --> 00:04:24,820 Before we get there, I want the notion of a folded state. 83 00:04:28,210 --> 00:04:31,740 So this is all pretty normal if you're an origamist. 84 00:04:31,740 --> 00:04:34,890 This is really something only the mathematicians use, 85 00:04:34,890 --> 00:04:36,630 this term, folded state. 86 00:04:40,040 --> 00:04:46,425 Formally, this is what we refer to as the finished product. 87 00:04:49,800 --> 00:04:54,460 This pinwheel is a folded state of this piece of paper. 88 00:04:54,460 --> 00:04:55,470 Clear? 89 00:04:55,470 --> 00:04:58,600 Defining that is a real pain, and you can look at chapter 11, 90 00:04:58,600 --> 00:04:59,910 I think, in the textbook. 91 00:04:59,910 --> 00:05:01,240 Maybe I'll talk about it at some point, 92 00:05:01,240 --> 00:05:02,656 but I don't want to get distracted 93 00:05:02,656 --> 00:05:04,540 by how you define what a folded state is. 94 00:05:04,540 --> 00:05:05,956 But you're not allowed to stretch, 95 00:05:05,956 --> 00:05:07,800 you're not allowed to tear. 96 00:05:07,800 --> 00:05:10,350 It should be a nice, valid folding. 97 00:05:10,350 --> 00:05:12,130 You can't cross yourself. 98 00:05:12,130 --> 00:05:15,530 So that's this notion of folded state. 99 00:05:15,530 --> 00:05:21,960 And if you take a folded state and then unfold it-- 100 00:05:21,960 --> 00:05:24,190 this is something you can do with a folded state-- 101 00:05:24,190 --> 00:05:25,420 you get a crease pattern. 102 00:05:30,420 --> 00:05:35,860 So I can take this pinwheel, unfold it, 103 00:05:35,860 --> 00:05:38,760 and I can see which lines are creased 104 00:05:38,760 --> 00:05:40,720 to make that pinwheel happen. 105 00:05:40,720 --> 00:05:47,540 So that's the crease pattern of a folded state. 106 00:05:47,540 --> 00:05:50,240 Today especially, and a lot of the field 107 00:05:50,240 --> 00:05:54,560 is concerned with the idea of flat foldings, which 108 00:05:54,560 --> 00:06:04,480 are a particular kind of folded state that lives in the plane, 109 00:06:04,480 --> 00:06:05,325 lying in the plane. 110 00:06:09,029 --> 00:06:11,070 Of course, it doesn't literally lie on the plane. 111 00:06:11,070 --> 00:06:13,590 There's lots of layers stacked on top of each other, 112 00:06:13,590 --> 00:06:15,370 but geometrically, everything has 113 00:06:15,370 --> 00:06:19,460 been collapsed onto one slightly thickened plane here. 114 00:06:19,460 --> 00:06:21,660 So this is a flat origami. 115 00:06:21,660 --> 00:06:26,840 This is also a flat origami, the trivial kind. 116 00:06:26,840 --> 00:06:28,400 That's a flat folding. 117 00:06:28,400 --> 00:06:34,520 And when this is possible, we call that crease pattern 118 00:06:34,520 --> 00:06:35,205 flat foldable. 119 00:06:47,850 --> 00:06:53,380 And finally, we get to the idea of mountains and valleys. 120 00:06:53,380 --> 00:07:05,620 So a mountain crease is when the bottom sides 121 00:07:05,620 --> 00:07:17,600 touch, and a valley crease is when the top sides touch. 122 00:07:21,890 --> 00:07:23,820 So remember my distinguished sides. 123 00:07:23,820 --> 00:07:25,840 I'll call the colored side the top, 124 00:07:25,840 --> 00:07:28,510 so then this is a mountain crease over here 125 00:07:28,510 --> 00:07:31,120 because I'm bringing the two white sides together, 126 00:07:31,120 --> 00:07:33,420 and where's a valley? 127 00:07:33,420 --> 00:07:37,270 I guess this is a valley, where I'm bringing the two colored 128 00:07:37,270 --> 00:07:39,550 sides, the two top sides, together. 129 00:07:39,550 --> 00:07:42,060 Together, that makes this pattern, 130 00:07:42,060 --> 00:07:45,620 and hopefully, that matches up with here. 131 00:07:45,620 --> 00:07:46,660 I mean, it's symmetric. 132 00:07:46,660 --> 00:07:47,951 You could flip everything over. 133 00:07:47,951 --> 00:07:49,965 You'd just get the inverted color pattern. 134 00:07:49,965 --> 00:07:52,466 It doesn't really matter which side you defined to be which, 135 00:07:52,466 --> 00:07:54,006 but the point is to distinguish which 136 00:07:54,006 --> 00:07:55,710 are mountains versus which are valleys. 137 00:07:55,710 --> 00:07:58,630 You get different parity, it looks 138 00:07:58,630 --> 00:08:00,640 like, I have matched up there. 139 00:08:00,640 --> 00:08:04,260 So let's fold this, just for some experience, 140 00:08:04,260 --> 00:08:06,590 if you've never made one before. 141 00:08:06,590 --> 00:08:10,970 Take your square piece of paper, and to get started, 142 00:08:10,970 --> 00:08:12,590 let's say with the white side up, 143 00:08:12,590 --> 00:08:15,910 just fold along the two diagonals, 144 00:08:15,910 --> 00:08:18,890 the valley folds, bringing the two white sides together. 145 00:08:21,660 --> 00:08:26,334 So you fold along one diagonal, then you unfold, then 146 00:08:26,334 --> 00:08:27,750 you fold along the other diagonal. 147 00:08:37,650 --> 00:08:40,150 How are we doing? 148 00:08:40,150 --> 00:08:41,154 Couple diagonals. 149 00:08:41,154 --> 00:08:42,159 I have the big sheet. 150 00:08:42,159 --> 00:08:43,200 I should be going slower. 151 00:08:45,860 --> 00:08:47,830 So now, I want to fold that inner square, 152 00:08:47,830 --> 00:08:49,810 and we've conveniently marked the center point. 153 00:08:49,810 --> 00:08:53,040 So I'm going to fold each of the four edges of the square 154 00:08:53,040 --> 00:08:55,410 to the center, and make sure to line up 155 00:08:55,410 --> 00:08:58,395 the left and right edges, and that guarantees that I 156 00:08:58,395 --> 00:09:01,470 fold parallel to that bottom edge. 157 00:09:01,470 --> 00:09:03,010 In this case, I just want to crease 158 00:09:03,010 --> 00:09:05,620 the middle half of that line. 159 00:09:05,620 --> 00:09:07,370 So I'm not creasing all the way. 160 00:09:07,370 --> 00:09:10,090 I'm just creasing in the middle half 161 00:09:10,090 --> 00:09:12,740 here to make that one edge of the square, 162 00:09:12,740 --> 00:09:15,865 and then I turn and I repeat that four times. 163 00:09:30,630 --> 00:09:32,680 Switch sides for different visibility. 164 00:09:45,930 --> 00:09:48,660 This is not the usual way to teach this model, 165 00:09:48,660 --> 00:09:52,550 but I think it's the mathematically appropriate way. 166 00:09:52,550 --> 00:09:55,700 I'm trying to make that crease pattern as quickly as I can. 167 00:09:55,700 --> 00:09:57,680 I did it with the white side up because I'm 168 00:09:57,680 --> 00:09:59,502 folding everything valley, which is easier. 169 00:09:59,502 --> 00:10:01,210 In reality, I want these to be mountains. 170 00:10:01,210 --> 00:10:04,810 When I turn them over, they're all mountains, and I'm happy. 171 00:10:04,810 --> 00:10:08,762 We made some extra creases here, which we didn't need to. 172 00:10:08,762 --> 00:10:09,470 How are we doing? 173 00:10:09,470 --> 00:10:11,810 How many people are done with those? 174 00:10:11,810 --> 00:10:14,140 How many people are not? 175 00:10:14,140 --> 00:10:15,310 Couple people. 176 00:10:15,310 --> 00:10:18,860 Sarah, hurry up. 177 00:10:18,860 --> 00:10:22,130 Now comes the fun part, and this is 178 00:10:22,130 --> 00:10:24,390 to give you an idea of what general origami folds can 179 00:10:24,390 --> 00:10:24,890 be like. 180 00:10:24,890 --> 00:10:27,610 I want to simultaneously fold all those creases 181 00:10:27,610 --> 00:10:31,310 that I put in except for the center, where 182 00:10:31,310 --> 00:10:32,890 it's supposed to be uncreased. 183 00:10:32,890 --> 00:10:37,260 So I'm going to bring those four edges of the square in, 184 00:10:37,260 --> 00:10:39,130 and if you do it right, you'll end up 185 00:10:39,130 --> 00:10:42,100 with these four flaps sticking out of the top. 186 00:10:42,100 --> 00:10:43,540 It's not going to lie flat. 187 00:10:43,540 --> 00:10:45,070 We don't have enough creases in yet. 188 00:10:45,070 --> 00:10:48,060 But we have these four flaps of the pinwheel, 189 00:10:48,060 --> 00:10:50,140 and then you just get to choose, do I 190 00:10:50,140 --> 00:10:52,250 flip them over one way or the other? 191 00:10:52,250 --> 00:10:54,860 If you flip them over as much as they go in that direction-- 192 00:10:54,860 --> 00:10:58,865 I'll try to match the figure. 193 00:10:58,865 --> 00:11:00,700 I'll put this guy over at the left, 194 00:11:00,700 --> 00:11:08,290 crease him in, put this guy down at that end, and this guy 195 00:11:08,290 --> 00:11:14,150 over to the right, and this guy up. 196 00:11:14,150 --> 00:11:16,460 I get my pinwheel. 197 00:11:16,460 --> 00:11:19,510 If that doesn't work out for you, you can try again. 198 00:11:19,510 --> 00:11:22,550 It's not so hard, but it gives you some idea 199 00:11:22,550 --> 00:11:24,480 that putting all these creases in at once 200 00:11:24,480 --> 00:11:27,730 is kind of necessary for this to happen. 201 00:11:27,730 --> 00:11:30,480 In particular, if you look at the crease pattern 202 00:11:30,480 --> 00:11:33,100 of this model, there is no fold that 203 00:11:33,100 --> 00:11:35,910 goes all the way through the square, 204 00:11:35,910 --> 00:11:38,892 so you can't make a first fold and then 205 00:11:38,892 --> 00:11:40,100 make a second fold and so on. 206 00:11:40,100 --> 00:11:42,429 You really have to do this all at once. 207 00:11:42,429 --> 00:11:44,970 There's many ways to do it, some of which are more intuitive, 208 00:11:44,970 --> 00:11:49,100 but I thought this simultaneous collapse is the coolest. 209 00:11:49,100 --> 00:11:51,585 It's nice and fourfold symmetric as you do it, I guess, 210 00:11:51,585 --> 00:11:52,940 if you do it right. 211 00:11:52,940 --> 00:11:56,410 I think this even folds rigidly. 212 00:11:56,410 --> 00:11:59,200 That's to give you some intuition, 213 00:11:59,200 --> 00:12:01,495 some real, practical origami experience. 214 00:12:05,710 --> 00:12:08,130 And you get an idea of what a crease pattern is, 215 00:12:08,130 --> 00:12:09,850 when a mountain valley is. 216 00:12:09,850 --> 00:12:11,740 I have two more terms and then we 217 00:12:11,740 --> 00:12:13,805 can go to math, some real stuff. 218 00:12:20,400 --> 00:12:23,430 There's two notions that involve mountains and valleys. 219 00:12:23,430 --> 00:12:31,884 One is a mountain-valley assignment, 220 00:12:31,884 --> 00:12:33,550 and this is something that goes together 221 00:12:33,550 --> 00:12:35,360 with the crease pattern. 222 00:12:35,360 --> 00:12:39,859 And this specifies which creases are mountains 223 00:12:39,859 --> 00:12:40,775 and which are valleys. 224 00:12:48,902 --> 00:12:51,490 Did you all get it? 225 00:12:51,490 --> 00:12:52,200 Good. 226 00:12:52,200 --> 00:12:53,330 The pinwheel, I mean. 227 00:13:00,890 --> 00:13:03,550 So you could think of this as a coloring of the edges, 228 00:13:03,550 --> 00:13:04,260 whatever. 229 00:13:04,260 --> 00:13:07,590 In this case, I'm simultaneously using color red 230 00:13:07,590 --> 00:13:10,200 to denote mountains and blue to denote valleys, 231 00:13:10,200 --> 00:13:12,710 and also, the standard in origami world, 232 00:13:12,710 --> 00:13:15,060 although it's a bit annoying, is dot dash 233 00:13:15,060 --> 00:13:18,135 means mountain and dash means valley. 234 00:13:18,135 --> 00:13:19,010 That's pretty common. 235 00:13:19,010 --> 00:13:22,740 You'll see that throughout. 236 00:13:22,740 --> 00:13:32,850 That was invented in the 1950s, I guess, by Yoshizawa, 237 00:13:32,850 --> 00:13:34,162 maybe earlier. 238 00:13:34,162 --> 00:13:35,120 Somewhere around there. 239 00:13:37,660 --> 00:13:39,630 And then if I take a crease pattern together 240 00:13:39,630 --> 00:13:41,046 with a mountain-valley assignment, 241 00:13:41,046 --> 00:13:43,620 I'm going to call this a mountain-valley pattern. 242 00:13:43,620 --> 00:13:47,332 That's the picture that I'm really drawing here. 243 00:13:47,332 --> 00:13:49,790 Sometimes it's useful to think of these as coming together, 244 00:13:49,790 --> 00:13:51,330 sometimes as separate. 245 00:13:51,330 --> 00:13:54,090 You're going to see this terminology all over the place, 246 00:13:54,090 --> 00:13:58,340 so I figure I should define it. 247 00:14:07,730 --> 00:14:09,970 And sometimes, m is mountain, v is valley. 248 00:14:14,330 --> 00:14:16,560 Now we can do some fun stuff. 249 00:14:21,007 --> 00:14:22,590 I showed you this pinwheel to give you 250 00:14:22,590 --> 00:14:25,220 an idea of more complicated folds, 251 00:14:25,220 --> 00:14:26,950 the general origami folds. 252 00:14:26,950 --> 00:14:30,140 But today, I want to focus on relatively simple kind 253 00:14:30,140 --> 00:14:32,000 of folds, which is called simple folds. 254 00:14:37,460 --> 00:14:39,860 Unlike this thing where I have to simultaneously collapse 255 00:14:39,860 --> 00:14:43,510 many creases, a simple fold just involves one crease. 256 00:14:43,510 --> 00:14:55,540 It is a fold along a single line, 257 00:14:55,540 --> 00:14:59,970 and we fold it by plus or minus 180 degrees. 258 00:14:59,970 --> 00:15:02,440 So depending which way you're counting, 259 00:15:02,440 --> 00:15:05,520 plus 180 is maybe valley, minus 180 is mountain. 260 00:15:08,400 --> 00:15:11,850 If I start with a flat folding, after I make a simple fold, 261 00:15:11,850 --> 00:15:13,806 I will again have a flat folding. 262 00:15:13,806 --> 00:15:15,180 What we're interested in is, what 263 00:15:15,180 --> 00:15:17,040 can you make by a sequence of simple folds? 264 00:15:17,040 --> 00:15:18,710 I start from a square paper, say, 265 00:15:18,710 --> 00:15:20,640 and I make a sequence of simple folds. 266 00:15:20,640 --> 00:15:22,880 At all times, I will be flat folding. 267 00:15:22,880 --> 00:15:24,852 It's easy to get there. 268 00:15:24,852 --> 00:15:27,310 The pinwheel is something you could not make in this model. 269 00:15:27,310 --> 00:15:29,370 It's a restricted model, but it's still 270 00:15:29,370 --> 00:15:30,500 surprisingly powerful. 271 00:15:30,500 --> 00:15:32,000 You still get a universality result. 272 00:15:32,000 --> 00:15:35,900 Even with simple folds, you can make any flat thing. 273 00:15:35,900 --> 00:15:37,910 And if you relax this constraint and don't 274 00:15:37,910 --> 00:15:40,490 say it's plus or minus 180, then you 275 00:15:40,490 --> 00:15:43,020 can actually make any 3D thing you want. 276 00:15:43,020 --> 00:15:45,500 So that's the first thing we will prove today, 277 00:15:45,500 --> 00:15:48,510 and that's in the origami design world. 278 00:15:48,510 --> 00:15:50,550 And then we'll go to origami foldability. 279 00:15:50,550 --> 00:15:51,841 I'll give you a crease pattern. 280 00:15:51,841 --> 00:15:55,020 I want to know, can it be folded by a sequence of simple folds? 281 00:15:55,020 --> 00:15:58,580 That's a much harder problem, but some cases are easy. 282 00:16:21,330 --> 00:16:25,490 So this the universality result I mentioned in lecture one. 283 00:16:25,490 --> 00:16:27,530 This is going to be the inefficient version, 284 00:16:27,530 --> 00:16:29,770 but we're going to prove that everything 285 00:16:29,770 --> 00:16:32,636 can be folded from a square paper. 286 00:16:32,636 --> 00:16:34,260 I'm going to be more precise about what 287 00:16:34,260 --> 00:16:35,134 I mean by everything. 288 00:16:47,420 --> 00:16:51,260 So I'm going to imagine there are a bunch of polygons in 3D 289 00:16:51,260 --> 00:16:52,810 and they're somehow joined together 290 00:16:52,810 --> 00:16:54,530 into one connected mass. 291 00:16:54,530 --> 00:16:56,420 This could be a polyhedron like a cube, 292 00:16:56,420 --> 00:16:58,370 it could be a polyhedron like a bunny, 293 00:16:58,370 --> 00:17:01,510 it could be some crazy, thorny mess, 294 00:17:01,510 --> 00:17:03,105 but each piece locally is a polygon, 295 00:17:03,105 --> 00:17:04,771 and you just stick a whole bunch of them 296 00:17:04,771 --> 00:17:08,290 together to some connected thing. 297 00:17:08,290 --> 00:17:15,079 Plus you get to specify what color, what side of the paper 298 00:17:15,079 --> 00:17:18,439 you want to be visible on each of those polygons. 299 00:17:25,240 --> 00:17:30,270 You can even do that on each side of the polygon 300 00:17:30,270 --> 00:17:30,770 if you want. 301 00:17:35,970 --> 00:18:14,600 And you can fold that thing from a big enough piece of paper, 302 00:18:14,600 --> 00:18:18,590 such as a big enough square, a big enough rectangle, whatever 303 00:18:18,590 --> 00:18:20,232 you feel like. 304 00:18:20,232 --> 00:18:22,990 Bicolor paper here refers to the idea 305 00:18:22,990 --> 00:18:26,560 that your paper is different colors 306 00:18:26,560 --> 00:18:29,110 on one side and the other, so white on one side, 307 00:18:29,110 --> 00:18:31,704 colored on the other side, white on one side, 308 00:18:31,704 --> 00:18:33,120 black on the other side, whatever. 309 00:18:33,120 --> 00:18:40,350 We usually try to draw that with the folded over corner. 310 00:18:43,850 --> 00:18:47,410 There we go, bicolor paper. 311 00:18:47,410 --> 00:18:51,980 So for example, for fun, here's how 312 00:18:51,980 --> 00:18:55,026 you could imagine this design by John Montroll. 313 00:18:55,026 --> 00:18:57,400 This is folded from a square of paper, white on one side, 314 00:18:57,400 --> 00:18:58,191 black on the other. 315 00:18:58,191 --> 00:19:00,100 It's a zebra. 316 00:19:00,100 --> 00:19:01,990 And you could imagine the input to that 317 00:19:01,990 --> 00:19:04,610 problem was a bunch of polygons. 318 00:19:04,610 --> 00:19:07,150 There's some white polygons, some black polygons. 319 00:19:07,150 --> 00:19:10,120 Together, they form a connected union, 320 00:19:10,120 --> 00:19:12,340 and that could be the input. 321 00:19:12,340 --> 00:19:14,770 This could be an output to how you fold your square 322 00:19:14,770 --> 00:19:16,044 to make that color pattern. 323 00:19:16,044 --> 00:19:18,210 Of course, if you gave this input to that algorithm, 324 00:19:18,210 --> 00:19:22,070 you'll get a much uglier folding than the one designed 325 00:19:22,070 --> 00:19:23,594 by a human, but that's the idea. 326 00:19:23,594 --> 00:19:25,010 You could also imagine you're just 327 00:19:25,010 --> 00:19:26,840 given a polygon you want to make. 328 00:19:26,840 --> 00:19:29,120 You say, I want to make a horse, a flat horse. 329 00:19:29,120 --> 00:19:31,270 Then you could give that to this algorithm 330 00:19:31,270 --> 00:19:33,750 and it will tell you how to fold a horse. 331 00:19:33,750 --> 00:19:35,940 That's the general idea. 332 00:19:40,650 --> 00:19:42,836 So how do we prove that? 333 00:19:42,836 --> 00:19:43,835 Want to make everything. 334 00:19:56,250 --> 00:19:58,860 So this is an early result in computational origami design, 335 00:19:58,860 --> 00:20:02,565 I think from 1998, probably. 336 00:20:02,565 --> 00:20:08,100 Trial version here is in 2000. 337 00:20:08,100 --> 00:20:10,840 Any ideas how to prove this? 338 00:20:10,840 --> 00:20:13,709 How can we make anything? 339 00:20:13,709 --> 00:20:15,250 Got to get you thinking a little bit. 340 00:20:21,570 --> 00:20:25,120 We have a subtle hint. 341 00:20:25,120 --> 00:20:30,210 So the idea is to take a really long strip, 342 00:20:30,210 --> 00:20:33,410 much longer than this one, and fold it, and just 343 00:20:33,410 --> 00:20:37,770 wrap around your polyhedron over and over and over, 344 00:20:37,770 --> 00:20:40,810 and eventually make whatever you want. 345 00:20:40,810 --> 00:20:42,800 So it's going to be a little flimsy, 346 00:20:42,800 --> 00:20:45,730 but mathematically, it'll work perfectly. 347 00:20:45,730 --> 00:20:51,930 So first thing to do is fold your paper, whatever 348 00:20:51,930 --> 00:20:58,235 shape it is, down to a long, narrow strip, so 349 00:20:58,235 --> 00:20:59,455 a long, narrow rectangle. 350 00:21:10,150 --> 00:21:12,530 Conveniently, you have your strips of paper already, 351 00:21:12,530 --> 00:21:13,530 but if you started with a square, 352 00:21:13,530 --> 00:21:15,446 the first thing you'd do is accordion pleat it 353 00:21:15,446 --> 00:21:18,390 down to just one edge of the square. 354 00:21:18,390 --> 00:21:19,139 Crazy already. 355 00:21:19,139 --> 00:21:21,180 If you started with a square, this is a bad idea. 356 00:21:21,180 --> 00:21:24,110 If you started with a strip, it's actually pretty good. 357 00:21:24,110 --> 00:21:27,520 It's probably close to optimal. 358 00:21:27,520 --> 00:21:31,150 But now, how do I actually, say, wrap everything with the strip? 359 00:21:31,150 --> 00:21:33,037 Let's be more formal about that. 360 00:21:33,037 --> 00:21:34,620 The first thing I'm going to do, which 361 00:21:34,620 --> 00:21:39,150 is a common trick in computational geometry, 362 00:21:39,150 --> 00:21:41,960 it's like, well, polygons are confusing. 363 00:21:41,960 --> 00:21:42,980 I like triangles better. 364 00:21:42,980 --> 00:21:44,220 They're much simpler. 365 00:21:44,220 --> 00:21:46,200 So just triangulate all the polygons. 366 00:21:46,200 --> 00:21:49,420 If I have some crazy polygon, imagine 367 00:21:49,420 --> 00:21:53,560 that's some piece of the zebra, I'll 368 00:21:53,560 --> 00:21:57,935 just subdivide it into triangles. 369 00:21:57,935 --> 00:21:59,560 There's standard algorithms to do this. 370 00:21:59,560 --> 00:22:00,768 I don't want to go into that. 371 00:22:00,768 --> 00:22:04,100 You can just add a bunch of these diagonals to the polygon, 372 00:22:04,100 --> 00:22:06,820 just keep adding them as long as they don't cross each other. 373 00:22:06,820 --> 00:22:09,250 When you're done, it will be triangulated. 374 00:22:09,250 --> 00:22:12,018 You can do that efficiently, linear time, even. 375 00:22:14,881 --> 00:22:16,630 That's a little bit easier to think about. 376 00:22:16,630 --> 00:22:20,030 Now I just have to figure out how to wrap one triangle 377 00:22:20,030 --> 00:22:23,566 and then how to combine those wrappings together somehow. 378 00:22:23,566 --> 00:22:25,440 Then I could go through triangle by triangle, 379 00:22:25,440 --> 00:22:29,960 visit everything, cover everything I need to cover. 380 00:22:29,960 --> 00:22:33,240 So that's what we're going to do. 381 00:22:36,880 --> 00:22:41,730 Cover each of these triangles in some order. 382 00:22:41,730 --> 00:22:44,630 We're going to cover each one at least once, 383 00:22:44,630 --> 00:22:48,180 possibly more if we're not careful. 384 00:22:48,180 --> 00:22:49,960 For starters, we'll be not careful. 385 00:22:56,650 --> 00:23:07,160 And we're going to do this by using a zigzag path 386 00:23:07,160 --> 00:23:15,140 parallel to the next edge starting 387 00:23:15,140 --> 00:23:16,620 from the opposite corner. 388 00:23:16,620 --> 00:23:19,470 So some technical details. 389 00:23:19,470 --> 00:23:24,800 Much easier to see in a picture, so let me draw you a picture. 390 00:23:32,540 --> 00:23:35,550 Let's say we want to visit this triangle first, 391 00:23:35,550 --> 00:23:37,067 and after we visit this triangle, 392 00:23:37,067 --> 00:23:38,650 let's say we've planned out that we're 393 00:23:38,650 --> 00:23:40,380 going to visit this triangle next. 394 00:23:40,380 --> 00:23:43,300 After that one, we're going to go to this triangle. 395 00:23:43,300 --> 00:23:46,490 So this is what I'd call the next edge, because it's 396 00:23:46,490 --> 00:23:48,910 the edge connecting to the next triangle. 397 00:23:48,910 --> 00:23:51,310 So I'd like to zigzag parallel to this edge, 398 00:23:51,310 --> 00:23:54,480 so I want to go back and forth like that. 399 00:23:54,480 --> 00:23:58,230 And then it says I want to start at the opposite corner, which 400 00:23:58,230 --> 00:24:00,220 is this one. 401 00:24:00,220 --> 00:24:01,720 So I'm going to start here, and I'll 402 00:24:01,720 --> 00:24:03,136 draw this a little more precisely. 403 00:24:06,740 --> 00:24:09,090 I'm going to take my strip and just 404 00:24:09,090 --> 00:24:14,770 go this way for a little while, and turn around and come back 405 00:24:14,770 --> 00:24:19,590 this way, and then turn around, and we're 406 00:24:19,590 --> 00:24:23,420 going to see how to do turning around in a moment, so 407 00:24:23,420 --> 00:24:24,580 something like that. 408 00:24:29,560 --> 00:24:33,430 Now I've got to start planning a little bit, because I want 409 00:24:33,430 --> 00:24:36,210 to do this triangle in the same way, meaning I want to go back 410 00:24:36,210 --> 00:24:39,380 and forth parallel to the next edge 411 00:24:39,380 --> 00:24:41,520 starting at the opposite corner. 412 00:24:41,520 --> 00:24:44,960 So that means next thing I want to do is start at this corner, 413 00:24:44,960 --> 00:24:47,700 so I need to set this up so that I don't end here. 414 00:24:47,700 --> 00:24:49,130 I'd like to end here. 415 00:24:49,130 --> 00:24:52,900 You can do that just by doing one more iteration. 416 00:24:52,900 --> 00:24:55,170 If I end up with the wrong parity-- here 417 00:24:55,170 --> 00:24:57,285 it looks like I won't too much. 418 00:24:59,950 --> 00:25:01,280 Here it's a little ugly. 419 00:25:01,280 --> 00:25:03,890 Got a little bit of uncovered portion of the triangle, 420 00:25:03,890 --> 00:25:07,330 so you need a slightly different turn here. 421 00:25:07,330 --> 00:25:10,640 Then I'll just come and sort of overlap myself. 422 00:25:10,640 --> 00:25:11,150 That's OK. 423 00:25:11,150 --> 00:25:13,500 I just have to cover it at least once. 424 00:25:13,500 --> 00:25:17,010 And then I do some kind of turn here, 425 00:25:17,010 --> 00:25:19,522 and then I'm going to do this. 426 00:25:19,522 --> 00:25:21,230 So there's obviously some questions here. 427 00:25:21,230 --> 00:25:23,290 How do I make these 180 degree turns? 428 00:25:23,290 --> 00:25:24,930 How do I make an arbitrary turn? 429 00:25:24,930 --> 00:25:27,310 How do I turn even when I'm allowed to overlap myself? 430 00:25:27,310 --> 00:25:29,101 I'm not going to worry about that too much, 431 00:25:29,101 --> 00:25:30,360 basically the same. 432 00:25:30,360 --> 00:25:33,550 Basically, how do I turn? 433 00:25:33,550 --> 00:25:34,220 How do I turn? 434 00:25:34,220 --> 00:25:40,440 Well, let's take our strips and do some turning, 435 00:25:40,440 --> 00:25:46,140 and I have a slide to follow along. 436 00:25:46,140 --> 00:25:52,970 I have a big strip, a little bit too big. 437 00:25:52,970 --> 00:25:56,020 So the idea is you have your strip, 438 00:25:56,020 --> 00:25:58,960 you're gone merrily along, and at some point you decide, 439 00:25:58,960 --> 00:26:02,470 oh, I really need to make a 132 degree turn. 440 00:26:02,470 --> 00:26:04,624 I mean, it could be anything because here, we've 441 00:26:04,624 --> 00:26:05,790 got to make some crazy turn. 442 00:26:05,790 --> 00:26:07,510 We don't know what the amount is. 443 00:26:07,510 --> 00:26:09,480 So the turn gadget is always the same. 444 00:26:09,480 --> 00:26:12,659 You start out by making a mountain fold perpendicular 445 00:26:12,659 --> 00:26:13,200 to the strip. 446 00:26:16,340 --> 00:26:17,985 So just fold in half, so to speak. 447 00:26:17,985 --> 00:26:19,485 Of course, half of this is connected 448 00:26:19,485 --> 00:26:21,220 to everything you built already. 449 00:26:21,220 --> 00:26:22,970 The other half, you don't know. 450 00:26:22,970 --> 00:26:27,010 And then you take the back layer and fold it to whatever angle 451 00:26:27,010 --> 00:26:27,510 you want. 452 00:26:27,510 --> 00:26:31,740 You have this degree of freedom wherever you want to put it. 453 00:26:31,740 --> 00:26:34,370 So for example, if we wanted to make an obtuse angle. 454 00:26:34,370 --> 00:26:37,290 I'm always going to fold incident to that bottom corner. 455 00:26:40,170 --> 00:26:43,340 It's a little hard to handle, but you get the idea. 456 00:26:43,340 --> 00:26:47,160 That's a nice, perfect turn right there. 457 00:26:47,160 --> 00:26:49,080 Looks even cooler on my side. 458 00:26:49,080 --> 00:26:52,790 You just see the angular bisector or something. 459 00:26:52,790 --> 00:26:54,770 So that's a decent obtuse angle. 460 00:26:54,770 --> 00:26:58,450 Things are little bit uglier when you make an acute angle, 461 00:26:58,450 --> 00:27:01,110 something like this. 462 00:27:01,110 --> 00:27:03,290 You could make whatever angle you want. 463 00:27:03,290 --> 00:27:04,790 You should have enough strip to make 464 00:27:04,790 --> 00:27:09,060 several turns if you feel like it. 465 00:27:09,060 --> 00:27:12,122 If I make an acute angle like this, I get some overhang. 466 00:27:12,122 --> 00:27:14,330 So I have what I want, which is the strip coming here 467 00:27:14,330 --> 00:27:17,920 and the strip coming here at the extremes, 468 00:27:17,920 --> 00:27:20,980 but over in this corner I have some extra material, 469 00:27:20,980 --> 00:27:23,120 and it could be a lot. 470 00:27:23,120 --> 00:27:25,320 But whatever it is, I'll just rid of it 471 00:27:25,320 --> 00:27:27,610 by folding it underneath repeatedly. 472 00:27:27,610 --> 00:27:30,390 In this case, one fold, it's gone. 473 00:27:30,390 --> 00:27:32,270 In general, when I make that fold, 474 00:27:32,270 --> 00:27:36,220 it might come out over here and I have to wrap around that cone 475 00:27:36,220 --> 00:27:38,840 several times. 476 00:27:38,840 --> 00:27:41,680 There's probably an example of that here. 477 00:27:41,680 --> 00:27:43,850 This figure in the middle version, 478 00:27:43,850 --> 00:27:47,090 you start just the same by folding a super acute angle. 479 00:27:47,090 --> 00:27:49,740 I fold that corner over and it overhangs a little bit 480 00:27:49,740 --> 00:27:51,690 on the bottom, but it got tinier. 481 00:27:51,690 --> 00:27:53,440 And if you just wrap it around the corner, 482 00:27:53,440 --> 00:27:56,330 eventually it will disappear and you have exactly what you want, 483 00:27:56,330 --> 00:27:58,538 which is a strip coming horizontally and then a strip 484 00:27:58,538 --> 00:28:02,230 coming off diagonally at whatever angle you wanted. 485 00:28:02,230 --> 00:28:03,624 So that's really easy. 486 00:28:03,624 --> 00:28:05,040 That's what we call a turn gadget. 487 00:28:13,680 --> 00:28:15,915 So we do this using turn gadgets. 488 00:28:19,970 --> 00:28:23,020 Now, you can't actually turn with 180 degree angle here, 489 00:28:23,020 --> 00:28:25,050 which is why in this picture, what I 490 00:28:25,050 --> 00:28:30,210 drew was a 90 degree turn and then another 90 degree turn. 491 00:28:30,210 --> 00:28:32,320 You can do that however you want. 492 00:28:32,320 --> 00:28:33,930 That's one way. 493 00:28:33,930 --> 00:28:37,390 In fact, the way I've shown it, it would be orthogonal, 494 00:28:37,390 --> 00:28:41,200 but doesn't really matter. 495 00:28:41,200 --> 00:28:44,350 These turns are not much harder. 496 00:28:44,350 --> 00:28:47,310 You just do one or two of those turns to get aligned. 497 00:28:47,310 --> 00:28:50,510 And you can decide here, do I turn around to the left 498 00:28:50,510 --> 00:28:54,409 or do I turn around to the right, and then I zigzag? 499 00:28:54,409 --> 00:28:56,450 Depending on which way I go, I will end up either 500 00:28:56,450 --> 00:28:58,950 at this corner or this corner, and if my next triangle's 501 00:28:58,950 --> 00:29:01,350 up here, then I want to end up at this corner. 502 00:29:01,350 --> 00:29:03,330 If my next triangle's over here, then I 503 00:29:03,330 --> 00:29:06,140 want to end up at this corner. 504 00:29:06,140 --> 00:29:08,190 You plan out ahead of time what order you're 505 00:29:08,190 --> 00:29:10,840 going to visit the triangles, just decide. 506 00:29:10,840 --> 00:29:13,090 Technically, it's like the traveling salesman problem, 507 00:29:13,090 --> 00:29:15,120 but you don't have to be efficient at the moment. 508 00:29:15,120 --> 00:29:16,630 We're not worrying about efficiency. 509 00:29:16,630 --> 00:29:19,100 Just visit triangle after triangle, keep going to one 510 00:29:19,100 --> 00:29:21,260 you haven't visited before. 511 00:29:21,260 --> 00:29:22,510 It could be far away from you. 512 00:29:22,510 --> 00:29:24,230 Maybe you have to go over many triangles you visited 513 00:29:24,230 --> 00:29:25,460 before, but eventually you'll get to one 514 00:29:25,460 --> 00:29:27,900 you haven't visited until you're done, and then boom, 515 00:29:27,900 --> 00:29:29,630 you've made anything. 516 00:29:29,630 --> 00:29:31,710 Amazing. 517 00:29:31,710 --> 00:29:33,830 We haven't quite proved the theorem. 518 00:29:33,830 --> 00:29:38,270 At this point, we can make any connected union of polygons. 519 00:29:38,270 --> 00:29:40,110 This is one polygon, but if you're 520 00:29:40,110 --> 00:29:44,650 in 3D, when you cross over from one polygon to the next, 521 00:29:44,650 --> 00:29:47,464 you might have to bend it at some angle that's not flat, 522 00:29:47,464 --> 00:29:49,130 but then you can make any 3D thing, too. 523 00:29:49,130 --> 00:29:49,857 Yeah? 524 00:29:49,857 --> 00:29:52,425 AUDIENCE: When you wrap around the triangle whose edge type 525 00:29:52,425 --> 00:29:54,760 is the bottom, how do you do that when 526 00:29:54,760 --> 00:29:56,400 the top left triangle is there? 527 00:29:56,400 --> 00:29:57,270 PROFESSOR: Sorry. 528 00:29:57,270 --> 00:29:58,971 When I wrap around here? 529 00:29:58,971 --> 00:30:01,572 AUDIENCE: No, the triangle whose edge is the bottom edge. 530 00:30:01,572 --> 00:30:02,405 PROFESSOR: This one? 531 00:30:02,405 --> 00:30:03,030 AUDIENCE: Yeah. 532 00:30:03,030 --> 00:30:04,919 How do you do that, given that there's 533 00:30:04,919 --> 00:30:07,620 the triangle in the top left? 534 00:30:07,620 --> 00:30:10,444 PROFESSOR: Oh, I see. 535 00:30:10,444 --> 00:30:11,860 The question is, what do I do here 536 00:30:11,860 --> 00:30:14,050 because I'm going to overlap into this triangle again? 537 00:30:14,050 --> 00:30:14,716 AUDIENCE: Right. 538 00:30:14,716 --> 00:30:15,390 PROFESSOR: Yeah. 539 00:30:15,390 --> 00:30:17,223 So you have to be a little bit careful here, 540 00:30:17,223 --> 00:30:19,890 and there's one other gadget I need to mention here, 541 00:30:19,890 --> 00:30:24,090 which is the hide gadget, I think it's called. 542 00:30:27,477 --> 00:30:29,310 Because I didn't fold exactly this triangle, 543 00:30:29,310 --> 00:30:31,180 I folded too much. 544 00:30:31,180 --> 00:30:35,040 At this point when I have the strip here, and maybe a little 545 00:30:35,040 --> 00:30:37,140 bit out of the way, I'd like to get rid 546 00:30:37,140 --> 00:30:40,470 of all this messy stuff, just tuck it underneath. 547 00:30:40,470 --> 00:30:44,220 So I'm going to fold along this line, fold all this stuff back 548 00:30:44,220 --> 00:30:48,300 behind, and that might still be a mess because it could stick 549 00:30:48,300 --> 00:30:50,540 out over here or somewhere else. 550 00:30:50,540 --> 00:30:53,260 So let me tell you about the hide gadget a little bit, 551 00:30:53,260 --> 00:30:54,340 maybe over here. 552 00:31:02,030 --> 00:31:04,950 You have to be a little careful when you do that exactly, 553 00:31:04,950 --> 00:31:07,000 but it's not that hard. 554 00:31:13,000 --> 00:31:23,770 So in general, the hide gadget is 555 00:31:23,770 --> 00:31:26,300 you have some convex polygon. 556 00:31:26,300 --> 00:31:28,870 Here, it's always going to be a triangle, actually, 557 00:31:28,870 --> 00:31:31,610 but this would work for any convex polygon. 558 00:31:31,610 --> 00:31:35,020 You have some folding which contains that polygon, 559 00:31:35,020 --> 00:31:37,180 but it might go outside all over the place. 560 00:31:37,180 --> 00:31:38,519 That's our situation. 561 00:31:38,519 --> 00:31:39,435 We want this triangle. 562 00:31:39,435 --> 00:31:41,320 We have a folding that includes the triangle 563 00:31:41,320 --> 00:31:44,290 but it has all this junk on the outside. 564 00:31:44,290 --> 00:31:47,460 All you need to do is repeatedly fold along 565 00:31:47,460 --> 00:31:50,590 the extension of each of these edges, 566 00:31:50,590 --> 00:31:53,180 and you can prove-- I'm not going to do it here 567 00:31:53,180 --> 00:31:55,100 because it's a little bit messy-- 568 00:31:55,100 --> 00:31:57,960 but as you fold that thing, obviously 569 00:31:57,960 --> 00:32:00,640 the area that you go outside gets smaller, 570 00:32:00,640 --> 00:32:03,440 and you do make steady progress and eventually, 571 00:32:03,440 --> 00:32:05,817 after finitely many folds, you will have reduced 572 00:32:05,817 --> 00:32:07,400 this thing down to the convex polygon. 573 00:32:07,400 --> 00:32:10,570 Just keep folding mountain fold along each of those edges. 574 00:32:13,280 --> 00:32:16,524 And I think there's even a pseudo polynomial bound 575 00:32:16,524 --> 00:32:18,190 on the number of folds you need to make, 576 00:32:18,190 --> 00:32:20,690 although I won't go into that here. 577 00:32:24,032 --> 00:32:25,740 So that's what you do with this triangle. 578 00:32:25,740 --> 00:32:30,940 Then you fold the next one, and then you apply the hide gadget 579 00:32:30,940 --> 00:32:32,110 to that triangle. 580 00:32:32,110 --> 00:32:36,562 When you do that, there is this triangle in the way. 581 00:32:36,562 --> 00:32:38,270 Technically, I guess, that wouldn't quite 582 00:32:38,270 --> 00:32:41,100 be simple folds because you're folding along this line. 583 00:32:41,100 --> 00:32:43,600 You need to tuck it in between the layers. 584 00:32:43,600 --> 00:32:45,870 You have to avoid collision with this triangle 585 00:32:45,870 --> 00:32:47,990 and go underneath this one. 586 00:32:47,990 --> 00:32:48,490 Hm. 587 00:32:48,490 --> 00:32:50,305 I hadn't thought about that. 588 00:32:50,305 --> 00:32:52,180 I'd like to say this works with simple folds. 589 00:32:56,070 --> 00:32:58,340 It sort of does in that each fold you make, 590 00:32:58,340 --> 00:33:00,490 you're just folding along one line at a time, 591 00:33:00,490 --> 00:33:03,560 but you do actually collide in the middle. 592 00:33:03,560 --> 00:33:05,100 It's not so ideal. 593 00:33:05,100 --> 00:33:09,050 Origami folds, this definitely works. 594 00:33:09,050 --> 00:33:11,820 I wonder if there's a way to do it with simple folds now. 595 00:33:11,820 --> 00:33:12,320 Interesting. 596 00:33:15,916 --> 00:33:16,416 Hm. 597 00:33:16,416 --> 00:33:17,392 I don't know. 598 00:33:17,392 --> 00:33:21,296 AUDIENCE: Can you undo the corner gadget 599 00:33:21,296 --> 00:33:24,224 to separate the triangles, hide the folds, 600 00:33:24,224 --> 00:33:26,955 and then put them back? 601 00:33:26,955 --> 00:33:29,450 PROFESSOR: The question is, can I pull them apart, do them 602 00:33:29,450 --> 00:33:31,130 separately, and push them back together? 603 00:33:31,130 --> 00:33:33,588 Pulling them apart and pushing them back together would not 604 00:33:33,588 --> 00:33:34,850 be simple folds, I think. 605 00:33:34,850 --> 00:33:35,820 Maybe you could do it. 606 00:33:35,820 --> 00:33:37,403 AUDIENCE: You just type a [INAUDIBLE]. 607 00:33:37,403 --> 00:33:38,430 PROFESSOR: Yeah, OK. 608 00:33:38,430 --> 00:33:39,215 Interesting. 609 00:33:39,215 --> 00:33:40,340 We should think about that. 610 00:33:40,340 --> 00:33:42,950 I'm surprised this is not known, but I'm pretty sure 611 00:33:42,950 --> 00:33:46,190 now that it's not known. 612 00:33:46,190 --> 00:33:47,750 Maybe first open problem session, 613 00:33:47,750 --> 00:33:50,157 we can solve that in a few minutes. 614 00:33:50,157 --> 00:33:50,865 Should be doable. 615 00:33:56,700 --> 00:33:58,630 There's one more gadget we need if we 616 00:33:58,630 --> 00:34:01,760 want to get the two color pattern. 617 00:34:05,260 --> 00:34:07,390 This polygon maybe is all one color. 618 00:34:07,390 --> 00:34:10,260 Maybe it's all the colored side of the paper. 619 00:34:10,260 --> 00:34:13,489 But then at some point, we might transition over 620 00:34:13,489 --> 00:34:19,747 to another polygon, and maybe that's the other color. 621 00:34:19,747 --> 00:34:21,830 So all we're going to do is we're, again, visiting 622 00:34:21,830 --> 00:34:23,290 these triangles in some order. 623 00:34:23,290 --> 00:34:27,370 Whenever we go between a black triangle and a white triangle, 624 00:34:27,370 --> 00:34:30,630 we are going to apply a color reversal 625 00:34:30,630 --> 00:34:37,089 gadget, which is very simple. 626 00:34:42,280 --> 00:34:44,989 I forgot to color one side of the strip, 627 00:34:44,989 --> 00:34:50,500 so bear with me for a second, and feel free to color 628 00:34:50,500 --> 00:34:51,250 in your own strip. 629 00:34:54,687 --> 00:34:56,659 Probably killing this microphone right here. 630 00:35:00,840 --> 00:35:03,070 Purple on one side, white on the other, 631 00:35:03,070 --> 00:35:06,490 although it's so transparent, it looks almost the same. 632 00:35:06,490 --> 00:35:07,960 The idea is very simple. 633 00:35:07,960 --> 00:35:09,620 Where are we starting from? 634 00:35:09,620 --> 00:35:12,970 We're starting from the not purple side. 635 00:35:12,970 --> 00:35:15,130 Then we make a mountain fold. 636 00:35:15,130 --> 00:35:16,920 It's just like a turn gadget. 637 00:35:16,920 --> 00:35:20,320 In fact, it's just like you're turning 90 degrees. 638 00:35:20,320 --> 00:35:24,400 We fold up like this, and then we just immediately make 639 00:35:24,400 --> 00:35:28,840 another 90 degree turn without the first fold, 640 00:35:28,840 --> 00:35:34,960 and we get a sharp transition straight along the strip 641 00:35:34,960 --> 00:35:39,130 from light purple to dark purple, it looks like, 642 00:35:39,130 --> 00:35:43,389 but from the white side to the purple side. 643 00:35:43,389 --> 00:35:45,680 Right when you transition between any pair of triangles 644 00:35:45,680 --> 00:35:48,000 of the opposite color, you just apply this gadget. 645 00:35:48,000 --> 00:35:49,680 You make three more folds, and boom, 646 00:35:49,680 --> 00:35:52,287 you've got your color reversal. 647 00:35:52,287 --> 00:35:53,620 AUDIENCE: Can you do that again? 648 00:35:53,620 --> 00:35:53,970 PROFESSOR: Do it again? 649 00:35:53,970 --> 00:35:54,470 All right. 650 00:35:54,470 --> 00:35:56,040 So first I make a mountain fold. 651 00:35:58,740 --> 00:36:00,690 Now both sides are white. 652 00:36:00,690 --> 00:36:05,680 Then I fold this guy up, so now I have a regular 90 degree 653 00:36:05,680 --> 00:36:09,007 turn, and then I fold this guy back. 654 00:36:09,007 --> 00:36:11,590 That's the tricky part because you can't see what's happening. 655 00:36:11,590 --> 00:36:13,330 Let me flip it over for you. 656 00:36:13,330 --> 00:36:20,080 So I'm just folding the top two layers here 657 00:36:20,080 --> 00:36:26,412 along that 45 degree angle, and then I get that transition. 658 00:36:26,412 --> 00:36:28,120 On this side, it's still purple and white 659 00:36:28,120 --> 00:36:29,161 but it's a little uglier. 660 00:36:31,720 --> 00:36:32,600 It's on the slide. 661 00:36:32,600 --> 00:36:35,720 You can try again. 662 00:36:35,720 --> 00:36:38,100 It doesn't say here, but fold just the back layers. 663 00:36:43,720 --> 00:36:44,790 All right? 664 00:36:44,790 --> 00:36:47,070 Questions? 665 00:36:47,070 --> 00:36:50,270 So that's how you can visit all these polygons, 666 00:36:50,270 --> 00:36:54,150 do color transitions, you can do it in 3D, make any polyhedron. 667 00:36:54,150 --> 00:36:56,680 A little crazy, but the point is you 668 00:36:56,680 --> 00:36:59,810 do need to do some algorithmic construction to really see 669 00:36:59,810 --> 00:37:01,410 this method work. 670 00:37:01,410 --> 00:37:02,632 You need those three gadgets. 671 00:37:02,632 --> 00:37:04,340 You need to be sure that you can actually 672 00:37:04,340 --> 00:37:05,589 combine them in the right way. 673 00:37:05,589 --> 00:37:08,680 And I'm hand waving a little bit some of these details. 674 00:37:08,680 --> 00:37:11,330 You should be more careful, but it works. 675 00:37:14,950 --> 00:37:17,612 It's obviously horribly inefficient, 676 00:37:17,612 --> 00:37:19,945 but we can make it at least a little bit more efficient. 677 00:37:19,945 --> 00:37:21,070 Let me tell you about that. 678 00:37:39,640 --> 00:37:54,000 So we can achieve something I call pseudo efficiency, which 679 00:37:54,000 --> 00:38:00,200 is if you are allowed to start with any rectangle 680 00:38:00,200 --> 00:38:10,080 you want, so in particular, if we 681 00:38:10,080 --> 00:38:13,150 start with a long strip of paper like you did here, 682 00:38:13,150 --> 00:38:14,940 then you can make this super efficient. 683 00:38:23,580 --> 00:38:30,260 The area of the piece of paper that we start with 684 00:38:30,260 --> 00:38:40,256 can be equal to the surface area of the thing 685 00:38:40,256 --> 00:38:41,130 that we want to fold. 686 00:38:44,030 --> 00:38:46,390 If we could achieve this, we would be really happy. 687 00:38:46,390 --> 00:38:47,840 That means we have zero wastage. 688 00:38:47,840 --> 00:38:49,810 I can't get zero wastage, but I can 689 00:38:49,810 --> 00:38:52,205 get just a little tiny bit of wastage. 690 00:38:55,280 --> 00:39:00,290 So we write epsilon for a very tiny result, very tiny error. 691 00:39:00,290 --> 00:39:01,791 So epsilon does have to be positive, 692 00:39:01,791 --> 00:39:03,290 we have to have some wastage, but it 693 00:39:03,290 --> 00:39:04,510 can be arbitrarily small. 694 00:39:04,510 --> 00:39:07,850 Basically, the narrower and longer you make your strip, 695 00:39:07,850 --> 00:39:09,600 the closer the area of your piece of paper 696 00:39:09,600 --> 00:39:12,780 gets to the surface area of your target shape, which is the best 697 00:39:12,780 --> 00:39:14,590 you can hope for. 698 00:39:14,590 --> 00:39:17,630 So with this crazy proviso, which is completely impractical 699 00:39:17,630 --> 00:39:21,270 we get a super practical outcome. 700 00:39:21,270 --> 00:39:22,960 And the summation is not very practical, 701 00:39:22,960 --> 00:39:25,439 but it's kind of nifty, and I particularly 702 00:39:25,439 --> 00:39:26,480 like the proof technique. 703 00:39:26,480 --> 00:39:35,650 It uses a very powerful tool in computational geometry, so let 704 00:39:35,650 --> 00:39:36,400 me show it to you. 705 00:39:42,260 --> 00:39:47,400 So the central issue is that while visiting a triangle, 706 00:39:47,400 --> 00:39:48,860 we actually do pretty efficiently. 707 00:39:48,860 --> 00:39:51,390 Except for the very last strip, where 708 00:39:51,390 --> 00:39:55,950 we might overlap ourselves, except for these little corner 709 00:39:55,950 --> 00:39:58,110 turn gadgets, and also the material that's 710 00:39:58,110 --> 00:40:00,407 used up in the gadgets themselves, 711 00:40:00,407 --> 00:40:01,740 We're actually really efficient. 712 00:40:01,740 --> 00:40:04,590 We're covering this triangle almost one for one. 713 00:40:04,590 --> 00:40:06,470 There's a little bit of garbage at the edge 714 00:40:06,470 --> 00:40:09,010 and a little bit of garbage at this edge, 715 00:40:09,010 --> 00:40:11,720 but you could imagine as the strip gets smaller, 716 00:40:11,720 --> 00:40:14,990 this wastage gets smaller, because it's essentially 717 00:40:14,990 --> 00:40:18,120 this length times the width of the strip up 718 00:40:18,120 --> 00:40:19,400 to constant factors. 719 00:40:19,400 --> 00:40:21,810 So as the strip gets narrower, it gets closer and closer 720 00:40:21,810 --> 00:40:24,834 to this thing times 0, so this wastage 721 00:40:24,834 --> 00:40:26,000 gets closer and closer to 0. 722 00:40:26,000 --> 00:40:28,660 Same thing over here, same thing over here. 723 00:40:28,660 --> 00:40:30,290 The real problem at this point is 724 00:40:30,290 --> 00:40:33,460 that we visit some triangles possibly many times. 725 00:40:33,460 --> 00:40:37,210 I was totally free about how I visited triangles. 726 00:40:37,210 --> 00:40:40,690 Maybe I go here and then over here and then back over here. 727 00:40:40,690 --> 00:40:42,510 I may be very wasteful. 728 00:40:42,510 --> 00:40:44,580 So I just want to avoid that wastage. 729 00:40:44,580 --> 00:40:47,795 Just visit each triangle once, exactly once. 730 00:40:47,795 --> 00:40:50,880 How could you do it exactly once? 731 00:40:50,880 --> 00:40:54,680 With a great idea called Hamiltonian refinement. 732 00:40:57,980 --> 00:41:00,000 I've used this idea many times. 733 00:41:00,000 --> 00:41:03,026 This was the first time I used it. 734 00:41:03,026 --> 00:41:04,330 It's very cool. 735 00:41:04,330 --> 00:41:07,280 I think it goes back to '97 or something. 736 00:41:07,280 --> 00:41:10,365 So you have some triangulation. 737 00:41:14,830 --> 00:41:17,907 I'm going to keep it relatively simple. 738 00:41:17,907 --> 00:41:20,240 And I'd really like to visit each triangle exactly once. 739 00:41:20,240 --> 00:41:21,656 So you think, OK, maybe I'll start 740 00:41:21,656 --> 00:41:25,880 here, then I'll go over here, and over here, and uh oh. 741 00:41:25,880 --> 00:41:27,550 Then I'm stuck. 742 00:41:27,550 --> 00:41:29,290 I can't get to the other triangles 743 00:41:29,290 --> 00:41:33,260 without revisiting an existing triangle. 744 00:41:33,260 --> 00:41:35,630 It's impossible to find what we call a Hamiltonian path. 745 00:41:35,630 --> 00:41:38,690 Hamiltonian path will visit each triangle exactly once. 746 00:41:38,690 --> 00:41:40,866 But who cares about these triangles? 747 00:41:40,866 --> 00:41:42,740 I just made these triangles to make it easier 748 00:41:42,740 --> 00:41:44,170 to think about a polygon. 749 00:41:44,170 --> 00:41:45,670 I could actually cut these triangles 750 00:41:45,670 --> 00:41:50,610 into smaller triangles and ideally make it Hamiltonian. 751 00:41:50,610 --> 00:41:53,940 So here's the idea. 752 00:41:53,940 --> 00:41:57,850 There's this idea called the dual graph, which is also 753 00:41:57,850 --> 00:42:01,020 pretty central in computational geometry, 754 00:42:01,020 --> 00:42:04,180 where I'm going to make a little dot for every triangle 755 00:42:04,180 --> 00:42:06,610 inside each triangle, and then I'm 756 00:42:06,610 --> 00:42:08,720 going to connect those dots whenever 757 00:42:08,720 --> 00:42:11,430 there is two triangles that share an edge. 758 00:42:15,620 --> 00:42:17,960 This is called the dual graph. 759 00:42:17,960 --> 00:42:22,380 These parts are called the dual graph of the triangulation. 760 00:42:22,380 --> 00:42:24,130 And in this picture, I've got a nice, kind 761 00:42:24,130 --> 00:42:25,880 of tree-shaped graph. 762 00:42:25,880 --> 00:42:27,980 In general, you might get cycles, but then 763 00:42:27,980 --> 00:42:29,966 just throw away edges until you get a tree. 764 00:42:29,966 --> 00:42:31,590 I really like to think of it as a tree, 765 00:42:31,590 --> 00:42:35,210 and if I throw in enough edges, it will be a tree. 766 00:42:35,210 --> 00:42:40,920 Then my idea is I'd really like to just visit everything 767 00:42:40,920 --> 00:42:44,850 in this order, walking around the tree. 768 00:42:47,990 --> 00:42:49,500 I can go around in a cycle and visit 769 00:42:49,500 --> 00:42:52,474 each of these little polygons exactly once. 770 00:42:52,474 --> 00:42:54,015 Now, these polygons aren't triangles, 771 00:42:54,015 --> 00:42:57,790 which is kind of annoying, so I just triangulate. 772 00:42:57,790 --> 00:43:03,520 So you could just triangulate like this, I guess. 773 00:43:03,520 --> 00:43:07,930 If you want to be excessive and simpler, 774 00:43:07,930 --> 00:43:10,320 from every one of these dots, you 775 00:43:10,320 --> 00:43:12,710 can cut to the midpoint of the three edges 776 00:43:12,710 --> 00:43:14,740 and also cut to the vertices. 777 00:43:14,740 --> 00:43:17,090 For every triangle, I just cut to the three vertices 778 00:43:17,090 --> 00:43:20,320 and cut to the midpoints. 779 00:43:20,320 --> 00:43:24,160 You do that, and you are guaranteed. 780 00:43:24,160 --> 00:43:27,620 This gets a little messy, but this is theoretical. 781 00:43:30,590 --> 00:43:34,160 You will increase the number of triangles by a factor of six. 782 00:43:34,160 --> 00:43:38,060 Each triangle gets replaced by six little smaller triangles. 783 00:43:38,060 --> 00:43:40,980 It covers the same region, and yet now it's Hamiltonian. 784 00:43:40,980 --> 00:43:43,770 Now I can visit these triangles and visit each triangle exactly 785 00:43:43,770 --> 00:43:47,010 once, and that's how you get pseudo efficiency. 786 00:43:47,010 --> 00:43:50,231 Pretty easy once you know this trick. 787 00:43:50,231 --> 00:43:50,730 Questions? 788 00:43:53,932 --> 00:43:55,840 AUDIENCE: Why is it called pseudo efficiency? 789 00:43:55,840 --> 00:43:57,390 PROFESSOR: Well, I made that up. 790 00:43:57,390 --> 00:43:59,420 Why is it called pseudo efficiency? 791 00:43:59,420 --> 00:44:02,550 Because it's not really efficient. 792 00:44:02,550 --> 00:44:05,960 It is efficient, I agree, but only 793 00:44:05,960 --> 00:44:07,930 with this crazy assumption that you're 794 00:44:07,930 --> 00:44:10,430 allowed to start with an arbitrarily narrow strip, 795 00:44:10,430 --> 00:44:12,850 and that's not really practical. 796 00:44:12,850 --> 00:44:15,610 The practical origami problem is I start with a square. 797 00:44:15,610 --> 00:44:20,070 How big of a square do I need to fold this shape? 798 00:44:20,070 --> 00:44:25,000 And that's something we'll talk about probably next week. 799 00:44:25,000 --> 00:44:27,320 That's been studied but it's a lot harder, 800 00:44:27,320 --> 00:44:30,115 so I'm starting with the easy stuff. 801 00:44:30,115 --> 00:44:32,698 AUDIENCE: Does this necessarily keep the mountains and valleys 802 00:44:32,698 --> 00:44:34,330 that you defined at the beginning? 803 00:44:34,330 --> 00:44:36,871 PROFESSOR: Will this keep the mountains and valleys the same? 804 00:44:38,962 --> 00:44:41,420 The input didn't specify where the mountains valleys should 805 00:44:41,420 --> 00:44:41,920 be. 806 00:44:41,920 --> 00:44:45,370 It just said you have to make these polygons. 807 00:44:45,370 --> 00:44:49,680 Every fold you make in this construction will stay, 808 00:44:49,680 --> 00:44:52,904 but you may make more folds that go on top of existing folds. 809 00:44:52,904 --> 00:44:55,320 I think all the mountains and valleys that you define here 810 00:44:55,320 --> 00:44:58,710 will remain mountains or valleys as we specified, but keep 811 00:44:58,710 --> 00:45:02,100 adding creases until you finish. 812 00:45:02,100 --> 00:45:04,100 Yeah? 813 00:45:04,100 --> 00:45:06,850 AUDIENCE: Are you just doing a 2D case 814 00:45:06,850 --> 00:45:09,334 of what we call the 3D pyramid there? 815 00:45:09,334 --> 00:45:09,833 [INAUDIBLE] 816 00:45:13,440 --> 00:45:15,450 PROFESSOR: The question is, what about 3D? 817 00:45:15,450 --> 00:45:18,900 This dual graph idea does work in 3D. 818 00:45:18,900 --> 00:45:22,020 I drew a 2D picture because it's a lot easier to draw, 819 00:45:22,020 --> 00:45:26,067 but you could imagine, for example, you have a cube. 820 00:45:26,067 --> 00:45:27,900 I'm not going to be able to draw everything, 821 00:45:27,900 --> 00:45:31,810 but you can still form a dual graph. 822 00:45:31,810 --> 00:45:33,365 I should have triangulated the cube, 823 00:45:33,365 --> 00:45:35,490 but in fact, you can do this for any kind of graph. 824 00:45:35,490 --> 00:45:40,277 I'm making a dot for every face of the thing 825 00:45:40,277 --> 00:45:41,110 that I want to make. 826 00:45:41,110 --> 00:45:43,185 Each of these polygons, I make a dot. 827 00:45:43,185 --> 00:45:45,540 So in fact, if I finished this picture, 828 00:45:45,540 --> 00:45:51,940 I would get the dual graph is the octahedron. 829 00:45:51,940 --> 00:45:55,630 It would look something like this topologically. 830 00:45:55,630 --> 00:45:57,940 It doesn't have to be a geometric thing. 831 00:45:57,940 --> 00:45:59,764 It works just as well in 3D. 832 00:45:59,764 --> 00:46:01,430 It actually still works even if you just 833 00:46:01,430 --> 00:46:03,900 say, for every triangle, just cut 834 00:46:03,900 --> 00:46:07,382 to the midpoints of the three edges and to the vertices. 835 00:46:07,382 --> 00:46:08,840 Then that thing will be Hamiltonian 836 00:46:08,840 --> 00:46:11,380 and you can walk around it by doing this dual 837 00:46:11,380 --> 00:46:13,890 and spanning tree construction. 838 00:46:13,890 --> 00:46:15,390 I won't go into it exactly, but that 839 00:46:15,390 --> 00:46:18,440 could be done in linear time. 840 00:46:18,440 --> 00:46:20,440 I'm still only making a two dimensional surface. 841 00:46:20,440 --> 00:46:22,290 I'm not trying to make a 3D solid here, 842 00:46:22,290 --> 00:46:24,990 so that's why locally, it's two dimensional. 843 00:46:24,990 --> 00:46:27,060 Locally, this just works. 844 00:46:27,060 --> 00:46:29,450 Another question? 845 00:46:29,450 --> 00:46:33,602 AUDIENCE: So it seems that if you could also just stick with 846 00:46:33,602 --> 00:46:37,204 the original construction, and when you cannot find 847 00:46:37,204 --> 00:46:41,890 the Hamiltonian path around the triangles, [INAUDIBLE], 848 00:46:41,890 --> 00:46:45,710 you don't cover them again, you just get a narrow strip that 849 00:46:45,710 --> 00:46:48,410 runs through them and just rigidly [INAUDIBLE]? 850 00:46:48,410 --> 00:46:49,660 PROFESSOR: That's a neat idea. 851 00:46:49,660 --> 00:46:51,920 So instead of doing this Hamiltonian refinement, 852 00:46:51,920 --> 00:46:54,320 you could visit the triangles in any order, 853 00:46:54,320 --> 00:46:56,250 but then to get from one triangle 854 00:46:56,250 --> 00:46:58,410 to some distant triangle, instead 855 00:46:58,410 --> 00:47:01,260 of covering all the triangles in the middle which you've already 856 00:47:01,260 --> 00:47:03,900 covered, which is excessive, just go 857 00:47:03,900 --> 00:47:08,249 as straight there as possible. 858 00:47:08,249 --> 00:47:10,540 Locally, you'd think of that as being a one dimensional 859 00:47:10,540 --> 00:47:12,810 wastage instead of a two dimensional wastage. 860 00:47:12,810 --> 00:47:16,520 So probably you can get the same result by that technique. 861 00:47:16,520 --> 00:47:18,130 I hadn't thought of that before. 862 00:47:18,130 --> 00:47:22,780 We should check it, but I think that should work. 863 00:47:22,780 --> 00:47:25,600 This is kind of cooler, just because it uses a fancy tool, 864 00:47:25,600 --> 00:47:28,504 but I imagine with a little more analysis, 865 00:47:28,504 --> 00:47:30,295 you could prove that that works because you 866 00:47:30,295 --> 00:47:31,836 know there aren't too many triangles. 867 00:47:34,420 --> 00:47:34,920 Good. 868 00:47:38,070 --> 00:47:40,370 I think I'm going to skip seem placement 869 00:47:40,370 --> 00:47:42,450 and move on to foldability. 870 00:47:42,450 --> 00:47:44,840 This was a little intro to origami design. 871 00:47:44,840 --> 00:47:47,840 Obviously not practical yet, but it's something. 872 00:47:51,576 --> 00:47:52,575 It's nice and universal. 873 00:47:52,575 --> 00:47:53,625 You can make everything. 874 00:47:56,710 --> 00:47:59,230 So let me tell you a little bit about the other side 875 00:47:59,230 --> 00:48:02,900 of the world, which is, instead of a target shape, 876 00:48:02,900 --> 00:48:04,580 what if I gave you a crease pattern? 877 00:48:04,580 --> 00:48:08,790 We just want to know, does this crease pattern fold? 878 00:48:08,790 --> 00:48:14,100 I give you the crease pattern of that pinwheel. 879 00:48:14,100 --> 00:48:15,880 I say, does this fold into anything? 880 00:48:15,880 --> 00:48:17,860 Does it fold into a flat origami? 881 00:48:17,860 --> 00:48:19,670 Does it fold into something else? 882 00:48:19,670 --> 00:48:21,700 Can you fold it with simple folds? 883 00:48:21,700 --> 00:48:23,540 Can you fold it with origami folds? 884 00:48:23,540 --> 00:48:25,560 Those are all interesting questions. 885 00:48:25,560 --> 00:48:31,290 We're going to start with a very simple form of it, which 886 00:48:31,290 --> 00:48:34,880 is when the piece of paper is one dimensional, which 887 00:48:34,880 --> 00:48:37,750 is almost what you were thinking of here. 888 00:48:37,750 --> 00:48:40,390 It's like strip folding, but I'm going to furthermore require 889 00:48:40,390 --> 00:48:43,590 that all the folds I make are perpendicular to the strip 890 00:48:43,590 --> 00:48:45,140 direction. 891 00:48:45,140 --> 00:48:51,160 So I'm always going to fold like this and then like this. 892 00:48:51,160 --> 00:48:53,320 So it's going to remain a strip. 893 00:48:53,320 --> 00:48:55,120 It'll just be a smaller strip. 894 00:48:55,120 --> 00:48:57,300 I know it seems kind of boring, but it's 895 00:48:57,300 --> 00:48:58,970 something we understand really well, 896 00:48:58,970 --> 00:49:03,030 so that's why I'm going to tell you about it. 897 00:49:03,030 --> 00:49:05,430 And then it relates to locally what 898 00:49:05,430 --> 00:49:07,550 happens at one vertex of the crease pattern, 899 00:49:07,550 --> 00:49:12,250 but we'll get to that next class. 900 00:49:12,250 --> 00:49:16,290 So one dimensional flat folding. 901 00:49:16,290 --> 00:49:18,000 I don't have them anymore here, but I'm 902 00:49:18,000 --> 00:49:20,300 going to redefine "piece of paper" and all these things 903 00:49:20,300 --> 00:49:22,730 because I want to be even simpler. 904 00:49:26,830 --> 00:49:30,481 So piece of paper now is going to be a one dimensional line 905 00:49:30,481 --> 00:49:30,980 segment. 906 00:49:36,300 --> 00:49:43,440 A crease is going to be a point on that segment. 907 00:49:43,440 --> 00:49:45,697 In general, a crease is one dimension smaller 908 00:49:45,697 --> 00:49:47,530 than the piece of paper you're working with. 909 00:49:47,530 --> 00:49:48,720 You could actually define "piece of paper" 910 00:49:48,720 --> 00:49:51,070 to be a five dimensional solid if you wanted. 911 00:49:51,070 --> 00:49:53,130 Then creases would be four dimensional flats, 912 00:49:53,130 --> 00:49:54,820 they're called. 913 00:49:54,820 --> 00:49:55,790 It's not a hyperplane. 914 00:49:55,790 --> 00:49:57,410 Four dimensional flat. 915 00:49:57,410 --> 00:50:00,660 That's what they're called. 916 00:50:00,660 --> 00:50:03,080 Or if you wanted to do curved creases in five dimensions, 917 00:50:03,080 --> 00:50:07,492 then it's a four dimensional surface somewhere in there. 918 00:50:07,492 --> 00:50:11,670 We won't do 5D origami in this class. 919 00:50:11,670 --> 00:50:15,980 Flat folding now means that your folding lies on a line. 920 00:50:15,980 --> 00:50:17,730 Of course, it will always lie in the plane 921 00:50:17,730 --> 00:50:21,452 if you're folding a line segment in the plane. 922 00:50:21,452 --> 00:50:23,660 So generally, we take a piece of paper one dimension, 923 00:50:23,660 --> 00:50:26,600 we fold it in a space that's one higher dimension, 924 00:50:26,600 --> 00:50:28,794 so we're going to be folding this thing in 2D. 925 00:50:28,794 --> 00:50:30,710 We use increases that are one lower dimension. 926 00:50:30,710 --> 00:50:33,234 Here, zero dimensional points. 927 00:50:33,234 --> 00:50:35,150 If we want it to be flat, it should in the end 928 00:50:35,150 --> 00:50:37,191 lie in the same dimension the piece of paper did. 929 00:50:40,060 --> 00:50:42,010 It's great because I can just draw. 930 00:50:42,010 --> 00:50:43,580 There's my piece of paper. 931 00:50:43,580 --> 00:50:46,470 Here are some creases. 932 00:50:46,470 --> 00:50:47,870 They're spread out however. 933 00:50:47,870 --> 00:50:49,890 Maybe they're marked mountain and valley, 934 00:50:49,890 --> 00:50:53,080 and I'm not going to use dashes and dot dashes here 935 00:50:53,080 --> 00:50:56,070 because they're points, so I'll write 936 00:50:56,070 --> 00:50:57,960 m's and v's, something like that. 937 00:50:57,960 --> 00:50:59,910 And I didn't check, but this either 938 00:50:59,910 --> 00:51:01,625 will be flat foldable or not. 939 00:51:06,940 --> 00:51:09,680 I think it is. 940 00:51:09,680 --> 00:51:10,960 How do I know that it is? 941 00:51:10,960 --> 00:51:13,430 Because we have a great characterization that tells you 942 00:51:13,430 --> 00:51:16,289 when a mountain valley pattern is flat foldable. 943 00:51:16,289 --> 00:51:17,830 Now, the first question you might ask 944 00:51:17,830 --> 00:51:20,400 is, what about crease patterns if I 945 00:51:20,400 --> 00:51:22,980 don't specify mountains and valleys? 946 00:51:22,980 --> 00:51:25,610 Then you can always fold anything. 947 00:51:25,610 --> 00:51:28,720 If I just gave you a segment and I put creases 948 00:51:28,720 --> 00:51:34,170 in some crazy pattern, all I do is alternate mountain, 949 00:51:34,170 --> 00:51:37,110 valley, mountain, valley, mountain, valley, mountain, 950 00:51:37,110 --> 00:51:40,910 valley, and I'm going to regret making such a big example. 951 00:51:40,910 --> 00:51:47,800 And I just go mountain, valley, mountain-- it alternates. 952 00:51:47,800 --> 00:51:49,450 You get some zigzag. 953 00:51:49,450 --> 00:51:52,300 And this thing is not going to collide with itself. 954 00:51:52,300 --> 00:51:53,410 It just works. 955 00:51:53,410 --> 00:51:55,740 In fact, you could even fold it one step at a time left 956 00:51:55,740 --> 00:51:58,140 to right by simple folds, and you'll 957 00:51:58,140 --> 00:52:00,889 get a nice flat folding of that thing. 958 00:52:00,889 --> 00:52:02,430 So every crease pattern can be folded 959 00:52:02,430 --> 00:52:04,840 with some mountain-valley assignment, namely 960 00:52:04,840 --> 00:52:08,560 the alternating accordion pleat, but what if I give you 961 00:52:08,560 --> 00:52:11,920 the mountain-valley assignment? 962 00:52:11,920 --> 00:52:16,500 Sometimes I can do it, sometimes I can't. 963 00:52:16,500 --> 00:52:23,260 So here's an example where you can't, two valleys 964 00:52:23,260 --> 00:52:26,160 with some really big segments on either side. 965 00:52:26,160 --> 00:52:29,530 So when you try to fold that, it looks something like that, 966 00:52:29,530 --> 00:52:31,700 and if one of these was really short, one of them 967 00:52:31,700 --> 00:52:34,610 could tuck inside the other, but because they're both long, 968 00:52:34,610 --> 00:52:36,657 they have to collide. 969 00:52:36,657 --> 00:52:38,490 That's the sort of thing we're worried about 970 00:52:38,490 --> 00:52:40,948 and we'd like to detect that efficiently with an algorithm. 971 00:52:55,510 --> 00:52:57,020 So let me tell you how. 972 00:52:57,020 --> 00:53:08,840 We're going to use two different operations, 973 00:53:08,840 --> 00:53:10,760 and both of these operations will, in fact, 974 00:53:10,760 --> 00:53:15,000 be performable using simple folds. 975 00:53:15,000 --> 00:53:16,500 What we're going to end up proving 976 00:53:16,500 --> 00:53:19,160 is that a one dimensional mountain valley 977 00:53:19,160 --> 00:53:22,320 pattern is flat foldable if and only if it's flat 978 00:53:22,320 --> 00:53:25,060 foldable via a sequence of simple folds. 979 00:53:25,060 --> 00:53:28,490 So these simple folds that we've been arguing about 980 00:53:28,490 --> 00:53:32,110 are universally powerful for 1D paper. 981 00:53:32,110 --> 00:53:34,200 They are not universally powerful for 2D paper 982 00:53:34,200 --> 00:53:37,620 because you cannot fold this pinwheel with simple folds. 983 00:53:37,620 --> 00:53:39,680 There's no first fold to make. 984 00:53:39,680 --> 00:53:42,680 But in one dimension, turns out there is. 985 00:53:42,680 --> 00:53:47,050 It's not obvious, but it's true. 986 00:53:47,050 --> 00:53:50,480 And here, we're going to make two very simple kinds 987 00:53:50,480 --> 00:53:52,070 of simple folds. 988 00:53:52,070 --> 00:53:54,650 The first one is called an end fold, which 989 00:53:54,650 --> 00:53:59,150 is you take the last, or the very first crease, 990 00:53:59,150 --> 00:54:03,490 and you just fold it over. 991 00:54:03,490 --> 00:54:04,410 That's an n fold. 992 00:54:04,410 --> 00:54:09,540 Now, for this to be valid in the simple folding world, 993 00:54:09,540 --> 00:54:12,940 to not get stuck, I'm going to require 994 00:54:12,940 --> 00:54:20,100 that this portion has no creases inside it. 995 00:54:20,100 --> 00:54:22,360 So it could have a crease right at the boundary, 996 00:54:22,360 --> 00:54:25,660 but there are no creases in that region. 997 00:54:25,660 --> 00:54:28,730 So when I fold this thing over, I don't cover anything up. 998 00:54:28,730 --> 00:54:31,670 No creases get covered up. 999 00:54:31,670 --> 00:54:34,840 This in some sense is clearly a good thing to do. 1000 00:54:34,840 --> 00:54:36,069 This may not exist. 1001 00:54:36,069 --> 00:54:38,110 If you find such a crease and you make this fold, 1002 00:54:38,110 --> 00:54:39,950 now you can think of this piece of paper 1003 00:54:39,950 --> 00:54:41,330 as sort of glued up there. 1004 00:54:41,330 --> 00:54:44,680 You can think of just you cut off one part of the paper. 1005 00:54:44,680 --> 00:54:47,540 You made a problem even easier. 1006 00:54:47,540 --> 00:54:49,705 So this is clearly a safe thing to do. 1007 00:54:49,705 --> 00:54:51,540 It should make your folding easier, 1008 00:54:51,540 --> 00:54:53,580 but usually it doesn't exist. 1009 00:54:53,580 --> 00:54:55,150 In that case, we make what's called 1010 00:54:55,150 --> 00:55:00,270 a crimp, which is we have our segment. 1011 00:55:00,270 --> 00:55:03,890 We take two creases with different mountain-valley 1012 00:55:03,890 --> 00:55:04,410 assignments. 1013 00:55:04,410 --> 00:55:06,576 Maybe the first one's mountain, second one's valley, 1014 00:55:06,576 --> 00:55:09,890 but they're different, and then we fold those two creases. 1015 00:55:12,890 --> 00:55:14,180 Something like that. 1016 00:55:14,180 --> 00:55:16,056 So the paper get smaller again, but here it's 1017 00:55:16,056 --> 00:55:18,346 a little less clear what's happening because there were 1018 00:55:18,346 --> 00:55:20,050 creases over here, creases over here. 1019 00:55:20,050 --> 00:55:21,390 They got closer to each other. 1020 00:55:21,390 --> 00:55:22,670 Maybe that's a problem. 1021 00:55:22,670 --> 00:55:23,640 Maybe it messes you up. 1022 00:55:23,640 --> 00:55:24,515 Turns out it doesn't. 1023 00:55:27,310 --> 00:55:31,760 The other thing, again, we require, if this is distance x, 1024 00:55:31,760 --> 00:55:36,750 we require that this region of length x and this region 1025 00:55:36,750 --> 00:55:38,384 of length x is empty of creases. 1026 00:55:38,384 --> 00:55:39,300 There's nothing there. 1027 00:55:39,300 --> 00:55:41,280 You could have things on the boundary 1028 00:55:41,280 --> 00:55:43,870 but nothing interior to here, nothing interior to here, 1029 00:55:43,870 --> 00:55:45,750 and nothing interior to here. 1030 00:55:45,750 --> 00:55:48,460 Just two increases with sort of a nice, safe neighborhood 1031 00:55:48,460 --> 00:55:50,680 around them so that when I make these two 1032 00:55:50,680 --> 00:55:54,550 folds, this whole region, I didn't cover anything up 1033 00:55:54,550 --> 00:55:55,050 in there. 1034 00:55:55,050 --> 00:55:57,250 There's nothing in this Zorro mark. 1035 00:56:00,370 --> 00:56:02,680 That's the definition of an n fold and a crimp. 1036 00:56:02,680 --> 00:56:05,070 Now, maybe these things don't exist. 1037 00:56:05,070 --> 00:56:07,860 I claim if they don't exist, you are not flat foldable 1038 00:56:07,860 --> 00:56:09,260 no matter what you do. 1039 00:56:09,260 --> 00:56:11,080 These are enough to make everything 1040 00:56:11,080 --> 00:56:12,050 that is flat foldable. 1041 00:56:15,172 --> 00:56:16,255 We're going to prove that. 1042 00:56:37,710 --> 00:56:43,870 We're going to characterize flat foldability by saying, 1043 00:56:43,870 --> 00:56:48,989 if I gave you some mountain-valley pattern, 1044 00:56:48,989 --> 00:56:50,655 and I want to know, is it flat foldable? 1045 00:56:53,990 --> 00:56:55,530 Remember, a mountain-valley pattern 1046 00:56:55,530 --> 00:56:59,080 is a crease pattern together with m's and v's written 1047 00:56:59,080 --> 00:57:01,210 on each crease. 1048 00:57:01,210 --> 00:57:03,920 I claim it is flat foldable if and only 1049 00:57:03,920 --> 00:57:19,320 if there is a sequence of crimps and end folds. 1050 00:57:24,299 --> 00:57:26,840 So this is saying that crimps and end folds are all you need. 1051 00:57:26,840 --> 00:57:30,120 In fact, we're going to show that you can make any crimp 1052 00:57:30,120 --> 00:57:32,100 and any end fold, anything that looks valid, 1053 00:57:32,100 --> 00:57:35,220 anything that has these empty regions here, 1054 00:57:35,220 --> 00:57:38,480 just greedily keep doing one after the other 1055 00:57:38,480 --> 00:57:40,980 without regard to what's going to happen in the future. 1056 00:57:40,980 --> 00:57:43,850 If you get stuck, then your thing in the first place 1057 00:57:43,850 --> 00:57:44,812 was not flat foldable. 1058 00:57:44,812 --> 00:57:46,770 If you finish, obviously, it was flat foldable. 1059 00:57:46,770 --> 00:57:48,220 You've folded it. 1060 00:57:48,220 --> 00:57:51,270 So it doesn't matter even what order you do these operations. 1061 00:57:51,270 --> 00:57:54,390 I won't write that here, but we will prove that. 1062 00:57:54,390 --> 00:57:56,600 And there's another characterization 1063 00:57:56,600 --> 00:57:58,160 which we use to prove this one. 1064 00:57:58,160 --> 00:58:00,750 This is the one we care about. 1065 00:58:00,750 --> 00:58:04,920 This one's a little bit easier to think about. 1066 00:58:04,920 --> 00:58:06,700 I call it the mingling property. 1067 00:58:06,700 --> 00:58:09,680 It says mountains and valleys hang out with each other, 1068 00:58:09,680 --> 00:58:10,180 basically. 1069 00:58:14,791 --> 00:58:16,040 That's why I call it mingling. 1070 00:58:19,880 --> 00:58:21,210 So what does it mean? 1071 00:58:32,710 --> 00:58:35,990 If I take a whole bunch of v's in a row, 1072 00:58:35,990 --> 00:58:42,194 or I take a whole bunch m's in a row, as many as I can, 1073 00:58:42,194 --> 00:58:44,610 that means they're surrounded by creases of the other type 1074 00:58:44,610 --> 00:58:47,137 or possibly the edge of the paper. 1075 00:58:47,137 --> 00:58:49,470 But if I take a whole bunch of v's, then the next crease 1076 00:58:49,470 --> 00:58:53,310 and the previous crease are m's, let's say, 1077 00:58:53,310 --> 00:59:03,170 then the adjacent crease of the other type, 1078 00:59:03,170 --> 00:59:08,430 or possibly the end of the paper, 1079 00:59:08,430 --> 00:59:14,470 and I want this to be true on at least one side. 1080 00:59:14,470 --> 00:59:16,220 This is, again, one of those things that's 1081 00:59:16,220 --> 00:59:19,150 much easier in a picture, but I will write down the words 1082 00:59:19,150 --> 00:59:21,530 first. 1083 00:59:21,530 --> 00:59:38,910 Is nearer than the adjacent m or v. Sorry, backwards. 1084 00:59:38,910 --> 00:59:42,560 I'm trying to use parallel construction here 1085 00:59:42,560 --> 00:59:46,491 but I got it inverted for my notes. 1086 00:59:46,491 --> 00:59:47,490 Let me draw the picture. 1087 00:59:47,490 --> 00:59:51,520 You have a segment. 1088 00:59:51,520 --> 00:59:55,990 Suppose I take some maximal sequence of m's all in a row. 1089 00:59:58,750 --> 01:00:01,434 So maybe on one side, maybe the next thing 1090 01:00:01,434 --> 01:00:02,850 here is just the edge of the paper 1091 01:00:02,850 --> 01:00:03,974 so there's no crease there. 1092 01:00:07,650 --> 01:00:11,270 If this was it, then I would say this thing is not 1093 01:00:11,270 --> 01:00:14,930 mingling because this distance is bigger 1094 01:00:14,930 --> 01:00:18,380 than this one and this distance is bigger than this one. 1095 01:00:18,380 --> 01:00:20,210 So these m's are really clustered together. 1096 01:00:20,210 --> 01:00:22,992 They're not mingling with the edges, so that's bad. 1097 01:00:22,992 --> 01:00:24,450 And in fact, this thing is not flat 1098 01:00:24,450 --> 01:00:26,860 foldable for the same reason that one is not. 1099 01:00:26,860 --> 01:00:30,810 But if, for example, I put a valley crease right here, 1100 01:00:30,810 --> 01:00:33,340 then this is closer than that. 1101 01:00:33,340 --> 01:00:35,510 I mean this distance is smaller than that one. 1102 01:00:35,510 --> 01:00:38,170 So I call that mingling because at least on the right side, 1103 01:00:38,170 --> 01:00:39,630 the m's are mingling with the v's. 1104 01:00:39,630 --> 01:00:41,560 On the left side, they're not mingling, 1105 01:00:41,560 --> 01:00:43,400 but right side's better than nothing. 1106 01:00:43,400 --> 01:00:45,490 I need this to hold on at least one side. 1107 01:00:45,490 --> 01:00:47,380 Could hold on both. 1108 01:00:47,380 --> 01:00:52,405 I claim this thing is flat foldable. 1109 01:00:52,405 --> 01:00:55,060 Is that true? 1110 01:00:55,060 --> 01:00:56,490 Yeah. 1111 01:00:56,490 --> 01:00:58,650 Maybe. 1112 01:00:58,650 --> 01:00:59,650 It doesn't look so true. 1113 01:01:09,067 --> 01:01:11,275 I think I probably need some condition on the v also. 1114 01:01:17,250 --> 01:01:19,040 Well, we'll see what happens in the proof. 1115 01:01:19,040 --> 01:01:20,505 That's more exciting. 1116 01:01:20,505 --> 01:01:22,380 Usually the proof is easier than the example. 1117 01:01:37,500 --> 01:01:38,230 Oh, I see why. 1118 01:01:44,140 --> 01:01:45,130 There's a problem here. 1119 01:01:45,130 --> 01:01:49,262 I think I need to say "forever." 1120 01:01:49,262 --> 01:01:50,720 I'll say what I mean by that later. 1121 01:01:54,540 --> 01:01:56,220 The proof will still work, but it's not 1122 01:01:56,220 --> 01:01:57,178 such a pretty property. 1123 01:02:05,370 --> 01:02:10,090 The proof is in three parts, basically. 1124 01:02:10,090 --> 01:02:19,880 First part is if you're flat foldable, 1125 01:02:19,880 --> 01:02:21,595 then you are mingling. 1126 01:02:26,740 --> 01:02:28,740 Certainly, if I want these all to be equivalent, 1127 01:02:28,740 --> 01:02:32,277 it should be true that the first one implies the third one. 1128 01:02:32,277 --> 01:02:33,860 This is actually really easy to prove. 1129 01:02:39,320 --> 01:02:41,300 Suppose it were not mingling. 1130 01:02:41,300 --> 01:02:43,110 Suppose I had some crease pattern that 1131 01:02:43,110 --> 01:02:44,190 violates that constraint. 1132 01:02:44,190 --> 01:02:46,600 What that means is I have a bunch of m's here, 1133 01:02:46,600 --> 01:02:49,460 I don't know how they're spread out. 1134 01:02:49,460 --> 01:02:51,200 Then I know that there's a big gap 1135 01:02:51,200 --> 01:02:55,586 until the next crease, something like that. 1136 01:02:55,586 --> 01:02:57,460 Well, let's think about-- there aren't really 1137 01:02:57,460 --> 01:02:59,280 that many ways to fold a whole bunch of m's 1138 01:02:59,280 --> 01:03:02,230 in a row or a whole bunch of v's that's symmetric. 1139 01:03:02,230 --> 01:03:04,757 You pretty much have to spiral around. 1140 01:03:04,757 --> 01:03:06,465 You're always turning the same direction. 1141 01:03:11,460 --> 01:03:13,952 If it's flat foldable, which is what we're assuming here, 1142 01:03:13,952 --> 01:03:15,410 it's got to look kind of like that. 1143 01:03:15,410 --> 01:03:17,326 In fact, it has two ends, so it could actually 1144 01:03:17,326 --> 01:03:19,470 make two spirals. 1145 01:03:19,470 --> 01:03:21,690 But if you think about what happens there, 1146 01:03:21,690 --> 01:03:28,880 this last part at the end of the spirals, that's this segment. 1147 01:03:28,880 --> 01:03:32,980 That's when you possibly turn around the other direction. 1148 01:03:32,980 --> 01:03:34,690 Maybe this is the valley. 1149 01:03:34,690 --> 01:03:36,360 But take a look at what's going on here. 1150 01:03:36,360 --> 01:03:39,650 This segment is tucked inside the spiral. 1151 01:03:39,650 --> 01:03:41,220 If it's super long like it is here, 1152 01:03:41,220 --> 01:03:44,670 if it's longer than the previous portion-- let me label this. 1153 01:03:44,670 --> 01:03:53,250 This is x, this is x, this is y, this is y. 1154 01:03:53,250 --> 01:03:54,800 That looks really bad. 1155 01:03:54,800 --> 01:03:56,134 x has to be smaller than y. 1156 01:03:56,134 --> 01:03:57,800 Otherwise, it's going to penetrate right 1157 01:03:57,800 --> 01:04:01,210 through that corner. 1158 01:04:01,210 --> 01:04:05,310 In this case with the double spiral, double spiral all 1159 01:04:05,310 --> 01:04:07,750 the way, what does it mean? 1160 01:04:10,360 --> 01:04:12,750 I guess no one knows double rainbow. 1161 01:04:16,310 --> 01:04:18,270 Bad stuff. 1162 01:04:18,270 --> 01:04:20,810 In the double spiral case, in fact, this end 1163 01:04:20,810 --> 01:04:22,680 must be shorter than this one and this one 1164 01:04:22,680 --> 01:04:24,500 must be shorter than this one. 1165 01:04:24,500 --> 01:04:27,320 So in this case, you must be mingling on both sides, 1166 01:04:27,320 --> 01:04:29,430 but it doesn't have to be that way. 1167 01:04:29,430 --> 01:04:32,040 If it's just a single spiral, this last segment 1168 01:04:32,040 --> 01:04:33,040 can be really long. 1169 01:04:33,040 --> 01:04:33,680 That's OK. 1170 01:04:33,680 --> 01:04:36,950 It's much longer than this one, but the inner part 1171 01:04:36,950 --> 01:04:39,510 of the spiral must always be smaller 1172 01:04:39,510 --> 01:04:41,540 than the next segment of the spiral, 1173 01:04:41,540 --> 01:04:43,210 and that's exactly what mingling says. 1174 01:04:43,210 --> 01:04:45,790 Could be on the left, could be on the right, you don't know. 1175 01:04:45,790 --> 01:04:48,120 But on at least one of the sides, you must be mingling. 1176 01:04:48,120 --> 01:04:50,790 And therefore, if you're going to have a flat folding at all, 1177 01:04:50,790 --> 01:04:52,430 you must be mingling. 1178 01:04:52,430 --> 01:04:53,120 Clear? 1179 01:04:53,120 --> 01:04:53,640 Question? 1180 01:04:53,640 --> 01:04:56,490 AUDIENCE: When you say a maximal sequence, what do you mean? 1181 01:04:56,490 --> 01:04:57,940 PROFESSOR: Maximal means that you 1182 01:04:57,940 --> 01:04:59,990 can't make it any bigger locally. 1183 01:04:59,990 --> 01:05:02,090 I can't add any more creases and still 1184 01:05:02,090 --> 01:05:04,000 have a contiguous bunch of mountains. 1185 01:05:04,000 --> 01:05:05,640 It's a good question. 1186 01:05:05,640 --> 01:05:08,840 In mathematics, there's maximal and maximum. 1187 01:05:08,840 --> 01:05:12,400 Maximal means locally biggest, maximum means globally biggest. 1188 01:05:12,400 --> 01:05:14,540 So I don't mean take the biggest sequence 1189 01:05:14,540 --> 01:05:16,400 of contiguous m's you can find. 1190 01:05:16,400 --> 01:05:19,120 I just mean take some contiguous sequence of m's that you 1191 01:05:19,120 --> 01:05:22,790 can't make any bigger on either side. 1192 01:05:22,790 --> 01:05:25,070 I just mean that it's surrounded by valleys. 1193 01:05:25,070 --> 01:05:31,530 I want this to hold for every contiguous sequence of m's. 1194 01:05:31,530 --> 01:05:34,280 That's one of those funny math technical terms. 1195 01:05:34,280 --> 01:05:36,970 Other questions? 1196 01:05:36,970 --> 01:05:38,540 So that's the first thing. 1197 01:05:38,540 --> 01:05:41,205 Somehow I want to connect all these properties together. 1198 01:05:44,890 --> 01:05:53,780 The next part is that if you're mingling, 1199 01:05:53,780 --> 01:05:56,270 either there's an end fold or there's a crimp. 1200 01:06:02,950 --> 01:06:06,160 At least one of those two things must be possible. 1201 01:06:06,160 --> 01:06:08,470 So this would be good news because if I'm 1202 01:06:08,470 --> 01:06:10,320 flat foldable, then I'm mingling. 1203 01:06:10,320 --> 01:06:12,750 If I'm mingling, then there's a fold I can make. 1204 01:06:12,750 --> 01:06:13,889 That seems like progress. 1205 01:06:13,889 --> 01:06:15,930 Then I have to show that I don't mess anything up 1206 01:06:15,930 --> 01:06:18,770 when I make that fold. 1207 01:06:18,770 --> 01:06:21,219 But if I could keep going in a circle like this, 1208 01:06:21,219 --> 01:06:23,010 then I would keep being able to make folds, 1209 01:06:23,010 --> 01:06:24,670 and then eventually I would fold the whole thing 1210 01:06:24,670 --> 01:06:26,440 because there's only end folds at the end. 1211 01:06:26,440 --> 01:06:27,640 I'll have made them all. 1212 01:06:33,270 --> 01:06:36,715 It's not that much trickier, I guess. 1213 01:06:36,715 --> 01:06:39,675 It involves parentheses. 1214 01:06:39,675 --> 01:06:40,800 That's one way to write it. 1215 01:06:40,800 --> 01:06:41,425 Yeah, question? 1216 01:06:44,840 --> 01:06:47,290 AUDIENCE: So right now, the main property 1217 01:06:47,290 --> 01:06:53,750 is finding a way that [INAUDIBLE] two or more? 1218 01:06:53,750 --> 01:06:54,844 PROFESSOR: That's right. 1219 01:06:54,844 --> 01:06:57,364 AUDIENCE: And there is an example that [INAUDIBLE]? 1220 01:06:57,364 --> 01:06:58,030 PROFESSOR: Yeah. 1221 01:06:58,030 --> 01:06:59,600 Just ignore this. 1222 01:06:59,600 --> 01:07:02,640 It'll be easier to think of it that way. 1223 01:07:02,640 --> 01:07:05,170 I'm going to use mingling to prove this property, 1224 01:07:05,170 --> 01:07:08,240 but this is not equivalent to those properties. 1225 01:07:08,240 --> 01:07:11,200 You need to add "forever," but that's 1226 01:07:11,200 --> 01:07:13,015 too confusing at the moment. 1227 01:07:13,015 --> 01:07:14,390 This is a definition of mingling. 1228 01:07:14,390 --> 01:07:17,289 We're going to use it as a separate thing, 1229 01:07:17,289 --> 01:07:19,580 and I'm going to prove these two things are equivalent. 1230 01:07:19,580 --> 01:07:21,180 Then I'll be correct. 1231 01:07:21,180 --> 01:07:24,540 Correct is always a good thing. 1232 01:07:24,540 --> 01:07:32,720 So what I want to do is for each of these mingling sequences, 1233 01:07:32,720 --> 01:07:42,320 for each maximal sequence of m's or v's, I 1234 01:07:42,320 --> 01:07:45,840 want to write down two symbols. 1235 01:07:45,840 --> 01:07:50,170 I'm going to write a curved left parenthesis if it 1236 01:07:50,170 --> 01:08:00,520 is left mingling, meaning I have all these mountains, let's say. 1237 01:08:04,840 --> 01:08:06,440 Draw it in a reasonable way. 1238 01:08:06,440 --> 01:08:09,360 Then I'm happy on the left side. 1239 01:08:09,360 --> 01:08:14,137 I'm mingling with a v with an m fold on the left side. 1240 01:08:14,137 --> 01:08:16,470 I know it's either left mingling or it's right mingling. 1241 01:08:16,470 --> 01:08:18,760 I'm assuming mingling here. 1242 01:08:18,760 --> 01:08:22,290 I will write a square bracket if it's not. 1243 01:08:22,290 --> 01:08:25,319 Then I will write a curved right bracket 1244 01:08:25,319 --> 01:08:31,700 if you're right mingling, which is just 1245 01:08:31,700 --> 01:08:34,200 the same picture but on the other side. 1246 01:08:34,200 --> 01:08:37,160 And I'm going to write a closed square bracket if it's not. 1247 01:08:40,950 --> 01:08:45,783 Then I claim that parentheses sequence 1248 01:08:45,783 --> 01:08:46,824 has some nice properties. 1249 01:08:54,260 --> 01:08:58,160 So for example, this is a string that I get. 1250 01:08:58,160 --> 01:09:03,279 It might look like this, maybe that. 1251 01:09:03,279 --> 01:09:04,120 Who knows? 1252 01:09:04,120 --> 01:09:05,220 It looks like something. 1253 01:09:05,220 --> 01:09:08,714 I know that every pair, at least one of the two sides is curved. 1254 01:09:08,714 --> 01:09:10,880 I can't have them both be square because then you're 1255 01:09:10,880 --> 01:09:12,100 not mingling at all. 1256 01:09:12,100 --> 01:09:14,450 I know I'm always mingling left or mingling right, 1257 01:09:14,450 --> 01:09:15,790 so it's something. 1258 01:09:15,790 --> 01:09:17,910 Now, one thing that's really good 1259 01:09:17,910 --> 01:09:23,029 is I have no double square bracket. 1260 01:09:23,029 --> 01:09:24,790 That's impossible. 1261 01:09:24,790 --> 01:09:27,800 I also know that if I had something 1262 01:09:27,800 --> 01:09:30,479 like this, what does that mean? 1263 01:09:30,479 --> 01:09:38,000 It means I have a bunch of mountains over here, 1264 01:09:38,000 --> 01:09:41,279 then I have a very nearby valley, and then 1265 01:09:41,279 --> 01:09:45,260 a bunch of valleys over here, and I 1266 01:09:45,260 --> 01:09:48,600 know that the mountains are right mingling, which 1267 01:09:48,600 --> 01:09:55,850 means I know that x is bigger than y. 1268 01:09:55,850 --> 01:09:57,100 That's what this symbol means. 1269 01:09:57,100 --> 01:09:59,810 It means to the right of the m's, I've got a small gap, 1270 01:09:59,810 --> 01:10:02,580 and it means to the left of the v's, I also have a small gap. 1271 01:10:02,580 --> 01:10:06,325 That means y is also bigger than this z distance. 1272 01:10:09,190 --> 01:10:09,940 AUDIENCE: Smaller. 1273 01:10:09,940 --> 01:10:10,870 PROFESSOR: Thank you. 1274 01:10:13,520 --> 01:10:14,916 Smaller. 1275 01:10:14,916 --> 01:10:18,590 I'm too tempted to combine them together, 1276 01:10:18,590 --> 01:10:22,130 but it's a funny looking thing. y is smaller than both x and z. 1277 01:10:22,130 --> 01:10:25,130 So then what do I do? 1278 01:10:25,130 --> 01:10:25,630 Crimp. 1279 01:10:30,010 --> 01:10:32,620 That was sort of the definition of a crimp, 1280 01:10:32,620 --> 01:10:35,820 that at least if I took this region x and copied it 1281 01:10:35,820 --> 01:10:37,580 over here, those were empty of creases. 1282 01:10:37,580 --> 01:10:41,420 And now we're saying that there's a big gap here. 1283 01:10:41,420 --> 01:10:42,920 I guess technically it's bigger than 1284 01:10:42,920 --> 01:10:45,530 or equal to if you said all these things right. 1285 01:10:49,820 --> 01:10:54,060 I said "nearer," but I meant non-strictly nearer. 1286 01:10:54,060 --> 01:10:57,710 It's enough to do a crimp. 1287 01:10:57,710 --> 01:10:59,940 So if I find this, I'm happy. 1288 01:10:59,940 --> 01:11:01,920 The other thing that I'm really happy with 1289 01:11:01,920 --> 01:11:09,705 is if I have an open paren at the beginning, 1290 01:11:09,705 --> 01:11:13,140 a curved parenthesis, open parenthesis, 1291 01:11:13,140 --> 01:11:16,520 at the beginning of the string, or a right parenthesis 1292 01:11:16,520 --> 01:11:19,140 at the end of the string, because that 1293 01:11:19,140 --> 01:11:20,560 means I can do an end fold. 1294 01:11:26,060 --> 01:11:28,080 That means that you have a very short segment 1295 01:11:28,080 --> 01:11:30,230 at the very beginning so you can just fold it over. 1296 01:11:30,230 --> 01:11:32,188 This means you have a short segment at the end. 1297 01:11:32,188 --> 01:11:34,200 Just fold it over. 1298 01:11:34,200 --> 01:11:36,770 So as long as I can find one of these three things, 1299 01:11:36,770 --> 01:11:38,110 I am golden. 1300 01:11:38,110 --> 01:11:43,370 I have what I want, which was an end fold or a crimp. 1301 01:11:43,370 --> 01:11:46,390 How do I prove that I have one of those things? 1302 01:11:46,390 --> 01:11:47,990 The easy way is by contradiction. 1303 01:11:47,990 --> 01:11:50,880 Just imagine if you didn't have any one of these things, 1304 01:11:50,880 --> 01:11:53,400 it's pretty forced what you have to have in your string 1305 01:11:53,400 --> 01:11:56,380 because you know you must start with an open square bracket. 1306 01:11:56,380 --> 01:11:58,880 You also know that you cannot then immediately have a closed 1307 01:11:58,880 --> 01:12:03,207 square bracket, so you must have a closed paren. 1308 01:12:03,207 --> 01:12:05,540 Now I know that I cannot have an open paren because then 1309 01:12:05,540 --> 01:12:07,710 I'd have this, and that would be good for me. 1310 01:12:07,710 --> 01:12:10,360 So if I want to be bad, I have to have an open square bracket, 1311 01:12:10,360 --> 01:12:12,068 but then I'd have to have a closed paren, 1312 01:12:12,068 --> 01:12:14,600 and then I'd have to have an open square bracket, and so on. 1313 01:12:14,600 --> 01:12:18,100 In the end, you will have a closed paren at the end, 1314 01:12:18,100 --> 01:12:21,070 and that means you can make an end fold. 1315 01:12:21,070 --> 01:12:22,840 So either something failed in here. 1316 01:12:22,840 --> 01:12:25,280 Maybe I had two parens together-- then I 1317 01:12:25,280 --> 01:12:26,990 would have gotten a crimp-- or I can 1318 01:12:26,990 --> 01:12:28,770 keep going all the way through here 1319 01:12:28,770 --> 01:12:30,200 and get a closed paren at the end. 1320 01:12:30,200 --> 01:12:32,680 In any case, I get at least one of these three things 1321 01:12:32,680 --> 01:12:35,440 happening, and so I get either an end fold or a crimp 1322 01:12:35,440 --> 01:12:36,977 assuming I am mingling. 1323 01:12:36,977 --> 01:12:38,560 And instead of assuming I am mingling, 1324 01:12:38,560 --> 01:12:40,800 I can just assume that I'm flat foldable, 1325 01:12:40,800 --> 01:12:42,805 and then I get from this chain of implications 1326 01:12:42,805 --> 01:12:45,955 that there is an end fold or a crimp. 1327 01:12:45,955 --> 01:12:47,000 OK so far? 1328 01:12:47,000 --> 01:12:49,450 So that means I can make at least one fold. 1329 01:12:49,450 --> 01:12:52,070 If I am flat foldable, I can make at least one crimp or end 1330 01:12:52,070 --> 01:12:53,240 fold. 1331 01:12:53,240 --> 01:12:56,111 Now the question is, can I keep going? 1332 01:12:56,111 --> 01:12:57,610 For that, we need one more property. 1333 01:13:15,310 --> 01:13:17,870 If I make a crimp or an end fold, 1334 01:13:17,870 --> 01:13:21,680 I want to prove that that preserves flat foldability. 1335 01:13:32,605 --> 01:13:34,730 What that means is if I start with something that's 1336 01:13:34,730 --> 01:13:36,938 flat foldable and then I make a crimp or an end fold, 1337 01:13:36,938 --> 01:13:38,300 I will still be flat foldable. 1338 01:13:38,300 --> 01:13:40,000 This is the property I need because it 1339 01:13:40,000 --> 01:13:41,780 will let me close the chain. 1340 01:13:41,780 --> 01:13:43,530 I'm flat foldable, therefore I'm mingling, 1341 01:13:43,530 --> 01:13:46,380 therefore I have an end fold or a crimp, so I make it. 1342 01:13:46,380 --> 01:13:48,630 This property says that I will still be flat foldable, 1343 01:13:48,630 --> 01:13:50,088 therefore number one applies again. 1344 01:13:50,088 --> 01:13:52,560 I'm still mingling, I can still find a fold, 1345 01:13:52,560 --> 01:13:55,361 and I'm still flat foldable, so I'm still mingling. 1346 01:13:55,361 --> 01:13:56,360 I can still find a fold. 1347 01:13:56,360 --> 01:13:57,204 I can keep going. 1348 01:13:57,204 --> 01:13:58,620 I will never get stuck because I'm 1349 01:13:58,620 --> 01:14:00,550 guaranteed if I started flat foldable, 1350 01:14:00,550 --> 01:14:03,380 I will be flat foldable forever. 1351 01:14:03,380 --> 01:14:04,875 I will also be mingling forever. 1352 01:14:07,480 --> 01:14:09,670 I realize now but apparently have not 1353 01:14:09,670 --> 01:14:12,252 realized before, it's not enough to be mingling just 1354 01:14:12,252 --> 01:14:12,960 at the beginning. 1355 01:14:12,960 --> 01:14:15,350 You have to be mingling through every step there. 1356 01:14:15,350 --> 01:14:16,680 In fact, what we're showing is that you're 1357 01:14:16,680 --> 01:14:18,840 fat foldable at every step, which implies mingling, 1358 01:14:18,840 --> 01:14:20,214 but mingling is not quite enough. 1359 01:14:20,214 --> 01:14:23,620 You'd have to check it all the way through. 1360 01:14:23,620 --> 01:14:26,350 That's why I stated the theorem wrong. 1361 01:14:26,350 --> 01:14:30,470 How do we prove this theorem, this claim? 1362 01:14:30,470 --> 01:14:32,510 I'm going to do it proof by picture. 1363 01:14:32,510 --> 01:14:35,640 It's not always OK, but it will be OK. 1364 01:14:35,640 --> 01:14:36,230 Trust me. 1365 01:14:38,629 --> 01:14:40,295 We are assuming that we're flat foldable 1366 01:14:40,295 --> 01:14:42,460 and we're assuming there's a crimp or an end fold. 1367 01:14:42,460 --> 01:14:43,946 Actually, we know that that exists. 1368 01:14:43,946 --> 01:14:45,570 I'm going to think about the crimp case 1369 01:14:45,570 --> 01:14:47,780 because end fold is even simpler. 1370 01:14:50,700 --> 01:14:52,110 See, here's the thing. 1371 01:14:52,110 --> 01:14:56,940 I would really like to fold some crimp like this, let's say. 1372 01:14:56,940 --> 01:15:01,880 This is an m, this is a v. What I'm 1373 01:15:01,880 --> 01:15:04,420 claiming is I'm allowed to make that crimp first 1374 01:15:04,420 --> 01:15:07,450 and then do everything else, but conceivably, there's 1375 01:15:07,450 --> 01:15:10,290 some flat folding that doesn't make this crimp first. 1376 01:15:10,290 --> 01:15:11,830 I mean, it must fold this mountain 1377 01:15:11,830 --> 01:15:13,830 and it must fold this valley, but at some point. 1378 01:15:13,830 --> 01:15:15,380 We don't really know when. 1379 01:15:15,380 --> 01:15:17,390 Somehow it all simultaneously collapses 1380 01:15:17,390 --> 01:15:18,849 into some flat folding. 1381 01:15:18,849 --> 01:15:20,140 Now, this thing should be flat. 1382 01:15:20,140 --> 01:15:21,170 It should line in a line. 1383 01:15:21,170 --> 01:15:22,836 I'm stretching it out so you can see it. 1384 01:15:25,340 --> 01:15:27,740 It's a segment that we're folding, 1385 01:15:27,740 --> 01:15:30,630 so maybe it does this, some crazy thing. 1386 01:15:30,630 --> 01:15:34,880 Maybe this part does something crazy here, 1387 01:15:34,880 --> 01:15:36,740 goes there, whatever. 1388 01:15:36,740 --> 01:15:40,320 We know it's not crossing and it's sort of horizontal, 1389 01:15:40,320 --> 01:15:42,190 but it could be an ugly mess. 1390 01:15:42,190 --> 01:15:44,190 And in particular, in this picture, 1391 01:15:44,190 --> 01:15:47,860 it does not look like this crimp was done first-- maybe embolden 1392 01:15:47,860 --> 01:15:51,970 the crimp here-- because there's this stuff in the middle. 1393 01:15:51,970 --> 01:15:53,990 Now, we know that there's no creases in here. 1394 01:15:53,990 --> 01:15:59,340 We know the earliest we can turn is at the ends of the crimp. 1395 01:15:59,340 --> 01:16:02,240 But there could be stuff that comes in the middle here. 1396 01:16:02,240 --> 01:16:03,360 What do I do? 1397 01:16:03,360 --> 01:16:07,430 I just move the stuff out of the way. 1398 01:16:07,430 --> 01:16:08,910 Let me do this the right way. 1399 01:16:08,910 --> 01:16:11,310 Got to do this a little bit carefully, 1400 01:16:11,310 --> 01:16:13,769 and unfortunately, my picture is so messy, 1401 01:16:13,769 --> 01:16:14,810 hopefully you can see it. 1402 01:16:14,810 --> 01:16:18,850 So I'm going to take this part, which is inside the crimp, 1403 01:16:18,850 --> 01:16:23,070 I'm going to move it down here, just 1404 01:16:23,070 --> 01:16:26,470 lower those layers down to here. 1405 01:16:26,470 --> 01:16:30,990 Now I know locally, there's a fold right here. 1406 01:16:30,990 --> 01:16:33,330 And so if I take all the stuff that's in here 1407 01:16:33,330 --> 01:16:36,720 and move it down here, it will still be non-crossing. 1408 01:16:36,720 --> 01:16:42,550 And then I take this stuff and I move it up here. 1409 01:16:42,550 --> 01:16:45,017 Equivalently, you can think of me crushing this 1410 01:16:45,017 --> 01:16:46,850 down and crushing this up, but it's actually 1411 01:16:46,850 --> 01:16:48,690 important the direction these things go. 1412 01:16:48,690 --> 01:16:51,610 I could not put this stuff up here because I don't really 1413 01:16:51,610 --> 01:16:52,910 know how long this goes. 1414 01:16:52,910 --> 01:16:55,109 It's not valid for me to try to put it up. 1415 01:16:55,109 --> 01:16:56,400 I could collide with something. 1416 01:16:56,400 --> 01:16:58,330 But I know that there's a crease right here, 1417 01:16:58,330 --> 01:16:59,939 so I can put it right over here. 1418 01:16:59,939 --> 01:17:01,980 I can take this stuff and put it right over here. 1419 01:17:01,980 --> 01:17:04,188 This is a weird kind of manipulation of folded states 1420 01:17:04,188 --> 01:17:06,890 which we don't normally do, but here we need to do it. 1421 01:17:06,890 --> 01:17:09,340 After I've done that-- is there any hope 1422 01:17:09,340 --> 01:17:10,990 of me redrawing the picture? 1423 01:17:10,990 --> 01:17:13,500 Maybe. 1424 01:17:13,500 --> 01:17:15,950 After we've done that, we'll have the crimp. 1425 01:17:15,950 --> 01:17:18,930 It will be nice and tight. 1426 01:17:18,930 --> 01:17:27,020 And then we'll have this finger here, 1427 01:17:27,020 --> 01:17:32,640 and then it goes up here, up this anvil. 1428 01:17:32,640 --> 01:17:36,750 Then this part, instead of going inside, 1429 01:17:36,750 --> 01:17:46,630 actually goes in here, up there, something like that, 1430 01:17:46,630 --> 01:17:48,010 and that's the end. 1431 01:17:48,010 --> 01:17:49,910 Then this is the other side. 1432 01:17:49,910 --> 01:17:52,730 Other side wasn't too complicated, I think. 1433 01:17:52,730 --> 01:17:54,950 But you can check. 1434 01:17:54,950 --> 01:17:57,370 I guess now it's down here. 1435 01:17:57,370 --> 01:17:59,470 But you can really just do the shifting locally 1436 01:17:59,470 --> 01:18:00,900 and it doesn't mess anything up. 1437 01:18:00,900 --> 01:18:02,750 I'm going to wave my hands because it's 1438 01:18:02,750 --> 01:18:04,610 a bit messy to really prove that. 1439 01:18:04,610 --> 01:18:07,070 But if you do that, boom. 1440 01:18:07,070 --> 01:18:10,574 Now this crimp looks like it was actually done first. 1441 01:18:10,574 --> 01:18:12,240 You can see that this folded state could 1442 01:18:12,240 --> 01:18:14,980 have been made by first doing that crimp and then folding 1443 01:18:14,980 --> 01:18:16,470 the rest. 1444 01:18:16,470 --> 01:18:20,160 You can treat this as if the paper was fused together here, 1445 01:18:20,160 --> 01:18:22,941 and that will be a folded state of that smaller piece of paper. 1446 01:18:22,941 --> 01:18:25,440 So even though we were shifting creases closer to each other 1447 01:18:25,440 --> 01:18:27,529 and it looked a little scary what's happening, 1448 01:18:27,529 --> 01:18:29,070 we know from this construction if you 1449 01:18:29,070 --> 01:18:31,660 could fold the original thing, you can fold the thing even 1450 01:18:31,660 --> 01:18:33,390 when the crimp got fused together. 1451 01:18:33,390 --> 01:18:35,190 Therefore, crimps are always safe, 1452 01:18:35,190 --> 01:18:38,310 and by an even easier argument, end folds are always safe. 1453 01:18:38,310 --> 01:18:41,310 I already gave that argument, basically. 1454 01:18:41,310 --> 01:18:44,070 So a little crazy, but that proves 1455 01:18:44,070 --> 01:18:46,110 that you preserve flat foldability 1456 01:18:46,110 --> 01:18:50,930 by going in this chain n or n over 2 times, 1457 01:18:50,930 --> 01:18:53,600 you will fold everything if you were originally flat foldable. 1458 01:18:53,600 --> 01:18:55,933 If you're not, you will get stuck because otherwise, you 1459 01:18:55,933 --> 01:18:58,070 would find a folding, and so on. 1460 01:18:58,070 --> 01:18:58,570 Clear? 1461 01:19:03,630 --> 01:19:06,410 I'll just mention you can generalize 1462 01:19:06,410 --> 01:19:12,160 this to folding two dimensional maps with simple folds. 1463 01:19:12,160 --> 01:19:16,380 I should say, of course, an end fold is a simple fold. 1464 01:19:16,380 --> 01:19:18,567 A crimp is two simple folds. 1465 01:19:18,567 --> 01:19:20,150 You can first fold along the mountain, 1466 01:19:20,150 --> 01:19:22,733 you won't get any collision, and then fold back on the valley. 1467 01:19:22,733 --> 01:19:24,520 That's a simple fold. 1468 01:19:24,520 --> 01:19:26,520 So this is saying, in fact, something 1469 01:19:26,520 --> 01:19:28,880 is flat foldable in one dimension if 1470 01:19:28,880 --> 01:19:32,545 and only if it is flat foldable by a sequence of simple folds. 1471 01:19:32,545 --> 01:19:34,420 As long as you stick to crimps and end folds, 1472 01:19:34,420 --> 01:19:36,420 it doesn't matter what order you do them in. 1473 01:19:39,110 --> 01:19:42,340 That's not true as soon as you go to two dimensions. 1474 01:19:42,340 --> 01:19:55,580 So for example, just for fun, do I really 1475 01:19:55,580 --> 01:19:56,820 want to draw the whole thing? 1476 01:19:56,820 --> 01:19:57,778 I guess I've committed. 1477 01:20:02,737 --> 01:20:04,320 There's a couple examples in the notes 1478 01:20:04,320 --> 01:20:07,920 that you can take a look at, and I encourage you to make one. 1479 01:20:07,920 --> 01:20:11,326 This can be folded flat but not by simple folds, 1480 01:20:11,326 --> 01:20:13,700 and you can see that it can't be followed by simple folds 1481 01:20:13,700 --> 01:20:16,200 because there's no fold that goes all the way through. 1482 01:20:16,200 --> 01:20:18,450 This line is valley, mountain, mountain. 1483 01:20:18,450 --> 01:20:21,980 There's no line that's all mountains or all valleys. 1484 01:20:21,980 --> 01:20:25,810 But you can use this structure to understand at least which 1485 01:20:25,810 --> 01:20:29,030 rectangular maps with horizontal and vertical creases 1486 01:20:29,030 --> 01:20:31,230 can be folded flat by simple folds, 1487 01:20:31,230 --> 01:20:33,050 and the answer would be no here. 1488 01:20:33,050 --> 01:20:40,030 All you do is you say, well, I know if there's any fold, 1489 01:20:40,030 --> 01:20:42,350 I must have a mountain or a valley that 1490 01:20:42,350 --> 01:20:43,620 goes all the way through. 1491 01:20:43,620 --> 01:20:45,400 So Imagine here's your rectangle. 1492 01:20:45,400 --> 01:20:47,650 There's got to be something that's valley all the way. 1493 01:20:50,900 --> 01:20:56,510 Now, if you think about a crease that goes the other direction, 1494 01:20:56,510 --> 01:20:58,740 these are two valleys. 1495 01:20:58,740 --> 01:21:02,200 You can show, just by playing around with a little pattern 1496 01:21:02,200 --> 01:21:05,914 like this, you cannot have these two both be valleys 1497 01:21:05,914 --> 01:21:07,830 because that would be all four valleys is bad. 1498 01:21:07,830 --> 01:21:09,079 If you try that, doesn't work. 1499 01:21:09,079 --> 01:21:10,570 Or it also can't be both mountains. 1500 01:21:10,570 --> 01:21:12,920 You can't valley, mountain, valley, mountain in order. 1501 01:21:12,920 --> 01:21:13,955 That doesn't fold. 1502 01:21:13,955 --> 01:21:15,580 So in fact, these have to be different. 1503 01:21:15,580 --> 01:21:16,810 One's mountain, one's valley. 1504 01:21:16,810 --> 01:21:19,540 I don't care which is which, but I know that this line actually 1505 01:21:19,540 --> 01:21:21,140 cannot be folded by a simple fold. 1506 01:21:21,140 --> 01:21:23,970 It's got a valley and a mountain. 1507 01:21:23,970 --> 01:21:27,810 So in fact, the picture has to be I have a line like this. 1508 01:21:27,810 --> 01:21:30,640 Maybe I have a few lines that are mountains all the way 1509 01:21:30,640 --> 01:21:33,800 or valleys all the way. 1510 01:21:33,800 --> 01:21:36,690 And any horizontal line I can't fold. 1511 01:21:36,690 --> 01:21:38,930 I've got to finish folding these guys first. 1512 01:21:38,930 --> 01:21:41,530 That means I have a one dimensional problem. 1513 01:21:41,530 --> 01:21:43,790 So I apply this algorithm to see whether that one 1514 01:21:43,790 --> 01:21:45,720 dimensional thing folds. 1515 01:21:45,720 --> 01:21:48,741 Once I've folded all those vertical things, 1516 01:21:48,741 --> 01:21:50,240 solved that one dimensional problem, 1517 01:21:50,240 --> 01:21:51,927 I will then have a horizontal problem, 1518 01:21:51,927 --> 01:21:53,510 and then I'll have a vertical problem. 1519 01:21:53,510 --> 01:21:55,182 You just keep doing this, and you 1520 01:21:55,182 --> 01:21:56,640 can show in the same way you always 1521 01:21:56,640 --> 01:21:57,880 preserve flat foldability. 1522 01:21:57,880 --> 01:22:00,022 You just keep doing crimps and end folds, 1523 01:22:00,022 --> 01:22:01,480 and this will fold your map if it's 1524 01:22:01,480 --> 01:22:04,290 at all possible by simple folds, which most ones that 1525 01:22:04,290 --> 01:22:07,000 are in production are foldable by simple folds. 1526 01:22:07,000 --> 01:22:07,825 Question? 1527 01:22:07,825 --> 01:22:08,450 AUDIENCE: Yeah. 1528 01:22:08,450 --> 01:22:11,420 If we go back to the example where [INAUDIBLE]? 1529 01:22:15,034 --> 01:22:15,700 PROFESSOR: Yeah. 1530 01:22:21,890 --> 01:22:27,874 AUDIENCE: [INAUDIBLE] mountain, it would be reversed 1531 01:22:27,874 --> 01:22:30,224 and then it would be foldable [INAUDIBLE]. 1532 01:22:30,224 --> 01:22:30,890 PROFESSOR: Yeah. 1533 01:22:30,890 --> 01:22:31,770 This is good. 1534 01:22:31,770 --> 01:22:33,530 This is the case that could work. 1535 01:22:33,530 --> 01:22:35,560 This is foldable if you try it out. 1536 01:22:35,560 --> 01:22:38,040 You first told here, and then the valley 1537 01:22:38,040 --> 01:22:41,030 falls onto the mountain but it turns upside down as it folds, 1538 01:22:41,030 --> 01:22:43,400 and so they actually nest nicely inside each other. 1539 01:22:43,400 --> 01:22:45,180 It's confusing because it's inverting. 1540 01:22:45,180 --> 01:22:47,364 This is the one valley case, but my point 1541 01:22:47,364 --> 01:22:49,280 is you have to fold this before you fold that. 1542 01:22:52,120 --> 01:22:54,120 There's a little more details in the notes here. 1543 01:22:54,120 --> 01:22:56,620 You can make this into a really fast linear time algorithm 1544 01:22:56,620 --> 01:22:57,995 if you use fancy data structures, 1545 01:22:57,995 --> 01:23:01,880 but that's beyond this scope, and that 1546 01:23:01,880 --> 01:23:04,700 is the end of lecture two.