1 00:00:02,616 --> 00:00:03,490 PROFESSOR: All right. 2 00:00:03,490 --> 00:00:06,720 Welcome back to 6.849. 3 00:00:06,720 --> 00:00:12,620 Today we are going over TreeMaker, folding polyhedra, 4 00:00:12,620 --> 00:00:14,900 folding checkerboards, and Origamizer. 5 00:00:14,900 --> 00:00:16,500 Lots of stuff. 6 00:00:16,500 --> 00:00:18,150 And can't fit everything in today, 7 00:00:18,150 --> 00:00:20,340 but we'll be doing more Treemaker in some sense 8 00:00:20,340 --> 00:00:21,760 from a more practical perspective 9 00:00:21,760 --> 00:00:25,710 in the very next lecture video and next class, 10 00:00:25,710 --> 00:00:28,390 with a guest lecture by Jason Ku. 11 00:00:28,390 --> 00:00:30,440 So today we have a bunch of questions. 12 00:00:30,440 --> 00:00:33,460 We'll start with TreeMaker. 13 00:00:33,460 --> 00:00:35,132 So the first question is, wow, you 14 00:00:35,132 --> 00:00:37,340 can make all these crazy things with the tree method. 15 00:00:37,340 --> 00:00:38,910 Are they really uniaxial? 16 00:00:38,910 --> 00:00:41,990 And the answer is yes. 17 00:00:41,990 --> 00:00:43,580 And I think it's probably easiest 18 00:00:43,580 --> 00:00:47,080 to just go through some examples. 19 00:00:47,080 --> 00:00:49,880 These are all from Origami Design Secrets, this book 20 00:00:49,880 --> 00:00:53,760 by Robert Lang, which is on the recommended reading list. 21 00:00:53,760 --> 00:00:56,760 And I think there's even an electronic copy available 22 00:00:56,760 --> 00:00:57,960 through MIT Libraries. 23 00:00:57,960 --> 00:01:03,390 So you can read it online or there maybe one in the library. 24 00:01:03,390 --> 00:01:04,572 There's two editions. 25 00:01:04,572 --> 00:01:06,280 These figures are from the first edition. 26 00:01:06,280 --> 00:01:12,390 So here is a-- the goal is to design "Scorpion varileg." 27 00:01:12,390 --> 00:01:14,130 These are all Robert Lang designs. 28 00:01:14,130 --> 00:01:17,260 You come up with this stick figure, your tree. 29 00:01:17,260 --> 00:01:19,210 TreeMaker makes this crease pattern. 30 00:01:19,210 --> 00:01:23,130 You can also see the rivers here and the disks. 31 00:01:23,130 --> 00:01:26,069 And then you get this base. 32 00:01:26,069 --> 00:01:27,610 And then it folds into that scorpion. 33 00:01:27,610 --> 00:01:29,359 So while the scorpion doesn't look like it 34 00:01:29,359 --> 00:01:31,704 has a single axis-- scorpion, maybe. 35 00:01:31,704 --> 00:01:33,370 There's kind of an axis down the center. 36 00:01:33,370 --> 00:01:35,610 But it's not just this axis. 37 00:01:35,610 --> 00:01:37,110 It branches, then it branches again. 38 00:01:37,110 --> 00:01:40,490 You can make a tree and it's still uniaxial. 39 00:01:40,490 --> 00:01:44,960 So uniaxial refers to this axis in the base. 40 00:01:44,960 --> 00:01:48,050 All the flaps are attached to that one spot. 41 00:01:48,050 --> 00:01:50,210 So in this picture, it's vertical. 42 00:01:50,210 --> 00:01:53,280 Usually in the-- when we draw all the figures, 43 00:01:53,280 --> 00:01:57,560 we think of it as the horizontal plane, or horizontal line. 44 00:01:57,560 --> 00:02:02,250 So we imagine this is the floor. 45 00:02:02,250 --> 00:02:05,590 And then we have flaps attached to the floor. 46 00:02:05,590 --> 00:02:08,770 Something like this. 47 00:02:08,770 --> 00:02:10,690 So in that sense, when you flatten this base, 48 00:02:10,690 --> 00:02:12,635 everything lies along this axis. 49 00:02:12,635 --> 00:02:14,510 Even though, in terms of branching structure, 50 00:02:14,510 --> 00:02:16,120 you could have lots of branches. 51 00:02:16,120 --> 00:02:18,370 So that's to clarify what uniaxial means. 52 00:02:18,370 --> 00:02:22,940 Maybe not the best term, but I have a few more examples here. 53 00:02:22,940 --> 00:02:25,110 This hat includes a color reversal. 54 00:02:25,110 --> 00:02:27,860 Some crazy stick figure. 55 00:02:27,860 --> 00:02:29,329 And you get your pattern. 56 00:02:29,329 --> 00:02:30,620 Here's a somewhat simpler want. 57 00:02:30,620 --> 00:02:33,770 It makes a more three dimensional horse. 58 00:02:33,770 --> 00:02:36,846 This Alamo Stallion. 59 00:02:36,846 --> 00:02:38,720 So you can go through, and for each of these, 60 00:02:38,720 --> 00:02:43,340 I've just reverse engineered from this crease pattern what 61 00:02:43,340 --> 00:02:45,000 the stick figure must be. 62 00:02:45,000 --> 00:02:47,290 And you're going to do this on PSET2, 63 00:02:47,290 --> 00:02:50,750 which will be released in a few minutes. 64 00:02:50,750 --> 00:02:53,820 And so this uniaxial's to contrast from something, 65 00:02:53,820 --> 00:02:58,510 the kind of most famous non-uniaxial or biaxial origami 66 00:02:58,510 --> 00:03:02,790 bases, with Montroll's Dog Base, which looks like this. 67 00:03:02,790 --> 00:03:05,060 So it's very easy to turn this into all sorts of dogs. 68 00:03:05,060 --> 00:03:06,890 This is a wiener dog. 69 00:03:06,890 --> 00:03:11,970 But you have kind of one axis here and one axis here. 70 00:03:11,970 --> 00:03:15,130 Which is different. 71 00:03:15,130 --> 00:03:17,471 It's not the same kind of branching tree structure. 72 00:03:17,471 --> 00:03:19,970 Of course, you can make a dog using uniaxial method as well, 73 00:03:19,970 --> 00:03:23,220 but it's kind of a fun counter example of the things 74 00:03:23,220 --> 00:03:24,630 you might do. 75 00:03:24,630 --> 00:03:26,430 So that's uniaxial. 76 00:03:26,430 --> 00:03:31,370 Next question is, do people use TreeMaker and/or Origamizer 77 00:03:31,370 --> 00:03:32,710 in practice? 78 00:03:32,710 --> 00:03:35,500 And the short answer is no. 79 00:03:35,500 --> 00:03:38,260 I would say most origamists don't use software 80 00:03:38,260 --> 00:03:40,390 to design things. 81 00:03:40,390 --> 00:03:43,370 But they use a lot of the ideas in their own designs. 82 00:03:43,370 --> 00:03:45,290 So in particular, the tree method 83 00:03:45,290 --> 00:03:48,220 of origami design, which TreeMaker is implementing, 84 00:03:48,220 --> 00:03:51,940 almost every advanced origami designer uses. 85 00:03:51,940 --> 00:03:55,180 Not everyone, but most of the complicated-- most complex 86 00:03:55,180 --> 00:03:57,810 origami that you see uses the tree method at least 87 00:03:57,810 --> 00:03:59,380 to get started. 88 00:03:59,380 --> 00:04:03,050 Origamizer is much newer, so it's not used as much. 89 00:04:03,050 --> 00:04:05,840 Most of the cool designs-- I'll show you a bunch of both-- 90 00:04:05,840 --> 00:04:10,320 are by Tomohiro Tachi, who invented Origamizer. 91 00:04:10,320 --> 00:04:11,980 But I think it's still evolving. 92 00:04:11,980 --> 00:04:14,240 And hopefully, these more advanced techniques 93 00:04:14,240 --> 00:04:17,950 will catch on with time. 94 00:04:17,950 --> 00:04:21,339 And by now, you've seen lots of examples of the tree method. 95 00:04:21,339 --> 00:04:24,000 I'll show a few more from Robert Lang's website. 96 00:04:24,000 --> 00:04:28,050 This is a local delicacy. 97 00:04:28,050 --> 00:04:30,710 And these are all on langorigami.com, 98 00:04:30,710 --> 00:04:31,930 on his website. 99 00:04:31,930 --> 00:04:34,180 And we'll see even more examples in the next class. 100 00:04:34,180 --> 00:04:39,080 And Jason Ku is going to give an overview of various artists. 101 00:04:39,080 --> 00:04:41,460 And in some cases, they publish crease patterns. 102 00:04:41,460 --> 00:04:42,740 Not always. 103 00:04:42,740 --> 00:04:44,640 This is not quite the tree method. 104 00:04:44,640 --> 00:04:47,260 So this example was standard tree method. 105 00:04:47,260 --> 00:04:49,629 You've got disks and rivers, which aren't drawn here. 106 00:04:49,629 --> 00:04:50,670 You get a crease pattern. 107 00:04:50,670 --> 00:04:53,040 This could be done directly from TreeMaker. 108 00:04:53,040 --> 00:04:55,750 I don't know for sure whether it was. 109 00:04:55,750 --> 00:04:58,600 This pattern is not done with TreeMaker 110 00:04:58,600 --> 00:05:02,320 because it's not regular disks. 111 00:05:02,320 --> 00:05:05,469 This is the box pleating version of the tree method, which 112 00:05:05,469 --> 00:05:06,760 there's another question about. 113 00:05:06,760 --> 00:05:08,364 So we'll talk about that more. 114 00:05:08,364 --> 00:05:10,780 But you see all the creases here are horizontal, vertical, 115 00:05:10,780 --> 00:05:13,050 or diagonal, 45 degrees. 116 00:05:13,050 --> 00:05:15,050 That makes it a lot easier to fold, a lot easier 117 00:05:15,050 --> 00:05:16,091 to find reference points. 118 00:05:16,091 --> 00:05:17,660 You see it falls nicely on a grid. 119 00:05:17,660 --> 00:05:20,230 But still, you get an arbitrary tree structure. 120 00:05:23,160 --> 00:05:26,540 I think I have another example. 121 00:05:26,540 --> 00:05:31,319 This one also-- well, this has 22 and 1/2 degree folds. 122 00:05:31,319 --> 00:05:32,610 It's a little bit more general. 123 00:05:32,610 --> 00:05:36,780 Also very non-stick figure-like models, particularly. 124 00:05:36,780 --> 00:05:39,640 It's obviously called the "Fiddler Crab." 125 00:05:39,640 --> 00:05:40,140 No, sorry. 126 00:05:40,140 --> 00:05:42,250 I have the wrong title here. 127 00:05:42,250 --> 00:05:45,480 I will fix that. 128 00:05:45,480 --> 00:05:48,380 I forget this guy's name, but if anyone remembers. 129 00:05:48,380 --> 00:05:53,560 He wrote about the division between scientists and people 130 00:05:53,560 --> 00:05:55,575 who know literature as two different types 131 00:05:55,575 --> 00:05:56,950 of intellectuals that know almost 132 00:05:56,950 --> 00:05:57,991 nothing about each other. 133 00:05:57,991 --> 00:06:04,130 At least in the early 1900s when he was writing. 134 00:06:04,130 --> 00:06:06,280 And one more example. 135 00:06:06,280 --> 00:06:10,380 This is based on a 60 degree grid. 136 00:06:10,380 --> 00:06:15,720 So all the creases, I think, lie at 60 degrees to the axis, 137 00:06:15,720 --> 00:06:17,780 or horizontal. 138 00:06:17,780 --> 00:06:21,260 There might be some 30 degrees. 139 00:06:21,260 --> 00:06:23,980 So I'll talk a little bit about this alternate version 140 00:06:23,980 --> 00:06:26,490 of the tree method that doesn't just use arbitrary circles, 141 00:06:26,490 --> 00:06:31,130 but tries to stay on these nice grids. 142 00:06:31,130 --> 00:06:33,010 Here's just one more example. 143 00:06:33,010 --> 00:06:36,790 Jason Ku is the designer of this model called "Pan." 144 00:06:36,790 --> 00:06:39,710 and he'll be giving the guest lecture. 145 00:06:39,710 --> 00:06:41,190 So this is just one example. 146 00:06:41,190 --> 00:06:44,050 But tons and tons of advanced origami artists 147 00:06:44,050 --> 00:06:46,800 use tree method of origami design. 148 00:06:46,800 --> 00:06:50,390 And you get some pretty cool crease patterns from it. 149 00:06:50,390 --> 00:06:53,670 But I think most people do it by hand, usually 150 00:06:53,670 --> 00:06:55,890 on a computer using some drawing programs that 151 00:06:55,890 --> 00:07:00,000 can compute intersections and do things with high accuracy. 152 00:07:00,000 --> 00:07:04,220 And then they draw these pictures, then they fold them. 153 00:07:04,220 --> 00:07:06,174 And then you get your base and then 154 00:07:06,174 --> 00:07:07,340 you shape it into the model. 155 00:07:10,100 --> 00:07:13,630 So next I want to show you some practical examples of 156 00:07:13,630 --> 00:07:17,680 So there's a bunch on Tomohiro Tachi's Flickr site. 157 00:07:17,680 --> 00:07:18,890 This is one of the earliest. 158 00:07:18,890 --> 00:07:22,200 This is just making a negative curvature surface 159 00:07:22,200 --> 00:07:25,920 that curves in both ways like a saddle, called 160 00:07:25,920 --> 00:07:27,140 a hyperbolic paraboloid. 161 00:07:27,140 --> 00:07:28,860 We'll see more of those in the future. 162 00:07:28,860 --> 00:07:32,070 But this is folded from one square of paper. 163 00:07:32,070 --> 00:07:36,090 And there's tabs on the backside that you don't see here. 164 00:07:36,090 --> 00:07:39,315 Here he's making a kind of 3D bell curve. 165 00:07:44,692 --> 00:07:46,660 Here's a computer mouse. 166 00:07:46,660 --> 00:07:51,370 Apparently the scroll wheel works fine. [LAUGHS] 167 00:07:51,370 --> 00:07:53,136 Doesn't support USB, though. 168 00:07:53,136 --> 00:07:55,890 Only Bluetooth. 169 00:07:55,890 --> 00:07:59,480 We've got a nice mask. 170 00:07:59,480 --> 00:08:02,250 These are all-- you're given a 3D model, 171 00:08:02,250 --> 00:08:04,090 and you fold exactly that 3D model. 172 00:08:07,140 --> 00:08:08,830 This is a tetrapod. 173 00:08:08,830 --> 00:08:11,700 So here it sort of shows Tomohiro's background 174 00:08:11,700 --> 00:08:12,747 in architecture. 175 00:08:12,747 --> 00:08:14,330 If you haven't seen tetrapods, they're 176 00:08:14,330 --> 00:08:19,170 used to hold back the ocean and things from-- or hold 177 00:08:19,170 --> 00:08:20,780 from erosion and things like that. 178 00:08:20,780 --> 00:08:22,370 So usually made out of concrete. 179 00:08:22,370 --> 00:08:26,310 But here you can make them out of paper. 180 00:08:26,310 --> 00:08:27,940 Here's a flat design. 181 00:08:27,940 --> 00:08:32,990 So here the mesh, the triangles were all in a plane. 182 00:08:32,990 --> 00:08:36,270 And of course, you still get the tabs on the backside. 183 00:08:36,270 --> 00:08:38,900 But you make this very exact leaf. 184 00:08:38,900 --> 00:08:40,400 You can get the triangulation edges 185 00:08:40,400 --> 00:08:43,320 exactly where you want them to express 186 00:08:43,320 --> 00:08:50,310 these veins of the leaf. 187 00:08:50,310 --> 00:08:53,655 Here's what it looks like to fold one of these in practice. 188 00:08:53,655 --> 00:08:55,030 It's not quite a square of paper, 189 00:08:55,030 --> 00:08:56,790 but it could be a square. 190 00:08:56,790 --> 00:08:59,690 And this is actually folded at CSAIL. 191 00:08:59,690 --> 00:09:04,080 You might recognize some of the furniture. 192 00:09:04,080 --> 00:09:06,075 And so while Tomohiro's folding, he 193 00:09:06,075 --> 00:09:11,790 uses various devices-- paper clips and clips 194 00:09:11,790 --> 00:09:14,164 and so on-- to hold it in shape. 195 00:09:14,164 --> 00:09:15,830 Because until it's completely collapsed, 196 00:09:15,830 --> 00:09:17,630 it kind of want to open back up. 197 00:09:17,630 --> 00:09:21,390 So it's kind of like having a hundred hands at once. 198 00:09:21,390 --> 00:09:26,020 But when he's done, he'll take all of them off, 199 00:09:26,020 --> 00:09:29,380 and you'll get-- so this, I think, was about 10 hours 200 00:09:29,380 --> 00:09:32,510 and you get your bunny. 201 00:09:32,510 --> 00:09:35,320 Easy. 202 00:09:35,320 --> 00:09:37,920 Or this is the one we did a little bit later, 203 00:09:37,920 --> 00:09:39,240 just last year. 204 00:09:39,240 --> 00:09:43,560 This is a laser cut sheet of steel. 205 00:09:43,560 --> 00:09:46,290 And now, here we've cut out bigger holes 206 00:09:46,290 --> 00:09:49,610 so there aren't too many accumulation of layers. 207 00:09:49,610 --> 00:09:51,700 But essentially the same design. 208 00:09:51,700 --> 00:09:56,020 Somewhat coarser mesh of the bunny. 209 00:09:56,020 --> 00:09:59,110 And you have to wear gloves, otherwise you'll cut yourself. 210 00:09:59,110 --> 00:10:01,710 So we're perforating the metal at the creases. 211 00:10:01,710 --> 00:10:05,090 And then also takes about eight hours. 212 00:10:05,090 --> 00:10:06,760 A lot harder to fold steel. 213 00:10:09,390 --> 00:10:14,690 And in the end, you get your bunny. 214 00:10:14,690 --> 00:10:16,860 So in principle, out of any sheet of material, 215 00:10:16,860 --> 00:10:18,400 you can make any 3D shape you want. 216 00:10:18,400 --> 00:10:19,592 That's the exciting thing. 217 00:10:19,592 --> 00:10:21,300 So I think Origamizer is really powerful. 218 00:10:21,300 --> 00:10:23,880 Obviously, it's hard to fold. 219 00:10:23,880 --> 00:10:25,220 You need to be pretty advanced. 220 00:10:25,220 --> 00:10:28,250 But there's a lot of potential for designing 221 00:10:28,250 --> 00:10:30,650 very non-stick figure-like models. 222 00:10:33,190 --> 00:10:37,400 So that was TreeMaker and Origamizer in practice. 223 00:10:37,400 --> 00:10:41,010 So next question is about this box pleating, 224 00:10:41,010 --> 00:10:43,270 which is the horizontal, vertical, and 45 degree 225 00:10:43,270 --> 00:10:44,325 creases. 226 00:10:44,325 --> 00:10:48,820 And TreeMaker, is there some theory for this? 227 00:10:48,820 --> 00:10:49,820 And indeed there is. 228 00:10:49,820 --> 00:10:54,200 And we started working on it, me, Marty, and Rob Lang, 229 00:10:54,200 --> 00:10:56,290 I think during his first visit here, 230 00:10:56,290 --> 00:10:59,055 which was probably 2004 or something. 231 00:10:59,055 --> 00:11:01,420 A long time ago. 232 00:11:01,420 --> 00:11:04,190 And the best writeup of it currently 233 00:11:04,190 --> 00:11:07,071 is in Origami Design Secrets, especially Second Edition. 234 00:11:07,071 --> 00:11:07,570 So 235 00:11:07,570 --> 00:11:10,290 It has a chapter on basic tree theory, 236 00:11:10,290 --> 00:11:11,970 a chapter on box pleating in general. 237 00:11:11,970 --> 00:11:13,830 This is sort of classical box pleating. 238 00:11:13,830 --> 00:11:16,210 But then there's a chapter on mixing the two. 239 00:11:16,210 --> 00:11:18,380 Uniaxial box pleating and polygon packing. 240 00:11:18,380 --> 00:11:20,310 These kind of go together. 241 00:11:20,310 --> 00:11:21,910 So if you're interested in this stuff, 242 00:11:21,910 --> 00:11:26,430 you want to design something the way that the experts do, 243 00:11:26,430 --> 00:11:28,360 check out Origami Design Secrets. 244 00:11:28,360 --> 00:11:30,130 Or go to an origami convention where 245 00:11:30,130 --> 00:11:32,730 Rob was talking about this stuff. 246 00:11:32,730 --> 00:11:35,740 We are working on this giant manuscript. 247 00:11:35,740 --> 00:11:37,330 It's currently called the Mathematics 248 00:11:37,330 --> 00:11:40,410 of Origami Design, which is, in particular, trying 249 00:11:40,410 --> 00:11:43,860 to prove the tree method always works. 250 00:11:43,860 --> 00:11:46,300 And we'll generalize to things like this. 251 00:11:46,300 --> 00:11:49,950 But it's not finished, so we don't have a complete proof 252 00:11:49,950 --> 00:11:51,370 yet that it all works. 253 00:11:51,370 --> 00:11:54,880 Everything seems fine, but it's tedious to write it all down. 254 00:11:54,880 --> 00:11:56,670 So still working on it. 255 00:11:56,670 --> 00:11:59,170 But I thought I'd show you an example of box pleating 256 00:11:59,170 --> 00:12:02,960 uniaxial origami design from Origami Design Secrets. 257 00:12:02,960 --> 00:12:06,090 So this is sort of typical tree method of origami design. 258 00:12:06,090 --> 00:12:07,840 If you're not necessarily using TreeMaker, 259 00:12:07,840 --> 00:12:10,460 but you're doing it by hand, you think about, OK. 260 00:12:10,460 --> 00:12:15,670 Suppose I want to make this kind of insect, this stick figure. 261 00:12:15,670 --> 00:12:17,935 Maybe I realize, oh, it's kind of centrally symmetric, 262 00:12:17,935 --> 00:12:19,310 so I'd like to make the left half 263 00:12:19,310 --> 00:12:20,810 of the paper same as the right half. 264 00:12:20,810 --> 00:12:23,580 You can also express that in the TreeMaker software. 265 00:12:23,580 --> 00:12:25,350 And then you start thinking about where 266 00:12:25,350 --> 00:12:27,090 these leaves correspond to disks. 267 00:12:27,090 --> 00:12:29,089 Then you've got to have the corresponding rivers 268 00:12:29,089 --> 00:12:29,790 in between them. 269 00:12:29,790 --> 00:12:31,623 So you might start with this kind of layout, 270 00:12:31,623 --> 00:12:35,370 and then you try to blow it up until things can't be expanded 271 00:12:35,370 --> 00:12:37,270 anymore and there's lots of touching. 272 00:12:37,270 --> 00:12:38,880 So that would be the usual approach. 273 00:12:38,880 --> 00:12:40,879 Then from that, you could apply the tree method, 274 00:12:40,879 --> 00:12:42,530 get the crease pattern. 275 00:12:42,530 --> 00:12:45,780 With the box pleated version, essentially, instead of disks, 276 00:12:45,780 --> 00:12:47,440 you have squares. 277 00:12:47,440 --> 00:12:51,160 And instead of rivers, you have these orthogonal channels 278 00:12:51,160 --> 00:12:53,350 of constant width. 279 00:12:53,350 --> 00:12:55,670 And so I'll just wave my hands and say that happens. 280 00:12:55,670 --> 00:12:59,250 Now, you get these weird things-- you get these gaps. 281 00:12:59,250 --> 00:13:01,740 You tend to get more gaps in this way. 282 00:13:01,740 --> 00:13:04,050 But in this case, instead of just having a square, 283 00:13:04,050 --> 00:13:05,880 you can extend it out to be a rectangle. 284 00:13:05,880 --> 00:13:07,546 So these guys are actually going to fill 285 00:13:07,546 --> 00:13:11,650 in these holes like this. 286 00:13:11,650 --> 00:13:13,250 And then from this, you can start 287 00:13:13,250 --> 00:13:14,520 constructing a crease pattern. 288 00:13:14,520 --> 00:13:19,440 So you start with these ridge creases. 289 00:13:19,440 --> 00:13:21,749 And this is something like a generalization. 290 00:13:21,749 --> 00:13:24,290 It's actually something like a straight skeleton, which we'll 291 00:13:24,290 --> 00:13:25,748 be covering in a couple of lectures 292 00:13:25,748 --> 00:13:28,230 for a different purpose, where it was originally 293 00:13:28,230 --> 00:13:30,130 developed for origami purposes. 294 00:13:30,130 --> 00:13:34,060 But these are just lots to bisectors. 295 00:13:34,060 --> 00:13:37,260 And you'll have to see the general version later. 296 00:13:37,260 --> 00:13:39,390 And then you start putting in the hinge creases. 297 00:13:39,390 --> 00:13:44,610 These are these green lines. 298 00:13:44,610 --> 00:13:47,992 And they're perpendicular to these. 299 00:13:47,992 --> 00:13:50,880 These blue creases are going to be the floor. 300 00:13:50,880 --> 00:13:52,440 And so on, you fill it in. 301 00:13:52,440 --> 00:13:55,640 Eventually you get your complete crease pattern. 302 00:13:55,640 --> 00:13:57,830 So in a nutshell, that's how it works. 303 00:13:57,830 --> 00:14:01,580 You tend to get tabs that are a constant width like this. 304 00:14:01,580 --> 00:14:03,015 Constant height, I guess. 305 00:14:06,770 --> 00:14:08,730 And so, it's a general technique. 306 00:14:08,730 --> 00:14:11,300 It's a little bit much to explain here, but read the book 307 00:14:11,300 --> 00:14:12,681 if you want to see it. 308 00:14:12,681 --> 00:14:14,930 And I would say it's probably some of the most common. 309 00:14:14,930 --> 00:14:16,700 There's a lot of intuitive ways to do it, 310 00:14:16,700 --> 00:14:20,990 and then a lot of the details are worked out in that book. 311 00:14:20,990 --> 00:14:23,350 But it's quite common to design bases in this way, 312 00:14:23,350 --> 00:14:25,266 because it's just so much easier to fold them. 313 00:14:25,266 --> 00:14:28,006 You don't have to construct really weird angles. 314 00:14:28,006 --> 00:14:28,990 A lot cleaner. 315 00:14:28,990 --> 00:14:32,980 But a little bit less efficient, because instead 316 00:14:32,980 --> 00:14:35,350 of using a disk, which is the minimum amount of paper 317 00:14:35,350 --> 00:14:37,660 you need to make a flap, you're using a square. 318 00:14:37,660 --> 00:14:39,630 So you're wasting those little corners. 319 00:14:39,630 --> 00:14:43,190 But not that much more inefficient. 320 00:14:43,190 --> 00:14:46,940 Cool, so that was that method. 321 00:14:46,940 --> 00:14:50,320 Next question I have is about the triangulation algorithm, 322 00:14:50,320 --> 00:14:53,090 which I didn't even cover in lecture. 323 00:14:53,090 --> 00:14:59,160 So this was you set up-- you have your piece of paper. 324 00:14:59,160 --> 00:15:01,420 And you assign each leaf in this tree 325 00:15:01,420 --> 00:15:03,750 to some point in the piece of paper. 326 00:15:03,750 --> 00:15:06,440 And you satisfy the active path condition 327 00:15:06,440 --> 00:15:09,420 so each of these distances measured on the paper 328 00:15:09,420 --> 00:15:13,040 should be greater than or equal to the distance measured 329 00:15:13,040 --> 00:15:15,650 on the tree, which is the floor of the base. 330 00:15:18,360 --> 00:15:19,520 Great. 331 00:15:19,520 --> 00:15:21,510 And so if you happen to have some equalities, 332 00:15:21,510 --> 00:15:24,230 you draw in active paths. 333 00:15:24,230 --> 00:15:26,100 But what if you don't have any? 334 00:15:26,100 --> 00:15:27,990 Or you don't have enough? 335 00:15:27,990 --> 00:15:32,050 The tree method in its original form 336 00:15:32,050 --> 00:15:36,260 only works when the active paths decompose the piece of paper 337 00:15:36,260 --> 00:15:38,500 into convex polygons. 338 00:15:38,500 --> 00:15:40,580 This is what you'd like to have happen. 339 00:15:40,580 --> 00:15:43,170 Each of the faces here is a convex polygon. 340 00:15:43,170 --> 00:15:44,340 If that happens, great. 341 00:15:44,340 --> 00:15:47,279 You can use the universal molecule 342 00:15:47,279 --> 00:15:48,320 and you get your folding. 343 00:15:48,320 --> 00:15:50,450 If it doesn't happen, you have to modify 344 00:15:50,450 --> 00:15:52,210 your tree a little bit. 345 00:15:52,210 --> 00:15:55,230 There are a couple ways to do this in the TreeMaker. 346 00:15:55,230 --> 00:15:56,770 You can modify these edge lengths 347 00:15:56,770 --> 00:15:59,180 and try to modify them as little as possible 348 00:15:59,180 --> 00:16:01,050 so that things touch. 349 00:16:01,050 --> 00:16:03,290 But the easy way to prove that something 350 00:16:03,290 --> 00:16:06,440 is possible-- because remember, if we add a little bit 351 00:16:06,440 --> 00:16:09,344 to the base, we get an extra flap. 352 00:16:09,344 --> 00:16:10,510 That doesn't really hurt us. 353 00:16:10,510 --> 00:16:12,859 At the end, we can hide the flap. 354 00:16:12,859 --> 00:16:14,900 Just fold it against other flaps and just pretend 355 00:16:14,900 --> 00:16:16,070 it wasn't there. 356 00:16:16,070 --> 00:16:19,250 So as long as we can add flaps in order 357 00:16:19,250 --> 00:16:25,030 to make the active paths decompose into convex polygons, 358 00:16:25,030 --> 00:16:26,040 we're happy. 359 00:16:26,040 --> 00:16:30,580 And in fact, what we prove is that you can keep adding flaps 360 00:16:30,580 --> 00:16:32,200 until you get into triangles. 361 00:16:32,200 --> 00:16:34,390 And triangles are always convex. 362 00:16:34,390 --> 00:16:37,350 And so they make us happy. 363 00:16:37,350 --> 00:16:41,070 So we will end up adding a bunch of flaps in our tree 364 00:16:41,070 --> 00:16:43,524 in order to triangulate with active paths. 365 00:16:43,524 --> 00:16:44,940 And then in each of these, we just 366 00:16:44,940 --> 00:16:48,340 fill in a rabbit ear molecule. 367 00:16:48,340 --> 00:16:49,730 So that's the goal. 368 00:16:49,730 --> 00:16:52,970 Now let me tell you how the triangulation actually works. 369 00:16:56,810 --> 00:16:58,590 It was originally described by Lang. 370 00:16:58,590 --> 00:17:03,670 It's also in the textbook for this class. 371 00:17:03,670 --> 00:17:06,380 I'll try to give an abbreviated version here. 372 00:17:06,380 --> 00:17:09,900 It's a little bit technical, but here's the idea. 373 00:17:09,900 --> 00:17:14,920 So suppose-- all right. 374 00:17:14,920 --> 00:17:18,030 So we've assigned some leaves. 375 00:17:18,030 --> 00:17:22,367 And suppose we have some region that's not convex. 376 00:17:22,367 --> 00:17:23,200 It's not a triangle. 377 00:17:23,200 --> 00:17:25,490 Has more than three sides. 378 00:17:25,490 --> 00:17:29,949 Now, some of these edges may come from active paths, 379 00:17:29,949 --> 00:17:32,240 and some of them may be from the boundary of the paper. 380 00:17:32,240 --> 00:17:35,690 So maybe the paper is here. 381 00:17:35,690 --> 00:17:37,849 So this edge is not active. 382 00:17:37,849 --> 00:17:39,390 It's just the boundary of this region 383 00:17:39,390 --> 00:17:41,020 because it's the boundary of the paper. 384 00:17:41,020 --> 00:17:43,230 So there are two types of edges. 385 00:17:43,230 --> 00:17:47,430 But there should be, I guess, at least one active edge. 386 00:17:47,430 --> 00:17:50,700 And there's at least four edges total. 387 00:17:50,700 --> 00:17:55,020 So what I'm going to do is look at any of the active edges, 388 00:17:55,020 --> 00:17:58,660 and I'm going to imagine-- OK. 389 00:17:58,660 --> 00:18:01,900 So that active edge, this is in the piece of paper. 390 00:18:01,900 --> 00:18:03,920 But now I'm going to think about it in the tree. 391 00:18:03,920 --> 00:18:05,560 So the tree looks like something. 392 00:18:05,560 --> 00:18:08,780 We don't really know what. 393 00:18:08,780 --> 00:18:14,596 Any active path here corresponds to an active path 394 00:18:14,596 --> 00:18:16,220 in the tree of exactly the same length. 395 00:18:16,220 --> 00:18:18,740 So maybe it's from this leaf to this leaf. 396 00:18:18,740 --> 00:18:23,570 So it corresponds to-- use a color. 397 00:18:23,570 --> 00:18:27,380 This guy corresponds to this path. 398 00:18:27,380 --> 00:18:30,020 And the sum of those lengths should be equal to that length. 399 00:18:30,020 --> 00:18:36,080 It isn't, obviously, but there's a scale factor in between here. 400 00:18:36,080 --> 00:18:37,720 Lambda. 401 00:18:37,720 --> 00:18:38,220 OK. 402 00:18:38,220 --> 00:18:39,511 So here's what I'm going to do. 403 00:18:39,511 --> 00:18:44,870 I'm going to modify the tree by adding a new leaf somewhere 404 00:18:44,870 --> 00:18:47,310 off of this path. 405 00:18:47,310 --> 00:18:47,935 OK, where? 406 00:18:50,510 --> 00:18:56,090 For convenience, let's assume that this path length here 407 00:18:56,090 --> 00:18:58,160 is 1. 408 00:18:58,160 --> 00:19:04,290 Then I'm going to measure out some distance x here, 409 00:19:04,290 --> 00:19:09,680 which will leave a different distance, 1 minus x, here. 410 00:19:09,680 --> 00:19:14,530 And then I'm going to make this length, I guess, L. 411 00:19:14,530 --> 00:19:19,380 And I'm going to call this leaf L. Capital L. OK. 412 00:19:19,380 --> 00:19:20,950 This is a modification to a tree. 413 00:19:20,950 --> 00:19:21,650 I can do it. 414 00:19:21,650 --> 00:19:23,720 I can do it for any value of x between 0 and 1. 415 00:19:23,720 --> 00:19:27,640 And i can do it for any value of L greater or equal to 0. 416 00:19:27,640 --> 00:19:30,210 So what I'd like to do is design the tree 417 00:19:30,210 --> 00:19:32,490 so that this point ends up in an interesting place 418 00:19:32,490 --> 00:19:34,680 on the piece of paper. 419 00:19:34,680 --> 00:19:38,870 In fact, I claim that no matter where I draw L here, 420 00:19:38,870 --> 00:19:41,860 capital L, as my desired place. 421 00:19:41,860 --> 00:19:44,382 Now, what I'd really like is for-- I 422 00:19:44,382 --> 00:19:45,590 should give these guys names. 423 00:19:45,590 --> 00:19:51,740 This is called U, or say, in the notes, I call it V and W. 424 00:19:51,740 --> 00:19:57,190 So here we have W and V. What I would 425 00:19:57,190 --> 00:20:02,620 like is for these two paths to also be active. 426 00:20:02,620 --> 00:20:06,610 Meaning the lengths here match the lengths in the tree. 427 00:20:06,610 --> 00:20:10,260 I claim that no matter where I put L in the plane, 428 00:20:10,260 --> 00:20:14,480 anywhere in the plane, I can choose x and choose L 429 00:20:14,480 --> 00:20:17,710 so that these two lengths are exactly correct. 430 00:20:17,710 --> 00:20:18,870 Do you believe me? 431 00:20:21,660 --> 00:20:23,420 Not really. 432 00:20:23,420 --> 00:20:25,882 How many people think this is obvious? 433 00:20:25,882 --> 00:20:26,381 Good. 434 00:20:26,381 --> 00:20:29,339 A couple maybe. 435 00:20:29,339 --> 00:20:30,630 Let me give you a quick sketch. 436 00:20:30,630 --> 00:20:32,495 It's not that interesting, so I just 437 00:20:32,495 --> 00:20:33,620 want to do it very briefly. 438 00:20:37,040 --> 00:20:41,680 If you let x be a free parameter, but fix l, 439 00:20:41,680 --> 00:20:44,760 little l, then what this corresponds to 440 00:20:44,760 --> 00:20:55,870 is, in fact, an ellipse with V and W as foci of the ellipse. 441 00:20:55,870 --> 00:20:59,510 Because you have to hold the sum of these lengths fixed 442 00:20:59,510 --> 00:21:03,740 if you fix-- if you let x vary, this length plus this length 443 00:21:03,740 --> 00:21:07,630 is always going to be the same if you fix l. 444 00:21:07,630 --> 00:21:10,280 And so this is called the major axis, 445 00:21:10,280 --> 00:21:15,055 the sum of these two lengths is going to be like 1 plus 2l. 446 00:21:15,055 --> 00:21:17,440 And so if I have a point, basically, there's 447 00:21:17,440 --> 00:21:19,300 some ellipse of the appropriate size that 448 00:21:19,300 --> 00:21:22,600 passes through L. That will let me choose little l. 449 00:21:22,600 --> 00:21:25,357 And then as I vary x, I walk around the ellipse. 450 00:21:25,357 --> 00:21:26,940 So I just choose the appropriate value 451 00:21:26,940 --> 00:21:29,923 of x that gets me the desired point on the ellipse. 452 00:21:29,923 --> 00:21:32,116 That's it. 453 00:21:32,116 --> 00:21:34,490 The point is, the set of all ellipses with these two foci 454 00:21:34,490 --> 00:21:35,850 spans the entire plane. 455 00:21:35,850 --> 00:21:37,740 So wherever you want to put L, you 456 00:21:37,740 --> 00:21:39,400 can make those two paths active. 457 00:21:39,400 --> 00:21:41,800 OK, this is good, because it makes a little triangle. 458 00:21:41,800 --> 00:21:44,600 But of course, if I just add an arbitrary triangle, not very 459 00:21:44,600 --> 00:21:45,340 interesting. 460 00:21:45,340 --> 00:21:48,130 The good thing is I'm free to put L wherever I want. 461 00:21:48,130 --> 00:21:50,640 I'd really like to put L say here 462 00:21:50,640 --> 00:21:53,020 and draw active paths like that. 463 00:21:53,020 --> 00:21:55,120 Because then I'm kind of decomposing my polygon 464 00:21:55,120 --> 00:21:56,070 into triangles. 465 00:21:56,070 --> 00:21:58,260 That's not always possible. 466 00:21:58,260 --> 00:22:00,300 So let's just go through the cases. 467 00:22:00,300 --> 00:22:02,800 In all cases, we're going to simplify a polygon, 468 00:22:02,800 --> 00:22:04,820 make it have fewer sides. 469 00:22:04,820 --> 00:22:06,840 And that's on the next page. 470 00:22:09,600 --> 00:22:23,687 So the claim is-- well, the L. There's sort of three cases. 471 00:22:23,687 --> 00:22:25,270 Going to reorganize this a little bit. 472 00:22:39,500 --> 00:22:42,630 One thing that would be nice is I can place L-- maybe 473 00:22:42,630 --> 00:22:45,660 I'll draw another version of this. 474 00:22:45,660 --> 00:22:48,990 So I've got UV. 475 00:22:48,990 --> 00:22:52,290 I'll draw the active paths in red. 476 00:22:52,290 --> 00:22:54,090 So we know UV is active. 477 00:22:54,090 --> 00:22:56,810 We've got L. If I could somehow place L 478 00:22:56,810 --> 00:23:03,090 so that it's active with two other points, two other leaves. 479 00:23:03,090 --> 00:23:10,250 This is V and W. This is U and this is T. 480 00:23:10,250 --> 00:23:17,190 If I can do this, and the region looks something like that, 481 00:23:17,190 --> 00:23:18,970 then I'm happy. 482 00:23:18,970 --> 00:23:24,320 I'm going to put L there and add in these four edges. 483 00:23:24,320 --> 00:23:28,475 And essentially, if you look at any one of these regions that 484 00:23:28,475 --> 00:23:29,850 remains, there's a triangle here. 485 00:23:29,850 --> 00:23:31,030 That's definitely OK. 486 00:23:31,030 --> 00:23:32,640 These regions, each of them will have 487 00:23:32,640 --> 00:23:35,014 fewer edges than the original region. 488 00:23:35,014 --> 00:23:36,680 That's pretty easy to prove, because you 489 00:23:36,680 --> 00:23:39,810 have-- Because you're connecting to these four vertices, 490 00:23:39,810 --> 00:23:43,480 this region won't have this vertex or this vertex. 491 00:23:43,480 --> 00:23:46,830 So it has one added vertex and two removed vertices at least, 492 00:23:46,830 --> 00:23:48,880 W and T. So this one will be smaller. 493 00:23:48,880 --> 00:23:51,070 And this is symmetric for all of them. 494 00:23:51,070 --> 00:23:51,570 OK. 495 00:23:51,570 --> 00:23:53,020 So this would be a good case. 496 00:23:53,020 --> 00:23:55,480 So what I'm going to do is try to move L around. 497 00:23:55,480 --> 00:23:58,190 I'm going to start very close to VW 498 00:23:58,190 --> 00:24:00,780 and just start moving off the edge. 499 00:24:00,780 --> 00:24:02,420 Initially, nothing else will be active 500 00:24:02,420 --> 00:24:05,130 because it's basically right on top of VW. 501 00:24:05,130 --> 00:24:09,970 But as I move around, I might get another active path. 502 00:24:09,970 --> 00:24:10,870 OK. 503 00:24:10,870 --> 00:24:14,440 Maybe this is Case 1A. 504 00:24:14,440 --> 00:24:20,550 Before we get there, Case 1 is you get LU active. 505 00:24:20,550 --> 00:24:26,810 Suppose you just to get one additional active path. 506 00:24:26,810 --> 00:24:29,646 The other case is, nothing else becomes active. 507 00:24:29,646 --> 00:24:36,970 So here's V, here's W. Here's L. Here's U. So this is active, 508 00:24:36,970 --> 00:24:40,020 active, active, active. 509 00:24:40,020 --> 00:24:42,680 OK, what I'm going to do in this situation is, 510 00:24:42,680 --> 00:24:46,240 if I want to keep LU active, I can actually 511 00:24:46,240 --> 00:24:48,520 move L on a circle. 512 00:24:51,490 --> 00:24:57,570 As long as L stays on this circle, LU remains active. 513 00:24:57,570 --> 00:24:58,220 OK? 514 00:24:58,220 --> 00:25:00,320 So just move it along the circle. 515 00:25:00,320 --> 00:25:01,920 Now, there are two possibilities. 516 00:25:01,920 --> 00:25:04,120 It could be it becomes active with something else. 517 00:25:04,120 --> 00:25:06,550 Then I'm done. 518 00:25:06,550 --> 00:25:10,572 Or it could be it doesn't, which means it will hit the boundary. 519 00:25:10,572 --> 00:25:13,000 So it will hit a boundary point. 520 00:25:13,000 --> 00:25:15,240 So Case 1A-- so this is Case 1. 521 00:25:15,240 --> 00:25:17,130 You can make something active. 522 00:25:17,130 --> 00:25:18,880 When you move along the circle, either you 523 00:25:18,880 --> 00:25:21,447 make another thing active, or you won't and you 524 00:25:21,447 --> 00:25:22,155 hit the boundary. 525 00:25:24,950 --> 00:25:30,900 Case 1B is you hit the boundary. 526 00:25:30,900 --> 00:25:34,570 And then I claim you're also happy. 527 00:25:34,570 --> 00:25:43,340 So in that situation-- draw it one more time-- 528 00:25:43,340 --> 00:25:46,300 we've got L on the boundary. 529 00:25:46,300 --> 00:25:52,250 And we've got some path here, here, here, here. 530 00:25:52,250 --> 00:25:53,982 These are all active. 531 00:25:53,982 --> 00:25:57,350 So here's L. OK. 532 00:25:57,350 --> 00:25:59,210 I claim again, each of these regions 533 00:25:59,210 --> 00:26:03,206 has fewer vertices than before. 534 00:26:03,206 --> 00:26:05,340 Should be pretty obvious. 535 00:26:05,340 --> 00:26:17,733 This one is emitting W and this was U. Yeah. 536 00:26:17,733 --> 00:26:19,490 It's emitting W and it's emitting 537 00:26:19,490 --> 00:26:21,765 whatever's down here, if anything. 538 00:26:21,765 --> 00:26:23,414 Yeah, I think it's pretty obvious. 539 00:26:23,414 --> 00:26:24,830 You've got to check all the cases. 540 00:26:24,830 --> 00:26:26,410 There's a few details here. 541 00:26:26,410 --> 00:26:29,050 But it should be each of these regions has strictly 542 00:26:29,050 --> 00:26:31,811 fewer vertices than it did before. 543 00:26:31,811 --> 00:26:33,115 And so you're making progress. 544 00:26:33,115 --> 00:26:38,100 If You started with all faces having some number of edges, 545 00:26:38,100 --> 00:26:41,410 each time you do one of these steps, you decrease it. 546 00:26:41,410 --> 00:26:48,440 OK, the last case, Case 2, is you can't make anything active. 547 00:26:53,840 --> 00:26:56,070 This can happen, for example, if you're 548 00:26:56,070 --> 00:26:58,310 in the corner of the piece of paper 549 00:26:58,310 --> 00:27:03,897 and you have an active path. 550 00:27:03,897 --> 00:27:05,480 I guess that's a little less exciting. 551 00:27:05,480 --> 00:27:07,295 Maybe your piece of paper is this shape. 552 00:27:07,295 --> 00:27:09,070 Be a little bit weirder. 553 00:27:09,070 --> 00:27:12,162 So this will work for any convex piece of paper. 554 00:27:12,162 --> 00:27:13,370 So it's not quite a triangle. 555 00:27:13,370 --> 00:27:15,610 And you're moving L around here and you just 556 00:27:15,610 --> 00:27:17,030 can't get another active path. 557 00:27:17,030 --> 00:27:18,590 You have no vertices around. 558 00:27:18,590 --> 00:27:26,050 This is W, this is U-- sorry, this is V. And in this case, 559 00:27:26,050 --> 00:27:27,320 L can go anywhere here. 560 00:27:27,320 --> 00:27:28,600 It's free. 561 00:27:28,600 --> 00:27:30,950 In that case, I'm going to put L on one 562 00:27:30,950 --> 00:27:33,330 of the vertices of the paper. 563 00:27:33,330 --> 00:27:35,820 So I get this active and this active. 564 00:27:35,820 --> 00:27:40,167 And then I've decomposed that region into pieces by, 565 00:27:40,167 --> 00:27:41,500 this is what we call a diagonal. 566 00:27:41,500 --> 00:27:44,410 We're adding a vertex to vertex edge in this polygon. 567 00:27:44,410 --> 00:27:45,930 And that always decreases the number 568 00:27:45,930 --> 00:27:49,920 of sides in each of the subregions. 569 00:27:49,920 --> 00:27:53,660 So if I just keep doing this, taking any region of size 570 00:27:53,660 --> 00:27:55,570 larger than three, I will eventually 571 00:27:55,570 --> 00:27:58,430 reduce them all to have size three. 572 00:27:58,430 --> 00:28:00,460 So they'll all be triangles. 573 00:28:00,460 --> 00:28:04,450 So that's the triangulation algorithm in a nutshell. 574 00:28:04,450 --> 00:28:07,570 You can look at the textbook if you want to see it more detail, 575 00:28:07,570 --> 00:28:10,620 but it's pretty simple. 576 00:28:10,620 --> 00:28:13,264 This proves that something is possible. 577 00:28:13,264 --> 00:28:15,680 I don't think TreeMaker actually implements this algorithm 578 00:28:15,680 --> 00:28:18,170 specifically, but you can kind of do it by hand. 579 00:28:18,170 --> 00:28:19,510 So I have a little example here. 580 00:28:19,510 --> 00:28:20,900 This is TreeMaker. 581 00:28:20,900 --> 00:28:22,470 I drew this really weird tree. 582 00:28:22,470 --> 00:28:25,310 We call it a caterpillar tree because you would actually 583 00:28:25,310 --> 00:28:28,040 use it to make caterpillars. 584 00:28:28,040 --> 00:28:32,110 And if you just plug this into TreeMaker with all the lengths 585 00:28:32,110 --> 00:28:34,600 unit, it will give you this error message. 586 00:28:34,600 --> 00:28:36,160 I couldn't construct all the polygons 587 00:28:36,160 --> 00:28:37,950 because things weren't tight enough. 588 00:28:37,950 --> 00:28:39,866 And if you look carefully-- it's a little hard 589 00:28:39,866 --> 00:28:42,690 to see these colors-- but the light green edges, 590 00:28:42,690 --> 00:28:44,410 those are the active paths. 591 00:28:44,410 --> 00:28:47,810 And here-- so this was a triangle and it was happy. 592 00:28:47,810 --> 00:28:51,290 If you look at this green polygon-- maybe actually 593 00:28:51,290 --> 00:28:55,690 I'll draw with the tablet. 594 00:28:55,690 --> 00:29:05,560 So this thing is an active-- that's 595 00:29:05,560 --> 00:29:07,410 a region bounded by active paths. 596 00:29:07,410 --> 00:29:09,500 And probably the top edge is not an active path. 597 00:29:09,500 --> 00:29:10,540 That's just a boundary. 598 00:29:10,540 --> 00:29:11,870 But it's not convex. 599 00:29:11,870 --> 00:29:13,830 And so you're unhappy. 600 00:29:13,830 --> 00:29:15,880 And so I just kind of eyeball this and say, OK. 601 00:29:15,880 --> 00:29:18,680 Well, probably I should add another leaf here 602 00:29:18,680 --> 00:29:21,957 that would maybe add a disk that will probably fill that in. 603 00:29:21,957 --> 00:29:23,540 So you don't have to be super-precise. 604 00:29:23,540 --> 00:29:25,160 Here we had the very carefully place 605 00:29:25,160 --> 00:29:26,560 the disk to make things touch. 606 00:29:26,560 --> 00:29:28,310 But because TreeMaker's always just trying 607 00:29:28,310 --> 00:29:30,340 to blow things up and make them touch, 608 00:29:30,340 --> 00:29:32,600 it's quite a bit easier in practice. 609 00:29:32,600 --> 00:29:34,440 You just say, OK. 610 00:29:34,440 --> 00:29:39,575 I will add an extra leaf here, like that, to my tree. 611 00:29:39,575 --> 00:29:43,357 And then I hit Optimize, and boom, it works. 612 00:29:43,357 --> 00:29:44,440 In this case, I was lucky. 613 00:29:44,440 --> 00:29:46,850 In general, I might get a nonconvex region. 614 00:29:46,850 --> 00:29:50,800 I'd guess where to add another leaf, and it works. 615 00:29:50,800 --> 00:29:51,989 You can adjust. 616 00:29:51,989 --> 00:29:53,405 Did that leaf have to be that bit, 617 00:29:53,405 --> 00:29:55,030 or could I get away with a smaller one? 618 00:29:55,030 --> 00:29:57,339 But eventually, you will get a base. 619 00:29:57,339 --> 00:29:59,630 And if you want, you could carefully monitor this proof 620 00:29:59,630 --> 00:30:01,527 and actually always succeed. 621 00:30:01,527 --> 00:30:03,360 But in practice, it's usually not that hard. 622 00:30:06,310 --> 00:30:08,880 Questions about that? 623 00:30:08,880 --> 00:30:13,436 So that was the triangulation method in a nutshell. 624 00:30:13,436 --> 00:30:16,675 The next question is about the universal molecule. 625 00:30:16,675 --> 00:30:20,250 And we might spend some more time on this next class. 626 00:30:20,250 --> 00:30:22,540 But I thought we could actually look at this example, 627 00:30:22,540 --> 00:30:27,460 because it's got a bunch of different universal molecules. 628 00:30:27,460 --> 00:30:29,680 So I will continue to draw. 629 00:30:35,510 --> 00:30:41,621 So let me pick, let's say, this universal molecule here. 630 00:30:41,621 --> 00:30:42,495 It's a quadrilateral. 631 00:30:50,141 --> 00:30:50,640 All right. 632 00:30:50,640 --> 00:30:55,180 So those are four active paths in this case. 633 00:30:55,180 --> 00:30:59,040 And it corresponds to some tree. 634 00:30:59,040 --> 00:31:01,360 So I don't have handy here, but it's 635 00:31:01,360 --> 00:31:03,585 going to be a tree with four edges. 636 00:31:06,300 --> 00:31:08,980 With four leaves, sorry. 637 00:31:08,980 --> 00:31:11,590 Tree with four leaves is going to look something like this. 638 00:31:11,590 --> 00:31:13,900 This is a piece of the bigger tree, which 639 00:31:13,900 --> 00:31:17,900 you recall looked something like this. 640 00:31:17,900 --> 00:31:20,730 And we added one somewhere. 641 00:31:20,730 --> 00:31:22,910 But this particular quadrilateral 642 00:31:22,910 --> 00:31:27,070 is doing some particular subtree on four leaves. 643 00:31:27,070 --> 00:31:30,020 In particular, there are four leaves here, 644 00:31:30,020 --> 00:31:31,950 correspond to four leaves here. 645 00:31:31,950 --> 00:31:33,020 I don't know which ones. 646 00:31:33,020 --> 00:31:34,710 I don't really need to know. 647 00:31:34,710 --> 00:31:38,991 TreeMaker keeps track of it for me. 648 00:31:38,991 --> 00:31:41,240 And I happen to know that these four paths are active. 649 00:31:41,240 --> 00:31:44,710 Meaning the lengths in the plane here match exactly 650 00:31:44,710 --> 00:31:48,540 the lengths as measured along the tree. 651 00:31:48,540 --> 00:31:51,330 So what I do to make this work-- this 652 00:31:51,330 --> 00:31:54,240 is basically the floor of the molecule. 653 00:31:54,240 --> 00:31:59,320 Those four paths will all lie on the ground level. 654 00:31:59,320 --> 00:32:03,750 What I'm trying to do is slice higher and higher in the base 655 00:32:03,750 --> 00:32:06,010 to see what happens when I slice. 656 00:32:06,010 --> 00:32:11,640 And if you think about it, as you move up here, in the plane 657 00:32:11,640 --> 00:32:15,570 this corresponds to moving parallel to these edges. 658 00:32:15,570 --> 00:32:21,710 So I'm shrinking this polygon by parallel offsets. 659 00:32:21,710 --> 00:32:24,005 So let me draw one parallel offset. 660 00:32:28,330 --> 00:32:29,615 Approximately parallel. 661 00:32:33,580 --> 00:32:37,090 OK, after I shrink a little bit, might look like that. 662 00:32:37,090 --> 00:32:40,840 An interesting moment in this case is here. 663 00:32:40,840 --> 00:32:43,445 Let me draw this carefully. 664 00:32:43,445 --> 00:32:47,000 It should go-- this should go here. 665 00:32:50,430 --> 00:32:52,835 Ooh. 666 00:32:52,835 --> 00:32:54,615 Oops. 667 00:32:54,615 --> 00:32:55,720 I'm going to start over. 668 00:33:00,520 --> 00:33:02,417 This is actually already drawn for us. 669 00:33:02,417 --> 00:33:03,000 It's this guy. 670 00:33:08,115 --> 00:33:10,280 OK, this is a parallel offset. 671 00:33:10,280 --> 00:33:12,390 And in this case, something interesting 672 00:33:12,390 --> 00:33:16,450 happens, which is this path. 673 00:33:16,450 --> 00:33:18,940 So it turns out this middle edge, in this case, 674 00:33:18,940 --> 00:33:21,404 becomes active. 675 00:33:21,404 --> 00:33:22,820 So what that means is, originally, 676 00:33:22,820 --> 00:33:27,070 if you looked at these two points, these two leaves, 677 00:33:27,070 --> 00:33:29,680 they had the wrong length. 678 00:33:29,680 --> 00:33:30,180 OK? 679 00:33:30,180 --> 00:33:32,020 So that's going to correspond to something like these two 680 00:33:32,020 --> 00:33:32,519 leaves. 681 00:33:32,519 --> 00:33:34,460 They're opposite corners of the quad. 682 00:33:34,460 --> 00:33:38,920 And you measure the length along the tree, you get something. 683 00:33:38,920 --> 00:33:41,860 And the claim was, in the plane, it was too big. 684 00:33:41,860 --> 00:33:43,670 Now, as you do this parallel offset, 685 00:33:43,670 --> 00:33:46,520 as you shrink the polygon, at some point 686 00:33:46,520 --> 00:33:48,830 it will be just the right length. 687 00:33:48,830 --> 00:33:49,580 This could happen. 688 00:33:49,580 --> 00:33:51,230 Doesn't have to happen. 689 00:33:51,230 --> 00:33:52,890 If it happens, you have to stop. 690 00:33:52,890 --> 00:33:55,160 Because if you kept going, it would become too short. 691 00:33:55,160 --> 00:33:57,040 And we know every length here must 692 00:33:57,040 --> 00:34:01,660 be greater than or equal to length over here. 693 00:34:01,660 --> 00:34:04,175 So when it becomes equal, we have to stop, 694 00:34:04,175 --> 00:34:06,120 and we have to make this a crease. 695 00:34:06,120 --> 00:34:08,270 Making it a crease basically says, 696 00:34:08,270 --> 00:34:10,219 ah, this is exactly the right length. 697 00:34:10,219 --> 00:34:11,960 I have to split here. 698 00:34:11,960 --> 00:34:15,969 And where'd my chalk go? 699 00:34:15,969 --> 00:34:21,100 And I have to sort of grow now two different flaps 700 00:34:21,100 --> 00:34:22,179 from that point. 701 00:34:22,179 --> 00:34:24,590 The crease sort of makes it horizontal. 702 00:34:24,590 --> 00:34:27,050 And then we end up shrinking in two different parts. 703 00:34:27,050 --> 00:34:30,040 We shrink in this triangle, which gives us a rabbit ear. 704 00:34:30,040 --> 00:34:31,954 And we shrink up there in that triangle. 705 00:34:31,954 --> 00:34:33,120 Gives us another rabbit ear. 706 00:34:33,120 --> 00:34:35,389 So it will end up giving these angular bisectors. 707 00:34:35,389 --> 00:34:37,330 In general, as you do the shrinking, 708 00:34:37,330 --> 00:34:39,090 you watch where the vertices go. 709 00:34:39,090 --> 00:34:40,610 Those are your ridge creases. 710 00:34:40,610 --> 00:34:42,840 So this one went along an angular bisector. 711 00:34:42,840 --> 00:34:44,210 That one did as well. 712 00:34:44,210 --> 00:34:46,360 But once we do this split operation, 713 00:34:46,360 --> 00:34:49,350 because of this newly active path, which we also 714 00:34:49,350 --> 00:34:52,280 called a gusset in the universal molecule, 715 00:34:52,280 --> 00:34:55,379 then these vertices actually split and go in two directions. 716 00:34:55,379 --> 00:34:57,170 One for this thing angular bisector and one 717 00:34:57,170 --> 00:34:59,990 for that angular bisector. 718 00:34:59,990 --> 00:35:03,307 So in general, for a universal molecule, 719 00:35:03,307 --> 00:35:04,890 there, are two things that can happen. 720 00:35:17,532 --> 00:35:19,630 You can be shrinking your polygon 721 00:35:19,630 --> 00:35:22,370 and suddenly discover there's a newly active path. 722 00:35:22,370 --> 00:35:24,600 And then you have to divide it into two polygons 723 00:35:24,600 --> 00:35:29,060 and start shrinking those separately. 724 00:35:29,060 --> 00:35:32,950 Or vertices can disappear. 725 00:35:32,950 --> 00:35:35,020 So it could be you're just shrinking, shrinking, 726 00:35:35,020 --> 00:35:36,970 shrinking merrily along the way. 727 00:35:36,970 --> 00:35:38,620 And then suddenly, these two vertices 728 00:35:38,620 --> 00:35:41,110 collide with each other. 729 00:35:41,110 --> 00:35:43,100 So you get to ridge creases here. 730 00:35:43,100 --> 00:35:45,810 In this case, you just treat those two vertices now as one. 731 00:35:45,810 --> 00:35:48,800 So you'll start shrinking like this. 732 00:35:48,800 --> 00:35:52,382 And so these vertices will now start going that way. 733 00:35:52,382 --> 00:35:53,840 So in general, as you're shrinking, 734 00:35:53,840 --> 00:35:56,131 these are the only two types of events that can happen. 735 00:35:56,131 --> 00:35:58,990 Either two vertices merge, or you get a new diagonal here 736 00:35:58,990 --> 00:35:59,990 that becomes active. 737 00:36:03,520 --> 00:36:05,550 When that happens, you just divide. 738 00:36:05,550 --> 00:36:07,810 Now, this will work for any convex polygon 739 00:36:07,810 --> 00:36:10,800 and you get your molecule. 740 00:36:10,800 --> 00:36:13,370 It's hard to just draw a picture of this 741 00:36:13,370 --> 00:36:15,102 without using a computer tool or really 742 00:36:15,102 --> 00:36:16,560 keeping track of what you're doing. 743 00:36:16,560 --> 00:36:19,201 Because how do you tell when something becomes active? 744 00:36:19,201 --> 00:36:20,450 You have to look at your tree. 745 00:36:20,450 --> 00:36:21,400 You have to measure lengths. 746 00:36:21,400 --> 00:36:23,270 It's tricky to do without a computer tool. 747 00:36:23,270 --> 00:36:25,110 And that's why TreeMaker was made. 748 00:36:25,110 --> 00:36:27,174 But it can be done. 749 00:36:27,174 --> 00:36:28,840 Probably even easier with physical paper 750 00:36:28,840 --> 00:36:33,230 if you know exactly what lengths you're trying to match. 751 00:36:33,230 --> 00:36:36,810 So that was a quick overview of how universal molecule works 752 00:36:36,810 --> 00:36:37,840 in more detail. 753 00:36:42,170 --> 00:36:46,420 Next, I want to talk about a few different open problems. 754 00:36:46,420 --> 00:36:51,860 So one of them was-- these are called gift wrapping problems. 755 00:36:51,860 --> 00:36:55,500 So we have things like given a square, a unit square, 756 00:36:55,500 --> 00:36:58,210 let's say, what's the largest regular tetrahedron you 757 00:36:58,210 --> 00:36:59,080 can wrap? 758 00:36:59,080 --> 00:36:59,950 That's still open. 759 00:36:59,950 --> 00:37:02,283 All these problems are still open except for the squared 760 00:37:02,283 --> 00:37:04,840 to a cube which talked about in lecture. 761 00:37:04,840 --> 00:37:08,050 And I realize equilateral triangle to tetrahedron. 762 00:37:08,050 --> 00:37:10,161 That's also really easy. 763 00:37:10,161 --> 00:37:11,910 If you can do it without any paper wasted, 764 00:37:11,910 --> 00:37:12,872 just clearly optimal. 765 00:37:12,872 --> 00:37:14,580 But it would be very cool to study these. 766 00:37:14,580 --> 00:37:17,390 I think these would make a good project or open problem 767 00:37:17,390 --> 00:37:20,350 for the open problem session. 768 00:37:20,350 --> 00:37:24,600 Folding a given rectangle with given aspect ratio into a cube. 769 00:37:24,600 --> 00:37:25,704 Also open. 770 00:37:25,704 --> 00:37:27,370 Pretty much any version you can think of 771 00:37:27,370 --> 00:37:32,270 is open except the ones that we've already seen. 772 00:37:32,270 --> 00:37:36,130 Also wanted to mention for checkerboards, a fun problem, 773 00:37:36,130 --> 00:37:38,000 kind of in this spirit, is, what would 774 00:37:38,000 --> 00:37:41,250 be the best way to fold a two by two checkerboard? 775 00:37:41,250 --> 00:37:43,140 Even for a two by two, we have no idea 776 00:37:43,140 --> 00:37:44,860 how to argue any kind of lower bound 777 00:37:44,860 --> 00:37:47,620 about how bad you must do. 778 00:37:47,620 --> 00:37:50,480 That you can't just take a square and fold a two by two 779 00:37:50,480 --> 00:37:52,410 checkerboard of the same size. 780 00:37:52,410 --> 00:37:54,181 Surely you can't do that. 781 00:37:54,181 --> 00:37:55,680 But we don't know how to prove that. 782 00:37:55,680 --> 00:37:57,270 So it would be a nice target. 783 00:37:57,270 --> 00:37:58,700 We obviously have upper bounds. 784 00:37:58,700 --> 00:38:01,074 We have constructions that make two by two checkerboards. 785 00:38:01,074 --> 00:38:03,110 I think from a 3 by 3 grid, you can make a two 786 00:38:03,110 --> 00:38:04,480 by two checkerboard. 787 00:38:04,480 --> 00:38:06,580 But is that optimal? 788 00:38:06,580 --> 00:38:07,410 We have no idea. 789 00:38:10,720 --> 00:38:13,236 So a couple questions about the checkerboard folding. 790 00:38:13,236 --> 00:38:14,610 So we had this picture of how you 791 00:38:14,610 --> 00:38:16,487 would take a square root of paper-- you 792 00:38:16,487 --> 00:38:17,570 start with a square paper. 793 00:38:17,570 --> 00:38:21,040 You fold into this shape with these long slits. 794 00:38:21,040 --> 00:38:22,490 And then these guys fold over. 795 00:38:22,490 --> 00:38:24,440 And then there are also tab sticking up here, 796 00:38:24,440 --> 00:38:25,440 which are not drawn. 797 00:38:25,440 --> 00:38:27,835 And they fall over and give you the color reversal. 798 00:38:27,835 --> 00:38:29,960 Question was, how do you actually build this thing? 799 00:38:29,960 --> 00:38:31,126 This does not look uniaxial. 800 00:38:31,126 --> 00:38:34,140 And indeed, we do not use uniaxial techniques for this. 801 00:38:34,140 --> 00:38:36,220 We use a separate set of gadgets which 802 00:38:36,220 --> 00:38:37,665 are kind of based on pleating. 803 00:38:37,665 --> 00:38:39,790 And this will actually be very similar to something 804 00:38:39,790 --> 00:38:41,248 we see in a couple of lectures when 805 00:38:41,248 --> 00:38:45,410 we do box pleating to make cubes and stuff. 806 00:38:45,410 --> 00:38:47,950 But you just compose these gadgets. 807 00:38:47,950 --> 00:38:50,670 So this is kind of a general tab gadget. 808 00:38:50,670 --> 00:38:52,340 You collapse this crease pattern, 809 00:38:52,340 --> 00:38:53,710 and do a couple more folds. 810 00:38:53,710 --> 00:38:55,700 And you end up with this tab sticking out here, 811 00:38:55,700 --> 00:38:59,070 and you can flip it up or down. 812 00:38:59,070 --> 00:39:02,365 And you see that it has these pleats running off to the side. 813 00:39:02,365 --> 00:39:04,690 This mountain valley, mountain valley, and then 814 00:39:04,690 --> 00:39:06,850 valley mountain, valley mountain. 815 00:39:06,850 --> 00:39:10,580 And so those have to go all the way through the paper. 816 00:39:10,580 --> 00:39:14,070 And they come into this construction, which basically 817 00:39:14,070 --> 00:39:16,530 lets you make a big slit in the paper. 818 00:39:16,530 --> 00:39:18,770 Or you can also turn a corner. 819 00:39:18,770 --> 00:39:20,857 And so you just prove that these gadgets compose. 820 00:39:20,857 --> 00:39:22,440 I mean, it's pretty that they compose. 821 00:39:22,440 --> 00:39:25,150 You just have to analyze how much of the paper you're using. 822 00:39:25,150 --> 00:39:27,212 And it turns out to be really good. 823 00:39:27,212 --> 00:39:28,200 That a short version. 824 00:39:28,200 --> 00:39:33,070 If you want to try it out, fold some gadgets. 825 00:39:33,070 --> 00:39:37,760 Fun project idea from you guys is, 826 00:39:37,760 --> 00:39:40,120 given an image, sample at low resolution, 827 00:39:40,120 --> 00:39:41,350 make it black and white. 828 00:39:41,350 --> 00:39:43,450 And then come up with the crease pattern 829 00:39:43,450 --> 00:39:46,080 that would fold, by this checkerboard technique, 830 00:39:46,080 --> 00:39:50,210 into exactly that two color pixel pattern. 831 00:39:50,210 --> 00:39:52,440 You could make all sorts of useful things 832 00:39:52,440 --> 00:39:55,544 like Space Invaders and stuff like that. 833 00:39:55,544 --> 00:39:57,210 You'd never want to fold them, I think-- 834 00:39:57,210 --> 00:39:59,430 although maybe with low enough resolution you could fold them. 835 00:39:59,430 --> 00:40:01,513 But I think they'd be cool just as crease patterns 836 00:40:01,513 --> 00:40:02,650 by themselves. 837 00:40:02,650 --> 00:40:05,260 You input a different pattern-- I'd like web applet. 838 00:40:05,260 --> 00:40:06,790 You change the pixel pattern. 839 00:40:06,790 --> 00:40:08,800 And boom, it gives you the crease pattern. 840 00:40:08,800 --> 00:40:09,840 I think this would be a fun project 841 00:40:09,840 --> 00:40:11,298 if you're into implementing things. 842 00:40:11,298 --> 00:40:13,260 It's essentially the algorithm we already have, 843 00:40:13,260 --> 00:40:18,990 but it needs to be just written out in detail and coded up. 844 00:40:18,990 --> 00:40:20,575 OK. 845 00:40:20,575 --> 00:40:22,700 Here's what it looks like to fold an eight by eight 846 00:40:22,700 --> 00:40:23,949 checkerboard with this method. 847 00:40:23,949 --> 00:40:27,210 These are all by Robert Lang. 848 00:40:27,210 --> 00:40:30,155 So a lot of precreasing along this huge grid. 849 00:40:30,155 --> 00:40:32,826 I thinks it's roughly 48 by 48. 850 00:40:32,826 --> 00:40:34,200 And then some tape to hold things 851 00:40:34,200 --> 00:40:37,370 shut, mostly for photographing. 852 00:40:37,370 --> 00:40:40,770 And to collapse, now we're down by a factor of two or so. 853 00:40:40,770 --> 00:40:45,170 Now you can see the slits in one direction. 854 00:40:45,170 --> 00:40:47,960 I think the particular method being used here only 855 00:40:47,960 --> 00:40:49,461 has slits in one direction. 856 00:40:49,461 --> 00:40:49,960 Oh, no. 857 00:40:49,960 --> 00:40:52,251 Here we've got slits in the vertical direction as well. 858 00:40:52,251 --> 00:40:54,500 You could see the tabs sticking out here. 859 00:40:54,500 --> 00:40:56,180 They're going to fall down. 860 00:40:56,180 --> 00:40:58,640 There's some more of the tabs. 861 00:40:58,640 --> 00:41:01,100 Start folding over, and boom, you've 862 00:41:01,100 --> 00:41:04,170 got your eight by eight checkerboard. 863 00:41:04,170 --> 00:41:05,390 Robert Lang says, wow. 864 00:41:05,390 --> 00:41:09,180 That was not one of the easier things I've done. 865 00:41:09,180 --> 00:41:13,200 And this is folded from a 48 by 42 checkerboard. 866 00:41:13,200 --> 00:41:16,560 In principle, this method can go down to 36 plus epsilon 867 00:41:16,560 --> 00:41:20,360 by 36 plus epsilon, but it gets messier crease pattern-wise. 868 00:41:20,360 --> 00:41:25,260 And 36 would be better than the best known eight by eight board 869 00:41:25,260 --> 00:41:26,165 if you want seamless. 870 00:41:26,165 --> 00:41:27,840 Notice these are seamless squares. 871 00:41:27,840 --> 00:41:30,634 No crease lines through them. 872 00:41:30,634 --> 00:41:32,300 This one's not necessarily-- this is not 873 00:41:32,300 --> 00:41:35,190 better than known techniques, but it follows our algorithm. 874 00:41:35,190 --> 00:41:39,500 And that's what the crease pattern looks like. 875 00:41:39,500 --> 00:41:40,930 Cool. 876 00:41:40,930 --> 00:41:43,210 Next we go to Origamizer. 877 00:41:43,210 --> 00:41:45,740 So as I mentioned in lecture, there's 878 00:41:45,740 --> 00:41:47,140 two versions of Origamizer. 879 00:41:47,140 --> 00:41:50,210 There's the version that's implemented in the software, 880 00:41:50,210 --> 00:41:52,930 and there's a version that we're proving is always correct. 881 00:41:52,930 --> 00:41:55,221 These are different because the version in the software 882 00:41:55,221 --> 00:41:57,100 doesn't always work. 883 00:41:57,100 --> 00:41:59,480 And it remains that way because the software 884 00:41:59,480 --> 00:42:00,650 version is more practical. 885 00:42:03,480 --> 00:42:05,479 And also, the theoretical version 886 00:42:05,479 --> 00:42:06,520 is still a moving target. 887 00:42:06,520 --> 00:42:10,909 We change it every few weeks to fix part of the proof. 888 00:42:10,909 --> 00:42:12,700 It's almost done, but it's been almost done 889 00:42:12,700 --> 00:42:13,616 for a couple of years. 890 00:42:13,616 --> 00:42:14,890 So we're still working on it. 891 00:42:14,890 --> 00:42:16,806 Tomohiro was just visiting a couple weeks ago, 892 00:42:16,806 --> 00:42:18,210 as I mentioned. 893 00:42:18,210 --> 00:42:19,570 And we're closing in. 894 00:42:19,570 --> 00:42:24,870 I actually have with me the current draft of the paper. 895 00:42:24,870 --> 00:42:29,100 And it's not nearly as long as this one with the tree method, 896 00:42:29,100 --> 00:42:31,160 but it's still growing and working up, 897 00:42:31,160 --> 00:42:34,870 making sure all the details check out. 898 00:42:34,870 --> 00:42:37,020 And the theoretical version is kind of complicated, 899 00:42:37,020 --> 00:42:38,880 so I thought, in particular, given this question, 900 00:42:38,880 --> 00:42:40,421 we'll talk about the software version 901 00:42:40,421 --> 00:42:42,130 because it's actually a lot simpler. 902 00:42:42,130 --> 00:42:43,906 Only catch is, it doesn't always work. 903 00:42:43,906 --> 00:42:45,530 And it's described in this paper if you 904 00:42:45,530 --> 00:42:51,250 want to read it, by Tomohiro, about Origamizer 2010. 905 00:42:51,250 --> 00:42:53,640 And I have a few figures from that paper. 906 00:42:53,640 --> 00:42:55,860 Well, this is still from Tomohiro's Flickr. 907 00:42:55,860 --> 00:42:57,870 So we had this example where you wanted 908 00:42:57,870 --> 00:42:59,764 to fold this hyperbolic paraboloid. 909 00:42:59,764 --> 00:43:01,680 And here's what the crease pattern looks like. 910 00:43:01,680 --> 00:43:02,960 It's actually pretty simple. 911 00:43:02,960 --> 00:43:05,220 You've got these white polygons, which 912 00:43:05,220 --> 00:43:07,760 are polygons from the surface. 913 00:43:07,760 --> 00:43:09,630 That's what you need to fold. 914 00:43:09,630 --> 00:43:12,630 And the goal is to lay them out on the piece of paper-- 915 00:43:12,630 --> 00:43:15,120 this square is the piece of paper-- 916 00:43:15,120 --> 00:43:18,200 so that I can-- for example, I need 917 00:43:18,200 --> 00:43:20,330 to bring this polygon to touch this polygon. 918 00:43:20,330 --> 00:43:22,171 This edge has to touch this edge. 919 00:43:22,171 --> 00:43:23,670 Wouldn't it be great if I could just 920 00:43:23,670 --> 00:43:25,790 fold the bisector of those two edges, 921 00:43:25,790 --> 00:43:28,260 and this would come right onto here? 922 00:43:28,260 --> 00:43:30,010 Sometimes that happens, but for example, 923 00:43:30,010 --> 00:43:32,750 if this polygon is way down here, that won't happen. 924 00:43:32,750 --> 00:43:35,549 It will fold over and they won't align. 925 00:43:35,549 --> 00:43:36,340 There's two issues. 926 00:43:36,340 --> 00:43:38,150 First you have to get the angles to match. 927 00:43:38,150 --> 00:43:40,960 And also there's this vertical shifting. 928 00:43:40,960 --> 00:43:46,700 If you place the polygons in the plane in a good way, 929 00:43:46,700 --> 00:43:47,950 this will work. 930 00:43:47,950 --> 00:43:49,990 You always get alignment. 931 00:43:49,990 --> 00:43:54,720 And what Origamizer implements is nonconvex optimization, 932 00:43:54,720 --> 00:43:59,590 a constraint projection, to make all of the edges work. 933 00:43:59,590 --> 00:44:00,840 And sometimes that's possible. 934 00:44:00,840 --> 00:44:02,980 It's actually possible fairly often. 935 00:44:02,980 --> 00:44:05,130 And that's when the Origamizer software works. 936 00:44:05,130 --> 00:44:08,750 If it's not possible, you're screwed. 937 00:44:08,750 --> 00:44:10,920 This particular method won't work. 938 00:44:10,920 --> 00:44:13,250 But when it's possible, things are great. 939 00:44:13,250 --> 00:44:15,627 So essentially, you have to bring the edges together. 940 00:44:15,627 --> 00:44:17,710 Then you also have to bring the vertices together. 941 00:44:17,710 --> 00:44:21,130 Like these four vertices come together to a point. 942 00:44:21,130 --> 00:44:24,540 And so you get two kinds of gadgets, 943 00:44:24,540 --> 00:44:27,100 which are the vertex tucking molecule to bring vertices 944 00:44:27,100 --> 00:44:29,640 together, and edge tucking molecule 945 00:44:29,640 --> 00:44:30,820 to bring two edges together. 946 00:44:30,820 --> 00:44:32,920 Edge tucking molecule is a single crease. 947 00:44:32,920 --> 00:44:36,200 Trivial in this construction. 948 00:44:36,200 --> 00:44:38,500 Vertex tucking molecule is complicated. 949 00:44:38,500 --> 00:44:42,980 And in general, what happens is for these points, 950 00:44:42,980 --> 00:44:46,080 we construct what's called a Voronoi diagram, which 951 00:44:46,080 --> 00:44:48,590 is essentially, for each of these points, 952 00:44:48,590 --> 00:44:52,110 if you grow a disk at equal speeds around all of them, 953 00:44:52,110 --> 00:44:55,090 and when the disks meet, you stop them. 954 00:44:55,090 --> 00:44:57,010 So then these two disks will meet along 955 00:44:57,010 --> 00:44:58,890 this perpendicular bisector. 956 00:44:58,890 --> 00:45:02,650 And they'll keep growing until they kind of all die out. 957 00:45:02,650 --> 00:45:05,270 And you'll get this tree structure, in general. 958 00:45:05,270 --> 00:45:07,030 Those are your main creases. 959 00:45:07,030 --> 00:45:12,100 You follow that structure and you do some stuff. 960 00:45:12,100 --> 00:45:14,940 Essentially, what happens when you fold along this Voronoi 961 00:45:14,940 --> 00:45:19,140 diagram, you also have to add in these creases from the Voronoi 962 00:45:19,140 --> 00:45:20,980 diagram to the points. 963 00:45:20,980 --> 00:45:25,530 When you do that, you'll get a kind of mushrooming structure 964 00:45:25,530 --> 00:45:27,770 that has too much material. 965 00:45:27,770 --> 00:45:32,120 If you set it up right, all the angles of paper that you have 966 00:45:32,120 --> 00:45:35,360 are larger than what you need in the folded state, which 967 00:45:35,360 --> 00:45:37,440 would look like this. 968 00:45:37,440 --> 00:45:41,010 And so what you do are lots of little pleats to reduce 969 00:45:41,010 --> 00:45:41,740 the angle. . 970 00:45:41,740 --> 00:45:43,324 If you've got a big angle of paper, 971 00:45:43,324 --> 00:45:44,740 you can do a mountain and a valley 972 00:45:44,740 --> 00:45:46,370 and make it a smaller angle. 973 00:45:46,370 --> 00:45:48,210 And that's what all these creases are doing. 974 00:45:48,210 --> 00:45:51,570 You've got these pleats to reduce the angle here, 975 00:45:51,570 --> 00:45:55,380 reduce the amount of material here and here in order 976 00:45:55,380 --> 00:45:57,312 to match the 3D structure. 977 00:45:57,312 --> 00:45:58,770 So overall, what the algorithm does 978 00:45:58,770 --> 00:46:00,990 is first, given the surface, it constructs 979 00:46:00,990 --> 00:46:05,070 a suitable 3D structure of where to put all the extra tabs. 980 00:46:05,070 --> 00:46:08,419 Then it designs things so that the angles are bigger 981 00:46:08,419 --> 00:46:09,960 here than what they need to be there. 982 00:46:09,960 --> 00:46:12,250 This is, again, a constraint projection. 983 00:46:12,250 --> 00:46:17,430 Also constrains these edges to meet up perfectly. 984 00:46:17,430 --> 00:46:20,440 And then it just applies all these crimps-- 985 00:46:20,440 --> 00:46:23,550 it's easier in a computer than to see it here-- 986 00:46:23,550 --> 00:46:26,140 to make all the angles correct to match the 3D model. 987 00:46:26,140 --> 00:46:27,169 Yeah, question? 988 00:46:27,169 --> 00:46:28,765 AUDIENCE: So on the sheet metal one 989 00:46:28,765 --> 00:46:30,361 that you showed, do you just cut out 990 00:46:30,361 --> 00:46:31,750 the vertex shrinking [INAUDIBLE]? 991 00:46:31,750 --> 00:46:32,458 PROFESSOR: Right. 992 00:46:32,458 --> 00:46:36,520 So for the sheet metal bunny, we just cut out these polygons. 993 00:46:36,520 --> 00:46:40,060 Because otherwise, it's a mess and you get lots of layers. 994 00:46:40,060 --> 00:46:42,570 And the point was to make it of one sheet of material, 995 00:46:42,570 --> 00:46:44,460 but not necessarily a square. 996 00:46:44,460 --> 00:46:46,250 So if you have really thick material, 997 00:46:46,250 --> 00:46:48,124 recommend cutting those out. 998 00:46:48,124 --> 00:46:50,290 And we were laser cutting to score the lines anyway, 999 00:46:50,290 --> 00:46:52,330 so why not cut out some holes as well. 1000 00:46:52,330 --> 00:46:53,310 Yeah. 1001 00:46:53,310 --> 00:46:55,640 So that way, you just have edge tucking molecules, 1002 00:46:55,640 --> 00:46:56,556 which are really easy. 1003 00:46:56,556 --> 00:46:58,352 Mountain, valley, mountain. 1004 00:46:58,352 --> 00:46:59,935 And so it actually is pretty practical 1005 00:46:59,935 --> 00:47:04,310 to fold these things out of generalized sheets with holes. 1006 00:47:04,310 --> 00:47:08,700 So the non-software version of Origamizer 1007 00:47:08,700 --> 00:47:13,670 works the same way, except it does the full generality 1008 00:47:13,670 --> 00:47:16,480 of constructing this tuck proxy, where the tabs should go. 1009 00:47:16,480 --> 00:47:18,370 And that can get very messy in general. 1010 00:47:18,370 --> 00:47:19,737 It's just tedious to explain. 1011 00:47:19,737 --> 00:47:21,570 There are all these crazy spherical diagrams 1012 00:47:21,570 --> 00:47:24,010 that you saw in lecture. 1013 00:47:24,010 --> 00:47:26,770 But it's not that exciting. 1014 00:47:26,770 --> 00:47:29,390 The more interesting part is that the edge tucking molecules 1015 00:47:29,390 --> 00:47:31,920 can't just be a single crease anymore. 1016 00:47:31,920 --> 00:47:34,890 They have to, in general, kind of follow some path 1017 00:47:34,890 --> 00:47:37,050 to get to where they need to go. 1018 00:47:37,050 --> 00:47:37,790 But it's similar. 1019 00:47:37,790 --> 00:47:41,030 You just do a bunch of different folds. 1020 00:47:41,030 --> 00:47:43,180 You can sweep them one way or the other 1021 00:47:43,180 --> 00:47:46,430 in order to navigate this edge to be where it needs to be. 1022 00:47:46,430 --> 00:47:49,280 So again you place the faces in the plane. 1023 00:47:49,280 --> 00:47:51,890 In the mathematical version, you can place them 1024 00:47:51,890 --> 00:47:53,331 anywhere you want. 1025 00:47:53,331 --> 00:47:54,080 It doesn't matter. 1026 00:47:54,080 --> 00:47:56,140 And then you have to shrink them until they're 1027 00:47:56,140 --> 00:47:58,030 small enough that things work. 1028 00:47:58,030 --> 00:48:01,880 Then you route these paths to get from each edge 1029 00:48:01,880 --> 00:48:03,150 to each corresponding edge. 1030 00:48:06,201 --> 00:48:08,450 The edge tucking molecules are fairly straightforward. 1031 00:48:08,450 --> 00:48:11,390 The vertex tucking molecules become even messier. 1032 00:48:11,390 --> 00:48:14,830 We still use a Voronoi diagram in the end. 1033 00:48:14,830 --> 00:48:16,530 It's just there's a lot more points. 1034 00:48:16,530 --> 00:48:19,420 Here we're just using the corners of the triangles 1035 00:48:19,420 --> 00:48:21,560 as the source points for growing stuff. 1036 00:48:21,560 --> 00:48:23,060 In the general case, you have to add 1037 00:48:23,060 --> 00:48:24,880 lots of points in the middle too. 1038 00:48:24,880 --> 00:48:26,334 But it turns out not to matter. 1039 00:48:26,334 --> 00:48:27,750 You could throw in tons of points. 1040 00:48:27,750 --> 00:48:29,470 You could always fold a Voronoi diagram. 1041 00:48:29,470 --> 00:48:31,510 You get this mushroomy thing. 1042 00:48:31,510 --> 00:48:34,460 And then you can just fold away the parts you don't need. 1043 00:48:34,460 --> 00:48:36,750 You have to make them really small 1044 00:48:36,750 --> 00:48:38,977 in order to make these tabs not too big, 1045 00:48:38,977 --> 00:48:41,560 because if the tabs are too big, they collide with each other. 1046 00:48:41,560 --> 00:48:43,290 That already happened in the software version. 1047 00:48:43,290 --> 00:48:44,706 Even in the software version, this 1048 00:48:44,706 --> 00:48:47,400 might not work because this tab may be huge. 1049 00:48:47,400 --> 00:48:50,410 But you can-- in this case, you just add a few more 1050 00:48:50,410 --> 00:48:55,340 and make this pleat up and down, and you'll avoid collision. 1051 00:48:55,340 --> 00:48:58,440 So that was a quick, but a little bit more clear, 1052 00:48:58,440 --> 00:49:01,971 overview of Origamizer and how it works. 1053 00:49:01,971 --> 00:49:03,100 Any more questions? 1054 00:49:08,080 --> 00:49:09,280 Cool. 1055 00:49:09,280 --> 00:49:11,360 That's it for today.