WEBVTT 00:00:03.390 --> 00:00:05.270 PROFESSOR: All right, let's get started. 00:00:05.270 --> 00:00:08.290 If you haven't already, there's two handouts on the left, 00:00:08.290 --> 00:00:11.520 and you should take two pieces of paper. 00:00:11.520 --> 00:00:13.330 So we'll doing some actual origami folding. 00:00:13.330 --> 00:00:18.180 We'll be folding 6.849 today, just like this. 00:00:18.180 --> 00:00:22.500 It'll only take us eight hours or so. 00:00:22.500 --> 00:00:24.170 This is the Jenny and Eli folding 00:00:24.170 --> 00:00:25.430 you've seen on the poster. 00:00:25.430 --> 00:00:27.080 Pretty awesome. 00:00:27.080 --> 00:00:31.780 No, we'll be folding letters more like this. 00:00:31.780 --> 00:00:36.600 So thanks everyone for giving so many cool questions 00:00:36.600 --> 00:00:39.635 and comments, and the feedback is really helpful. 00:00:39.635 --> 00:00:41.760 I didn't know what I was going to cover this class. 00:00:41.760 --> 00:00:43.860 I had too many ideas, and your questions really 00:00:43.860 --> 00:00:45.974 helped narrow it down into exciting thing. 00:00:45.974 --> 00:00:47.390 So the structures going to be, I'm 00:00:47.390 --> 00:00:50.059 going to go through questions, so it's 00:00:50.059 --> 00:00:52.350 a funny kind of interactivity where there's a whole day 00:00:52.350 --> 00:00:53.933 in between where you ask the questions 00:00:53.933 --> 00:00:55.010 and where I answer them. 00:00:55.010 --> 00:00:57.460 But feel free to ask more questions, follow-up things. 00:00:57.460 --> 00:01:01.840 But there's a lot already here, so it should be fun. 00:01:01.840 --> 00:01:06.610 These are not questions but funnier comments. 00:01:06.610 --> 00:01:08.480 If you like double rainbow jokes, 00:01:08.480 --> 00:01:10.810 this lecture happened a couple months 00:01:10.810 --> 00:01:13.480 after the double rainbow fiasco. 00:01:13.480 --> 00:01:15.485 There's a lot more, as I recall. 00:01:15.485 --> 00:01:18.060 It was a running theme throughout the whole semester, 00:01:18.060 --> 00:01:20.940 so look out for that. 00:01:20.940 --> 00:01:22.411 I know this is entertaining. 00:01:22.411 --> 00:01:24.785 I'm getting used to listening to myself at double speeds. 00:01:27.380 --> 00:01:30.340 This class definitely is nice in the way 00:01:30.340 --> 00:01:32.450 that with a fairly simple technique 00:01:32.450 --> 00:01:34.175 you can prove a very powerful theorem. 00:01:36.294 --> 00:01:38.710 Sorry, I just remembered I need to push a different button 00:01:38.710 --> 00:01:40.830 here. 00:01:40.830 --> 00:01:43.830 That's hopefully a theme throughout the class, 00:01:43.830 --> 00:01:47.025 but definitely especially nice here. 00:01:53.530 --> 00:01:55.440 It is cool, though, strip folding, 00:01:55.440 --> 00:01:58.150 it was an open problem for a couple years, at least, 00:01:58.150 --> 00:02:00.280 of no one thinking, how do we fold any shape? 00:02:00.280 --> 00:02:01.880 That seems really tough. 00:02:01.880 --> 00:02:04.272 The point of the strip folding approach 00:02:04.272 --> 00:02:06.410 is that once you have the right idea, 00:02:06.410 --> 00:02:08.530 to start with a really long rectangle, somehow 00:02:08.530 --> 00:02:09.330 it becomes easy. 00:02:09.330 --> 00:02:11.621 There's still a lot of details in getting that to work, 00:02:11.621 --> 00:02:14.820 but it's kind of neat how that works out. 00:02:14.820 --> 00:02:17.100 Now we get to actual proposals. 00:02:17.100 --> 00:02:20.320 So folding practice, I was planning on doing this, 00:02:20.320 --> 00:02:23.030 but-- this is an explicit comment to that effect. 00:02:23.030 --> 00:02:26.590 So we're going to fold some letters of the alphabet. 00:02:26.590 --> 00:02:28.570 You have in your packet instructions 00:02:28.570 --> 00:02:36.280 for making the individual digits six, four, eight, and nine, 00:02:36.280 --> 00:02:38.040 which are all pretty easy. 00:02:38.040 --> 00:02:43.060 You also have a diagram for this crazy design called Typeset. 00:02:43.060 --> 00:02:45.140 This same folding can make any letter 00:02:45.140 --> 00:02:48.510 of the alphabet and any digit. 00:02:48.510 --> 00:02:50.640 So just to show you what they will 00:02:50.640 --> 00:02:57.020 look like these are my foldings of six, eight, four, and nine, 00:02:57.020 --> 00:02:59.440 according to the first set of diagrams. 00:02:59.440 --> 00:03:06.400 And then this is my folding of, I think the number six, 00:03:06.400 --> 00:03:09.945 out of Jason Ku's design. 00:03:13.140 --> 00:03:15.030 So if you wanted to reconfigure it, 00:03:15.030 --> 00:03:21.420 to-- I guess eight is kind of hard-- let me do four. 00:03:21.420 --> 00:03:28.340 OK, so for four we've gotta fold this guy under. 00:03:28.340 --> 00:03:33.340 Fold this here, this tab, I think goes back here. 00:03:35.870 --> 00:03:39.500 So the advanced origami folders in this class 00:03:39.500 --> 00:03:41.920 can definitely do Jason Ku's design, 00:03:41.920 --> 00:03:45.900 but it takes a little while. 00:03:45.900 --> 00:03:47.110 There, I've got a four. 00:03:49.820 --> 00:03:52.620 Got it? 00:03:52.620 --> 00:03:54.410 A little hard to hold in position, 00:03:54.410 --> 00:03:57.030 but, at least in theory, it will make any letter all out 00:03:57.030 --> 00:03:58.310 of one folding. 00:03:58.310 --> 00:04:00.570 You just have to move all the tabs around. 00:04:00.570 --> 00:04:05.200 So it's kind of neat, but it takes at least half an hour 00:04:05.200 --> 00:04:07.220 or so to fold that, unless you're really fast. 00:04:07.220 --> 00:04:10.480 So I would recommend-- pick one of these. 00:04:10.480 --> 00:04:12.090 Work in groups if you like. 00:04:12.090 --> 00:04:13.980 If you want you can form a group of four, 00:04:13.980 --> 00:04:16.399 and make six, eight, four, and nine. 00:04:16.399 --> 00:04:17.414 Follow the diagrams. 00:04:17.414 --> 00:04:20.269 This is an exercise of following diagrams, one of the other. 00:04:20.269 --> 00:04:23.680 We'll only have time to maybe make one digit each, 00:04:23.680 --> 00:04:27.250 but have fun with it. 00:04:27.250 --> 00:04:29.570 And if you have questions, raise your hand. 00:04:29.570 --> 00:04:36.550 I can tell you, the first step in six, eight, and nine, 00:04:36.550 --> 00:04:39.992 is to make an eight-by-eight grid. 00:04:39.992 --> 00:04:42.200 There's a lot of ways to make an eight-by-eight grid, 00:04:42.200 --> 00:04:46.310 but an easy one is shown here. 00:04:46.310 --> 00:04:48.490 So you take your sheet. 00:04:48.490 --> 00:04:50.780 You fold the bottom edge to the top edge-- 00:04:50.780 --> 00:04:53.355 I think they want to do it white side up-- so they're all 00:04:53.355 --> 00:04:53.855 valleys. 00:04:56.540 --> 00:04:57.640 We align those edges. 00:04:57.640 --> 00:05:01.120 You'll get a nice bisector. 00:05:01.120 --> 00:05:07.355 Then you repeat, folding the bottom edge to the middle. 00:05:07.355 --> 00:05:09.480 And once you do that, to save a little bit of time, 00:05:09.480 --> 00:05:12.310 you could then fold that new bottom edge to the middle 00:05:12.310 --> 00:05:13.100 again. 00:05:13.100 --> 00:05:15.947 That will do eighths on one side. 00:05:15.947 --> 00:05:17.780 It's a little bit inaccurate, because you're 00:05:17.780 --> 00:05:21.870 folding through two layers-- but it's a little faster. 00:05:21.870 --> 00:05:24.800 Time is of the essence. 00:05:24.800 --> 00:05:27.360 Then you do the same thing on the bottom, 00:05:27.360 --> 00:05:33.730 and you'll get eighths in one dimension. 00:05:33.730 --> 00:05:36.435 But then you have to fold it in eighths in the other dimension. 00:05:43.086 --> 00:05:44.960 Once you have eighths, it's like three steps. 00:05:44.960 --> 00:05:46.740 It's really easy. 00:05:46.740 --> 00:05:51.542 For six, eight, and nine-- Four uses a different approach, 00:05:51.542 --> 00:05:52.750 it's a little more free hand. 00:05:52.750 --> 00:05:55.620 If you want to be more creative, try the four. 00:05:55.620 --> 00:05:59.740 You've got to eyeball what looks and feels good for number four. 00:06:49.320 --> 00:06:50.820 So out of curiosity, how many people 00:06:50.820 --> 00:06:53.737 have an eight-by-eight grid at this point? 00:06:53.737 --> 00:06:54.320 Who wanted to? 00:06:54.320 --> 00:06:54.820 Cool. 00:07:00.917 --> 00:07:02.625 So that's the top half of these diagrams, 00:07:02.625 --> 00:07:04.400 the eight by eight grid. 00:07:04.400 --> 00:07:07.060 Then it's mostly folding up over individual edges 00:07:07.060 --> 00:07:10.570 and some corner faults, but they're 00:07:10.570 --> 00:07:14.000 all simple folds in this world. 00:07:14.000 --> 00:07:17.100 So you're folding through, I think, always all the layers. 00:07:17.100 --> 00:07:17.600 Oh, no. 00:07:17.600 --> 00:07:20.360 This is only folding through one layer. 00:07:20.360 --> 00:07:23.610 But these would always fall into the some layers, simple folds 00:07:23.610 --> 00:07:25.170 category. 00:07:25.170 --> 00:07:27.255 So for example, make a six. 00:07:30.751 --> 00:07:34.790 Fold this bottom edge up. 00:07:34.790 --> 00:07:37.475 Fold the left three squares over. 00:07:39.990 --> 00:07:44.680 Fold this corner up. 00:07:49.310 --> 00:07:50.725 Fold this corner down. 00:08:03.902 --> 00:08:13.585 And fold this down, this over. 00:08:22.200 --> 00:08:24.599 There's a really big six. 00:08:24.599 --> 00:08:27.140 And these numbers are all pretty much proportioned correctly. 00:08:27.140 --> 00:08:30.710 The four, you have to be-- it helps 00:08:30.710 --> 00:08:33.309 to have a reference model of one of the other digits 00:08:33.309 --> 00:08:37.059 to make it the right height, but it'll end up roughly correct, 00:08:37.059 --> 00:08:38.030 anyway. 00:08:38.030 --> 00:08:40.340 The eight's a little bit narrow in this design, 00:08:40.340 --> 00:08:42.909 slightly narrower than six and the nine. 00:08:42.909 --> 00:08:45.780 But otherwise they're nice compatible digits. 00:08:45.780 --> 00:08:48.180 There's a whole alphabet on the website that's 00:08:48.180 --> 00:08:53.250 linked from this slide, so you go check it out. 00:08:53.250 --> 00:08:54.135 Origami club. 00:08:56.800 --> 00:08:59.980 So it's kind of fun to think about font design and alphabet 00:08:59.980 --> 00:09:00.480 design. 00:09:00.480 --> 00:09:04.160 There's actually a lot of origami alphabets out there. 00:09:04.160 --> 00:09:05.440 This is one of the simplest. 00:09:05.440 --> 00:09:06.153 It has digits. 00:09:11.050 --> 00:09:12.065 Anyone have questions? 00:09:15.260 --> 00:09:19.155 How many people folded a letter, a digit? 00:09:19.155 --> 00:09:22.110 OK. 00:09:22.110 --> 00:09:24.870 You can keep folding, but maybe I will continue on. 00:09:24.870 --> 00:09:26.536 It's going to be a lot of fun crackling. 00:09:28.590 --> 00:09:31.305 Anyone folded this one? 00:09:31.305 --> 00:09:32.180 Anyone working on it? 00:09:34.820 --> 00:09:36.750 A bunch of people, cool. 00:09:36.750 --> 00:09:38.280 Let me know when you finish. 00:09:38.280 --> 00:09:40.490 It's kind of fun. 00:09:40.490 --> 00:09:42.005 It's not that hard. 00:09:42.005 --> 00:09:42.880 AUDIENCE: Yes, it is. 00:09:42.880 --> 00:09:45.370 PROFESSOR: I wanted to point at the OrigaMIT website. 00:09:45.370 --> 00:09:48.230 OrigaMIT is the origami club at MIT, and at the top 00:09:48.230 --> 00:09:49.920 you see a different alphabet. 00:09:49.920 --> 00:09:51.970 This is a four-fold alphabet designed 00:09:51.970 --> 00:09:54.610 by Jeannine Mosley who's an MIT alum 00:09:54.610 --> 00:09:58.230 and came to this class two years ago. 00:09:58.230 --> 00:10:01.190 And so that's the reference design in 2002. 00:10:01.190 --> 00:10:03.120 I'm not sure if it has digits, though. 00:10:03.120 --> 00:10:06.590 At least, the diagrams we found for the letters, 00:10:06.590 --> 00:10:07.820 do not also have digits. 00:10:07.820 --> 00:10:09.278 So I think an interesting challenge 00:10:09.278 --> 00:10:14.380 is to design a four-fold digit set 00:10:14.380 --> 00:10:16.020 to complement her letter set. 00:10:16.020 --> 00:10:20.480 If you're interested, that could be a cool project to work on. 00:10:20.480 --> 00:10:22.560 Folding design, minimal fold alphabets. 00:10:22.560 --> 00:10:26.200 You could try a three-fold alphabet, two-fold alphabet. 00:10:26.200 --> 00:10:28.640 Also on this website are the meeting schedules. 00:10:28.640 --> 00:10:30.950 It always Sundays, sometimes 2, sometimes 3 o'clock. 00:10:30.950 --> 00:10:31.950 You should check it out. 00:10:31.950 --> 00:10:33.900 And there's a convention coming up, 00:10:33.900 --> 00:10:38.230 our very own origami convention on October 27th. 00:10:38.230 --> 00:10:42.300 Jason Ku wanted me to remind you about that coming up. 00:10:42.300 --> 00:10:44.770 Even if you've never done origami before, 00:10:44.770 --> 00:10:46.750 other than today, you should check it out. 00:10:46.750 --> 00:10:52.070 It'll be fun, lots of different sessions, from simple models 00:10:52.070 --> 00:10:53.590 to complicated models. 00:10:53.590 --> 00:10:57.275 This is a cool design by Brain Chan, another MIT alum. 00:10:57.275 --> 00:11:00.400 It's one square paper folded into the Mens et Manus logo, 00:11:00.400 --> 00:11:03.730 but, here, instead of the oil lamp, 00:11:03.730 --> 00:11:06.450 you've got a little origami crane. 00:11:06.450 --> 00:11:07.130 Cool stuff. 00:11:09.840 --> 00:11:12.684 So we proceed on to other questions. 00:11:12.684 --> 00:11:13.850 This is a pretty simple one. 00:11:13.850 --> 00:11:19.190 At the top of the notes, it says folding, AKA silhouette folding 00:11:19.190 --> 00:11:22.770 and gift wrapping, it has a couple of references. 00:11:22.770 --> 00:11:25.680 So where do those terms come from, is the question. 00:11:25.680 --> 00:11:29.670 And one answer is it's the title of the paper, "Folding Flat 00:11:29.670 --> 00:11:32.120 Silhouettes and Wrapping Polyhedral Packages," 00:11:32.120 --> 00:11:33.940 but that's not the real answer. 00:11:33.940 --> 00:11:37.680 So this is the two of us and Joe Mitchell. 00:11:37.680 --> 00:11:40.110 It's also, I think, the introduction of the term 00:11:40.110 --> 00:11:44.390 computational origami, but those terms 00:11:44.390 --> 00:11:45.970 come from earlier references. 00:11:45.970 --> 00:11:50.010 So in that paper, there's a sentence, classic open question 00:11:50.010 --> 00:11:51.900 origami mathematics, and we don't really 00:11:51.900 --> 00:11:53.380 know where it came about. 00:11:53.380 --> 00:11:56.590 But it was first formally posed by Bern and Hayes 00:11:56.590 --> 00:11:58.940 in this SODA '96 paper, which we'll 00:11:58.940 --> 00:12:03.240 be talking about in the next lecture, lecture three. 00:12:03.240 --> 00:12:05.930 And this is a quote from their paper, 00:12:05.930 --> 00:12:08.530 "Is every simple polygon, when skilled sufficiently small, 00:12:08.530 --> 00:12:10.790 the silhouette of a flat origami?" 00:12:10.790 --> 00:12:12.920 The point of saying the word silhouette 00:12:12.920 --> 00:12:18.687 is that when you fold something, like this number six-- 00:12:18.687 --> 00:12:20.520 there's a whole bunch of layers, and there's 00:12:20.520 --> 00:12:22.020 a lot of complexity to this folding. 00:12:22.020 --> 00:12:25.190 By saying silhouette, we just mean, collapse all layers, 00:12:25.190 --> 00:12:27.180 ignore the coloring, and just take the outline. 00:12:27.180 --> 00:12:30.910 So the silhouette of this thing is a rectangle. 00:12:30.910 --> 00:12:33.210 And, in general, that's sort of the transformation 00:12:33.210 --> 00:12:35.418 to throw away the complexity of the folding, and say, 00:12:35.418 --> 00:12:36.830 I just care about the shape. 00:12:36.830 --> 00:12:39.411 Can I get the desired shape? 00:12:39.411 --> 00:12:41.660 There's some other interesting questions here, though, 00:12:41.660 --> 00:12:43.118 which haven't been fully addressed. 00:12:43.118 --> 00:12:44.875 How many creases are necessary to fold? 00:12:44.875 --> 00:12:47.060 We'll actually get to that in a later question. 00:12:47.060 --> 00:12:50.070 How thick must the origami be? 00:12:50.070 --> 00:12:52.712 The strip method shows this if you 00:12:52.712 --> 00:12:54.920 start from a rectangle of paper, the number of layers 00:12:54.920 --> 00:12:56.050 can be very small. 00:12:56.050 --> 00:12:59.760 I think two or three is enough for all the gadgets that we 00:12:59.760 --> 00:13:02.197 use, probably three . 00:13:02.197 --> 00:13:03.780 If you start from a square, though, we 00:13:03.780 --> 00:13:06.340 don't know the answer to that question. 00:13:06.340 --> 00:13:10.010 And in practice, folding through many layers is tough. 00:13:10.010 --> 00:13:12.970 So that is a silhouette problem. 00:13:12.970 --> 00:13:15.307 The gift wrapping problem, the motivation 00:13:15.307 --> 00:13:16.640 is you have a weird-shaped gift. 00:13:16.640 --> 00:13:19.540 You want to wrap it with a piece of paper. 00:13:19.540 --> 00:13:21.900 And this is posed to us in a talk 00:13:21.900 --> 00:13:25.230 by Jin Akiyama at a Canadian geometry conference. 00:13:25.230 --> 00:13:28.970 Jin Akiyama is a really cool guy. 00:13:28.970 --> 00:13:34.632 He has had for many, many years a mathematics TV show in Japan. 00:13:34.632 --> 00:13:37.090 And he's known throughout Japan, because everyone in school 00:13:37.090 --> 00:13:38.880 watches his videos. 00:13:38.880 --> 00:13:41.876 And it covers really interesting mathematics. 00:13:41.876 --> 00:13:43.250 Some of the results in this class 00:13:43.250 --> 00:13:45.520 are actually in his videos as well. 00:13:45.520 --> 00:13:48.630 It's mostly in Japanese, so it's a little hard for most of us 00:13:48.630 --> 00:13:49.130 watch. 00:13:49.130 --> 00:13:50.700 But there's some subtitled versions, 00:13:50.700 --> 00:13:51.709 and they're really fun. 00:13:51.709 --> 00:13:54.000 Maybe we can have a movie night, and watch one of them, 00:13:54.000 --> 00:13:56.840 if I get permission. 00:13:56.840 --> 00:13:58.690 So there isn't a great reference for that. 00:13:58.690 --> 00:14:00.290 I mean, he's written some papers about different kinds 00:14:00.290 --> 00:14:02.331 of wrapping problems, but mostly it was this talk 00:14:02.331 --> 00:14:05.530 that he gave in 1997, which is when I was just starting out 00:14:05.530 --> 00:14:07.270 in computational geometry. 00:14:07.270 --> 00:14:09.700 That's where the terms come from. 00:14:09.700 --> 00:14:11.250 A lot of words. 00:14:11.250 --> 00:14:12.860 Have you ever actually folded a model 00:14:12.860 --> 00:14:18.240 using this method of zigzagging and folding with the strip? 00:14:18.240 --> 00:14:20.620 Any real or sensible or pretty origami models, 00:14:20.620 --> 00:14:22.760 or is it purely for the sake of universality? 00:14:22.760 --> 00:14:26.290 My knee jerk reaction was, no way is this practical. 00:14:26.290 --> 00:14:28.150 This is just for universality. 00:14:28.150 --> 00:14:30.940 And the point of this theorem has always been, in my mind, 00:14:30.940 --> 00:14:32.780 to prove that everything is possible, 00:14:32.780 --> 00:14:35.020 but then the challenge is to find good foldings, 00:14:35.020 --> 00:14:36.280 for some notion of good. 00:14:36.280 --> 00:14:41.960 But, actually, there are a bunch of examples of strip folding, 00:14:41.960 --> 00:14:44.550 not a lot of different folds, not terribly many gadgets, 00:14:44.550 --> 00:14:48.460 but there's some cool things, especially with strip weaving. 00:14:48.460 --> 00:14:51.710 These are just a few examples of woven colored strips. 00:14:51.710 --> 00:14:54.480 You can make fun things like Space Invaders. 00:14:54.480 --> 00:14:57.470 You could weave together baskets, 00:14:57.470 --> 00:14:59.570 and wrap your packages, and so on. 00:15:02.150 --> 00:15:04.279 This is a little bit more origamic. 00:15:04.279 --> 00:15:05.820 So there were no real folds in there, 00:15:05.820 --> 00:15:07.630 except at the edges of the cube. 00:15:07.630 --> 00:15:09.130 This one is a modular origami. 00:15:09.130 --> 00:15:10.750 It involves a bunch of different folds 00:15:10.750 --> 00:15:13.240 to get all the pieces to lock together. 00:15:13.240 --> 00:15:16.460 Modular origami means you have a bunch of identical pieces. 00:15:16.460 --> 00:15:18.330 They kind of weave together through folding, 00:15:18.330 --> 00:15:20.610 and then you can make a nice little crown. 00:15:20.610 --> 00:15:22.040 This is a very classic model. 00:15:22.040 --> 00:15:23.520 You've probably seen at some point. 00:15:23.520 --> 00:15:27.220 Here that the paper's slit, and then it's woven. 00:15:27.220 --> 00:15:29.220 But there's some folds down here. 00:15:29.220 --> 00:15:32.120 Not a lot of folding, but strips are pretty neat. 00:15:32.120 --> 00:15:35.740 You can definitely use them in all sorts of different designs. 00:15:35.740 --> 00:15:41.060 Here's some more sculptural designing models 00:15:41.060 --> 00:15:44.320 from taking strips of paper. 00:15:44.320 --> 00:15:45.780 This has no glue in it, so I think 00:15:45.780 --> 00:15:51.020 there's more strips at the end locking this together. 00:15:51.020 --> 00:15:54.949 And this guy, Zachary Futterer, took 00:15:54.949 --> 00:15:56.740 a bunch of these kinds of units and started 00:15:56.740 --> 00:15:59.610 weaving them together to make really complicated shapes. 00:15:59.610 --> 00:16:04.240 So you can definitely do cool things with strip folding. 00:16:04.240 --> 00:16:07.520 And another common one around these days 00:16:07.520 --> 00:16:12.000 is taking gum or candy wrappers, and folding them 00:16:12.000 --> 00:16:14.300 down into little strips and weaving them together 00:16:14.300 --> 00:16:16.695 to make handbags and other things. 00:16:16.695 --> 00:16:18.320 This has become kind of a fashion trend 00:16:18.320 --> 00:16:20.760 over the last few years. 00:16:20.760 --> 00:16:24.080 So those are things you can do a strips. 00:16:24.080 --> 00:16:27.440 We have used it in one paper where the goal is actually 00:16:27.440 --> 00:16:31.580 to be efficient and use a small piece of paper, 00:16:31.580 --> 00:16:34.564 and not just prove some universality result. 00:16:34.564 --> 00:16:36.730 This is in our paper, folding a better checkerboard, 00:16:36.730 --> 00:16:40.260 which we'll talk about in two lectures, if I 00:16:40.260 --> 00:16:42.940 recall correctly, in more detail. 00:16:42.940 --> 00:16:44.490 But this is sort of a baseline. 00:16:44.490 --> 00:16:47.270 This is not the better method that's developed in this paper, 00:16:47.270 --> 00:16:49.590 but it's the starting point. 00:16:49.590 --> 00:16:52.400 You take a square, so this actually starts with a square. 00:16:52.400 --> 00:16:54.020 You do this pleating. 00:16:54.020 --> 00:16:56.370 And this is with bi-color paper. 00:16:56.370 --> 00:16:58.180 It's dark on one side, light on the other. 00:16:58.180 --> 00:17:01.270 You get this strip of squares in color pattern. 00:17:01.270 --> 00:17:03.740 And then you take that strip-- a huge number of layers 00:17:03.740 --> 00:17:05.569 in the middle, so it's not super practical, 00:17:05.569 --> 00:17:07.460 but it's actually pretty efficient in terms 00:17:07.460 --> 00:17:09.280 of how big a square you start with. 00:17:09.280 --> 00:17:12.720 To make an n-by-n, this is obviously not to scale. 00:17:12.720 --> 00:17:16.140 You need more squares here in order to make this thing. 00:17:16.140 --> 00:17:18.312 And then you just snake your path back and forth. 00:17:18.312 --> 00:17:20.020 You could use turn gadgets, or here we're 00:17:20.020 --> 00:17:21.910 just using 45 degree folds. 00:17:21.910 --> 00:17:25.030 And this is pretty close to what was believed 00:17:25.030 --> 00:17:26.910 to be the best way to fold a checkerboard, 00:17:26.910 --> 00:17:29.609 and then this paper shows how to do a factor of two better. 00:17:29.609 --> 00:17:30.900 So we'll talk about that later. 00:17:30.900 --> 00:17:33.240 But there are some uses for strip folding. 00:17:33.240 --> 00:17:34.710 This is a little bit theoretical, 00:17:34.710 --> 00:17:37.620 but it's actually pretty competitive against the best 00:17:37.620 --> 00:17:40.720 n-by-n checkerboard foldings in the origami world, 00:17:40.720 --> 00:17:43.690 like the one I showed last class. 00:17:43.690 --> 00:17:48.080 So that's practicality of strip folding. 00:17:48.080 --> 00:17:50.377 Next question's more about strip folding. 00:17:50.377 --> 00:17:52.710 There are a couple things that are in the lecture notes, 00:17:52.710 --> 00:17:55.180 the handwritten lecture notes, but were not even 00:17:55.180 --> 00:18:00.040 mentioned in the audio part of the lecture. 00:18:00.040 --> 00:18:03.840 So a few people asked, what are these things? 00:18:03.840 --> 00:18:06.570 Said pseudopolynomial upper bound. 00:18:06.570 --> 00:18:09.870 Pseudopolynomial is a fun term. 00:18:09.870 --> 00:18:12.200 Let me tell you a little bit about it. 00:18:12.200 --> 00:18:14.310 It's from the algorithms world, but even a lot 00:18:14.310 --> 00:18:16.220 of algorithms people don't know it. 00:18:16.220 --> 00:18:21.210 So let me tell you. 00:18:21.210 --> 00:18:24.760 So maybe first I should tell you about polynomial. 00:18:24.760 --> 00:18:27.510 In general, what these terms are about 00:18:27.510 --> 00:18:30.870 is measuring how fast an algorithm is. 00:18:30.870 --> 00:18:37.140 So the idea is you plot, conceptually, n, this 00:18:37.140 --> 00:18:38.360 is the problem size. 00:18:44.440 --> 00:18:48.304 If you wanted to fold an arbitrary polyhedron, 00:18:48.304 --> 00:18:49.970 the one way to think of the problem size 00:18:49.970 --> 00:18:51.960 is the number vertices, edges, and faces. 00:18:51.960 --> 00:18:55.000 Just the total number of things you're given as input, 00:18:55.000 --> 00:18:56.600 and then your output is whatever. 00:18:56.600 --> 00:18:59.300 But n is supposed to be the input problem size. 00:18:59.300 --> 00:19:02.590 And then on the y-axis, you want to plot 00:19:02.590 --> 00:19:05.860 the running time of your algorithm. 00:19:05.860 --> 00:19:08.750 So this is how long it takes to compute the way 00:19:08.750 --> 00:19:12.420 to fold your square paper into your desired shape, 00:19:12.420 --> 00:19:14.720 and generally this is going to increase. 00:19:14.720 --> 00:19:18.270 And the question is, does it increase in a reasonable way 00:19:18.270 --> 00:19:22.460 or in a crazy way that goes exponentially high? 00:19:22.460 --> 00:19:25.530 So you want to know, how does the running time grow with n? 00:19:25.530 --> 00:19:27.930 Polynomial is a sense of good growth, 00:19:27.930 --> 00:19:29.590 and it just means you grow, like, 00:19:29.590 --> 00:19:32.530 n to the c, where c is some constant. 00:19:36.750 --> 00:19:42.470 So ideally you'd have n-- or maybe you have n squared, 00:19:42.470 --> 00:19:44.490 or n cubed, or n to the fourth-- all these are 00:19:44.490 --> 00:19:47.250 considered good running times. 00:19:47.250 --> 00:19:49.440 Not quite as good as polynomial is pseudopolynomial. 00:19:57.940 --> 00:20:02.220 And I would conjecture, for this problem of folding an arbitrary 00:20:02.220 --> 00:20:07.290 given polyhedron, you cannot achieve a polynomial number 00:20:07.290 --> 00:20:08.320 of folds, let's say. 00:20:08.320 --> 00:20:09.650 So there are two things we could measure here, 00:20:09.650 --> 00:20:11.730 the running time of the algorithm-- we could measure, 00:20:11.730 --> 00:20:13.438 actually, three things-- we could measure 00:20:13.438 --> 00:20:16.700 the number of folds you make, the number operations you 00:20:16.700 --> 00:20:20.430 do on the paper, and a third thing would be scale factor. 00:20:20.430 --> 00:20:22.930 How big a square do I have to start with in order 00:20:22.930 --> 00:20:27.110 to make a desired polygon? 00:20:27.110 --> 00:20:35.430 And pseudopolynomial means n times r to the c. 00:20:35.430 --> 00:20:36.900 What's r? 00:20:36.900 --> 00:20:46.310 R is some geometric parameter, geometric ratio, in the input. 00:20:46.310 --> 00:20:49.990 And in particular for this problem what makes sense for r 00:20:49.990 --> 00:20:58.770 is basically the longest length divided by the shortest length. 00:20:58.770 --> 00:21:01.190 This is typically what r refers to. 00:21:01.190 --> 00:21:03.130 This will come up in later lectures as well. 00:21:06.220 --> 00:21:08.350 So for example, you take your entire shape, 00:21:08.350 --> 00:21:11.860 you measure the diameter of the shape, the two farthest points. 00:21:11.860 --> 00:21:13.100 That's your longest length. 00:21:13.100 --> 00:21:15.420 Shortest length would be-- maybe you 00:21:15.420 --> 00:21:18.260 have a triangle, something like this, 00:21:18.260 --> 00:21:20.690 in the target polyhedron you want to make. 00:21:20.690 --> 00:21:23.069 This would be your shortest distance. 00:21:23.069 --> 00:21:24.860 This is actually called the minimum feature 00:21:24.860 --> 00:21:28.300 size in computational geometry or the minimum altitude 00:21:28.300 --> 00:21:30.720 of any of your triangles. 00:21:30.720 --> 00:21:32.950 OK, so that's just some number. 00:21:32.950 --> 00:21:37.950 And you can have a triangle, which is super, super narrow. 00:21:37.950 --> 00:21:40.761 And so it's this ratio r could be arbitrarily large 00:21:40.761 --> 00:21:43.010 even though you only have three vertices, three edges, 00:21:43.010 --> 00:21:44.290 one face. 00:21:44.290 --> 00:21:46.756 So n and r not necessarily comparable, 00:21:46.756 --> 00:21:48.630 so that's why in pseudopolynomial we put them 00:21:48.630 --> 00:21:52.300 both together, and then we raise them to some constant power. 00:21:52.300 --> 00:21:54.720 That's a pseudopolynomial running time. 00:21:54.720 --> 00:21:58.510 So the question that's being posed here 00:21:58.510 --> 00:22:01.980 is, can you get a pseudopolynomial upper bound, 00:22:01.980 --> 00:22:04.109 and can you get a pseudopolynomial lower bound? 00:22:04.109 --> 00:22:05.650 And it doesn't say for what, but it's 00:22:05.650 --> 00:22:08.300 for all three problems-- running time, 00:22:08.300 --> 00:22:11.070 number of folds, scale factor. 00:22:11.070 --> 00:22:13.410 And not all of these are open. 00:22:13.410 --> 00:22:16.520 So in the original paper, there's 00:22:16.520 --> 00:22:19.830 this theorem that says lots of things-- you can fold anything. 00:22:19.830 --> 00:22:22.340 Then it says here, the folding requires a number of folds, 00:22:22.340 --> 00:22:27.480 polynomial of n and the ratio, r. 00:22:27.480 --> 00:22:30.940 So it already claims it there is a pseudopolynomial bound 00:22:30.940 --> 00:22:32.280 on the number of folds. 00:22:32.280 --> 00:22:35.370 It doesn't say what that pseudopolynomial bound is. 00:22:35.370 --> 00:22:37.250 Is it n times r? 00:22:37.250 --> 00:22:38.950 Is it n plus r? 00:22:38.950 --> 00:22:40.690 Is it n times r squared? 00:22:40.690 --> 00:22:45.110 I would guess one of the first two-- n times r or n plus r. 00:22:45.110 --> 00:22:47.280 So that's the upper bound question. 00:22:47.280 --> 00:22:49.420 Maybe we can work on this in a problem session. 00:22:49.420 --> 00:22:51.940 A lower bound question is, do you 00:22:51.940 --> 00:22:54.790 prove that you need some dependence, both on n and r? 00:22:54.790 --> 00:22:56.920 Which I would guess is pretty easy. 00:22:56.920 --> 00:22:58.910 If you want to take a square and fold it down 00:22:58.910 --> 00:23:00.510 to a really, really skinny triangle, 00:23:00.510 --> 00:23:04.540 I think you need at least r folds, roughly. 00:23:04.540 --> 00:23:06.900 And similarly you should need at least n folds, 00:23:06.900 --> 00:23:10.040 so there should be a lower bound like n plus r. 00:23:10.040 --> 00:23:12.210 But none of these have been written down explicitly, 00:23:12.210 --> 00:23:14.690 so that's what those open questions are. 00:23:14.690 --> 00:23:16.130 Then there was another slide, six, 00:23:16.130 --> 00:23:18.690 which was completely uncovered. 00:23:18.690 --> 00:23:20.080 Stop me if there are questions. 00:23:20.080 --> 00:23:23.890 I should maybe take a brief moment to breathe. 00:23:23.890 --> 00:23:29.640 So the next part of the lecture notes ask about seam placement. 00:23:29.640 --> 00:23:32.560 So seam placement has the following kind of issue. 00:23:32.560 --> 00:23:36.650 When you fold, like this number-- is this a six? 00:23:36.650 --> 00:23:39.040 This is a nine. 00:23:39.040 --> 00:23:40.510 Fold this number nine. 00:23:40.510 --> 00:23:43.400 In addition to seeing the color pattern, if you look closely 00:23:43.400 --> 00:23:45.090 there's also these kinds of seams. 00:23:45.090 --> 00:23:48.460 This white square is not just a white square. 00:23:48.460 --> 00:23:53.370 You can see on the top layer this crease line. 00:23:53.370 --> 00:23:55.880 And here there's a seam, here there's a seam. 00:23:55.880 --> 00:23:58.400 These are like visible lines. 00:23:58.400 --> 00:24:01.600 Of course, you have to have seams at the color transitions, 00:24:01.600 --> 00:24:03.920 but there's other seams as well. 00:24:03.920 --> 00:24:06.282 Maybe you want to minimize the seams you want to get, 00:24:06.282 --> 00:24:08.240 you want to place the scenes in a cool pattern. 00:24:11.010 --> 00:24:12.510 When you fold checkerboards, there's 00:24:12.510 --> 00:24:14.690 a such thing as a seamless checkerboard, where 00:24:14.690 --> 00:24:16.880 every square is a whole square paper. 00:24:16.880 --> 00:24:20.640 There's no visible crease lines on the top layer. 00:24:20.640 --> 00:24:23.890 So this is an extension of the universality result, 00:24:23.890 --> 00:24:26.830 to also get sort of universal seem placement. 00:24:26.830 --> 00:24:30.230 And what the original paper proves 00:24:30.230 --> 00:24:33.360 is that you can place the seams however 00:24:33.360 --> 00:24:36.020 you want, provided the seam regions, the regions 00:24:36.020 --> 00:24:39.930 between the seams, are convex polygons-- which 00:24:39.930 --> 00:24:41.160 is almost always the case. 00:24:41.160 --> 00:24:44.250 You look at a typical model-- here, 00:24:44.250 --> 00:24:48.140 the seam regions are all rectangles and triangles. 00:24:48.140 --> 00:24:51.680 So you could achieve exactly this seam pattern 00:24:51.680 --> 00:24:52.620 if you wanted it. 00:24:52.620 --> 00:24:54.620 You could also say, oh, here's a nice rectangle. 00:24:54.620 --> 00:24:56.570 I'll make that a seam region. 00:24:56.570 --> 00:24:59.330 Here's a nice rectangle, I'll make that same region. 00:24:59.330 --> 00:25:01.640 But you could not make the entire number nine 00:25:01.640 --> 00:25:05.940 here a seam region, because it's non convex. 00:25:05.940 --> 00:25:07.980 At least, you can't do it with this technique. 00:25:07.980 --> 00:25:10.260 We don't know, necessarily, whether this 00:25:10.260 --> 00:25:12.330 is possible by some other folding. 00:25:12.330 --> 00:25:16.780 I would guess no, but it is possible to make 00:25:16.780 --> 00:25:19.760 some non-convex seam regions. 00:25:19.760 --> 00:25:26.340 For example I could take this page and fold the corner over, 00:25:26.340 --> 00:25:30.120 and now I've got a non-convex seam region here. 00:25:30.120 --> 00:25:32.780 So some non-convexing regions are possible. 00:25:32.780 --> 00:25:38.360 Open question is-- if I give you a polygon-- 00:25:38.360 --> 00:25:40.050 we know every polygon's possible. 00:25:40.050 --> 00:25:43.004 Now I give you a polygon and I subdivide it into seam regions. 00:25:43.004 --> 00:25:44.170 Which of those are possible? 00:25:44.170 --> 00:25:46.750 Not everything is possible, I'm pretty sure, 00:25:46.750 --> 00:25:49.590 though I'm not sure we have a proof of that. 00:25:49.590 --> 00:25:52.780 Some things like this little heart shape are possible. 00:25:52.780 --> 00:25:53.960 Characterize. 00:25:53.960 --> 00:25:59.340 This another cool possible problem for problem session. 00:25:59.340 --> 00:26:00.250 Questions about that? 00:26:05.090 --> 00:26:07.920 I have a little bit about the proof of how this is done. 00:26:07.920 --> 00:26:11.040 If you wanted to just do convex regions. 00:26:11.040 --> 00:26:15.530 So the general approach here is you 00:26:15.530 --> 00:26:18.490 want to visit all of the regions in some order. 00:26:18.490 --> 00:26:20.726 This is called the tour. 00:26:20.726 --> 00:26:22.100 It's pretty easy to just-- I mean 00:26:22.100 --> 00:26:24.558 you're allowed to visit regions more than once, so you just 00:26:24.558 --> 00:26:28.310 keep going, keep trying to visit some unvisited seam region. 00:26:28.310 --> 00:26:31.360 When you visit a seam region, it's a convex polygon. 00:26:31.360 --> 00:26:34.290 So what we're going to do is make our strip fairly wide, 00:26:34.290 --> 00:26:37.150 actually, wide enough to completely cover 00:26:37.150 --> 00:26:39.400 that seam region. 00:26:39.400 --> 00:26:41.250 And then at this moment, we basically 00:26:41.250 --> 00:26:44.420 need to turn to do the next one. 00:26:44.420 --> 00:26:48.530 We know how to change the direction of the strip using 00:26:48.530 --> 00:26:50.470 a turn gadget. 00:26:50.470 --> 00:26:52.850 Then we have to change the width of the strip. 00:26:52.850 --> 00:26:53.990 Maybe it needs to be wider. 00:26:53.990 --> 00:26:56.080 Maybe it needs to be thinner. 00:26:56.080 --> 00:27:01.630 And then we need to shift the strip one way or the other. 00:27:01.630 --> 00:27:05.940 So if we just end here, we turn, we might be misaligned. 00:27:05.940 --> 00:27:09.380 We need to shift it over, expand it, then do the next one, 00:27:09.380 --> 00:27:13.860 then turn, then shift it over and set the right width. 00:27:13.860 --> 00:27:16.910 Keep going like that. 00:27:16.910 --> 00:27:17.520 OK. 00:27:17.520 --> 00:27:19.230 That's pretty messy and complicated, 00:27:19.230 --> 00:27:21.550 but you can do with these two gadgets. 00:27:21.550 --> 00:27:23.880 Strip width gadget-- you take a strip, 00:27:23.880 --> 00:27:28.750 and you could make it anywhere between 1/2 and 100% 00:27:28.750 --> 00:27:30.637 of its original width. 00:27:30.637 --> 00:27:33.220 So the idea is you start with a really wide strip, wide enough 00:27:33.220 --> 00:27:35.030 to cover all the polygons. 00:27:35.030 --> 00:27:37.840 Then you do this gadget and keep shrinking it by half 00:27:37.840 --> 00:27:40.041 until it's roughly the right size. 00:27:40.041 --> 00:27:41.540 And then when you're almost correct, 00:27:41.540 --> 00:27:43.420 you shrink it by a little bit more-- 00:27:43.420 --> 00:27:46.040 here shrinking to a third. 00:27:46.040 --> 00:27:48.560 And then you get your shrunken strip, 00:27:48.560 --> 00:27:51.680 and it happens right at the line that you specify. 00:27:51.680 --> 00:27:54.770 So you can basically on a dime shrink your strip, 00:27:54.770 --> 00:27:59.550 and then by doing the reverse can grow it back. 00:27:59.550 --> 00:28:03.260 This is maybe not with simple folds, though. 00:28:03.260 --> 00:28:05.290 And then the other gadget is a shift gadget, 00:28:05.290 --> 00:28:08.770 where you're at this position and you want to shift up. 00:28:08.770 --> 00:28:13.110 So that's pretty easy, you just do two turn gadgets. 00:28:13.110 --> 00:28:18.680 So that's at a high level how making a desired seam pattern 00:28:18.680 --> 00:28:20.460 works. 00:28:20.460 --> 00:28:23.580 Go on to the next question. 00:28:23.580 --> 00:28:27.210 A lot of people asked about this-- and this 00:28:27.210 --> 00:28:29.340 is an open problem that I mentioned orally. 00:28:29.340 --> 00:28:33.040 It's not written in the notes-- which is, can you actually 00:28:33.040 --> 00:28:37.380 do the things we said we can do with simple folds? 00:28:37.380 --> 00:28:40.280 So can you get a universal folding 00:28:40.280 --> 00:28:46.160 of a polygon, two-color pattern polyhedron, using simple folds? 00:28:46.160 --> 00:28:50.600 And I thought it'd be fun to actually work on this, here, 00:28:50.600 --> 00:28:53.484 live, because I think this is an easy problem. 00:28:53.484 --> 00:28:55.150 And there's a bunch of possible answers, 00:28:55.150 --> 00:28:58.850 and there are even two suggested ideas from the comments field. 00:28:58.850 --> 00:29:03.410 So let me just remind you of the issue, what's happening, 00:29:03.410 --> 00:29:06.980 and then I need your input, what's going to work here. 00:29:06.980 --> 00:29:11.890 So general picture for the strip method was we do one triangle, 00:29:11.890 --> 00:29:16.950 we end here, then we do a bunch of folds like this, 00:29:16.950 --> 00:29:20.870 and then we end here, maybe, and then we zigzag. 00:29:20.870 --> 00:29:24.520 And the trouble is we've already made this triangle over here. 00:29:24.520 --> 00:29:27.857 When we make this triangle, we have this excess stuff, 00:29:27.857 --> 00:29:29.440 which I haven't drawn very accurately. 00:29:29.440 --> 00:29:33.300 If you recall, it looks something like that. 00:29:33.300 --> 00:29:36.000 Or maybe even more like that if we use right angle turn 00:29:36.000 --> 00:29:37.730 gadgets. 00:29:37.730 --> 00:29:39.960 And then we want to fold it underneath 00:29:39.960 --> 00:29:41.790 and we're doing that, the way I said, 00:29:41.790 --> 00:29:44.700 with height gadget mountain folds. 00:29:44.700 --> 00:29:47.210 But the model's simple folds, which 00:29:47.210 --> 00:29:48.850 I should make more explicit. 00:29:54.250 --> 00:29:57.720 You're not allowed to collide during the motion. 00:29:57.720 --> 00:29:59.350 The idea with a simple fold is that you 00:29:59.350 --> 00:30:08.420 should be folding along one line segment, 00:30:08.420 --> 00:30:12.100 and you should fold by-- at least the model 00:30:12.100 --> 00:30:16.070 that we defined back in the day, I'll talk more about where 00:30:16.070 --> 00:30:20.600 this notion comes from-- you fold by plus or minus 180 00:30:20.600 --> 00:30:24.110 degrees, which means after you do the fold, 00:30:24.110 --> 00:30:26.050 you'll be flat again. 00:30:26.050 --> 00:30:29.745 And no collision during the motion. 00:30:36.767 --> 00:30:39.100 If we folded this triangle, and then we folded this one, 00:30:39.100 --> 00:30:41.900 and this stuff is on the top-- we can't mountain fold. 00:30:41.900 --> 00:30:44.920 That's not considered a simple fold. 00:30:44.920 --> 00:30:48.260 Now one proposal is, could we just 00:30:48.260 --> 00:30:53.530 make the next triangle underneath the previous one? 00:30:53.530 --> 00:30:55.890 A different proposal is valley fold. 00:30:55.890 --> 00:30:57.520 These are actually different proposals, 00:30:57.520 --> 00:30:59.920 because, especially for two color patterns, 00:30:59.920 --> 00:31:02.150 it'll make a difference. 00:31:02.150 --> 00:31:05.050 If we valley fold here there's going 00:31:05.050 --> 00:31:07.392 to be some junk on the front side, 00:31:07.392 --> 00:31:09.600 especially if you want to get a desired seam pattern. 00:31:09.600 --> 00:31:11.582 But maybe we'll leave seam patterns for later. 00:31:11.582 --> 00:31:13.040 If you want to get a color pattern, 00:31:13.040 --> 00:31:14.870 you might reveal some wrong color 00:31:14.870 --> 00:31:17.600 when you do that valley fold. 00:31:17.600 --> 00:31:19.870 So I haven't really thought about this idea yet. 00:31:19.870 --> 00:31:21.960 I think it might be good. 00:31:26.580 --> 00:31:30.940 The idea there would be-- so you've already 00:31:30.940 --> 00:31:34.040 made this triangle. 00:31:34.040 --> 00:31:35.350 You mountain fold everything. 00:31:35.350 --> 00:31:38.930 Now when you go and you do these zigzags, 00:31:38.930 --> 00:31:42.180 you want to be underneath everything that you've done. 00:31:42.180 --> 00:31:44.190 AUDIENCE: Can you fold the one you've already 00:31:44.190 --> 00:31:47.650 made, like as a unit, over the side? 00:31:47.650 --> 00:31:50.167 And then make it and then just put it back on top? 00:31:50.167 --> 00:31:50.750 PROFESSOR: OK. 00:31:50.750 --> 00:31:54.900 Different idea is you basically fold this out of the way, 00:31:54.900 --> 00:31:58.130 do this thing, height gadget, and then fold it back. 00:31:58.130 --> 00:31:59.810 Maybe. 00:31:59.810 --> 00:32:01.110 I've wondered about that, too. 00:32:01.110 --> 00:32:03.193 Is it's going to get a little challenging, though. 00:32:03.193 --> 00:32:05.680 In general, there's a huge set of triangles, 00:32:05.680 --> 00:32:08.680 so unless you can like go far away, make your triangle, 00:32:08.680 --> 00:32:12.630 and then plop it down-- maybe it's possible. 00:32:12.630 --> 00:32:16.780 I guess we can pursue that idea, but maybe first we 00:32:16.780 --> 00:32:19.280 should exhaust the easier ideas. 00:32:19.280 --> 00:32:21.800 I mean, that is definitely plausible that that's possible. 00:32:21.800 --> 00:32:25.080 Doing that with simple folds and not leaving any garbage 00:32:25.080 --> 00:32:26.580 is going to be a little challenging, 00:32:26.580 --> 00:32:29.450 but it might be doable. 00:32:29.450 --> 00:32:31.610 This to me is the simplest idea, so we should first 00:32:31.610 --> 00:32:32.590 see if it works. 00:32:32.590 --> 00:32:34.850 Does anyone see problems with this plan? 00:32:37.470 --> 00:32:39.480 I have some strips. 00:32:39.480 --> 00:32:41.320 We could think about what it means 00:32:41.320 --> 00:32:47.360 to be doing a turn gadget underneath here. 00:32:47.360 --> 00:32:50.480 So I don't know quite about, well, 00:32:50.480 --> 00:32:53.090 let's suppose we are already here. 00:32:53.090 --> 00:32:56.252 And now maybe I do some as the turn gadget 00:32:56.252 --> 00:32:57.210 goes to mountain folds. 00:32:59.800 --> 00:33:06.310 Then at 90 degrees, and then a valley fold like that. 00:33:06.310 --> 00:33:08.240 And then I do a mountain fold. 00:33:08.240 --> 00:33:11.180 I think they might be OK, because turn gadgets start 00:33:11.180 --> 00:33:13.860 with a mountain fold. 00:33:13.860 --> 00:33:15.790 So if you're underneath everything, 00:33:15.790 --> 00:33:17.990 that's going to avoid collision. 00:33:17.990 --> 00:33:20.880 And then the valley fold brings it back. 00:33:20.880 --> 00:33:24.100 Again, we're using just the space below everything 00:33:24.100 --> 00:33:26.095 that we see. 00:33:26.095 --> 00:33:28.220 So then we make that strip, we keep turning around, 00:33:28.220 --> 00:33:32.590 and then later on we're going to mountain fold this behind, 00:33:32.590 --> 00:33:34.075 somehow, to meet this edge. 00:33:37.800 --> 00:33:39.650 Seems OK. 00:33:39.650 --> 00:33:41.080 Yeah? 00:33:41.080 --> 00:33:42.119 Question? 00:33:42.119 --> 00:33:42.952 AUDIENCE: I'm sorry. 00:33:42.952 --> 00:33:44.270 I'm confused about something. 00:33:44.270 --> 00:33:47.210 When you're doing the turn gadget, 00:33:47.210 --> 00:33:51.130 while the paper is above the first triangle you go over, 00:33:51.130 --> 00:33:54.070 you're doing a bunch of turns where you're 00:33:54.070 --> 00:33:57.420 taking a piece of paper up another one, folding just that 00:33:57.420 --> 00:33:58.850 part Is that a s-- 00:33:58.850 --> 00:34:01.310 PROFESSOR: Going from one triangle next. 00:34:01.310 --> 00:34:01.810 Yeah. 00:34:01.810 --> 00:34:04.230 So we could think about that, too. 00:34:04.230 --> 00:34:05.560 It's very, you're right. 00:34:05.560 --> 00:34:08.164 I mean, I'm just looking at the turning around part 00:34:08.164 --> 00:34:10.080 for making a single triangle, but there's also 00:34:10.080 --> 00:34:12.446 the turn gadget going from here to here. 00:34:12.446 --> 00:34:14.070 It's actually slightly more complicated 00:34:14.070 --> 00:34:16.111 than we've covered in class, but not really much. 00:34:16.111 --> 00:34:18.350 It's just a slightly generalized turn gadget. 00:34:18.350 --> 00:34:21.170 So you're coming here, you basically want to turn around. 00:34:24.967 --> 00:34:26.550 Let's just think about a way to do it. 00:34:26.550 --> 00:34:32.354 You could imagine first doing a turn like this. 00:34:37.370 --> 00:34:39.540 It's not exactly a pure turn gadget. 00:34:39.540 --> 00:34:46.239 And then turning around to get to next place-- 00:34:46.239 --> 00:34:50.070 this is really hard to do on a blackboard. 00:34:50.070 --> 00:34:52.734 Strips just tend not to stay together well. 00:34:52.734 --> 00:34:54.984 OK, now we're going parallel to the correct direction, 00:34:54.984 --> 00:34:57.740 and then we turn back and forth. 00:34:57.740 --> 00:34:59.750 But each of those is just using turn gadgets. 00:34:59.750 --> 00:35:01.750 As long as a turn gadget works fine, 00:35:01.750 --> 00:35:04.655 a turn gadget is going to be a mountain fold, which 00:35:04.655 --> 00:35:06.720 is going to go behind everything, 00:35:06.720 --> 00:35:11.740 and then a valley fold to bring it to a desired direction. 00:35:11.740 --> 00:35:14.650 Those are all using everything behind the board. 00:35:14.650 --> 00:35:17.410 So it seems like all those operations are OK. 00:35:17.410 --> 00:35:19.240 Now we should also check the color reversal 00:35:19.240 --> 00:35:21.510 gadget, which-- it's funny thing. 00:35:21.510 --> 00:35:26.010 I remember everything I did before 2000 or so, 00:35:26.010 --> 00:35:29.240 so I still have memorized the color reversal gadget, at least 00:35:29.240 --> 00:35:29.820 I think so. 00:35:29.820 --> 00:35:33.770 I should probably color this piece of paper, 00:35:33.770 --> 00:35:35.550 so you can see the colors change. 00:35:35.550 --> 00:35:37.950 I don't remember anything I've done since 2000, 00:35:37.950 --> 00:35:41.530 but anything up to 2000 I'm OK. 00:35:41.530 --> 00:35:45.449 This is 1998, I think. 00:35:45.449 --> 00:35:47.490 I remember folding lots of-- this is ticker tape, 00:35:47.490 --> 00:35:49.360 they use it for or not ticker tape, 00:35:49.360 --> 00:35:52.190 but they use this for like adding machines. 00:35:52.190 --> 00:35:54.150 So I think it's a 90 degree mountain fold, 00:35:54.150 --> 00:35:57.680 then you fold up like that, and then 00:35:57.680 --> 00:36:00.639 you fold back down with a mountain fold like that. 00:36:00.639 --> 00:36:01.680 You get a color reversal. 00:36:01.680 --> 00:36:06.990 And all of those folds we're working behind my plane 00:36:06.990 --> 00:36:09.517 here, so should avoid collision with everything. 00:36:09.517 --> 00:36:11.350 I think you could do color reversal and turn 00:36:11.350 --> 00:36:13.370 gadgets behind. 00:36:13.370 --> 00:36:15.070 And so the suggestion works. 00:36:15.070 --> 00:36:17.600 Who made this suggestion? 00:36:17.600 --> 00:36:18.240 Good idea. 00:36:21.087 --> 00:36:23.420 Unless there are any objections, I think that will work. 00:36:23.420 --> 00:36:28.510 I had a different plan, which was to use this second idea 00:36:28.510 --> 00:36:32.050 and set up the turn gadgets so there is no-- when you fold 00:36:32.050 --> 00:36:35.550 this with the valley fold, there is no ugly colors. 00:36:35.550 --> 00:36:37.280 So you could maybe modify the turn gadget 00:36:37.280 --> 00:36:40.190 to be completely solidly colored on both sides, 00:36:40.190 --> 00:36:42.045 but I think this is much easier. 00:36:42.045 --> 00:36:44.170 It's probably why I didn't write it down the notes, 00:36:44.170 --> 00:36:45.270 but I'm not sure. 00:36:45.270 --> 00:36:45.986 Yeah? 00:36:45.986 --> 00:36:48.610 AUDIENCE: I think you could also just do a thing where you just 00:36:48.610 --> 00:36:50.901 take it far away, because you have a really long strip. 00:36:50.901 --> 00:36:54.806 So you can just take that strip, and go 00:36:54.806 --> 00:36:58.141 to where there are no things, fold it, and then take it out 00:36:58.141 --> 00:36:58.890 and put it on top. 00:36:58.890 --> 00:36:59.150 PROFESSOR: OK. 00:36:59.150 --> 00:37:00.358 How do you do that last part? 00:37:03.446 --> 00:37:07.817 AUDIENCE: Accordion fold it with the right lengths. 00:37:07.817 --> 00:37:09.900 PROFESSOR: So you have to do it with simple folds. 00:37:09.900 --> 00:37:12.110 That's the main, that's the challenge. 00:37:12.110 --> 00:37:14.642 So the idea is you're way out here, 00:37:14.642 --> 00:37:16.100 you have a triangle out here, which 00:37:16.100 --> 00:37:18.729 you want to bring over here. 00:37:18.729 --> 00:37:20.020 Maybe I should do it like this. 00:37:22.640 --> 00:37:24.930 You could do something like this, 00:37:24.930 --> 00:37:27.050 so now the triangle's over here. 00:37:27.050 --> 00:37:31.090 Then maybe you want to go almost all the way 00:37:31.090 --> 00:37:35.481 here and then fold it back, and then fold it forth, and back, 00:37:35.481 --> 00:37:37.730 until you get your triangle exactly where you want it. 00:37:37.730 --> 00:37:38.810 AUDIENCE: Figure out from the base-- 00:37:38.810 --> 00:37:39.480 PROFESSOR: It seems plausible. 00:37:39.480 --> 00:37:41.140 AUDIENCE: --where you need to fold it. 00:37:41.140 --> 00:37:43.970 PROFESSOR: My only concern would be when you do this thing, 00:37:43.970 --> 00:37:46.660 there might be a little corner. 00:37:46.660 --> 00:37:48.660 Depends how you fold this thing, and then you've 00:37:48.660 --> 00:37:50.100 got to hide that corner. 00:37:50.100 --> 00:37:52.910 And if there's triangles all around here, 00:37:52.910 --> 00:37:54.045 there may not be room. 00:37:54.045 --> 00:37:56.170 I mean, maybe if there's triangles all around here, 00:37:56.170 --> 00:37:59.130 it's OK to have that corner, but maybe the triangles 00:37:59.130 --> 00:38:00.670 are different colors. 00:38:00.670 --> 00:38:03.562 So I do believe that should be possible, 00:38:03.562 --> 00:38:05.520 but I think it is a little bit more complicated 00:38:05.520 --> 00:38:08.180 because you have to hide one last piece after you 00:38:08.180 --> 00:38:10.630 get in position. 00:38:10.630 --> 00:38:12.505 Anyway, I think there are at least three ways 00:38:12.505 --> 00:38:13.551 to solve this problem. 00:38:13.551 --> 00:38:14.050 Yeah? 00:38:14.050 --> 00:38:15.550 AUDIENCE: Are you allowed to unfold? 00:38:15.550 --> 00:38:17.091 PROFESSOR: Are you allowed to unfold? 00:38:17.091 --> 00:38:18.350 That's a good question. 00:38:18.350 --> 00:38:22.210 I don't remember whether the original model says 00:38:22.210 --> 00:38:23.890 whether you're allowed to unfold. 00:38:23.890 --> 00:38:27.170 So if there are two versions, simple folds and unfolds, 00:38:27.170 --> 00:38:29.530 or just simple folds. 00:38:29.530 --> 00:38:33.332 I don't think we actually said unfolding is allowed. 00:38:33.332 --> 00:38:35.540 Though we're definitely thinking about at some point, 00:38:35.540 --> 00:38:38.632 it's probably not in the model as defined. 00:38:38.632 --> 00:38:39.215 Any questions? 00:38:42.380 --> 00:38:44.350 What's next? 00:38:44.350 --> 00:38:46.357 This is the paper that introduced simple folds. 00:38:46.357 --> 00:38:47.940 It's called "When Can You Fold a Map?" 00:38:47.940 --> 00:38:51.920 because it originally was motivated by map folding. 00:38:51.920 --> 00:38:57.870 And it had a bunch of reasons for introducing simple folds, 00:38:57.870 --> 00:39:02.070 among them is this quote, which if you watched L1 was in there. 00:39:02.070 --> 00:39:05.500 I think the easiest way to refold a roadmap is differently 00:39:05.500 --> 00:39:07.930 and her goal was to make it easier 00:39:07.930 --> 00:39:11.100 to refold your roadmap correctly. 00:39:11.100 --> 00:39:13.770 So here's one quote from that paper as motivation. 00:39:13.770 --> 00:39:15.570 So it's origami motivation. 00:39:15.570 --> 00:39:17.920 But we're also wondering about applications, 00:39:17.920 --> 00:39:20.720 like sheet metal bending, cardboard folding, things 00:39:20.720 --> 00:39:23.660 like that where you want to manufacture things 00:39:23.660 --> 00:39:25.190 using a machine. 00:39:25.190 --> 00:39:28.610 And while origamists can do complicated folds, 00:39:28.610 --> 00:39:31.225 non-simple folds, to make art work, 00:39:31.225 --> 00:39:33.200 in practical manufacturing, you want 00:39:33.200 --> 00:39:35.140 to have the simplest possible machine. 00:39:35.140 --> 00:39:38.100 So if you can get away with just simple folds, as defined here, 00:39:38.100 --> 00:39:39.350 that would be great. 00:39:39.350 --> 00:39:42.780 Now you don't really need, some of these are maybe artificial. 00:39:42.780 --> 00:39:46.717 You probably don't need the 180 degree condition, 00:39:46.717 --> 00:39:49.050 because most of the things you want to fold aren't flat. 00:39:49.050 --> 00:39:52.567 We introduced that just to keep things simple mathematically. 00:39:52.567 --> 00:39:54.900 But you'd like to fold along just one segment at a time, 00:39:54.900 --> 00:39:55.840 ideally. 00:39:55.840 --> 00:39:57.670 You definitely don't want collision. 00:39:57.670 --> 00:39:59.410 You don't want material to hit things. 00:39:59.410 --> 00:40:01.110 Whereas in origami, you can do tucks, 00:40:01.110 --> 00:40:06.560 you can do things that are not simple folds. 00:40:06.560 --> 00:40:08.550 That's a lot harder with a machine 00:40:08.550 --> 00:40:10.900 that doesn't have any feedback. 00:40:10.900 --> 00:40:13.740 So here's a very simple machine. 00:40:13.740 --> 00:40:19.010 This is a brake folder. 00:40:19.010 --> 00:40:20.890 We actually have a brake folder in CC, 00:40:20.890 --> 00:40:23.490 although this one is Electrabrake, 00:40:23.490 --> 00:40:25.670 so this has an electric assist. 00:40:25.670 --> 00:40:30.380 So the idea is you slide your sheet in, and you hold here. 00:40:30.380 --> 00:40:32.570 You pull up, and, in this case, he's 00:40:32.570 --> 00:40:34.170 bending to a 90 degree angle. 00:40:34.170 --> 00:40:38.370 You can adjust it to different angles and so on. 00:40:38.370 --> 00:40:40.180 There are lots of automated machines, 00:40:40.180 --> 00:40:42.940 it's a little hard to get photos and videos of them, 00:40:42.940 --> 00:40:45.310 but they're based on this principle. 00:40:45.310 --> 00:40:48.340 Maybe you push in a v, and you end up 00:40:48.340 --> 00:40:49.640 with the crease in one spot. 00:40:49.640 --> 00:40:51.940 And you'd like to just make a sort of conveyor belt 00:40:51.940 --> 00:40:54.220 with lots of different pushes and pulls 00:40:54.220 --> 00:40:56.940 and do a bunch of simple folds, basically, 00:40:56.940 --> 00:40:59.137 except for this 180 degree constraint. 00:40:59.137 --> 00:41:00.720 And so we're just curious about what's 00:41:00.720 --> 00:41:02.530 possible by simple folds, and that led us 00:41:02.530 --> 00:41:04.420 into the map folding stuff, where 00:41:04.420 --> 00:41:06.370 it's fairly easy to characterize. 00:41:06.370 --> 00:41:10.790 Other things where it's harder, we'll see in lecture three. 00:41:10.790 --> 00:41:13.430 I thought I'd show you some examples of things 00:41:13.430 --> 00:41:15.670 people make with pretty much simple folds, 00:41:15.670 --> 00:41:17.870 other than this 180 condition. 00:41:17.870 --> 00:41:21.250 Out of things like-- this is folding wood. 00:41:21.250 --> 00:41:23.030 You take a sheet of material. 00:41:23.030 --> 00:41:24.410 You start bending these parts up, 00:41:24.410 --> 00:41:26.410 and you can make a little chair, a little table. 00:41:26.410 --> 00:41:28.159 And you could fold it back when you're not 00:41:28.159 --> 00:41:29.600 using your living room. 00:41:29.600 --> 00:41:31.760 You can hide everything. 00:41:31.760 --> 00:41:33.902 So you could imagine also having multiple sheets, 00:41:33.902 --> 00:41:35.610 and sometimes your room is a living room, 00:41:35.610 --> 00:41:38.140 other times it's-- whatever furniture you need, 00:41:38.140 --> 00:41:40.420 you just unfold the appropriate thing. 00:41:40.420 --> 00:41:42.602 That's the vision. 00:41:42.602 --> 00:41:44.060 Here's a cute little folding chair. 00:41:44.060 --> 00:41:46.480 There's a huge number of folding chairs, 00:41:46.480 --> 00:41:50.410 but this one is pretty much simple folds. 00:41:50.410 --> 00:41:52.010 The one thing I'm not sure whether it 00:41:52.010 --> 00:41:55.020 falls under simple folds is this fold. 00:41:55.020 --> 00:41:58.980 You do fold along one line, but it's in two different pieces. 00:41:58.980 --> 00:42:00.730 I'm not sure we'd call that a simple fold. 00:42:00.730 --> 00:42:03.350 There's, of course, lots of slits in the material here. 00:42:03.350 --> 00:42:05.584 But of course it has all the same advantages 00:42:05.584 --> 00:42:06.250 of simple folds. 00:42:06.250 --> 00:42:09.910 This is easy to execute one step at time. 00:42:09.910 --> 00:42:11.410 Here's some more complicated design. 00:42:11.410 --> 00:42:12.993 Some of these are computer renderings. 00:42:12.993 --> 00:42:14.100 Some of these are real. 00:42:14.100 --> 00:42:17.330 Again, taking furniture out of flat walls. 00:42:20.010 --> 00:42:21.330 And here's some table designs. 00:42:21.330 --> 00:42:24.119 These are sheet metal. 00:42:24.119 --> 00:42:24.785 I like this one. 00:42:24.785 --> 00:42:25.850 It's very simple. 00:42:25.850 --> 00:42:28.820 Take the square of sheet metal, put in some slips do, 00:42:28.820 --> 00:42:32.510 some very simple folds-- boom, you've got a table. 00:42:32.510 --> 00:42:36.470 This one's also pretty simple. 00:42:36.470 --> 00:42:39.060 Again, here we're folding along one line, 00:42:39.060 --> 00:42:40.540 but it's in two different pieces. 00:42:40.540 --> 00:42:41.610 So is that a simple fold? 00:42:41.610 --> 00:42:43.568 It's definitely harder to build such a machine, 00:42:43.568 --> 00:42:44.600 but it's doable. 00:42:44.600 --> 00:42:47.340 Here we have something that's definitely not a simple fold, 00:42:47.340 --> 00:42:49.810 but it's also fairly easy to execute. 00:42:49.810 --> 00:42:54.420 Using a roller, you can kind of curve one segment. 00:42:54.420 --> 00:42:56.055 I mean, when you go to reality, you 00:42:56.055 --> 00:42:57.860 can change the model all sorts of different ways 00:42:57.860 --> 00:42:59.318 and still have something practical. 00:42:59.318 --> 00:43:03.740 And no one rule set is gospel. 00:43:03.740 --> 00:43:07.200 But mathematically we have to hone in on at least one model 00:43:07.200 --> 00:43:10.010 at a time, and then we can see how changing the rules 00:43:10.010 --> 00:43:12.760 changes what you can make. 00:43:12.760 --> 00:43:14.300 OK, next question. 00:43:14.300 --> 00:43:17.020 This is actually about the definition of simple folds, 00:43:17.020 --> 00:43:20.359 so it was probably answered already. 00:43:20.359 --> 00:43:22.900 Is it allowed to bend the rest of paper to get it out the way 00:43:22.900 --> 00:43:23.770 and avoid collision? 00:43:23.770 --> 00:43:24.970 The answer is no. 00:43:24.970 --> 00:43:27.570 In simple folds, at least, you're 00:43:27.570 --> 00:43:29.899 only allowed to move that one segment. 00:43:29.899 --> 00:43:31.440 We have actually lately been thinking 00:43:31.440 --> 00:43:32.856 about a different model, where you 00:43:32.856 --> 00:43:37.120 do allow this, but simple folds, you can't move other parts. 00:43:37.120 --> 00:43:40.730 You can just move the single hinge that you're folding. 00:43:40.730 --> 00:43:44.824 And the end product has to be flat, yes, in our model. 00:43:44.824 --> 00:43:46.490 Though, it would be interesting to think 00:43:46.490 --> 00:43:48.781 without this condition, because you're doing 180 degree 00:43:48.781 --> 00:43:50.840 operations before you do the next one, 00:43:50.840 --> 00:43:52.364 you'll be flat at all times. 00:43:52.364 --> 00:43:53.780 1D or 2D, according to whether you 00:43:53.780 --> 00:43:58.280 started with a 1D piece of paper or 2D piece of paper. 00:43:58.280 --> 00:43:58.780 OK. 00:44:01.780 --> 00:44:04.410 The second half of the lecture was basically 00:44:04.410 --> 00:44:08.530 about proving, characterizing flat foldability 00:44:08.530 --> 00:44:11.110 of 1D segments. 00:44:11.110 --> 00:44:14.160 And it showed in particular that simple folds are universal, 00:44:14.160 --> 00:44:21.530 that if you have some mountain valley pattern, 00:44:21.530 --> 00:44:24.700 and it's foldable at all, if it's flat foldable, 00:44:24.700 --> 00:44:28.220 it will be flat foldable via simple folds. 00:44:28.220 --> 00:44:30.875 And in particular using crimps and n folds. 00:44:36.410 --> 00:44:37.860 And it was a bit of a messy prove, 00:44:37.860 --> 00:44:41.320 partly because I've made a mistake in lecture, as you saw. 00:44:41.320 --> 00:44:43.010 I kind of corrected for it on the fly, 00:44:43.010 --> 00:44:44.570 but it's maybe not the best written. 00:44:44.570 --> 00:44:48.660 So I wanted to go through a couple quick examples 00:44:48.660 --> 00:44:53.220 to make clear all the issues there. 00:44:53.220 --> 00:44:55.470 So here are the ones I prepared. 00:44:55.470 --> 00:44:58.530 We can certainly do more if it's still not clear. 00:45:02.660 --> 00:45:05.660 So here's a simple mountain-valley pattern, 00:45:05.660 --> 00:45:09.300 and it's got some long segments and, let's just 00:45:09.300 --> 00:45:11.110 say, equidistant segments here. 00:45:11.110 --> 00:45:13.320 Three valleys, then a mountain. 00:45:13.320 --> 00:45:18.510 So first question is, is this mingling? 00:45:18.510 --> 00:45:21.130 And then the ultimate question is, is it flat foldable? 00:45:28.140 --> 00:45:29.220 So is it mingling? 00:45:29.220 --> 00:45:32.680 Well, maybe you could answer for me. 00:45:32.680 --> 00:45:33.860 Just yes or no. 00:45:33.860 --> 00:45:34.950 50% chance. 00:45:37.910 --> 00:45:41.620 So maybe the definition of mingling is not super clear. 00:45:41.620 --> 00:45:42.990 Let me review it. 00:45:42.990 --> 00:45:45.760 So you look at each, I mean, generally 00:45:45.760 --> 00:45:47.410 of a sequence of mountains and valleys, 00:45:47.410 --> 00:45:49.301 you look at a chunk of all valleys, 00:45:49.301 --> 00:45:51.050 then you look at a chunk of all mountains, 00:45:51.050 --> 00:45:52.091 and chunk of all valleys. 00:45:52.091 --> 00:45:55.090 Here there's only two chunks-- three valleys and one mountain. 00:45:55.090 --> 00:45:57.760 And the definition is a little awkward for a single crease, 00:45:57.760 --> 00:46:00.160 but let's start with the valleys. 00:46:00.160 --> 00:46:04.860 The point is to check-- for the first segment between two 00:46:04.860 --> 00:46:10.050 valleys versus the segment just before it, which is bigger? 00:46:10.050 --> 00:46:14.030 And this is the bad case this is the non-mingling situation, 00:46:14.030 --> 00:46:16.550 because this thing is bigger than this. 00:46:16.550 --> 00:46:21.140 Strictly bigger, the notation we use in the lecture 00:46:21.140 --> 00:46:23.640 is an open square bracket. 00:46:23.640 --> 00:46:26.410 So square bracket meant that this is bigger. 00:46:26.410 --> 00:46:29.700 Round bracket would mean this is less than or equal to this. 00:46:29.700 --> 00:46:31.050 That's just the definition. 00:46:31.050 --> 00:46:33.010 Over on this side of the valleys, 00:46:33.010 --> 00:46:35.890 this length is equal to this length. 00:46:35.890 --> 00:46:38.240 So the last distance between two values 00:46:38.240 --> 00:46:40.140 is equal to the one right after it. 00:46:40.140 --> 00:46:45.820 And that's a good case, so we write a closed, round bracket. 00:46:45.820 --> 00:46:47.486 Then we have a sequence of mountains, 00:46:47.486 --> 00:46:48.860 and here it's a little confusing, 00:46:48.860 --> 00:46:51.490 but it's the same idea. 00:46:51.490 --> 00:46:53.159 So this is the very first mountain. 00:46:53.159 --> 00:46:55.450 You look at the length right after it versus the length 00:46:55.450 --> 00:46:58.070 right before it, and this is smaller. 00:46:58.070 --> 00:46:59.240 And that's a good case. 00:46:59.240 --> 00:47:03.280 We write an open, round bracket for this mountain group. 00:47:03.280 --> 00:47:04.430 And then same thing. 00:47:04.430 --> 00:47:06.390 Now we're comparing the same two distances, 00:47:06.390 --> 00:47:08.700 but it's now bad, because this one is strictly longer 00:47:08.700 --> 00:47:09.320 than this one. 00:47:09.320 --> 00:47:11.620 So we write a closed, square bracket. 00:47:11.620 --> 00:47:13.890 So that's the notation in this example. 00:47:13.890 --> 00:47:17.030 Any questions about that? 00:47:17.030 --> 00:47:19.140 So you just have to check-- in general, 00:47:19.140 --> 00:47:23.590 you have a whole group of valleys-- these are all 00:47:23.590 --> 00:47:26.510 valleys, or all mountains-- and you 00:47:26.510 --> 00:47:29.840 want to compare this one versus this one. 00:47:29.840 --> 00:47:32.760 And square or closed, according to which is bigger. 00:47:32.760 --> 00:47:36.010 And you want to look at the last one versus the after last one. 00:47:38.530 --> 00:47:40.070 So that's the notation. 00:47:40.070 --> 00:47:43.410 And the point of the proof was to argue that, either 00:47:43.410 --> 00:47:47.410 if you're going to be flat foldable at all, 00:47:47.410 --> 00:47:49.820 actually if you're mingling-- mingling meant 00:47:49.820 --> 00:47:52.570 that for each of these intervals, at least one 00:47:52.570 --> 00:47:54.780 of the sides was round. 00:47:54.780 --> 00:47:56.180 That was considered good. 00:47:56.180 --> 00:48:00.470 So this crease pattern is mingling, 00:48:00.470 --> 00:48:02.360 because there's two regions. 00:48:02.360 --> 00:48:03.590 This one has a round bracket. 00:48:03.590 --> 00:48:05.320 This one has a round bracket. 00:48:05.320 --> 00:48:08.740 And what we argue is it if you're mingling, 00:48:08.740 --> 00:48:11.140 which was necessary, if you're flat foldable, 00:48:11.140 --> 00:48:12.780 you have to be mingling. 00:48:12.780 --> 00:48:14.740 It's a necessary but not sufficient condition 00:48:14.740 --> 00:48:16.760 for flat foldability. 00:48:16.760 --> 00:48:19.300 If you're mingling, either you have a pattern 00:48:19.300 --> 00:48:22.140 like this, close round bracket, open round bracket. 00:48:22.140 --> 00:48:26.590 That's good because this is a crimp that you can do. 00:48:26.590 --> 00:48:27.470 You see it up here. 00:48:27.470 --> 00:48:29.070 This is a crimp. 00:48:29.070 --> 00:48:31.610 You valley fold, mountain fold, and you don't hide anything 00:48:31.610 --> 00:48:33.360 when you make that operation. 00:48:33.360 --> 00:48:35.820 Or there's an end fold, which corresponded to an open round 00:48:35.820 --> 00:48:37.236 bracket at the beginning or closed 00:48:37.236 --> 00:48:38.510 round bracket at the end. 00:48:38.510 --> 00:48:39.610 So here there's a crimp. 00:48:39.610 --> 00:48:42.260 Let's do the crimp. 00:48:42.260 --> 00:48:47.090 So when we do the crimp, let's keep this part of paper fixed. 00:48:47.090 --> 00:48:52.070 So this we go over to here, hopefully, 00:48:52.070 --> 00:48:56.470 then we valley fold, then we mountain fold, 00:48:56.470 --> 00:48:59.490 and we keep going from there. 00:48:59.490 --> 00:49:02.660 That segment is that segment. 00:49:02.660 --> 00:49:07.930 So we still have this valley. 00:49:07.930 --> 00:49:09.950 This was a valley we just folded. 00:49:09.950 --> 00:49:12.030 This is the mountain we just folded. 00:49:12.030 --> 00:49:14.472 Now conceptually, we just sort of fuse this back 00:49:14.472 --> 00:49:16.430 into the paper, because those creases are done. 00:49:16.430 --> 00:49:17.846 We don't need to think about them. 00:49:17.846 --> 00:49:22.160 The point is, in that region there were no extra creases. 00:49:22.160 --> 00:49:23.910 These round parentheses will guarantee 00:49:23.910 --> 00:49:27.530 there's nothing here, no creases here, here, or here. 00:49:27.530 --> 00:49:31.050 Could be creases farther away but by these inequalities, 00:49:31.050 --> 00:49:34.140 that this length is less than or equal to this one, 00:49:34.140 --> 00:49:38.220 and this length is less than or equal to this one. 00:49:38.220 --> 00:49:41.050 Sorry, greater than or equal to. 00:49:41.050 --> 00:49:42.430 Then you know this is OK. 00:49:47.430 --> 00:49:50.120 So there's two valleys left. 00:49:50.120 --> 00:49:52.799 So now we have two valleys, we have a long segment, 00:49:52.799 --> 00:49:53.590 and a long segment. 00:49:53.590 --> 00:49:55.970 And this is something that can't be made. 00:49:55.970 --> 00:49:58.470 Because there's no folded state of this thing, 00:49:58.470 --> 00:50:02.340 never mind simple folds, because it's going to cross like that. 00:50:02.340 --> 00:50:03.570 So this is not flat foldable. 00:50:06.340 --> 00:50:09.680 It's also not mingling, because if you 00:50:09.680 --> 00:50:11.842 look at these two valleys-- 00:50:11.842 --> 00:50:13.300 You look at the distance over here. 00:50:13.300 --> 00:50:15.160 It's bigger than this one, so that's bad, 00:50:15.160 --> 00:50:17.110 so you have an open square bracket. 00:50:17.110 --> 00:50:21.121 And this one is also bigger than this, imagine these as fused, 00:50:21.121 --> 00:50:22.620 and so it's a closed square bracket. 00:50:22.620 --> 00:50:25.140 And so this group of values is not mingling. 00:50:25.140 --> 00:50:26.930 So it's not mingling. 00:50:26.930 --> 00:50:29.160 So ultimately this pattern we started 00:50:29.160 --> 00:50:32.300 with is not flat foldable, because one of the things we 00:50:32.300 --> 00:50:36.279 proved is doing a crimp never changes flat foldability. 00:50:36.279 --> 00:50:37.570 It's always a safe thing to do. 00:50:40.302 --> 00:50:41.760 You might wonder, oh, maybe there's 00:50:41.760 --> 00:50:44.340 some other fold I could to do that eventually works, 00:50:44.340 --> 00:50:47.380 but we proved crimps are always safe to do. 00:50:47.380 --> 00:50:49.280 So we did it, and we got stock. 00:50:49.280 --> 00:50:51.220 That means this was not flat foldable, 00:50:51.220 --> 00:50:52.680 even though it was mingling. 00:50:52.680 --> 00:50:55.050 And so the mingling forever property just 00:50:55.050 --> 00:50:57.490 means, if it's mingling and then you do a crimp, 00:50:57.490 --> 00:50:59.185 and it's still mingling, and if you keep doing crimps, 00:50:59.185 --> 00:51:00.601 and it stays mingling all the way, 00:51:00.601 --> 00:51:01.947 then you were flat foldable. 00:51:01.947 --> 00:51:03.780 It's not a very satisfying characterization, 00:51:03.780 --> 00:51:07.660 but it is a thing. 00:51:07.660 --> 00:51:15.030 Maybe I'll do one more example where it works, 00:51:15.030 --> 00:51:18.110 so we're super clear. 00:51:18.110 --> 00:51:22.605 Whoa, we're out of time, so I won't do another example. 00:51:22.605 --> 00:51:23.480 All right, ambitious. 00:51:29.930 --> 00:51:31.800 I gotta work on my timing. 00:51:31.800 --> 00:51:32.300 There are a 00:51:32.300 --> 00:51:33.700 Couple other fun questions here. 00:51:33.700 --> 00:51:35.824 I would encourage you to read the notes about them. 00:51:35.824 --> 00:51:38.030 In particular, there's an algorithmic question. 00:51:38.030 --> 00:51:40.050 How do you actually compute this efficiently? 00:51:40.050 --> 00:51:43.030 You could do it very efficiently in linear time. 00:51:43.030 --> 00:51:47.320 So n, where c is one, just n time-- instead of n squared 00:51:47.320 --> 00:51:50.384 or something-- using a pretty simple idea. 00:51:50.384 --> 00:51:52.050 Basically just look for the first crimp, 00:51:52.050 --> 00:51:54.360 do it, and then see if there are crimps nearby, 00:51:54.360 --> 00:51:55.927 and keep going forward, and you can 00:51:55.927 --> 00:51:57.260 prove that it takes linear time. 00:52:00.220 --> 00:52:02.660 There is this fun question I enjoy thinking about. 00:52:02.660 --> 00:52:05.280 Can you make any mountain-valley pattern flat foldable 00:52:05.280 --> 00:52:06.260 by adding creases? 00:52:06.260 --> 00:52:07.197 The answer is yes. 00:52:07.197 --> 00:52:08.530 You can think of it as a puzzle. 00:52:08.530 --> 00:52:10.180 There is one proposed way to do it here. 00:52:10.180 --> 00:52:11.513 I have another one in the notes. 00:52:11.513 --> 00:52:13.592 You can think about it. 00:52:13.592 --> 00:52:15.050 And the last question, is what does 00:52:15.050 --> 00:52:17.300 it possibly mean to fold something in four dimensions? 00:52:17.300 --> 00:52:18.430 How do you imagine it? 00:52:18.430 --> 00:52:21.360 Hard to imagine, but you can think about it. 00:52:21.360 --> 00:52:23.390 You have a d-dimensional piece of paper, 00:52:23.390 --> 00:52:25.820 you fold it through d plus 1 dimensions. 00:52:25.820 --> 00:52:29.910 If you want it flat folded, it ends up back in d dimensions, 00:52:29.910 --> 00:52:32.890 and your creases are d minus 1 dimensional. 00:52:32.890 --> 00:52:34.660 And the rest you just have to visualize. 00:52:34.660 --> 00:52:36.565 I have one example folding a solid cube 00:52:36.565 --> 00:52:38.070 in half in the notes. 00:52:38.070 --> 00:52:39.432 That's certainly possible. 00:52:39.432 --> 00:52:41.140 That's not very well studied, and there's 00:52:41.140 --> 00:52:42.556 lots of interesting open questions 00:52:42.556 --> 00:52:44.380 about higher dimensional folding. 00:52:44.380 --> 00:52:47.860 Any questions before we go? 00:52:47.860 --> 00:52:48.950 All right. 00:52:48.950 --> 00:52:51.700 Watch lecture three, and please send your feedback. 00:52:51.700 --> 00:52:53.390 It was really helpful.