WEBVTT 00:00:03.400 --> 00:00:06.770 PROFESSOR: This class is talking about two lectures, both about 00:00:06.770 --> 00:00:09.760 protein folding, fixed angled chains, things like that. 00:00:09.760 --> 00:00:12.610 There's a few questions about them. 00:00:12.610 --> 00:00:15.330 One is mostly about these open problems, 00:00:15.330 --> 00:00:20.660 equilateral, equiangular, obtuse 3D chains, 00:00:20.660 --> 00:00:24.270 fixed angle open problem is, are they locked? 00:00:24.270 --> 00:00:26.850 So the question is about, what about any subset 00:00:26.850 --> 00:00:29.030 of those combinations? 00:00:29.030 --> 00:00:31.050 So this originally comes from an open problem 00:00:31.050 --> 00:00:35.800 I think posed in 1999, one of the first 3D linkage papers, 00:00:35.800 --> 00:00:42.980 and it asked whether equilateral universal joints can lock, 00:00:42.980 --> 00:00:45.140 and that's still open. 00:00:45.140 --> 00:00:47.830 For universal joints, these two constraints 00:00:47.830 --> 00:00:49.481 don't make a lot of sense because who 00:00:49.481 --> 00:00:50.980 cares if it's initially equiangular. 00:00:50.980 --> 00:00:54.140 As soon as you move it, it will no longer be equiangular. 00:00:54.140 --> 00:00:58.070 And obtuseness I don't think matters too much, 00:00:58.070 --> 00:01:00.310 although it potentially could. 00:01:00.310 --> 00:01:03.310 So for that problem, equilateral seems to be the core. 00:01:03.310 --> 00:01:07.470 For fixed angle, though, we conjecture 00:01:07.470 --> 00:01:11.800 that fixed angle equilateral is not enough from this example. 00:01:11.800 --> 00:01:13.750 It's still not proved to be locked. 00:01:13.750 --> 00:01:17.290 I don't know if it's hard, but it's probably tedious, 00:01:17.290 --> 00:01:21.630 so it hasn't been done. 00:01:21.630 --> 00:01:24.040 This was the crossed legs example. 00:01:24.040 --> 00:01:25.630 All the edge lengths are the same, 00:01:25.630 --> 00:01:29.670 and if you don't allow the touching part, 00:01:29.670 --> 00:01:31.500 then all the angles are also the same. 00:01:31.500 --> 00:01:34.220 So this is everything except the obtuse property 00:01:34.220 --> 00:01:39.470 and it's probably locked, so dropping obtuse is no good. 00:01:39.470 --> 00:01:43.330 The other things you could drop are equilateral or equiangular. 00:01:43.330 --> 00:01:46.950 This is if you drop equiangular. 00:01:49.870 --> 00:01:53.160 Fixed angle equilateral is not terribly constraining 00:01:53.160 --> 00:01:57.010 because you can simulate a long bar 00:01:57.010 --> 00:02:00.500 by having a lot of 180 degree angles. 00:02:00.500 --> 00:02:05.086 This is obtuse and it's equilateral 00:02:05.086 --> 00:02:06.460 but it's not equiangular and it's 00:02:06.460 --> 00:02:09.790 locked for a trivial reason. 00:02:09.790 --> 00:02:13.040 So this is dropping equiangular. 00:02:13.040 --> 00:02:17.120 If I drop equilateral, I can also 00:02:17.120 --> 00:02:18.930 just make a knitting needles example. 00:02:18.930 --> 00:02:23.152 I take a really long link, and this thing is basically string, 00:02:23.152 --> 00:02:24.110 so I don't really care. 00:02:24.110 --> 00:02:27.220 This connection can be done with a lot of obtuse angles, 00:02:27.220 --> 00:02:28.870 and then have a really long link. 00:02:28.870 --> 00:02:32.090 So dropping any of the three constraints 00:02:32.090 --> 00:02:34.860 makes it easy to lock, so for fixed angle chains, 00:02:34.860 --> 00:02:37.330 you need equilateral, equiangular, and obtuse. 00:02:37.330 --> 00:02:39.699 All these things together potentially mean 00:02:39.699 --> 00:02:41.240 you're not locked, but we don't know. 00:02:41.240 --> 00:02:42.510 That's the open problem. 00:02:42.510 --> 00:02:44.010 Of course, we don't necessarily need 00:02:44.010 --> 00:02:46.440 exactly equilateral or equiangular. 00:02:46.440 --> 00:02:50.790 Hopefully, within some small min to max ratio would be enough, 00:02:50.790 --> 00:02:52.050 but we don't know. 00:02:54.820 --> 00:03:00.480 Next question is, why did we model the ribosome as a cone? 00:03:00.480 --> 00:03:02.090 That seems rather simple. 00:03:02.090 --> 00:03:03.550 Is this realistic? 00:03:03.550 --> 00:03:08.330 And partly, when I taught the class in 2010, 00:03:08.330 --> 00:03:12.030 I didn't use any images I couldn't get permission for. 00:03:12.030 --> 00:03:15.580 This year I'm more lax. 00:03:15.580 --> 00:03:17.780 We'll ask for forgiveness instead of permission. 00:03:17.780 --> 00:03:20.880 This is a paper from Science 2000. 00:03:20.880 --> 00:03:22.910 This is what the ribosome actually looks like. 00:03:22.910 --> 00:03:25.680 There's many different figures of it, 00:03:25.680 --> 00:03:28.075 but I particularly like this one because it highlights 00:03:28.075 --> 00:03:31.570 a tunnel in the center of the ribosome. 00:03:31.570 --> 00:03:37.710 And so the idea is this is a machine for converting MRNA 00:03:37.710 --> 00:03:39.489 into your proteins, and the idea is 00:03:39.489 --> 00:03:40.780 the protein comes through here. 00:03:40.780 --> 00:03:42.100 There's a little bump in the tunnel. 00:03:42.100 --> 00:03:44.590 Some people conjecture this is where the amino acid gets 00:03:44.590 --> 00:03:46.670 attached, and then it feeds through here 00:03:46.670 --> 00:03:47.670 and starts spitting out. 00:03:47.670 --> 00:03:50.020 And as I said, there's barely enough room 00:03:50.020 --> 00:03:55.760 here for an alpha helix, so probably not too much folding 00:03:55.760 --> 00:03:59.510 happens inside and the folding should just happen over here. 00:03:59.510 --> 00:04:02.730 And the observation is if you have a reasonable size 00:04:02.730 --> 00:04:05.960 protein that's only going to be about this big, 00:04:05.960 --> 00:04:10.120 then there's this big, flat wall at the exit, 00:04:10.120 --> 00:04:14.640 so you have a plane there and this half space 00:04:14.640 --> 00:04:18.649 is more or less a big obstacle. 00:04:18.649 --> 00:04:23.090 It's the alpha cone model where alpha is 180, I think, 00:04:23.090 --> 00:04:25.110 if alpha is the half angle. 00:04:25.110 --> 00:04:26.500 So it's not really a cone. 00:04:26.500 --> 00:04:29.140 It's a cone that's been opened up all the way to a plane, 00:04:29.140 --> 00:04:32.490 but that's one of the situations that's handled by the theorems 00:04:32.490 --> 00:04:34.160 that we talked about. 00:04:34.160 --> 00:04:37.067 So that's why the cone model. 00:04:37.067 --> 00:04:38.650 We generalize to cones just because it 00:04:38.650 --> 00:04:41.390 works for general cones, but the real one 00:04:41.390 --> 00:04:44.462 is a sort of flat cone. 00:04:44.462 --> 00:04:45.670 That's where that comes from. 00:04:48.810 --> 00:04:51.500 I think actually we proved the theorem before we knew this, 00:04:51.500 --> 00:04:55.600 but then we looked it up and it was true. 00:04:55.600 --> 00:04:58.940 And the last question is about the lecture 21, 00:04:58.940 --> 00:05:01.660 which some of you may not have watched because it's optional, 00:05:01.660 --> 00:05:04.570 but there's this model called the HP model 00:05:04.570 --> 00:05:05.660 for protein folding. 00:05:05.660 --> 00:05:08.020 It's a model of protein energy and it 00:05:08.020 --> 00:05:11.230 says you have a chain of H and P nodes, 00:05:11.230 --> 00:05:13.850 and basically, the H nodes are attracted to each other 00:05:13.850 --> 00:05:16.280 and the P nodes don't care. 00:05:16.280 --> 00:05:19.570 And the model is that the H nodes are hydrophobic so they 00:05:19.570 --> 00:05:22.470 want to be next to each other so they're not next to water, 00:05:22.470 --> 00:05:24.200 which is surrounding the whole molecule, 00:05:24.200 --> 00:05:29.720 and the P nodes don't care, or they're hydrophilic. 00:05:29.720 --> 00:05:32.070 And it's known that if you have an HP string 00:05:32.070 --> 00:05:35.210 and you want to find the optimal folding into a 2D or a 3D 00:05:35.210 --> 00:05:38.370 structure, that's NP-hard, which is kind of weird 00:05:38.370 --> 00:05:39.970 because somehow nature does it, maybe 00:05:39.970 --> 00:05:43.100 because it found the easy instances by evolution, 00:05:43.100 --> 00:05:46.340 or maybe there's something we're missing. 00:05:46.340 --> 00:05:47.870 This model doesn't capture reality. 00:05:47.870 --> 00:05:49.020 We don't know. 00:05:49.020 --> 00:05:52.820 But it captures part of reality, as least as we observe it. 00:05:52.820 --> 00:05:54.320 Unfortunately, these hardness proofs 00:05:54.320 --> 00:05:58.270 are a little too complicated to cover here, 00:05:58.270 --> 00:06:01.360 but I can at least answer what are they reducing from. 00:06:01.360 --> 00:06:02.990 There's two proofs. 00:06:02.990 --> 00:06:08.242 I guess they were basically the same time, I think around 2001. 00:06:08.242 --> 00:06:11.600 The first one here is in 3D, and these 00:06:11.600 --> 00:06:13.280 are by two MIT professors. 00:06:16.400 --> 00:06:18.360 There's this big construction but the reduction 00:06:18.360 --> 00:06:19.570 is from bin packing. 00:06:19.570 --> 00:06:21.490 So you have a bunch of fixed size bins 00:06:21.490 --> 00:06:23.975 and you have a bunch of items of varying sizes 00:06:23.975 --> 00:06:25.350 and you just want to fit them all 00:06:25.350 --> 00:06:28.750 in using the fewest bins possible. 00:06:28.750 --> 00:06:32.720 And the rough idea of the construction of this one, this 00:06:32.720 --> 00:06:35.220 is how an individual number is represented. 00:06:35.220 --> 00:06:37.215 The rough idea is you have this big cube. 00:06:41.650 --> 00:06:43.590 You fill the sides with stuff to protect 00:06:43.590 --> 00:06:45.930 from the outside boundary. 00:06:45.930 --> 00:06:49.840 The insides are just used for connections, 00:06:49.840 --> 00:06:53.650 and then the front face here is this stuff. 00:06:53.650 --> 00:06:57.964 You construct these bins, and you 00:06:57.964 --> 00:07:00.130 construct numbers which have to fit inside the bins. 00:07:00.130 --> 00:07:01.900 That's the rough idea. 00:07:01.900 --> 00:07:04.230 The details are complicated. 00:07:04.230 --> 00:07:10.330 The 2D proof by several people is 00:07:10.330 --> 00:07:13.660 from Hamiltonicity in maximum degree-4 graphs. 00:07:13.660 --> 00:07:15.450 So you have a graph for every vertex that 00:07:15.450 --> 00:07:16.783 has at most four incident edges. 00:07:16.783 --> 00:07:19.610 You want to find a Hamiltonian cycle, I think. 00:07:22.250 --> 00:07:25.960 It's much harder to see this picture, I would say. 00:07:25.960 --> 00:07:29.530 There's no one diagram that summarizes it. 00:07:29.530 --> 00:07:31.630 This is roughly the construction, 00:07:31.630 --> 00:07:36.265 which is quite complicated, and I'll just leave it at that. 00:07:36.265 --> 00:07:37.640 That's what they're reduced from. 00:07:37.640 --> 00:07:39.098 I think an interesting open problem 00:07:39.098 --> 00:07:42.730 would be to find a simple or cleaner proof of these results. 00:07:42.730 --> 00:07:44.460 Now that it's known that they're hard, 00:07:44.460 --> 00:07:46.120 it's probably easy or hardness proofs. 00:07:46.120 --> 00:07:48.036 There's also some open questions from lecture, 00:07:48.036 --> 00:07:51.880 like is it APX hard? 00:07:51.880 --> 00:07:55.380 Is it approximable to a 1 plus epsilon factor or 1 minus 00:07:55.380 --> 00:07:57.130 epsilon factor for any epsilon or is there 00:07:57.130 --> 00:07:58.910 some limit to approximability? 00:07:58.910 --> 00:08:04.930 We saw a nice approximation 4/3 whatever, 3/4 whatever, 00:08:04.930 --> 00:08:07.990 but can you do better? 00:08:07.990 --> 00:08:10.410 Still open. 00:08:10.410 --> 00:08:16.367 So that was the questions, and then this is a question 00:08:16.367 --> 00:08:17.950 that you usually ask in every lecture. 00:08:17.950 --> 00:08:19.574 You didn't actually ask it on this one, 00:08:19.574 --> 00:08:21.820 but I copied and pasted. 00:08:21.820 --> 00:08:25.810 So there's some interesting progress 00:08:25.810 --> 00:08:29.220 from this class two years ago, and I think in the open problem 00:08:29.220 --> 00:08:32.559 session initially, probably also a class project 00:08:32.559 --> 00:08:35.789 related to this part. 00:08:35.789 --> 00:08:39.220 So this is back in lecture 20. 00:08:39.220 --> 00:08:42.539 We had this proof, which is kind of fun, 00:08:42.539 --> 00:08:45.540 that flattening a fixed angle chain, 00:08:45.540 --> 00:08:48.500 deciding whether there was a flat folded state, 00:08:48.500 --> 00:08:51.860 is weakly NP hard, weakly meaning 00:08:51.860 --> 00:08:54.560 it depended a lot on what these numbers look like 00:08:54.560 --> 00:08:56.410 and it was a reduction from partitions. 00:08:56.410 --> 00:08:59.940 You had to split up the numbers into two equal parts, 00:08:59.940 --> 00:09:02.360 and if you did, then this key would fit in the slot 00:09:02.360 --> 00:09:03.610 and you're OK. 00:09:03.610 --> 00:09:08.840 If you didn't, the key would collide with something here. 00:09:08.840 --> 00:09:11.400 So this means the problem is hard 00:09:11.400 --> 00:09:14.100 if your edge lengths are vastly different. 00:09:14.100 --> 00:09:17.810 They would have to differ by a ratio of exponential 00:09:17.810 --> 00:09:20.709 in the number of edges for this to really be hard. 00:09:20.709 --> 00:09:23.000 So a natural question is, well, what if all the lengths 00:09:23.000 --> 00:09:25.970 are equal, equilateral chains? 00:09:25.970 --> 00:09:30.410 Turns out that is still hard, and this is a paper just 00:09:30.410 --> 00:09:33.330 published last year with Sarah Eisenstadt, who 00:09:33.330 --> 00:09:35.340 took the class then. 00:09:35.340 --> 00:09:38.140 And there's a bunch of results here. 00:09:38.140 --> 00:09:41.770 All of these are NP hardness results, strong NP hardness, 00:09:41.770 --> 00:09:43.890 so it doesn't depend on the numbers. 00:09:43.890 --> 00:09:46.070 We've got flattening results. 00:09:46.070 --> 00:09:48.027 There's also min flat span and max flat span. 00:09:48.027 --> 00:09:49.610 Once you have that flattening is hard, 00:09:49.610 --> 00:09:51.090 these are pretty easy to show hard, 00:09:51.090 --> 00:09:54.090 so I won't talk about them so much. 00:09:54.090 --> 00:09:57.890 But basically, we consider different ranges of angles 00:09:57.890 --> 00:09:59.850 and what you might allow. 00:10:03.180 --> 00:10:05.780 In some cases, we can get perfect equilateral chains. 00:10:05.780 --> 00:10:08.400 In some cases, they have to range between, say, 1 and 2 00:10:08.400 --> 00:10:09.010 in length. 00:10:13.655 --> 00:10:15.760 A natural question is obtuse angles. 00:10:15.760 --> 00:10:16.920 We don't quite know that. 00:10:16.920 --> 00:10:22.580 The best we have is 60 degrees minus epsilon and larger. 00:10:22.580 --> 00:10:25.240 This is nice for orthogonal chains, 00:10:25.240 --> 00:10:27.560 but it's not for chains, it's for trees. 00:10:27.560 --> 00:10:33.421 So still some open questions here, but lots of hardness 00:10:33.421 --> 00:10:33.920 results. 00:10:33.920 --> 00:10:36.419 I thought I'd show you roughly what the hardness proofs look 00:10:36.419 --> 00:10:38.340 like because they're kind of fun. 00:10:38.340 --> 00:10:40.030 They all follow this kind of structure. 00:10:40.030 --> 00:10:42.839 This is a gadget, and it's a fixed angle chain. 00:10:42.839 --> 00:10:44.880 Here I'm going to show everything with 90 degrees 00:10:44.880 --> 00:10:46.510 because it's easier to think about. 00:10:46.510 --> 00:10:54.210 And it's kind of like-- I'm thinking plunger, 00:10:54.210 --> 00:10:55.810 but it's something like that. 00:10:55.810 --> 00:10:58.475 You can decide which parts get pushed in 00:10:58.475 --> 00:11:04.310 and which parts get popped out, and this L shape can basically 00:11:04.310 --> 00:11:08.450 shift left and right to three different places 00:11:08.450 --> 00:11:14.190 if I did it right-- this one, this one, and this one. 00:11:14.190 --> 00:11:16.850 So that's a useful construction, as you might imagine, 00:11:16.850 --> 00:11:26.170 and the idea is you take that and you add on these guys. 00:11:26.170 --> 00:11:31.750 So I've got another kind of plungey thing like this, 00:11:31.750 --> 00:11:37.890 and this guy can move left and right to various extents. 00:11:37.890 --> 00:11:39.410 That's not very intuitive. 00:11:42.920 --> 00:11:46.920 So then this connects to these little guys, 00:11:46.920 --> 00:11:49.320 and these can just flip up or down 00:11:49.320 --> 00:11:54.400 if it's a flat embedding of a fixed angle chain. 00:11:54.400 --> 00:12:00.120 And so the idea is you have three of these pokey elements 00:12:00.120 --> 00:12:01.560 and they can attach. 00:12:01.560 --> 00:12:04.800 They bump into these different things. 00:12:04.800 --> 00:12:07.490 And if it's up, then this can be up, 00:12:07.490 --> 00:12:12.130 but when they're down-- like this guy is currently down, 00:12:12.130 --> 00:12:15.250 but it could be pushed up-- then this guy 00:12:15.250 --> 00:12:17.100 must be flipped down like that. 00:12:17.100 --> 00:12:21.000 That's the rough idea of how these parts fit together. 00:12:21.000 --> 00:12:26.050 And so then this is how you end up building some kind of three 00:12:26.050 --> 00:12:27.020 sat problem. 00:12:27.020 --> 00:12:29.880 So you have some variables. 00:12:29.880 --> 00:12:33.810 They can either be true or false. 00:12:33.810 --> 00:12:36.720 And the idea is that the true guys are on the top, 00:12:36.720 --> 00:12:38.680 the bottom guys are on the bottom, 00:12:38.680 --> 00:12:41.740 and there's some complicated interaction between them 00:12:41.740 --> 00:12:44.232 to make sure that they can't both be true and false. 00:12:44.232 --> 00:12:45.940 Basically, you want these things to point 00:12:45.940 --> 00:12:49.160 in whenever possible because then these guys can go down. 00:12:49.160 --> 00:12:51.140 Otherwise, you would collide. 00:12:51.140 --> 00:12:53.500 And if this guy's in, the corresponding guys 00:12:53.500 --> 00:12:55.770 in the bottom must be down, and that's 00:12:55.770 --> 00:12:58.550 enforced by this long thing, which is either 00:12:58.550 --> 00:13:00.254 completely down or completely up. 00:13:00.254 --> 00:13:01.920 So when it's completely down, these guys 00:13:01.920 --> 00:13:03.378 all have to be down, which is going 00:13:03.378 --> 00:13:06.994 to be a problem down below if things aren't satisfied. 00:13:06.994 --> 00:13:09.160 When it's up, all of these guys would have to be up. 00:13:09.160 --> 00:13:11.182 Some of them could be up if you feel like it, 00:13:11.182 --> 00:13:12.640 but really, you'd probably put them 00:13:12.640 --> 00:13:14.590 all down whenever you can because then you 00:13:14.590 --> 00:13:17.536 can take these guys and stick them down. 00:13:17.536 --> 00:13:18.910 So this is how you set variables, 00:13:18.910 --> 00:13:20.640 and then the other things I showed 00:13:20.640 --> 00:13:23.340 were clauses, essentially. 00:13:23.340 --> 00:13:26.100 At least one of those pins had to be pushed down 00:13:26.100 --> 00:13:30.470 and you don't know which one, and that's the hard part. 00:13:30.470 --> 00:13:32.930 Rough sketch of how this proof looks. 00:13:35.590 --> 00:13:40.740 One thing is here, for this to work, 00:13:40.740 --> 00:13:44.360 these edges can't be spinnable. 00:13:44.360 --> 00:13:46.670 When I go here, I must immediately go back down. 00:13:46.670 --> 00:13:50.939 I can't flip it. 00:13:50.939 --> 00:13:51.980 So these are rigid edges. 00:13:51.980 --> 00:13:53.900 That's not really the original problem, 00:13:53.900 --> 00:13:55.330 but you can simulate rigid edges, 00:13:55.330 --> 00:13:57.310 and this is where the angles get small. 00:13:57.310 --> 00:14:00.160 You can simulate rigid edges with a very sharp zigzag 00:14:00.160 --> 00:14:02.260 because if you ever flipped one of these, 00:14:02.260 --> 00:14:04.450 it would collide with the previous edge. 00:14:04.450 --> 00:14:07.680 So for sufficiently sharp angles, 00:14:07.680 --> 00:14:12.390 you can force parts to stay straight in this funny way, 00:14:12.390 --> 00:14:15.650 which is not great, but it's one way to force it. 00:14:15.650 --> 00:14:19.160 We can also do it with trees or various other techniques, 00:14:19.160 --> 00:14:22.370 but that's how the proof looks. 00:14:22.370 --> 00:14:28.750 So that is flattening fixed angle chains, strongly NP hard. 00:14:28.750 --> 00:14:34.870 We go next to one more topic for fun. 00:14:34.870 --> 00:14:37.335 This is the topic of flips. 00:14:41.130 --> 00:14:44.590 I guess I have an image for this. 00:14:44.590 --> 00:14:46.500 This is a problem posed by Paul Erdos, 00:14:46.500 --> 00:14:48.670 I think when he was a student. 00:14:48.670 --> 00:14:50.190 He was a very famous mathematician. 00:14:50.190 --> 00:14:53.500 1935, he asked this question. 00:14:53.500 --> 00:14:56.950 This is the whole question as originally written. 00:14:56.950 --> 00:15:01.049 You have a polygon and you take the convex hull of the polygon, 00:15:01.049 --> 00:15:03.340 and you take these regions which are in the convex hull 00:15:03.340 --> 00:15:04.339 but outside the polygon. 00:15:04.339 --> 00:15:05.720 We call those pockets. 00:15:05.720 --> 00:15:08.140 And you imagine flipping those outside, 00:15:08.140 --> 00:15:11.430 reflecting through that supporting line, 00:15:11.430 --> 00:15:14.520 that tangent on the outside, so then you get this polygon. 00:15:14.520 --> 00:15:16.440 Here you get two pockets. 00:15:16.440 --> 00:15:20.430 Maybe you flip both of them, and then you get a convex polygon. 00:15:20.430 --> 00:15:22.970 Now you have no pockets, and so you're done. 00:15:22.970 --> 00:15:26.040 And the question is as posed. 00:15:26.040 --> 00:15:28.950 Prove that after a finite number of such steps, 00:15:28.950 --> 00:15:31.770 the polygon would become convex. 00:15:31.770 --> 00:15:34.980 So you might call this a conjecture by Erdos 00:15:34.980 --> 00:15:37.240 that it is finite. 00:15:37.240 --> 00:15:39.920 He'd never published a proof. 00:15:39.920 --> 00:15:42.120 There's one issue with the problem as stated. 00:15:42.120 --> 00:15:43.860 You can't actually flip both pockets 00:15:43.860 --> 00:15:47.940 at once if you want to avoid collisions. 00:15:47.940 --> 00:15:49.860 The relation to linkages is we imagine 00:15:49.860 --> 00:15:52.910 these flips could actually happen by a rotation, 00:15:52.910 --> 00:15:54.410 but you could also just imagine them 00:15:54.410 --> 00:15:57.350 as reflecting instantaneously. 00:15:57.350 --> 00:15:58.520 There's this issue. 00:15:58.520 --> 00:16:00.870 If you flip two pockets at once, you 00:16:00.870 --> 00:16:03.730 might have collisions afterwards. 00:16:03.730 --> 00:16:06.580 The way this problem has been interpreted by most people 00:16:06.580 --> 00:16:08.810 is don't flip them all at once. 00:16:08.810 --> 00:16:10.400 Just flip them one at a time, and then 00:16:10.400 --> 00:16:13.420 you guarantee, because this is a supporting line, 00:16:13.420 --> 00:16:14.970 a tangent line, of the convex hull, 00:16:14.970 --> 00:16:18.060 when you flip one pocket out, it will go to the other side 00:16:18.060 --> 00:16:20.000 so it can't intersect the rest of the polygon. 00:16:20.000 --> 00:16:25.940 So you would avoid collision if you do one flip at a time. 00:16:25.940 --> 00:16:30.420 Now, this was observed, I guess, a few years later 00:16:30.420 --> 00:16:32.020 by Bela de Sz. 00:16:32.020 --> 00:16:40.980 Nagy, and he also published the first proof. 00:16:40.980 --> 00:16:45.390 Before we get there, a weird property 00:16:45.390 --> 00:16:48.760 and why you might worry about this being finite or infinite 00:16:48.760 --> 00:16:50.740 is if you take this quadrilateral 00:16:50.740 --> 00:16:54.270 and make this very narrow, this edge very small 00:16:54.270 --> 00:16:56.980 relative to the horizontal edge, then 00:16:56.980 --> 00:17:01.410 you can require arbitrarily many flips to convexify. 00:17:01.410 --> 00:17:05.479 Even for n equals 4, you could require a million flips. 00:17:05.479 --> 00:17:07.020 So definitely there's some dependence 00:17:07.020 --> 00:17:11.223 on the ratio between the longest length and the smallest length, 00:17:11.223 --> 00:17:12.764 though in open problem, what we still 00:17:12.764 --> 00:17:15.210 don't know is whether you can bound the number of flips 00:17:15.210 --> 00:17:17.380 in terms of n and that ratio. 00:17:17.380 --> 00:17:18.380 Is it pseudo-polynomial? 00:17:18.380 --> 00:17:19.119 Who knows? 00:17:25.770 --> 00:17:27.670 So it turns out it's always finite, 00:17:27.670 --> 00:17:31.020 and there have been many proofs over the years of this result. 00:17:31.020 --> 00:17:34.700 It's kind of been rediscovered many times. 00:17:34.700 --> 00:17:40.360 Nagy solved it originally in 1939, and he cited Erdos. 00:17:40.360 --> 00:17:42.110 These two guys didn't cite anyone, 00:17:42.110 --> 00:17:45.590 so they may have come up with the problem independently. 00:17:45.590 --> 00:17:48.770 This paper, these are two Russian proofs. 00:17:48.770 --> 00:17:53.175 Then this paper cited-- they knew about everything. 00:17:53.175 --> 00:17:55.000 They cited all of them. 00:17:55.000 --> 00:17:56.737 They cited Reshetnyak. 00:17:56.737 --> 00:17:58.820 I don't think they necessarily knew about Yusupov. 00:18:01.620 --> 00:18:04.230 Independently, Kaluza posed the problem in 1981 00:18:04.230 --> 00:18:08.720 and Wegner solved it, and then to finally clean it up, 00:18:08.720 --> 00:18:13.740 Grunbaum, who we saw from un-un-unfoldable, 00:18:13.740 --> 00:18:15.519 knew about everything. 00:18:15.519 --> 00:18:16.810 Presumably, this is everything. 00:18:16.810 --> 00:18:18.860 Of course, we might have missed one. 00:18:18.860 --> 00:18:23.290 But he knew all the above, came up with his own proof. 00:18:23.290 --> 00:18:26.250 And then Godfried Touissant, father 00:18:26.250 --> 00:18:30.100 of computational geometry, one of them, knew about these 00:18:30.100 --> 00:18:33.420 and came up with yet another, simpler proof. 00:18:33.420 --> 00:18:35.900 So the story of this is kind of fun. 00:18:35.900 --> 00:18:42.070 When I taught this class for the very first time, 2003 or so, 00:18:42.070 --> 00:18:43.710 I thought, OK, cool. 00:18:43.710 --> 00:18:45.170 This is a classic theorem. 00:18:45.170 --> 00:18:47.140 Everyone should know it. 00:18:47.140 --> 00:18:50.680 So I thought I'd cover the latest proof that's presumably 00:18:50.680 --> 00:18:55.490 the best, so I covered Godfried's proof. 00:18:55.490 --> 00:18:58.655 I was a little unhappy with it, but I finished writing down 00:18:58.655 --> 00:19:01.030 the proof, and then one of the students raised their hand 00:19:01.030 --> 00:19:03.420 and asked, is that really right? 00:19:03.420 --> 00:19:05.120 Can you do that in step two? 00:19:05.120 --> 00:19:07.080 And the answer was no, you can't do that. 00:19:07.080 --> 00:19:10.130 It basically skipped a step. 00:19:10.130 --> 00:19:11.200 And so I thought, oh gee. 00:19:11.200 --> 00:19:13.900 And so I corresponded with O'Rourke and Godfried 00:19:13.900 --> 00:19:16.665 that weekend and was like, is something missing here? 00:19:16.665 --> 00:19:18.790 Maybe we should go back to some of the other proofs 00:19:18.790 --> 00:19:20.780 because we've got lots to choose from. 00:19:20.780 --> 00:19:25.110 So we went to the original, which is a very short proof. 00:19:25.110 --> 00:19:28.410 It's only one page long, maybe one and a half pages. 00:19:28.410 --> 00:19:29.620 And so this is fun. 00:19:29.620 --> 00:19:33.160 This is 1939, the way mathematics used to be done. 00:19:33.160 --> 00:19:37.009 And it's funny because Grunbaum's proof was based 00:19:37.009 --> 00:19:38.550 on Nagy's proof and Toussaint's proof 00:19:38.550 --> 00:19:40.250 was based on Grunbaum's proof, so in the end, 00:19:40.250 --> 00:19:41.680 these proofs were very similar. 00:19:41.680 --> 00:19:45.370 In fact, they differed exactly in the one step, 00:19:45.370 --> 00:19:46.690 which was kind of omitted here. 00:19:46.690 --> 00:19:47.990 So I thought, OK great. 00:19:47.990 --> 00:19:52.010 Here we have a fill-in on how to do it. 00:19:52.010 --> 00:19:54.710 And so the next class, I went up and I presented it. 00:19:54.710 --> 00:19:55.790 I was really happy. 00:19:55.790 --> 00:19:58.350 Isn't this cool, 1939 mathematics? 00:19:58.350 --> 00:20:00.650 It's really awesome, and it's one line 00:20:00.650 --> 00:20:02.230 that filled in this step. 00:20:02.230 --> 00:20:05.450 The same student raises his hand and is like, 00:20:05.450 --> 00:20:07.660 I don't think that's true. 00:20:07.660 --> 00:20:10.675 So now the step was filled in but it was wrong. 00:20:10.675 --> 00:20:12.550 So it turns out actually most of these proofs 00:20:12.550 --> 00:20:16.000 are wrong, but not all of them, fortunately, 00:20:16.000 --> 00:20:17.830 so the theorem is still true. 00:20:17.830 --> 00:20:21.900 So that one was wrong, this one was wrong, this one was wrong, 00:20:21.900 --> 00:20:23.620 this one skipped a step that was key, 00:20:23.620 --> 00:20:27.310 this one essentially also skipped the step that was key. 00:20:27.310 --> 00:20:31.130 So in the end, there's two correct proofs, Reshetnyak 00:20:31.130 --> 00:20:37.260 and Bing and Kazarinoff, which was surprising. 00:20:37.260 --> 00:20:43.860 So I thought I'd show you one of the errors in the Nagy proof. 00:20:43.860 --> 00:20:45.940 There's one other interesting feature here, 00:20:45.940 --> 00:20:48.710 which is, why wasn't this discovered until our class? 00:20:48.710 --> 00:20:52.650 The student was Blaise Gassand, and so then we wrote a paper 00:20:52.650 --> 00:20:56.510 about it and we have our own proof, of course. 00:20:56.510 --> 00:20:59.280 Now, some people may have realized 00:20:59.280 --> 00:21:01.400 there was an error in some of the proofs. 00:21:01.400 --> 00:21:04.370 Bing and Kazarinoff, one of the correct proofs, a very nice 00:21:04.370 --> 00:21:08.565 one, wrote-- here's the original Russian sentence. 00:21:08.565 --> 00:21:10.340 The English translation is, "the proof 00:21:10.340 --> 00:21:13.930 of this theorem given by Nagy is incorrect," 00:21:13.930 --> 00:21:17.280 which leaves something to be desired. 00:21:17.280 --> 00:21:19.890 And Grunbaum, who can read Russian-- 00:21:19.890 --> 00:21:23.680 we had to get them translated-- mentioned this. 00:21:23.680 --> 00:21:26.380 So he noticed that point. "They remarked that Nagy's proof is 00:21:26.380 --> 00:21:29.680 invalid but there's no basis for this claim." 00:21:29.680 --> 00:21:37.920 Then it remained undiscovered until 2005 or something. 00:21:37.920 --> 00:21:39.690 Kind of funny. 00:21:39.690 --> 00:21:41.790 Good thing there's so many proofs to choose from. 00:21:41.790 --> 00:21:44.000 So this is Nagy's original proof. 00:21:44.000 --> 00:21:47.150 You can see the example where flipping two pockets 00:21:47.150 --> 00:21:48.555 simultaneously causes a crossing. 00:21:51.727 --> 00:21:53.310 Don't read the whole thing, obviously, 00:21:53.310 --> 00:21:55.760 but there's one sentence here which is, 00:21:55.760 --> 00:22:00.270 if you take a polygon, call it p0, and you flip a pocket 00:22:00.270 --> 00:22:02.810 and you get p1, and you flip a pocket and you get p2, 00:22:02.810 --> 00:22:06.010 and you take the convex hull of p and you get c0, 00:22:06.010 --> 00:22:09.640 and you take the convex hull of p1 and you get c1, p2, 00:22:09.640 --> 00:22:10.520 you get c2. 00:22:10.520 --> 00:22:13.215 And then you interleave these polygons 00:22:13.215 --> 00:22:16.020 so that the polygon is a convex hull, next polygon is a convex 00:22:16.020 --> 00:22:18.910 hull, because you're always flipping out, 00:22:18.910 --> 00:22:24.680 each of these polygons obviously contains the foregoing ones. 00:22:24.680 --> 00:22:26.825 And so that seemed really nice and we thought, 00:22:26.825 --> 00:22:28.230 oh, this is so elegant. 00:22:28.230 --> 00:22:32.076 They used this to prove that the limit of the p's is convex 00:22:32.076 --> 00:22:33.450 because the limit of the p's then 00:22:33.450 --> 00:22:36.760 would be the limit of the c's because it was interleaving. 00:22:36.760 --> 00:22:38.670 But it's not true that these things 00:22:38.670 --> 00:22:42.490 contain the previous ones because if you have two pockets 00:22:42.490 --> 00:22:44.640 and you flip one of them, it's this one 00:22:44.640 --> 00:22:46.270 pocket versus multiple pockets issue. 00:22:46.270 --> 00:22:48.036 If you flip one of them, that will not 00:22:48.036 --> 00:22:49.660 contain the convex hull of the original 00:22:49.660 --> 00:22:51.940 because you haven't flipped them all. 00:22:51.940 --> 00:22:54.300 If you flipped all the pockets, than you would contain 00:22:54.300 --> 00:22:56.560 the convex hull of the original, but that 00:22:56.560 --> 00:22:58.030 would end up arguing that some flip 00:22:58.030 --> 00:23:01.030 sequences work, not all of them. 00:23:01.030 --> 00:23:04.010 So this is annoying, and I think this 00:23:04.010 --> 00:23:07.310 is where I run out of slides. 00:23:07.310 --> 00:23:15.520 But we have some time, so I can give you a sketch of the proof. 00:23:15.520 --> 00:23:19.660 This is the Bing and Kazarinoff proof or our version of it. 00:23:22.200 --> 00:23:25.890 Give you an idea of how this works. 00:23:25.890 --> 00:23:28.730 It's an easy proof, it's just easy also to get it wrong 00:23:28.730 --> 00:23:31.080 or to skip one of the steps. 00:23:31.080 --> 00:23:34.281 I will just do a proof by picture, I think. 00:23:36.990 --> 00:23:41.580 So suppose you have a polygon on the plane, 00:23:41.580 --> 00:23:45.980 and let's say you flip a pocket. 00:23:45.980 --> 00:23:59.210 So this would look something like that. 00:23:59.210 --> 00:24:02.850 A little hard to do a reflection. 00:24:02.850 --> 00:24:04.360 So this is the reflective polygon, 00:24:04.360 --> 00:24:09.850 and if we look at each of the vertices over here, 00:24:09.850 --> 00:24:15.570 these guys don't move, these guys moved over here, 00:24:15.570 --> 00:24:17.590 reflecting through that line. 00:24:17.590 --> 00:24:19.000 Observation one. 00:24:19.000 --> 00:24:26.340 If I take some point x interior to the polygon, 00:24:26.340 --> 00:24:31.470 then these points that move get farther away from x. 00:24:31.470 --> 00:24:32.330 Why? 00:24:32.330 --> 00:24:37.980 Because if you look at this line here, 00:24:37.980 --> 00:24:40.830 let's say you look at a vertex and where it goes. 00:24:40.830 --> 00:24:43.444 This line is the Voronoi diagram of those two points. 00:24:43.444 --> 00:24:45.610 This is the perpendicular bisector of this and this. 00:24:45.610 --> 00:24:47.580 That's the meaning of reflection. 00:24:47.580 --> 00:24:49.580 So that means everything to the left of the line 00:24:49.580 --> 00:24:52.332 here is closer to this point than that point. 00:24:52.332 --> 00:24:53.790 Everything to the right of the line 00:24:53.790 --> 00:24:55.960 is closer to this point than that point. 00:24:55.960 --> 00:24:57.856 Now x, which is interior to the polygon, 00:24:57.856 --> 00:25:00.230 must be to the left of the line because the whole polygon 00:25:00.230 --> 00:25:02.190 is to the left of the line. 00:25:02.190 --> 00:25:05.705 So distance from x to this vertex increases. 00:25:09.750 --> 00:25:12.180 I mean, x will remain inside because as you flip, 00:25:12.180 --> 00:25:13.900 you only get bigger. 00:25:13.900 --> 00:25:15.970 So if I take a point x and I look at the distance 00:25:15.970 --> 00:25:18.340 to some vertex, it can only monotonically increase. 00:25:18.340 --> 00:25:20.390 It also can't get arbitrarily large 00:25:20.390 --> 00:25:23.790 because the maximum it could possibly be 00:25:23.790 --> 00:25:26.632 is half the perimeter of the polygon. 00:25:26.632 --> 00:25:28.340 The perimeter of the polygon is preserved 00:25:28.340 --> 00:25:31.260 so it can only stretch so far. 00:25:31.260 --> 00:25:35.170 So if you look at this distance, it's monotonically increasing 00:25:35.170 --> 00:25:37.170 and it's bounded because it can never get bigger 00:25:37.170 --> 00:25:38.727 than the perimeter of the polygon. 00:25:38.727 --> 00:25:39.810 Therefore, it has a limit. 00:25:39.810 --> 00:25:41.340 That distance has a limit because 00:25:41.340 --> 00:25:47.070 monotone bounded sequences always have a unique limit. 00:25:47.070 --> 00:25:47.680 So cool. 00:25:47.680 --> 00:25:51.120 Distance from x to some vertex has a limit. 00:25:51.120 --> 00:25:54.100 Well, I'm going to do this for three different points, x, that 00:25:54.100 --> 00:25:58.330 lie on some non-degenerate triangle, so not all on a line. 00:25:58.330 --> 00:26:00.960 That means the three distances from these points 00:26:00.960 --> 00:26:04.150 to this vertex all converge to some limit. 00:26:04.150 --> 00:26:06.210 And therefore, that point converges 00:26:06.210 --> 00:26:08.300 to a limit, namely the intersection of those three 00:26:08.300 --> 00:26:11.290 circles centered at those points. 00:26:11.290 --> 00:26:12.890 So this proves that the polygon has 00:26:12.890 --> 00:26:16.800 a limit because every vertex has a limiting point. 00:26:16.800 --> 00:26:17.880 Cool. 00:26:17.880 --> 00:26:21.270 Now the tricky part is to argue that limit is convex, 00:26:21.270 --> 00:26:24.140 and this is where everyone had an issue. 00:26:27.880 --> 00:26:32.770 The first thing we argue is that the angles converge to a limit. 00:26:32.770 --> 00:26:35.230 This is kind of a technicality because we 00:26:35.230 --> 00:26:36.730 know the points converge to a limit, 00:26:36.730 --> 00:26:39.390 so surely the angle does. 00:26:39.390 --> 00:26:43.476 The only issue is, well, if all the points converge 00:26:43.476 --> 00:26:45.850 to the same point, then the angle would not have a limit, 00:26:45.850 --> 00:26:47.510 but that's easy to argue can't happen 00:26:47.510 --> 00:26:49.670 because these edge links are preserved. 00:26:49.670 --> 00:26:53.059 So I will just skip that one. 00:26:53.059 --> 00:26:54.850 It follows from some of the things we said. 00:26:59.310 --> 00:27:00.860 Now we get to the fun part. 00:27:00.860 --> 00:27:02.060 This polygon has a limit. 00:27:02.060 --> 00:27:04.380 The angles have limits. 00:27:04.380 --> 00:27:07.010 So I want to look at the limiting angles. 00:27:07.010 --> 00:27:09.070 I want to in particular look at the vertices that 00:27:09.070 --> 00:27:13.900 move because if you flip an infinite number of times, that 00:27:13.900 --> 00:27:17.280 means some vertex must move an infinite number of times, 00:27:17.280 --> 00:27:19.330 because every time you flip, somebody moves. 00:27:19.330 --> 00:27:20.830 These guys are considered not moving 00:27:20.830 --> 00:27:23.070 even though they're kind of involved in the flip. 00:27:23.070 --> 00:27:28.470 So if you look at the moved guys, what is their angle? 00:27:28.470 --> 00:27:31.380 So before, there was some angle here, 00:27:31.380 --> 00:27:36.170 and afterwards, the interior angle is the reverse. 00:27:36.170 --> 00:27:39.040 If this was reflex before, it's convex now. 00:27:39.040 --> 00:27:42.990 If there were a convex angle over here like this, 00:27:42.990 --> 00:27:48.220 it would become reflex over here. 00:27:48.220 --> 00:27:51.450 You alternate between being less than 180 00:27:51.450 --> 00:27:55.620 and greater than 180 every time you move. 00:27:55.620 --> 00:28:01.230 So if you have a vertex that is moving infinitely many times, 00:28:01.230 --> 00:28:05.830 its angle must alternate between less than 180 convex 00:28:05.830 --> 00:28:08.960 and greater than 180 reflex infinitely many times. 00:28:08.960 --> 00:28:11.760 If that happens and you have a unique limit angle, 00:28:11.760 --> 00:28:15.570 your limit angle must be 180. 00:28:15.570 --> 00:28:17.550 That's interesting. 00:28:17.550 --> 00:28:26.110 If our vertex moves infinitely many times, 00:28:26.110 --> 00:28:42.317 then its limit angle equals 180, must be flat. 00:28:42.317 --> 00:28:44.150 Very close to a contradiction at this point. 00:28:49.090 --> 00:28:52.930 These guys are going to end up looking like this. 00:28:52.930 --> 00:28:56.690 Well, let's look at the other guys. 00:28:56.690 --> 00:28:58.425 So we have some limit polygon. 00:28:58.425 --> 00:29:00.550 In the limit polygon, there are some flat vertices, 00:29:00.550 --> 00:29:03.020 but there must also be some non-flat vertices. 00:29:03.020 --> 00:29:05.720 You can't just go straight and hope to close a cycle. 00:29:05.720 --> 00:29:07.920 So there's maybe some convex ones like this. 00:29:07.920 --> 00:29:11.350 There may be some reflex ones. 00:29:11.350 --> 00:29:14.570 At some point, these vertices must 00:29:14.570 --> 00:29:19.300 stop moving because everyone who moves infinitely many times 00:29:19.300 --> 00:29:20.890 has a limit angle flat. 00:29:20.890 --> 00:29:23.154 So anybody who is-- we call it "pointed" 00:29:23.154 --> 00:29:25.570 here, although it's a little different from pointed pseudo 00:29:25.570 --> 00:29:26.810 triangulations. 00:29:26.810 --> 00:29:29.200 Anyone who's not a flat angle must 00:29:29.200 --> 00:29:31.040 stop moving after finite time. 00:29:31.040 --> 00:29:34.810 So let's go to that time when all of these guys 00:29:34.810 --> 00:29:35.680 have stopped moving. 00:29:40.020 --> 00:29:41.590 So your limit polygon is something. 00:29:41.590 --> 00:29:43.960 We don't know whether it's convex or whatever. 00:29:43.960 --> 00:29:45.830 It could have many flat angles. 00:29:45.830 --> 00:29:47.750 It's got to have at least one of them 00:29:47.750 --> 00:29:50.590 if we assume there's something infinite here. 00:29:50.590 --> 00:29:53.180 This is the limit polygon. 00:29:53.180 --> 00:29:55.880 Now, at finite time, we know that these guys 00:29:55.880 --> 00:30:00.320 have stopped moving, meaning we're done with those guys. 00:30:03.040 --> 00:30:07.170 I've drawn the limit here, but also the finite thing, which 00:30:07.170 --> 00:30:08.910 must be on the inside, right? 00:30:08.910 --> 00:30:14.490 This must be something like this. 00:30:14.490 --> 00:30:18.810 Somehow it's going to flip and reach the infinitely many times 00:30:18.810 --> 00:30:21.980 and reach this in the limit. 00:30:21.980 --> 00:30:23.500 Hm. 00:30:23.500 --> 00:30:27.270 That looks weird. 00:30:27.270 --> 00:30:30.770 So the way to argue this in the clean way 00:30:30.770 --> 00:30:36.210 is if you look at the convex hull of the limit, 00:30:36.210 --> 00:30:39.355 let's say, convex hull of the limit is this. 00:30:42.320 --> 00:30:45.120 And look at the convex hull of this finite time 00:30:45.120 --> 00:30:47.750 when the squared vertices have stopped moving. 00:30:47.750 --> 00:30:49.950 Convex hull will be-- well, it's got 00:30:49.950 --> 00:30:52.650 to be at least this because these guys are already there. 00:30:52.650 --> 00:30:55.666 The convex hull is defined by the square points. 00:30:55.666 --> 00:30:57.290 You don't care about the flat vertices. 00:30:57.290 --> 00:30:58.930 That won't affect the convex hull. 00:30:58.930 --> 00:31:02.000 So that means the convex hull equals the limit convex 00:31:02.000 --> 00:31:04.850 hull at this finite time. 00:31:04.850 --> 00:31:06.887 Now you're about to do another flip, which 00:31:06.887 --> 00:31:08.720 means you're going to go outside that convex 00:31:08.720 --> 00:31:12.500 hull and contradiction. 00:31:12.500 --> 00:31:13.920 Is that clear? 00:31:13.920 --> 00:31:16.430 Maybe go through that part one more time. 00:31:16.430 --> 00:31:19.750 After finite time, when all of the non-flat vertices 00:31:19.750 --> 00:31:23.400 have stopped moving, we have reached the final convex hull. 00:31:23.400 --> 00:31:26.120 That means you can't do any more flips because every flip makes 00:31:26.120 --> 00:31:28.440 the convex hull bigger, so you actually 00:31:28.440 --> 00:31:30.706 had to stop at that time. 00:31:30.706 --> 00:31:31.830 That's kind of a fun proof. 00:31:31.830 --> 00:31:34.300 The key is really this part, that if a vertex flips 00:31:34.300 --> 00:31:37.377 infinitely many times, then that limit angle must be flat, 00:31:37.377 --> 00:31:39.710 and so they really don't participate in the convex hull. 00:31:39.710 --> 00:31:43.170 This is the Bing and Kazarinoff key idea. 00:31:43.170 --> 00:31:44.170 AUDIENCE: Two questions. 00:31:44.170 --> 00:31:47.470 Does anyone do this for 3D? 00:31:47.470 --> 00:31:49.320 PROFESSOR: 3D is a good. 00:31:49.320 --> 00:31:52.040 People have tried to define flips for 3D, 00:31:52.040 --> 00:31:56.380 and I think there's never really been a successful definition. 00:31:56.380 --> 00:31:58.030 AUDIENCE: [INAUDIBLE]? 00:31:58.030 --> 00:32:00.024 PROFESSOR: Yeah, and exactly how to flip it. 00:32:00.024 --> 00:32:01.940 I mean, you can define pocket in the same way, 00:32:01.940 --> 00:32:04.960 but then the boundary won't be a single plane. 00:32:04.960 --> 00:32:09.180 It'll be some convex cap, and so flipping, you 00:32:09.180 --> 00:32:11.100 can't really just reflect. 00:32:11.100 --> 00:32:13.820 It's kind of annoying. 00:32:13.820 --> 00:32:16.380 AUDIENCE: The other question was, 00:32:16.380 --> 00:32:18.145 you have a sequence of simple operations 00:32:18.145 --> 00:32:21.120 that takes you from non-convex to convex. 00:32:21.120 --> 00:32:23.489 Does anybody use that in a proof to say, 00:32:23.489 --> 00:32:25.530 these things are preserved across the operations? 00:32:25.530 --> 00:32:27.805 You said perimeter was preserved. 00:32:27.805 --> 00:32:32.000 What about shortest paths? 00:32:32.000 --> 00:32:34.080 PROFESSOR: So is this useful for something? 00:32:34.080 --> 00:32:36.650 Shortest paths are certainly not preserved. 00:32:36.650 --> 00:32:38.020 Edge lengths are preserved. 00:32:38.020 --> 00:32:40.478 Of course, we know how to do that with the carpenter's rule 00:32:40.478 --> 00:32:42.900 theorem, just staying in 2D, so it's not that exciting. 00:32:42.900 --> 00:32:46.594 But the operations are definitely a lot simpler. 00:32:46.594 --> 00:32:48.260 I think the easy answer to your question 00:32:48.260 --> 00:32:52.760 is there are many natural, simple moves, 00:32:52.760 --> 00:32:54.810 and this is the first one that people considered, 00:32:54.810 --> 00:32:56.268 but actually, there are a lot more, 00:32:56.268 --> 00:32:58.300 and I'm going to talk about those. 00:32:58.300 --> 00:33:03.110 The main application I know for this stuff is basically, 00:33:03.110 --> 00:33:07.900 people wanted to generate random closed walks in 3D typically, 00:33:07.900 --> 00:33:10.200 and so they wanted to find a small set of operations 00:33:10.200 --> 00:33:12.190 they could just perform randomly and hope 00:33:12.190 --> 00:33:14.220 that that was a rapidly mixing Markov chain, 00:33:14.220 --> 00:33:16.600 so eventually, you'd have a kind of random thing. 00:33:16.600 --> 00:33:18.830 I don't think there are any rapid mixing 00:33:18.830 --> 00:33:20.410 results, at least that I'm aware of. 00:33:23.020 --> 00:33:25.500 In order to hope to get the space randomly, 00:33:25.500 --> 00:33:27.850 you would at least have to be able to make anything. 00:33:27.850 --> 00:33:32.044 So for that, the question is, can you convexify anything, 00:33:32.044 --> 00:33:34.210 because if you can convexify anything, then at least 00:33:34.210 --> 00:33:37.000 you can make anything, more or less. 00:33:37.000 --> 00:33:39.680 So that's where these questions come from. 00:33:39.680 --> 00:33:41.240 We focus more on the universality, 00:33:41.240 --> 00:33:43.750 but I think what people care about is this random generation 00:33:43.750 --> 00:33:44.250 business. 00:33:46.704 --> 00:33:48.620 To that end, of course, for random generation, 00:33:48.620 --> 00:33:50.670 you don't really care about crossings. 00:33:50.670 --> 00:33:52.510 And so another fun extension which 00:33:52.510 --> 00:33:54.880 was in, I think, our paper for the first time, 00:33:54.880 --> 00:33:58.860 although there's a weaker version in Grunbaum's paper, 00:33:58.860 --> 00:34:01.570 if you start with a non-crossing, this proof 00:34:01.570 --> 00:34:03.200 still works. 00:34:03.200 --> 00:34:06.640 You need to add a little more to the argument. 00:34:06.640 --> 00:34:10.199 Either you decrease the number of crossings, which can only 00:34:10.199 --> 00:34:15.901 happen a finite number of times, or this kind of stuff works. 00:34:15.901 --> 00:34:18.359 You can make crossing polygons, too, which is kind of cool. 00:34:22.389 --> 00:34:24.989 So that's the end of basic flips. 00:34:24.989 --> 00:34:31.620 Then we have something called a flip turn, which is kind of fun 00:34:31.620 --> 00:34:34.040 and in some ways better behaved, and gets to your question 00:34:34.040 --> 00:34:35.560 of what else is preserved. 00:34:35.560 --> 00:34:39.590 So let's say you have a pocket like this. 00:34:39.590 --> 00:34:42.460 So normally with a flip, we would reflect. 00:34:42.460 --> 00:34:46.679 With a flip turn, you reflect and then also flip this way, 00:34:46.679 --> 00:34:54.810 so it's the same as rotating 180 degrees about the center. 00:34:54.810 --> 00:34:57.139 In addition to preserving perimeter, 00:34:57.139 --> 00:35:00.310 this preserves the edge directions. 00:35:00.310 --> 00:35:03.040 This matches this, this matches this, this matches this. 00:35:03.040 --> 00:35:05.050 It does not preserve the edge order, however. 00:35:05.050 --> 00:35:06.630 Here, we preserve the edge order. 00:35:06.630 --> 00:35:09.970 Here, we've reversed the order of those three edges. 00:35:09.970 --> 00:35:12.190 So all this is really doing is permuting 00:35:12.190 --> 00:35:14.270 the sequence of edges. 00:35:14.270 --> 00:35:17.150 The edge directions and the edge lengths are all the same. 00:35:17.150 --> 00:35:21.900 So this means you could make at most n factorial moves here. 00:35:21.900 --> 00:35:24.220 Immediately, it's different from this situation 00:35:24.220 --> 00:35:26.124 where even for n equals 4, you could 00:35:26.124 --> 00:35:27.290 have arbitrarily many moves. 00:35:27.290 --> 00:35:28.760 Here, it's at most n factorial. 00:35:28.760 --> 00:35:30.320 That was the original bound. 00:35:30.320 --> 00:35:33.040 It turns out every polygon convexifies 00:35:33.040 --> 00:35:37.640 after order n squared flip turns. 00:35:37.640 --> 00:35:40.630 It's a fun proof but I don't have time to cover it here. 00:35:43.150 --> 00:35:46.040 That's flip turns. 00:35:46.040 --> 00:35:50.550 Then we go to deflations. 00:35:50.550 --> 00:35:52.470 This is the inverse of a flip. 00:35:52.470 --> 00:35:56.070 So suppose you take a polygon and you do an operation 00:35:56.070 --> 00:35:59.310 that, if flipped, would result in the original polygon, 00:35:59.310 --> 00:36:01.960 exactly the opposite of a flip. 00:36:01.960 --> 00:36:05.420 This was conjectured to also finish after finite time, 00:36:05.420 --> 00:36:06.715 but in fact it doesn't. 00:36:06.715 --> 00:36:10.880 If you take any quadrilateral satisfying Kawasaki, 00:36:10.880 --> 00:36:13.110 so if you add up the opposite edge lengths 00:36:13.110 --> 00:36:18.170 and they're equal-- so 6 plus 3 is 9, 4 plus 5 is 9-- 00:36:18.170 --> 00:36:22.070 then there it is at least a flat limit possibly. 00:36:22.070 --> 00:36:24.410 That's the Kawasaki thing. 00:36:24.410 --> 00:36:27.710 And in fact, it will converge to that flat limit 00:36:27.710 --> 00:36:30.334 and it will take infinite time to get there. 00:36:30.334 --> 00:36:31.250 So that's kind of fun. 00:36:31.250 --> 00:36:33.600 It gets very hard to draw the pictures. 00:36:33.600 --> 00:36:37.950 This is really the main example of an infinitely deflating 00:36:37.950 --> 00:36:38.770 polygon. 00:36:38.770 --> 00:36:43.710 We have a paper called "Deflating the Pentagon," which 00:36:43.710 --> 00:36:50.050 got some fun political views at some point. 00:36:50.050 --> 00:36:53.780 It's about a pentagon, five sides. 00:36:53.780 --> 00:36:56.560 And essentially, unless you have a flat vertex, in which case 00:36:56.560 --> 00:36:59.365 you are quadrilateral, there is no infinitely deflating 00:36:59.365 --> 00:37:01.740 pentagon, which means you can deflate the pentagon always 00:37:01.740 --> 00:37:02.620 in finite time. 00:37:05.130 --> 00:37:06.920 It is open for hexagons and higher 00:37:06.920 --> 00:37:08.820 whether there is another example different 00:37:08.820 --> 00:37:11.060 from the quadrilateral. 00:37:11.060 --> 00:37:12.950 Then there's the idea of a pop. 00:37:12.950 --> 00:37:15.500 This is an even simpler operation than a flip. 00:37:15.500 --> 00:37:18.350 You just take two edges, like here we're 00:37:18.350 --> 00:37:24.680 taking these two edges, and, ignoring crossings, 00:37:24.680 --> 00:37:26.890 you just flip as if that were a pocket lid, 00:37:26.890 --> 00:37:28.260 so if this were a pocket. 00:37:28.260 --> 00:37:29.380 Then you get this polygon. 00:37:29.380 --> 00:37:31.150 Now here, you can be forced to get crossings. 00:37:31.150 --> 00:37:33.220 No matter how you flip, you might get a crossing. 00:37:33.220 --> 00:37:36.320 Still, we're wondering-- who cares about crossings-- 00:37:36.320 --> 00:37:39.840 are pops enough to make anything, or to convexify? 00:37:39.840 --> 00:37:41.350 And the answer is no. 00:37:41.350 --> 00:37:43.750 There's this set of polygons called alternating polygons 00:37:43.750 --> 00:37:46.040 where the vertices alternate between the x-axis 00:37:46.040 --> 00:37:47.370 and the y-axis. 00:37:47.370 --> 00:37:50.320 And you can prove that no matter what pop you do, 00:37:50.320 --> 00:37:53.270 you are still an alternating polygon, 00:37:53.270 --> 00:37:55.270 and alternating polygons, you can also prove, 00:37:55.270 --> 00:37:57.900 are never convex, so you're stuck. 00:37:57.900 --> 00:38:03.280 This was open for many years but finally solved a few years ago. 00:38:03.280 --> 00:38:07.690 There's also pop turns, which is where you take two edges 00:38:07.690 --> 00:38:11.320 and you do a 180 degree rotation like this. 00:38:11.320 --> 00:38:13.910 And there, we can prove if you allow crossings, 00:38:13.910 --> 00:38:15.870 you can convexify any polygon. 00:38:15.870 --> 00:38:18.420 I don't have a figure of that because it's just an algorithm. 00:38:18.420 --> 00:38:21.840 We haven't actually run it on a nice example. 00:38:21.840 --> 00:38:24.560 If you avoid crossings, we can characterize when it's possible 00:38:24.560 --> 00:38:26.160 and when it's impossible. 00:38:26.160 --> 00:38:28.557 And those are pretty much all the simple operations 00:38:28.557 --> 00:38:29.890 that at least have been studied. 00:38:29.890 --> 00:38:34.330 There's probably more to think about, but that's it, 00:38:34.330 --> 00:38:37.530 and that's the end of my part of the class.