WEBVTT 00:00:10.538 --> 00:00:14.420 PROFESSOR: I wanted to make a few comments on the third 00:00:14.420 --> 00:00:17.530 problem set, which was a funny puzzle. 00:00:17.530 --> 00:00:20.320 And it was intended to be entertaining and get you 00:00:20.320 --> 00:00:23.320 thinking about things related to symmetry 00:00:23.320 --> 00:00:25.250 in an amusing context. 00:00:25.250 --> 00:00:26.640 And there were two problems. 00:00:26.640 --> 00:00:32.030 The first problem had a platter on which there were 00:00:32.030 --> 00:00:37.650 arranged cherries with either two or three berries to a 00:00:37.650 --> 00:00:40.940 spray, and apples and pears. 00:00:40.940 --> 00:00:43.300 And they were either black or white, which didn't make then 00:00:43.300 --> 00:00:45.130 particularly appetizing. 00:00:45.130 --> 00:00:51.140 But in any case, there were three platters and the fruits 00:00:51.140 --> 00:00:53.020 marched around on the platter. 00:00:53.020 --> 00:00:57.760 And the puzzle was, what would be the fourth 00:00:57.760 --> 00:01:00.280 platter in the sequence? 00:01:00.280 --> 00:01:04.090 This didn't involve symmetry in any sense of the word. 00:01:04.090 --> 00:01:09.120 But yet, what was going on from picture to picture was a 00:01:09.120 --> 00:01:13.410 mapping of the motifs from one location to another. 00:01:13.410 --> 00:01:18.430 Not just rotation or translation or anything nice 00:01:18.430 --> 00:01:21.870 in the crystallographic, but still they were being moved 00:01:21.870 --> 00:01:24.010 from one location to another. 00:01:24.010 --> 00:01:27.300 And just about everybody got the right 00:01:27.300 --> 00:01:31.420 answer just by deduction. 00:01:31.420 --> 00:01:34.430 But the reason I liked that problem, and the reason I gave 00:01:34.430 --> 00:01:38.030 it to you, is that it introduced another sort of 00:01:38.030 --> 00:01:40.660 transformation. 00:01:40.660 --> 00:01:44.000 One of the pairs of fruits not only change locations, but 00:01:44.000 --> 00:01:47.150 they switched from black to white as you went from one 00:01:47.150 --> 00:01:49.260 position to the other. 00:01:49.260 --> 00:01:50.880 And that is another sort of 00:01:50.880 --> 00:01:53.250 transformation we can introduce. 00:01:53.250 --> 00:01:56.250 And this is something we could call color symmetry. 00:01:56.250 --> 00:02:01.230 And I would observe that this is something that you are 00:02:01.230 --> 00:02:06.830 already very familiar with in patterns, in particular, in 00:02:06.830 --> 00:02:13.760 the pattern that is represented by a checkerboard 00:02:13.760 --> 00:02:18.140 where you have black squares and white squares. 00:02:18.140 --> 00:02:23.450 This location here is a bona fide four-fold axis, as is 00:02:23.450 --> 00:02:26.010 this location here. 00:02:26.010 --> 00:02:30.670 But there other locations, such as the corners of the 00:02:30.670 --> 00:02:35.180 black and white squares, where, indeed, there is a 90 00:02:35.180 --> 00:02:38.390 degree rotation that takes this square and transforms it 00:02:38.390 --> 00:02:39.360 into this square. 00:02:39.360 --> 00:02:41.680 But in doing that transformation, you change it 00:02:41.680 --> 00:02:45.360 from a black one to a white one, and then from a white one 00:02:45.360 --> 00:02:48.750 to a black one again, back to a white one. 00:02:48.750 --> 00:02:52.962 And that is a color symmetry. 00:02:52.962 --> 00:02:55.880 You could make it red and green if you like to be a 00:02:55.880 --> 00:02:57.280 little more attractive. 00:02:57.280 --> 00:03:01.190 Similarly, there are mirror lines like this that are true 00:03:01.190 --> 00:03:02.240 mirror lines. 00:03:02.240 --> 00:03:04.030 The pattern is left invariant. 00:03:04.030 --> 00:03:07.140 There's another locus such as this one here where when you 00:03:07.140 --> 00:03:11.050 reflect you go from a black one to white one and then back 00:03:11.050 --> 00:03:14.600 to a black one when you do the operation twice. 00:03:14.600 --> 00:03:17.230 So there are black-white mirror planes as well as 00:03:17.230 --> 00:03:18.830 regular mirror planes. 00:03:18.830 --> 00:03:22.160 There are two-fold axes but all of the two-fold axes are 00:03:22.160 --> 00:03:26.390 black-white axes, white to black, black back to white. 00:03:26.390 --> 00:03:30.070 So that's a new sort of transformation that we can 00:03:30.070 --> 00:03:32.540 develop if we wanted to, switching the color. 00:03:35.370 --> 00:03:37.910 Does this is have any utility in the physical sciences? 00:03:37.910 --> 00:03:40.510 Be nice for wallpaper designers, but does it have 00:03:40.510 --> 00:03:42.360 utility in physical sciences? 00:03:42.360 --> 00:03:43.990 And the answer is yes. 00:03:43.990 --> 00:03:47.760 Because what is going on here is that there is a binary 00:03:47.760 --> 00:03:50.500 character assigned to each of the motifs. 00:03:50.500 --> 00:03:52.990 So any time the atom or whatever you have, the 00:03:52.990 --> 00:03:57.240 molecule, has some sort of binary characteristic that it 00:03:57.240 --> 00:04:00.840 can exist in two distinct states, the arrangement of 00:04:00.840 --> 00:04:05.080 that sort of motif requires a black-white symmetry. 00:04:05.080 --> 00:04:09.080 And one example in crystals that requires this sort of 00:04:09.080 --> 00:04:12.610 symmetry is the case of a magnetic structure-- 00:04:12.610 --> 00:04:15.480 and I think I mentioned this last time-- where the magnetic 00:04:15.480 --> 00:04:17.760 moment can either point up or down. 00:04:17.760 --> 00:04:20.500 Well, saying that you have an atom with spin up and an atom 00:04:20.500 --> 00:04:23.470 with spin down is the same thing as saying, in terms of 00:04:23.470 --> 00:04:27.140 transformations, having a black atom or a white atom. 00:04:27.140 --> 00:04:33.270 It's a binary character that switches from atom to atom. 00:04:33.270 --> 00:04:36.314 Could you have three color symmetries? 00:04:36.314 --> 00:04:36.746 Yeah. 00:04:36.746 --> 00:04:37.830 You can. 00:04:37.830 --> 00:04:44.020 There is a Dutch artist named Maurits Escher who spent much 00:04:44.020 --> 00:04:48.280 of his, career deducing patterns, and did this in a 00:04:48.280 --> 00:04:50.080 purely intuitive way. 00:04:50.080 --> 00:04:52.100 I mean you look at these patterns and say, wow, that 00:04:52.100 --> 00:04:54.470 guy is some crystallographer. 00:04:54.470 --> 00:04:58.310 But I heard him, while he was still alive, speak twice. 00:04:58.310 --> 00:05:01.700 And he describes it in metaphysical terms. 00:05:01.700 --> 00:05:06.330 This is the twoness merging with the fourness, and just 00:05:06.330 --> 00:05:08.580 completely intuitive. 00:05:08.580 --> 00:05:09.790 But his patterns are gorgeous. 00:05:09.790 --> 00:05:12.650 And he has a number of very nice ones that are three-color 00:05:12.650 --> 00:05:13.790 symmetries. 00:05:13.790 --> 00:05:16.410 And maybe later on in the term, I'll bring in a 00:05:16.410 --> 00:05:18.520 projector and show you examples of some of these. 00:05:18.520 --> 00:05:19.770 They're very entertaining. 00:05:22.280 --> 00:05:22.520 All right. 00:05:22.520 --> 00:05:25.420 So the purpose of that first puzzle was to introduce you to 00:05:25.420 --> 00:05:26.670 black-white symmetries. 00:05:28.990 --> 00:05:32.390 Then the second puzzle on the sheet was a pretty dumb thing. 00:05:32.390 --> 00:05:36.310 There was a stack of blocks and you said how many-- 00:05:36.310 --> 00:05:40.070 I asked how many blocks are there in the stack? 00:05:40.070 --> 00:05:42.250 And you could count them up, 1, 2, 3, 4. 00:05:42.250 --> 00:05:44.790 And If you did it right, you came out with 32. 00:05:44.790 --> 00:05:46.570 So that was pretty dumb. 00:05:46.570 --> 00:05:51.090 Well, this was intended as an example of how one can use 00:05:51.090 --> 00:05:56.610 symmetry in the solution of a physical problem. 00:05:56.610 --> 00:05:59.140 Let's suppose that you were a cowboy and you had to go out 00:05:59.140 --> 00:06:02.230 and buy horseshoes for a herd of horses. 00:06:02.230 --> 00:06:04.660 Would you count the number of feet? 00:06:04.660 --> 00:06:04.960 No. 00:06:04.960 --> 00:06:08.924 You'd count the number of horses and multiply by four. 00:06:08.924 --> 00:06:09.750 OK. 00:06:09.750 --> 00:06:12.061 So there you'd be using symmetry. 00:06:12.061 --> 00:06:15.826 Now The point of this problem really was-- 00:06:15.826 --> 00:06:17.535 You could have had this one. 00:06:17.535 --> 00:06:20.600 We have just the exact number seats. 00:06:20.600 --> 00:06:27.092 The point of this problem was that you have to, in assigning 00:06:27.092 --> 00:06:31.140 a symmetry to a system, usually make physical 00:06:31.140 --> 00:06:32.290 assumptions. 00:06:32.290 --> 00:06:35.470 When you do a derivation, as we've been doing for the last 00:06:35.470 --> 00:06:38.490 several days, we say this is a four-fold access, and there's 00:06:38.490 --> 00:06:40.410 no ambiguity whatsoever. 00:06:40.410 --> 00:06:43.110 Because it's my ball game, and I want it to be a four-fold 00:06:43.110 --> 00:06:44.720 axis, 00:06:44.720 --> 00:06:47.480 But when you're given a pattern or given a crystal and 00:06:47.480 --> 00:06:50.500 you have to assign a symmetry to that crystal, you 00:06:50.500 --> 00:06:53.840 invariably have to make assumptions. 00:06:53.840 --> 00:06:56.410 The person buying the horseshoes, for example, has 00:06:56.410 --> 00:07:00.390 to assume that five- and six-legged horses are pretty 00:07:00.390 --> 00:07:07.350 rare articles and that a horse with only three or two legs is 00:07:07.350 --> 00:07:09.905 going to be of little utility to the rancher. 00:07:09.905 --> 00:07:12.680 It Would probably have a limited existence at the 00:07:12.680 --> 00:07:13.960 ranchers expense. 00:07:13.960 --> 00:07:19.240 So you're assuming that all horse have four legs. 00:07:19.240 --> 00:07:23.110 In looking at a crystal, though, things are not that 00:07:23.110 --> 00:07:24.460 easy to decide. 00:07:24.460 --> 00:07:28.140 And let me give you a few really practical examples. 00:07:28.140 --> 00:07:31.490 A lot of mineral crystals grow in fissures in the rock. 00:07:31.490 --> 00:07:34.970 And let's suppose we've got a crack, and bubbling through 00:07:34.970 --> 00:07:39.050 this crack is a solution that contains silica. 00:07:39.050 --> 00:07:41.690 And as the solution cools off, it will deposit 00:07:41.690 --> 00:07:43.140 a crystal of quartz. 00:07:43.140 --> 00:07:47.010 And this crystal of quartz would probably grow looking 00:07:47.010 --> 00:07:49.370 something like this. 00:07:49.370 --> 00:07:51.440 And you would knock the crystal out of the rock. 00:07:51.440 --> 00:07:54.075 And you would say, what is the symmetry of this crystal? 00:07:56.890 --> 00:07:59.890 It doesn't appear to be anything hexagonal about it. 00:08:02.420 --> 00:08:06.990 But what's happened is this crystal if it had developed in 00:08:06.990 --> 00:08:11.660 a uniform environment would have had a hexagonal shape. 00:08:11.660 --> 00:08:15.040 But because the nutrients are flowing in from one direction, 00:08:15.040 --> 00:08:17.990 it tends to elongate in that direction. 00:08:17.990 --> 00:08:22.640 So how could you assign the symmetry to that crystal that 00:08:22.640 --> 00:08:24.840 you dug out of the rock? 00:08:24.840 --> 00:08:28.330 Unfortunately, crystals do not come with a legend on the 00:08:28.330 --> 00:08:37.409 bottom that says quartz SIO2 space group P3 subscript 121, 00:08:37.409 --> 00:08:38.960 made in USA. 00:08:38.960 --> 00:08:42.860 You have to yourself assign the symmetry. 00:08:42.860 --> 00:08:45.500 And what you would do is you would look at this face and 00:08:45.500 --> 00:08:48.050 this face and you say, gee, these angles, when I measure 00:08:48.050 --> 00:08:51.720 them, all turn out to be 120 degrees. 00:08:51.720 --> 00:08:54.610 This face has a luster on it that looks like the 00:08:54.610 --> 00:08:56.190 luster on this face. 00:08:56.190 --> 00:08:59.360 This face has little edge pits on it, and they point in the 00:08:59.360 --> 00:09:02.820 same way as the edge pits on this face. 00:09:02.820 --> 00:09:06.190 I can measure the electrical conductivity and find that it 00:09:06.190 --> 00:09:09.830 varies in a fashion consistent with a hexagonal crystal. 00:09:09.830 --> 00:09:13.390 And you would do any physical test that you decided you 00:09:13.390 --> 00:09:14.430 would need to make. 00:09:14.430 --> 00:09:17.650 And then when you were all done, you would say everything 00:09:17.650 --> 00:09:20.810 that I can measure about this crystal other than it's shape 00:09:20.810 --> 00:09:28.060 is invariant to a 120 degree rotation. 00:09:28.060 --> 00:09:34.100 Quartz has a hexagonal shape very often, but actually only 00:09:34.100 --> 00:09:38.120 this face, this face, and this face are symmetrically 00:09:38.120 --> 00:09:38.760 equivalent. 00:09:38.760 --> 00:09:41.350 And there's only a three-fold axis in there. 00:09:41.350 --> 00:09:44.550 And if you look very carefully at this crystal, you could see 00:09:44.550 --> 00:09:46.130 some difference there, too. 00:09:46.130 --> 00:09:50.085 The neighboring faces very often have striations, very 00:09:50.085 --> 00:09:53.210 faint striations on one face, no striations on the 00:09:53.210 --> 00:09:54.890 neighboring face. 00:09:54.890 --> 00:09:56.540 So you find out all that can. 00:09:56.540 --> 00:09:59.550 You have to make some assumptions, and then on that 00:09:59.550 --> 00:10:02.860 basis decide the symmetry. 00:10:02.860 --> 00:10:03.920 Let me give you another example. 00:10:03.920 --> 00:10:04.960 This is a real one. 00:10:04.960 --> 00:10:08.210 And one of the things that it's fun to do is to grow 00:10:08.210 --> 00:10:09.780 crystals from solution. 00:10:09.780 --> 00:10:13.040 The Science Museum has little kits that you can buy that 00:10:13.040 --> 00:10:16.150 have some powder that you could heat up in water and 00:10:16.150 --> 00:10:20.420 then put it in a beaker and put a string in there, maybe 00:10:20.420 --> 00:10:23.480 with a little crumb of the stuff that you dissolved. 00:10:23.480 --> 00:10:26.340 And one of the crystals that is nice for 00:10:26.340 --> 00:10:30.410 this purpose is alum. 00:10:30.410 --> 00:10:31.790 It's a cubic crystal. 00:10:31.790 --> 00:10:36.810 And if you let the stuff set there, it will form a nice, 00:10:36.810 --> 00:10:39.063 lovely equiaxed cubed. 00:10:39.063 --> 00:10:43.190 Science Museum usually puts a pinch of another salt in there 00:10:43.190 --> 00:10:46.330 so the crystal is not only pretty and shiny, it's purple 00:10:46.330 --> 00:10:48.725 or it's the orange. 00:10:48.725 --> 00:10:52.830 Now, unless you attach this little seed crystal carefully, 00:10:52.830 --> 00:10:56.230 there would be a good chance that it would fall off and 00:10:56.230 --> 00:10:57.850 land on the bottom of the beaker. 00:10:57.850 --> 00:11:01.180 When that happened, your crystal would not have the 00:11:01.180 --> 00:11:03.670 shape of a cube. 00:11:03.670 --> 00:11:08.060 It would have a shape in which these two top edges had length 00:11:08.060 --> 00:11:13.970 L and these faces on the side were exactly half that length. 00:11:13.970 --> 00:11:16.560 And what's happening is that this crystal is growing in an 00:11:16.560 --> 00:11:18.335 unconstrained environment. 00:11:18.335 --> 00:11:21.640 The nutrient is coming on and crystallizing from all 00:11:21.640 --> 00:11:22.820 directions. 00:11:22.820 --> 00:11:27.830 In this case, nutrient can't get in from the directions 00:11:27.830 --> 00:11:29.270 that are below the crystal. 00:11:29.270 --> 00:11:32.780 So the growth in that direction is eliminated, and 00:11:32.780 --> 00:11:35.100 this height is exactly half that at 00:11:35.100 --> 00:11:37.880 the edge of the crystal. 00:11:37.880 --> 00:11:41.480 So that's sort of a very specialized case of a crystal 00:11:41.480 --> 00:11:43.225 growing in a constrained environment. 00:11:47.160 --> 00:11:50.110 I'm digressing now, but let me make a few more remarks. 00:11:50.110 --> 00:11:52.485 The shape that a crystal has-- 00:11:55.930 --> 00:11:59.170 and the shape is in crystallography designated by 00:11:59.170 --> 00:12:00.460 a special term. 00:12:00.460 --> 00:12:01.710 It's called habit. 00:12:06.450 --> 00:12:10.960 Crystals that undergo plastic deformation never die, they 00:12:10.960 --> 00:12:12.280 just develop bad habits. 00:12:15.020 --> 00:12:15.590 Oh, come on. 00:12:15.590 --> 00:12:18.070 This is a tough crowd. 00:12:18.070 --> 00:12:19.510 I thought that was pretty funny. 00:12:19.510 --> 00:12:20.810 Anyway. 00:12:20.810 --> 00:12:25.420 The habit that a crystal has can depend very much on 00:12:25.420 --> 00:12:29.730 impurities in the solution or melt from which it grows. 00:12:29.730 --> 00:12:33.340 And let me suppose I start with a little crystal that 00:12:33.340 --> 00:12:35.650 looks like this. 00:12:35.650 --> 00:12:39.640 And there are two different kinds of faces on there. 00:12:39.640 --> 00:12:42.540 And let's suppose one of them grows rapidly and one of them 00:12:42.540 --> 00:12:44.060 grows slowly. 00:12:44.060 --> 00:12:48.050 Which face do you think would be the one that eventually 00:12:48.050 --> 00:12:51.000 forms the boundaries to the crystal volume? 00:12:51.000 --> 00:12:54.714 The fast-growing face or the slow-growing face? 00:12:54.714 --> 00:12:55.670 AUDIENCE: Fast-growing face. 00:12:55.670 --> 00:12:58.640 PROFESSOR: Fast, slow are both possibilities. 00:12:58.640 --> 00:13:00.320 Let's do a time lapse experiment. 00:13:00.320 --> 00:13:02.840 Let's suppose that this is the fast-growing face. 00:13:02.840 --> 00:13:06.070 And, after a certain amount of time these faces would have 00:13:06.070 --> 00:13:09.260 all advanced to this location. 00:13:09.260 --> 00:13:10.820 And this is the slow-growing face. 00:13:10.820 --> 00:13:14.210 And after the same amount of time, it will have advanced to 00:13:14.210 --> 00:13:16.550 this location. 00:13:16.550 --> 00:13:18.650 After the next time increment, this one would 00:13:18.650 --> 00:13:20.100 have grown up to here. 00:13:20.100 --> 00:13:22.270 This would only have grown that far. 00:13:22.270 --> 00:13:25.620 And you could see the only thing that's gonna be left is 00:13:25.620 --> 00:13:27.350 the slow-growing face. 00:13:27.350 --> 00:13:29.480 So it's sort of counter intuitive. 00:13:29.480 --> 00:13:32.740 The faces that survive to bound the crystal as rational 00:13:32.740 --> 00:13:35.430 planes are the slow-growing faces, not the 00:13:35.430 --> 00:13:36.680 rapidly growing faces. 00:13:39.050 --> 00:13:42.670 This is of great use to the people who want to control 00:13:42.670 --> 00:13:45.330 crystal shape. 00:13:45.330 --> 00:13:51.020 And this is very often in very every day, ordinary contexts. 00:13:51.020 --> 00:13:53.890 The people who grow table salt, for example, want to 00:13:53.890 --> 00:13:57.360 grow a crystal that has nice pointy edges so when it comes 00:13:57.360 --> 00:13:59.880 down on your tomato, it doesn't bounce off. 00:13:59.880 --> 00:14:02.110 It sticks to it. 00:14:02.110 --> 00:14:03.930 So the shape of table salt makes a 00:14:03.930 --> 00:14:06.260 difference in how it behaves. 00:14:06.260 --> 00:14:09.250 And the way you change the shape is to find something 00:14:09.250 --> 00:14:12.760 that will be absorbed preferentially on one set of 00:14:12.760 --> 00:14:17.060 faces that you would like to have be the faces that appear 00:14:17.060 --> 00:14:18.470 on the crystal. 00:14:18.470 --> 00:14:22.980 And that will do the job for you. 00:14:22.980 --> 00:14:27.070 I'll close with just one final example of this in real, every 00:14:27.070 --> 00:14:29.230 day industry or technology. 00:14:29.230 --> 00:14:35.790 I once had a phone call from a company that made salt for 00:14:35.790 --> 00:14:38.240 sprinkling on roads in the winter time, particularly 00:14:38.240 --> 00:14:41.830 around New England when the roads tend to ice up. 00:14:41.830 --> 00:14:46.070 And in an earlier era it, was politically correct to use 00:14:46.070 --> 00:14:47.760 sodium chloride. 00:14:47.760 --> 00:14:51.760 But you wanted the sodium chloride, if you could, to 00:14:51.760 --> 00:14:55.270 have an acicular, a needle-like, shape to improve 00:14:55.270 --> 00:14:57.680 its flow properties so you didn't waste a lot when you 00:14:57.680 --> 00:15:00.120 shook it out of the back of the truck. 00:15:00.120 --> 00:15:04.680 And the material that they use for that-- you think rock salt 00:15:04.680 --> 00:15:06.450 in the environment is bad-- 00:15:06.450 --> 00:15:11.580 they used a cyanide compound to modify the growth habit and 00:15:11.580 --> 00:15:13.430 make it needle-like. 00:15:13.430 --> 00:15:15.970 And they wanted to get away from that before they were 00:15:15.970 --> 00:15:22.330 really dragged into some sort of trial by the EPA. 00:15:22.330 --> 00:15:27.420 And did I know of something that could change the shape of 00:15:27.420 --> 00:15:29.025 the salt crystals for them? 00:15:29.025 --> 00:15:30.720 And the answer to that question was quite 00:15:30.720 --> 00:15:31.560 straightforward. 00:15:31.560 --> 00:15:32.450 No. 00:15:32.450 --> 00:15:34.100 That was the end of the conversation. 00:15:34.100 --> 00:15:36.760 But it was an interesting example of how surface 00:15:36.760 --> 00:15:39.090 chemistry can change crystal shape. 00:15:43.790 --> 00:15:44.260 OK. 00:15:44.260 --> 00:15:46.480 So all that long-winded explanation was a 00:15:46.480 --> 00:15:49.130 justification of the meaning of those 00:15:49.130 --> 00:15:51.490 two final two puzzles. 00:15:51.490 --> 00:15:54.010 And the final thing I have to say about them is that these 00:15:54.010 --> 00:15:57.810 puzzles were lifted from a puzzle book 00:15:57.810 --> 00:16:00.030 published by MENSA. 00:16:00.030 --> 00:16:02.790 Have any of you heard of MENSA? 00:16:02.790 --> 00:16:07.880 MENSA is an association of self-declared geniuses. 00:16:07.880 --> 00:16:10.120 So this was a puzzle book for geniuses. 00:16:10.120 --> 00:16:14.200 And you all did horribly well on that third problem set. 00:16:14.200 --> 00:16:16.810 Everybody at MIT is bright. 00:16:16.810 --> 00:16:20.410 But you folks are not only bright, you are geniuses 00:16:20.410 --> 00:16:24.294 because you cracked the MENSA puzzles. 00:16:24.294 --> 00:16:25.270 OK. 00:16:25.270 --> 00:16:27.080 Enough enjoyment and fun and games. 00:16:27.080 --> 00:16:31.070 Let's get serious, back down to what we were doing earlier. 00:16:31.070 --> 00:16:35.320 We had established in earlier discussion that there are a 00:16:35.320 --> 00:16:43.800 very limited number of ways in which we can arrange a mirror 00:16:43.800 --> 00:16:46.040 plane and a rotation axis about one 00:16:46.040 --> 00:16:47.690 fixed point in space. 00:16:47.690 --> 00:16:51.440 And these, accordingly, are called the point groups. 00:16:51.440 --> 00:16:54.640 And in particular, they are the two-dimensional 00:16:54.640 --> 00:16:56.215 crystallographic point groups. 00:17:03.100 --> 00:17:08.859 We had rotation axes by themselves, 1, 2, 3, 4, and 6. 00:17:08.859 --> 00:17:11.730 And these are the only ones we need worry about because these 00:17:11.730 --> 00:17:14.720 are the only rotational symmetries that are compatible 00:17:14.720 --> 00:17:16.369 with translation. 00:17:16.369 --> 00:17:17.679 Then we had a mirror plane. 00:17:20.790 --> 00:17:24.240 And then there was no reason why we could not combine a 00:17:24.240 --> 00:17:27.319 mirror plane with these rotational symmetries. 00:17:27.319 --> 00:17:31.500 And in doing so, we used a theorem that said if you take 00:17:31.500 --> 00:17:35.590 two reflection operations, sigma 1 and sigma 2-- and 00:17:35.590 --> 00:17:38.510 again, I use sigma to represent an individual 00:17:38.510 --> 00:17:41.350 operation of reflection-- 00:17:41.350 --> 00:17:45.540 and a mirror plane is the locus of operations of 00:17:45.540 --> 00:17:47.415 reflection, and then reflecting back again, there 00:17:47.415 --> 00:17:49.200 are two operations involved. 00:17:49.200 --> 00:17:53.920 And if these are combined at some angle mu, sigma 1 00:17:53.920 --> 00:17:58.480 followed by sigma 2, when they are separated by an angle mu 00:17:58.480 --> 00:18:02.940 is the same as the net transformation of a rotation 00:18:02.940 --> 00:18:06.300 about their point of intersection through twice the 00:18:06.300 --> 00:18:09.780 angle which the mirror planes are combined. 00:18:09.780 --> 00:18:11.560 And that led us-- 00:18:11.560 --> 00:18:14.730 in addition to a one-fold axis, a two-fold access, a 00:18:14.730 --> 00:18:18.260 three-fold axis, a four-fold axis, and a six-fold axis-- 00:18:18.260 --> 00:18:22.440 let us combine reflection with these axes to get symmetry 00:18:22.440 --> 00:18:26.080 combinations that we named according to the symmetry 00:18:26.080 --> 00:18:29.130 elements which were present in the final combination. 00:18:29.130 --> 00:18:34.990 We used two mirror planes if the mirror planes that arose 00:18:34.990 --> 00:18:38.410 where independent. 00:18:38.410 --> 00:18:41.770 Independent in the sense that the two-fold axis never turned 00:18:41.770 --> 00:18:42.750 one into another. 00:18:42.750 --> 00:18:45.750 And independent in the sense that if we draw a 00:18:45.750 --> 00:18:50.970 representative motif, the way those motifs were disposed 00:18:50.970 --> 00:18:53.900 about one mirror plane was different from the way they 00:18:53.900 --> 00:18:56.730 were arranged about the other one. 00:18:56.730 --> 00:19:00.830 For a four-fold axis, we could add mirror planes. 00:19:00.830 --> 00:19:03.830 And they would have to be at half the angle of the 00:19:03.830 --> 00:19:07.380 four-fold axis. 00:19:07.380 --> 00:19:09.450 And as we saw, it was kind of a mouthful 00:19:09.450 --> 00:19:10.700 to call this 4MMMMMMMM. 00:19:13.110 --> 00:19:16.210 There were just two kinds of mirror planes, so this one was 00:19:16.210 --> 00:19:18.240 called 4MM. 00:19:18.240 --> 00:19:23.300 6MM, analogously, was a six-fold axis with mirror 00:19:23.300 --> 00:19:26.490 planes separated by 30 degrees, half the throw of the 00:19:26.490 --> 00:19:28.400 six-fold axis. 00:19:28.400 --> 00:19:35.370 And the one that I left out is 3, not m m but 3M because a 00:19:35.370 --> 00:19:41.540 three-fold axis has mirror planes 00:19:41.540 --> 00:19:43.750 separated by 60 degrees. 00:19:43.750 --> 00:19:49.190 And the same thing goes on at each of these mirror planes, 00:19:49.190 --> 00:19:52.730 pair of objects hanging at one end of the mirror plane, the 00:19:52.730 --> 00:19:54.730 other end is unadorned. 00:19:54.730 --> 00:19:56.180 So they all behave the same. 00:19:56.180 --> 00:19:58.060 There's only one kind of mirror plane. 00:19:58.060 --> 00:20:01.530 The three-fold axis maps one mirror plane into another, and 00:20:01.530 --> 00:20:03.360 so they have to be the same. 00:20:03.360 --> 00:20:11.130 So therefore, only one M appears in the symbol. 00:20:11.130 --> 00:20:17.050 Independent of this, we had, when we combined reflection or 00:20:17.050 --> 00:20:21.230 rotation with a lattice, found that there were a very limited 00:20:21.230 --> 00:20:24.990 number of lattices in two dimensions, the oblique 00:20:24.990 --> 00:20:33.560 lattice which had T1 not equal to T2 and T1 inclined to T2 by 00:20:33.560 --> 00:20:36.690 some general angle. 00:20:36.690 --> 00:20:41.600 The next degree of specialization was for a 00:20:41.600 --> 00:20:44.580 two-fold axis-- so either a one-fold or a two-fold axis 00:20:44.580 --> 00:20:46.622 could fit in here. 00:20:46.622 --> 00:20:51.960 For a three-fold axis or a six-fold axis, we had to have 00:20:51.960 --> 00:20:56.390 a lattice in which the two translations were identical to 00:20:56.390 --> 00:21:01.230 one another, identical in magnitude and identical in the 00:21:01.230 --> 00:21:05.060 way the atoms were ranged relative to these 00:21:05.060 --> 00:21:06.300 translations. 00:21:06.300 --> 00:21:10.740 And the angle between T1 over T2 had to be 00:21:10.740 --> 00:21:13.560 identically 120 degrees. 00:21:13.560 --> 00:21:17.920 And by convention, we take the larger of two angles, 120 00:21:17.920 --> 00:21:21.806 rather than 60, to define the inter axial angle. 00:21:21.806 --> 00:21:27.130 A four-fold axis required a net that was exactly square, 00:21:27.130 --> 00:21:30.240 not approximately square but exactly square. 00:21:30.240 --> 00:21:33.930 So T1 and T2 had to be identical in magnitude. 00:21:33.930 --> 00:21:38.960 The angle between them had to be exactly 90 degrees. 00:21:38.960 --> 00:21:40.300 And close is no cigar. 00:21:40.300 --> 00:21:43.700 It has to be exactly 90 degrees because that angle is 00:21:43.700 --> 00:21:45.700 generated by symmetry. 00:21:45.700 --> 00:21:48.930 And then at the place in which I'm gonna pick up today in 00:21:48.930 --> 00:21:53.975 just a moment is the case of a mirror plane and reflection, 00:21:53.975 --> 00:21:57.970 we saw depending on how we arranged the translations 00:21:57.970 --> 00:22:02.090 relative to the reflection plane, could give us either a 00:22:02.090 --> 00:22:05.290 lattice in the shape of a rectangle-- 00:22:05.290 --> 00:22:11.480 and here T1 and T2 made an angle of exactly 90 degrees. 00:22:11.480 --> 00:22:15.210 But the magnitude of T1 could be anything it liked relative 00:22:15.210 --> 00:22:20.400 to the magnitude of T2, and then a double cell, which had 00:22:20.400 --> 00:22:24.140 the shape of a diamond. 00:22:24.140 --> 00:22:28.820 But it was operationally much to our advantage to pick a 00:22:28.820 --> 00:22:33.200 double cell which had a right angle in it. 00:22:33.200 --> 00:22:36.940 And the reason for that was that it is a nightmare to do 00:22:36.940 --> 00:22:40.750 calculations of inter-atomic distances and angles or angles 00:22:40.750 --> 00:22:44.040 between faces in an oblique system. 00:22:44.040 --> 00:22:47.980 The right angle makes these calculations very simple. 00:22:47.980 --> 00:22:51.270 And the fact that this cell tells you twice as much of the 00:22:51.270 --> 00:22:55.330 area as you really need to know about is a 00:22:55.330 --> 00:22:56.650 small price to pay. 00:22:56.650 --> 00:22:59.810 So this is the rectangular net, as 00:22:59.810 --> 00:23:01.550 we'll call it in words. 00:23:04.830 --> 00:23:06.480 And this is one that we'll call the 00:23:06.480 --> 00:23:07.730 centered rectangular net. 00:23:14.690 --> 00:23:15.170 OK. 00:23:15.170 --> 00:23:17.660 So let me pause here and suck in air and see if there are 00:23:17.660 --> 00:23:19.766 any questions. 00:23:19.766 --> 00:23:21.590 This is where we left off. 00:23:21.590 --> 00:23:24.500 And now having everything spread out on the board, we're 00:23:24.500 --> 00:23:26.320 going to start to make combinations and 00:23:26.320 --> 00:23:27.310 continue that process. 00:23:27.310 --> 00:23:28.090 Yes, sir. 00:23:28.090 --> 00:23:29.743 AUDIENCE: So the centered rectangular has the same exact 00:23:29.743 --> 00:23:31.961 [? intercept ?] as the rectangular but the only thing 00:23:31.961 --> 00:23:34.180 you're doing is you're deriving it based on symmetry 00:23:34.180 --> 00:23:36.645 that you're using like a diamond kind of thing to get 00:23:36.645 --> 00:23:37.138 the center one? 00:23:37.138 --> 00:23:37.640 PROFESSOR: Yeah. 00:23:37.640 --> 00:23:41.760 So all these arose when we did it slowly and systematically. 00:23:41.760 --> 00:23:45.930 We said let's let this be the location of the mirror plane. 00:23:45.930 --> 00:23:49.210 What happens if we combine it with a translation? 00:23:49.210 --> 00:23:51.820 We can take one lattice point on the mirror plane since 00:23:51.820 --> 00:23:53.740 there's no unique lattice point. 00:23:53.740 --> 00:23:55.530 And then the mirror plane is gonna reflect 00:23:55.530 --> 00:23:57.710 this over to here. 00:23:57.710 --> 00:23:59.340 So here's a first translation. 00:23:59.340 --> 00:24:02.170 I've got a second non-colinear translation. 00:24:02.170 --> 00:24:03.420 Wham-o. 00:24:03.420 --> 00:24:04.530 Instant lattice. 00:24:04.530 --> 00:24:09.720 So what I'm doing is taking this diamond shape and I am 00:24:09.720 --> 00:24:13.530 taking one translation here-- let's put some labels on this. 00:24:13.530 --> 00:24:16.870 Let's call this T1 and this T2. 00:24:16.870 --> 00:24:21.930 So this translation here, call it T1 prime, is my original 00:24:21.930 --> 00:24:24.490 translation T1 plus T2. 00:24:24.490 --> 00:24:30.778 In this translation, T2 is the negative of the-- 00:24:30.778 --> 00:24:31.004 Yeah. 00:24:31.004 --> 00:24:34.660 It's gonna be minus T1 plus T2. 00:24:34.660 --> 00:24:37.130 So I've taken linear combinations of the two 00:24:37.130 --> 00:24:38.950 vectors of the diamond-shaped cell. 00:24:41.590 --> 00:24:46.870 And again, this is contrary to the rules we have that say 00:24:46.870 --> 00:24:48.980 pick the shortest two translations. 00:24:48.980 --> 00:24:50.010 Why? 00:24:50.010 --> 00:24:52.295 The final larger area of volume then you need. 00:24:52.295 --> 00:24:56.990 And the answer to that is occasionally the provision of 00:24:56.990 --> 00:25:00.970 a coordinate system, which is far easier to work in, is a 00:25:00.970 --> 00:25:04.375 small price to pay for that redundancy. 00:25:04.375 --> 00:25:04.800 OK. 00:25:04.800 --> 00:25:06.770 And then the primitive rectangular net, just to 00:25:06.770 --> 00:25:09.855 remind you again, we said is there any case in which this 00:25:09.855 --> 00:25:11.100 is not true? 00:25:11.100 --> 00:25:15.380 And that case was if you take T1 exactly perpendicular to 00:25:15.380 --> 00:25:19.200 the mirror plane and then you define only a lattice row. 00:25:19.200 --> 00:25:22.110 So that gives you a second choice for T2. 00:25:22.110 --> 00:25:24.960 You can't pick it anywhere you like, otherwise it gets 00:25:24.960 --> 00:25:27.970 reflected across just as in the first case, and you have 00:25:27.970 --> 00:25:31.800 two translations that are not compatible. 00:25:31.800 --> 00:25:35.850 And the only way that they are compatible is that if you make 00:25:35.850 --> 00:25:38.870 the translation T1 straddle the mirror plane. 00:25:38.870 --> 00:25:40.810 And that's what we've already got up here. 00:25:40.810 --> 00:25:42.870 So that's how we got the rectangular net. 00:25:42.870 --> 00:25:46.320 We could get a second distinct sort of lattice only if this 00:25:46.320 --> 00:25:49.460 translation was exactly in the plane of the mirror line. 00:25:49.460 --> 00:25:49.880 Yes, sir. 00:25:49.880 --> 00:25:53.039 AUDIENCE: But the number and kind of symmetry elements are 00:25:53.039 --> 00:25:54.740 the same for both, right? 00:25:54.740 --> 00:25:56.684 PROFESSOR: Exactly. 00:25:56.684 --> 00:25:58.640 AUDIENCE: Uh, so-- 00:25:58.640 --> 00:26:01.670 PROFESSOR: So there's a curious sort of duality here. 00:26:01.670 --> 00:26:06.610 Here in the case of a hexagonal net, this is just a 00:26:06.610 --> 00:26:10.030 lattice of a very specialized shape. 00:26:10.030 --> 00:26:12.790 And does that have a six-fold axis in it? 00:26:12.790 --> 00:26:14.870 No, not unless I decide to put one in. 00:26:14.870 --> 00:26:17.660 I could put in a three-fold axis, alternatively. 00:26:17.660 --> 00:26:20.700 So here, I have one lattice that can accept two different 00:26:20.700 --> 00:26:21.900 kinds of symmetry. 00:26:21.900 --> 00:26:23.950 Here I've got the reverse situation. 00:26:23.950 --> 00:26:27.500 I have two distinct kind of lattices, both of which are 00:26:27.500 --> 00:26:31.860 happy and content with the same symmetry on them. 00:26:31.860 --> 00:26:35.560 So there's going to be a far greater number of combinations 00:26:35.560 --> 00:26:39.180 of lattice with symmetry than simply a one-to-one 00:26:39.180 --> 00:26:43.600 correspondence between lattice types and point group types. 00:26:49.040 --> 00:26:49.440 OK. 00:26:49.440 --> 00:26:50.690 Any other questions? 00:26:53.120 --> 00:26:58.460 All right so let me now shift down to low gear and remind 00:26:58.460 --> 00:27:03.930 you of one thing that we did last time. 00:27:03.930 --> 00:27:14.060 We asked what happens if I can combine a rotation operation, 00:27:14.060 --> 00:27:18.690 a alpha, with a translation? 00:27:18.690 --> 00:27:21.510 Call this T1. 00:27:21.510 --> 00:27:25.900 Sounds like something I already did in showing that 00:27:25.900 --> 00:27:30.490 the values of alpha are restricted to values. 00:27:30.490 --> 00:27:34.480 But what I'm going to do now is use something that looks as 00:27:34.480 --> 00:27:38.220 though it's a similar starting point to arrive at a different 00:27:38.220 --> 00:27:41.270 result that we saw last time. 00:27:41.270 --> 00:27:47.760 I'm going to deliberately look at a line that is alpha over 2 00:27:47.760 --> 00:27:52.410 on one side of the perpendicular to T1. 00:27:52.410 --> 00:27:58.730 And then the operation of the a alpha is going to take this 00:27:58.730 --> 00:28:01.930 translation with a lattice point of necessity at the end 00:28:01.930 --> 00:28:06.530 of it, and it's going to move it over to here, alpha over 2, 00:28:06.530 --> 00:28:10.540 on the other side of the translation of the 00:28:10.540 --> 00:28:12.470 perpendicular. 00:28:12.470 --> 00:28:15.850 Then I'm going to pick this up with the translation T1 and 00:28:15.850 --> 00:28:21.050 move it and everything along it to a location like this. 00:28:21.050 --> 00:28:28.280 And my question now is, what is a alpha followed by the 00:28:28.280 --> 00:28:31.805 translation T1? 00:28:31.805 --> 00:28:34.540 The handedness of the objects have not changed. 00:28:34.540 --> 00:28:35.900 So this has to be either 00:28:35.900 --> 00:28:39.530 translation or another rotation. 00:28:39.530 --> 00:28:43.510 And what we can very quickly show is if upon dropping a 00:28:43.510 --> 00:28:48.050 perpendicular down to the original translation, these 00:28:48.050 --> 00:28:49.310 lines are parallel. 00:28:49.310 --> 00:28:50.800 This line cuts across it. 00:28:50.800 --> 00:28:54.470 So this angle is alpha over 2. 00:28:54.470 --> 00:28:58.730 And this line is inclined to the perpendicular 00:28:58.730 --> 00:29:00.140 by alpha over 2. 00:29:00.140 --> 00:29:05.580 So this angle in here is also alpha over 2. 00:29:05.580 --> 00:29:11.900 So the answer is that if I rotate from one location to 00:29:11.900 --> 00:29:16.420 another by rotation alpha and then pick up that motif or the 00:29:16.420 --> 00:29:22.710 entire space and translate it by the original translation T1 00:29:22.710 --> 00:29:26.010 to get a third one-- this is number 1, right-handed, say, 00:29:26.010 --> 00:29:28.040 number 2, right-handed. 00:29:28.040 --> 00:29:29.150 This is number 3. 00:29:29.150 --> 00:29:30.470 Stays right-handed. 00:29:30.470 --> 00:29:32.750 Has to be related by a rotation. 00:29:32.750 --> 00:29:37.540 And we next ask, what point is the point about which the 00:29:37.540 --> 00:29:39.230 rotation occurs? 00:29:39.230 --> 00:29:40.110 Then we used-- 00:29:40.110 --> 00:29:44.240 what at the time seemed like a trivial observation-- 00:29:44.240 --> 00:29:47.830 that a rotation axis in space or a rotation point in two 00:29:47.830 --> 00:29:51.080 dimensions is the locus that is left 00:29:51.080 --> 00:29:55.500 unmoved by the rotation. 00:29:55.500 --> 00:30:00.460 So if rotating from here and translating over to here is 00:30:00.460 --> 00:30:03.770 equivalent to a rotation and if the locus of the rotation 00:30:03.770 --> 00:30:08.530 has to be the point that's left on moved, bingo. 00:30:08.530 --> 00:30:11.370 That's where the rotation occurs. 00:30:11.370 --> 00:30:14.260 And it's through the same angle, so we can label this an 00:30:14.260 --> 00:30:16.870 operation b alpha. 00:30:16.870 --> 00:30:21.050 And we know exactly where that point is gonna be. 00:30:21.050 --> 00:30:24.080 It's gonna be along the perpendicular bisector of the 00:30:24.080 --> 00:30:25.300 original translation. 00:30:25.300 --> 00:30:30.180 And it's gonna be up a distance x which is T over 2 00:30:30.180 --> 00:30:33.185 times the cotangent of alpha over 2. 00:30:33.185 --> 00:30:38.490 So now we've got another, what I call, combination theorems. 00:30:38.490 --> 00:30:42.260 And again, this is nothing more than knowing how to 00:30:42.260 --> 00:30:46.490 complete the product of two operations in establishing the 00:30:46.490 --> 00:30:48.130 group multiplication table. 00:30:48.130 --> 00:30:51.970 so let me write it down in the form of a combination theorem. 00:30:51.970 --> 00:30:59.060 This says that a alpha followed by a translation is 00:30:59.060 --> 00:31:01.570 another rotation-- 00:31:01.570 --> 00:31:04.190 in the same sense, both counter 00:31:04.190 --> 00:31:06.550 clockwise in this example-- 00:31:06.550 --> 00:31:10.280 another rotation b alpha in the same sense about a point 00:31:10.280 --> 00:31:15.550 that is always T over 2 times the cotangent of alpha over 2 00:31:15.550 --> 00:31:18.310 along the perpendicular bisector. 00:31:18.310 --> 00:31:20.870 So that's how you would go about making an entry in the 00:31:20.870 --> 00:31:24.050 group multiplication table. 00:31:24.050 --> 00:31:24.260 Sorry. 00:31:24.260 --> 00:31:24.890 I put you off. 00:31:24.890 --> 00:31:25.316 AUDIENCE: Yeah. 00:31:25.316 --> 00:31:27.840 Can you just, like, I just want to see what you did 00:31:27.840 --> 00:31:30.830 actually to the motif in that drawing. 00:31:30.830 --> 00:31:31.310 PROFESSOR: OK. 00:31:31.310 --> 00:31:35.520 Rather than taking an abstraction, an arbitrary line 00:31:35.520 --> 00:31:40.210 that is at alpha over 2, I put a motif in the space that 00:31:40.210 --> 00:31:44.240 doesn't necessarily hang on this particular locus that I 00:31:44.240 --> 00:31:45.550 drew for reference. 00:31:45.550 --> 00:31:49.650 And the rotation through alpha would move it to here. 00:31:49.650 --> 00:31:51.600 It's lagging behind this first line. 00:31:51.600 --> 00:31:54.550 It lagged behind the second one by the same amount. 00:31:54.550 --> 00:31:57.850 Then I pick it up and I move it by the same translation and 00:31:57.850 --> 00:32:02.780 that slides it over here, still canted over to the left. 00:32:02.780 --> 00:32:06.260 And now what I claim is that I get from the first one to the 00:32:06.260 --> 00:32:07.170 third one-- 00:32:07.170 --> 00:32:09.740 and actually should have raised your hand on a 00:32:09.740 --> 00:32:10.470 different matter. 00:32:10.470 --> 00:32:12.540 This is out in front of the translation. 00:32:12.540 --> 00:32:15.380 So number 3 should sit here, also to the right to the 00:32:15.380 --> 00:32:16.720 translation. 00:32:16.720 --> 00:32:19.590 And the way I get from 1 to 3 in one shot is by 00:32:19.590 --> 00:32:20.919 rotating alpha b. 00:32:23.913 --> 00:32:26.408 OK. 00:32:26.408 --> 00:32:31.840 Now, a cautionary note just so you do not use your new power 00:32:31.840 --> 00:32:33.330 recklessly. 00:32:33.330 --> 00:32:47.710 Let me emphasize that this is an escalation in operations, 00:32:47.710 --> 00:32:49.183 not in symmetry elements. 00:32:59.260 --> 00:33:04.990 And the reason for that is that a rotation axis, in 00:33:04.990 --> 00:33:08.190 general, contains a number of different rotations. 00:33:08.190 --> 00:33:12.840 An n-fold axis has n different rotation operations implied in 00:33:12.840 --> 00:33:15.760 it, including the identity operation of 00:33:15.760 --> 00:33:17.850 rotating 360 degrees. 00:33:17.850 --> 00:33:21.460 So if I say I'm gonna drop a four-fold access in a lattice, 00:33:21.460 --> 00:33:24.570 what I'm doing is dropping into the lattice a 90 degree 00:33:24.570 --> 00:33:27.430 rotation, a 180 degree rotation, and 00:33:27.430 --> 00:33:30.180 a 270 degree rotation. 00:33:30.180 --> 00:33:35.500 And the location of the new operations depends on alpha. 00:33:35.500 --> 00:33:38.050 So these new operations are not gonna pop 00:33:38.050 --> 00:33:39.560 up at the same location. 00:33:39.560 --> 00:33:43.660 They're gonna be sprinkled over different locations. 00:33:43.660 --> 00:33:46.740 So as an abstraction, that may be hard to appreciate. 00:33:46.740 --> 00:33:49.960 But we're going to straight away derive a 00:33:49.960 --> 00:33:52.560 couple of more additions. 00:33:52.560 --> 00:33:56.320 And then when we see what happens upon adding a rotation 00:33:56.320 --> 00:34:00.570 axis to a lattice in a couple of nontrivial cases, I'll just 00:34:00.570 --> 00:34:03.710 summarize the results and assume that you'll be able to 00:34:03.710 --> 00:34:06.450 derive them on your own. 00:34:06.450 --> 00:34:09.120 The last time we had got around to doing just one of 00:34:09.120 --> 00:34:11.460 these combinations. 00:34:11.460 --> 00:34:16.730 And we ask what happens when you combine with a translation 00:34:16.730 --> 00:34:20.290 a rotation operation a pi. 00:34:20.290 --> 00:34:24.760 So we'll take some motif that sets up here, right-handed, 00:34:24.760 --> 00:34:29.960 rotate it by 180 degrees to get a second one, and then 00:34:29.960 --> 00:34:35.020 pick that up and translate it by T. So I'll get a third one 00:34:35.020 --> 00:34:39.650 sitting down like this of the same handedness. 00:34:39.650 --> 00:34:41.750 And the question now is how do I get from 00:34:41.750 --> 00:34:43.449 1 to number 3 directly? 00:34:43.449 --> 00:34:45.020 Well, let's use our theorem. 00:34:45.020 --> 00:34:48.010 Our theorem says we should go a distance x along the 00:34:48.010 --> 00:34:52.989 perpendicular bisector, which is half the magnitude of T 00:34:52.989 --> 00:34:57.310 times the cotangent of pi over 2. 00:34:57.310 --> 00:35:00.120 The cotangent of pi over 2 is zero. 00:35:00.120 --> 00:35:01.760 So we go up a distance x. 00:35:01.760 --> 00:35:04.860 That's equal to 0, all in the perpendicular bisector, which 00:35:04.860 --> 00:35:06.680 means we stay put. 00:35:06.680 --> 00:35:08.860 And-- 00:35:08.860 --> 00:35:13.060 in this class, there is always truth in advertising. 00:35:13.060 --> 00:35:18.580 Around this locus here is a rotation operation b pi, which 00:35:18.580 --> 00:35:22.820 takes the first one into the third one. 00:35:22.820 --> 00:35:25.720 OK. 00:35:25.720 --> 00:35:30.450 And now, lickety split, I will derive again-- and I apologize 00:35:30.450 --> 00:35:33.500 for going fast because I did do it last time. 00:35:33.500 --> 00:35:37.730 Let's ask what happens when we combine a two-fold axis with a 00:35:37.730 --> 00:35:40.160 parallelogram net? 00:35:40.160 --> 00:35:44.030 So we're taking a two-fold axis plus a parallelogram net. 00:35:47.020 --> 00:35:49.070 In this two dimensional lattice there are two 00:35:49.070 --> 00:35:54.880 translations, T1 and T2, of arbitrary magnitude with some 00:35:54.880 --> 00:35:56.960 angle between them that's arbitrary. 00:35:56.960 --> 00:36:01.440 And to this lattice point, I'll add a two-fold axis. 00:36:01.440 --> 00:36:06.880 A two-fold axis involves the presence of two operations, 00:36:06.880 --> 00:36:10.490 the identity operation, and that we can forget about, and 00:36:10.490 --> 00:36:14.520 then there's the operation a pi. 00:36:14.520 --> 00:36:19.250 By combine a pi with T1, I'll get an operation b pi that 00:36:19.250 --> 00:36:20.910 sits at the midpoint of T1. 00:36:20.910 --> 00:36:28.340 If I combine a pi with T2, we get an operation c pi at the 00:36:28.340 --> 00:36:29.900 midpoint of T2. 00:36:29.900 --> 00:36:37.540 If I combine a pi with T1 plus T2, I'll get another operation 00:36:37.540 --> 00:36:40.680 d pi at the center of the cell. 00:36:40.680 --> 00:36:44.540 And we don't have to ask what happens when we go three 00:36:44.540 --> 00:36:46.160 translations over. 00:36:46.160 --> 00:36:49.730 There's gonna be another operation of 180 degree 00:36:49.730 --> 00:36:51.340 rotation on that translation. 00:36:51.340 --> 00:36:55.420 But we really only need bother about what goes on on the 00:36:55.420 --> 00:36:58.720 edges and in the interior of one unit cell, because 00:36:58.720 --> 00:37:02.340 whatever's there has to be repeated by translation. 00:37:02.340 --> 00:37:15.270 So, in making these additions, we need consider only 00:37:15.270 --> 00:37:16.740 independent translations-- 00:37:16.740 --> 00:37:19.110 that is, not related to one another by 00:37:19.110 --> 00:37:20.660 the rotational symmetry-- 00:37:20.660 --> 00:37:28.050 independent translations that terminate 00:37:28.050 --> 00:37:29.300 within the unit cell. 00:37:40.910 --> 00:37:45.860 And that means for the addition of a two-fold axis, I 00:37:45.860 --> 00:37:48.570 need do only what I have, in fact, just done. 00:37:48.570 --> 00:37:57.850 I want to consider a pi with T1, T2, and T1 plus T2. 00:38:02.180 --> 00:38:07.010 If I have the operation a pi or b pi or c pi or d pi at 00:38:07.010 --> 00:38:15.720 these four locations, that is all I need have present to say 00:38:15.720 --> 00:38:19.250 that I have, in addition to the two-fold axis that I added 00:38:19.250 --> 00:38:23.390 to the lattice point, a two-fold axis halfway along 00:38:23.390 --> 00:38:28.230 T1, a two-fold access halfway along T2, and another one 00:38:28.230 --> 00:38:30.680 smack in the middle of the cell. 00:38:30.680 --> 00:38:32.140 These are lattice points. 00:38:32.140 --> 00:38:36.050 So those axes have to be repeated by translations, this 00:38:36.050 --> 00:38:40.560 one from here to here, this one from here, this one from 00:38:40.560 --> 00:38:41.600 here to here. 00:38:41.600 --> 00:38:45.240 So I'll have two-fold axes in the middle of all of the edges 00:38:45.240 --> 00:38:51.150 of the cell plus this fourth one right in the middle. 00:38:51.150 --> 00:38:56.610 So this is an example of a two dimensional space group. 00:38:56.610 --> 00:39:01.470 It's a symmetry that acts on all of space. 00:39:01.470 --> 00:39:10.400 And the names that we will give to these combinations is 00:39:10.400 --> 00:39:14.820 a symbol for the symmetry element that we added, in this 00:39:14.820 --> 00:39:16.930 case a two-fold axis. 00:39:16.930 --> 00:39:21.480 And then we'll specify the nature of the lattice to which 00:39:21.480 --> 00:39:22.630 we've added it. 00:39:22.630 --> 00:39:25.930 And all that I have to say is that the lattice is primitive 00:39:25.930 --> 00:39:30.200 because you're all aware at this point that a two-fold 00:39:30.200 --> 00:39:33.540 axis requires only a primitive, a 00:39:33.540 --> 00:39:34.990 parallelogram net. 00:39:34.990 --> 00:39:37.770 The only place we'll need a special symbol is when we add 00:39:37.770 --> 00:39:40.610 a mirror plane to either the rectangular net or the 00:39:40.610 --> 00:39:41.910 centered rectangular net. 00:39:41.910 --> 00:39:45.450 And then we'll have to specify which type of lattice of that 00:39:45.450 --> 00:39:48.023 shape, primitive or centered, we're dealing with. 00:39:48.023 --> 00:39:48.790 Yes, sir. 00:39:48.790 --> 00:39:50.511 AUDIENCE: So all you did with that derivation there is prove 00:39:50.511 --> 00:39:52.115 that you don't have to worry about any 00:39:52.115 --> 00:39:54.039 rotations outside of that? 00:39:54.039 --> 00:39:54.520 PROFESSOR: That's correct. 00:39:54.520 --> 00:39:55.482 AUDIENCE: That's correct? 00:39:55.482 --> 00:39:57.410 PROFESSOR: Yep. 00:39:57.410 --> 00:39:59.640 Which is very fortunate 'cause there's an infinite number 00:39:59.640 --> 00:40:02.730 lurking outside the boundaries of the cell. 00:40:02.730 --> 00:40:05.300 I'll show you another general truth. 00:40:05.300 --> 00:40:07.650 What does a pattern look like? 00:40:07.650 --> 00:40:12.780 The pattern looks exactly like the pattern of a two-fold axis 00:40:12.780 --> 00:40:16.530 with that pattern of motifs hung at every lattice point. 00:40:23.780 --> 00:40:29.250 These new two-fold axes don't do any further repetition of 00:40:29.250 --> 00:40:31.610 the motifs. 00:40:31.610 --> 00:40:34.040 They don't do any repetition of the motif. 00:40:34.040 --> 00:40:38.770 They just express relation between things that you get 00:40:38.770 --> 00:40:41.820 when you take this pair and then repeat it by the 00:40:41.820 --> 00:40:43.250 translations. 00:40:43.250 --> 00:40:46.500 So this two-fold access, for example, relates this to this. 00:40:46.500 --> 00:40:49.460 This two-fold axis relates this one to this one, and this 00:40:49.460 --> 00:40:51.130 one to this one, and so on. 00:40:51.130 --> 00:40:54.280 They're just relations that exist between the things that 00:40:54.280 --> 00:40:57.260 you get when you add the pattern of a two-fold axis to 00:40:57.260 --> 00:40:59.090 a lattice point. 00:40:59.090 --> 00:41:02.420 So another generalization of what we're doing here that 00:41:02.420 --> 00:41:17.100 turns out to be valid, the pattern of motifs for a 00:41:17.100 --> 00:41:43.460 particular plane group is nothing more than the pattern 00:41:43.460 --> 00:41:44.760 of the symmetry element-- 00:41:51.540 --> 00:41:55.670 symmetry element or symmetry elements, sometimes we'll be 00:41:55.670 --> 00:41:56.920 adding more than one-- 00:42:08.890 --> 00:42:12.330 that we've added to a lattice point. 00:42:21.090 --> 00:42:25.330 So in other words, if we take 2MM and drop that into a 00:42:25.330 --> 00:42:27.700 rectangular lattice, the pattern is simply gonna be 00:42:27.700 --> 00:42:33.400 these four atoms related by 2MM hung at every corner of 00:42:33.400 --> 00:42:34.385 the lattice. 00:42:34.385 --> 00:42:37.700 If you add it to a centered lattice, these would be hung 00:42:37.700 --> 00:42:39.890 at every corner of the lattice, and also at the 00:42:39.890 --> 00:42:41.330 centered lattice point. 00:42:41.330 --> 00:42:42.790 So it's that simple. 00:42:42.790 --> 00:42:45.490 If you look at some of these high symmetry space groups or 00:42:45.490 --> 00:42:48.050 plane groups, wow, there's symmetry all over the place. 00:42:48.050 --> 00:42:49.960 You say, how can I draw a pattern for that? 00:42:49.960 --> 00:42:50.550 Easy. 00:42:50.550 --> 00:42:53.060 Just do what the point group does and drop it in at the 00:42:53.060 --> 00:42:55.300 lattice points. 00:42:55.300 --> 00:42:59.530 So that turns out to be a universal feature of all space 00:42:59.530 --> 00:43:00.780 groups and plane groups. 00:43:03.187 --> 00:43:03.670 OK. 00:43:03.670 --> 00:43:05.320 Somebody had a question over here and I cut you off. 00:43:05.320 --> 00:43:05.750 Yeah. 00:43:05.750 --> 00:43:06.180 AUDIENCE: Yeah. 00:43:06.180 --> 00:43:08.871 On that two [INAUDIBLE] diagram, wouldn't you want to 00:43:08.871 --> 00:43:11.642 use four of those because you said it was-- you don't 00:43:11.642 --> 00:43:14.255 concern ones that are-- you're only concerned independent 00:43:14.255 --> 00:43:15.690 ones, the ones you can't get by-- 00:43:15.690 --> 00:43:17.090 PROFESSOR: You're absolutely right. 00:43:17.090 --> 00:43:19.070 And let's look at this pattern again. 00:43:19.070 --> 00:43:21.610 Here are a pair that are hanging close to 00:43:21.610 --> 00:43:23.680 this two-fold axis. 00:43:23.680 --> 00:43:26.910 Here's a pair hanging close to the two-fold axis, very 00:43:26.910 --> 00:43:30.520 different arrangement, different arrangement relative 00:43:30.520 --> 00:43:32.780 to this two-fold axis, different arrangements 00:43:32.780 --> 00:43:34.260 relative to this two-fold axis. 00:43:34.260 --> 00:43:36.660 So what we're saying is-- and this is the way, in fact, we 00:43:36.660 --> 00:43:37.680 derive them-- 00:43:37.680 --> 00:43:41.640 that these four two-fold axes are all distinct. 00:43:41.640 --> 00:43:46.210 But the ones that are along the edges are just related by 00:43:46.210 --> 00:43:47.800 translation. 00:43:47.800 --> 00:43:49.420 So why do I have to put them in? 00:43:49.420 --> 00:43:52.240 AUDIENCE: You just put them in to just show the symmetry? 00:43:52.240 --> 00:43:52.710 PROFESSOR: Yeah. 00:43:52.710 --> 00:43:53.490 Exactly. 00:43:53.490 --> 00:43:57.080 Exactly. 'Cause if I say this is a lattice and I have 00:43:57.080 --> 00:44:03.900 two-fold axes here, that's an incomplete representation 00:44:03.900 --> 00:44:05.980 because if these are translations, I jolly well 00:44:05.980 --> 00:44:09.290 better have the same thing at all the lattice points. 00:44:09.290 --> 00:44:12.230 I'm just emphasizing the property of a lattice which is 00:44:12.230 --> 00:44:16.930 understood to be present in all of these combinations. 00:44:16.930 --> 00:44:19.330 OK. 00:44:19.330 --> 00:44:20.580 Other questions? 00:44:26.380 --> 00:44:30.060 And I would remind you that a reasonable question or request 00:44:30.060 --> 00:44:31.890 would be, could you do that all again, 00:44:31.890 --> 00:44:33.800 please, at half speed? 00:44:33.800 --> 00:44:35.700 And I'd be happy to comply. 00:44:35.700 --> 00:44:38.500 But really this is ground that we covered last time. 00:44:38.500 --> 00:44:40.100 And I'm just reinforcing it. 00:44:40.100 --> 00:44:42.650 So now that we've run out of time, we can go on to 00:44:42.650 --> 00:44:43.910 something new. 00:44:43.910 --> 00:44:46.940 Let me do, in the couple minutes remaining, let me make 00:44:46.940 --> 00:44:52.130 one more addition just to show you how things change when you 00:44:52.130 --> 00:44:54.930 add a rotation axis that has more than 00:44:54.930 --> 00:44:58.745 one rotation operation. 00:44:58.745 --> 00:45:01.970 Let me ask what happens when you add a 00:45:01.970 --> 00:45:05.430 four-fold axis to a lattice. 00:45:05.430 --> 00:45:10.050 We've seen that a four-fold axis can coexist with no 00:45:10.050 --> 00:45:15.940 lattice other than 1, which is dimensionally square and which 00:45:15.940 --> 00:45:22.940 has the same thing hanging about, two translations that 00:45:22.940 --> 00:45:27.400 are exactly 90 degrees apart. 00:45:27.400 --> 00:45:32.030 So what we're doing is taking a four-fold axis and drop it 00:45:32.030 --> 00:45:36.030 in at the corner lattice point and then step back before all 00:45:36.030 --> 00:45:37.872 hell breaks looks. 00:45:37.872 --> 00:45:40.510 Now, first thing we're gonna do is to get the four-fold 00:45:40.510 --> 00:45:44.980 axis at every corner lattice point. 00:45:44.980 --> 00:45:48.970 And then we're gonna use our theorem, taking into account 00:45:48.970 --> 00:45:52.910 that when I say I'm adding a four-fold axis, I am adding 00:45:52.910 --> 00:46:01.010 the operations a pi over 2, a pi, a3 pi over 2. 00:46:01.010 --> 00:46:06.490 But let me call that, instead, a minus pi over 2, the same 00:46:06.490 --> 00:46:08.900 thing and a little easier to deal with. 00:46:08.900 --> 00:46:11.860 And then the operation a2 pi, which is the same as the 00:46:11.860 --> 00:46:13.020 identity operation. 00:46:13.020 --> 00:46:14.950 And that's dull and uninteresting, so we won't 00:46:14.950 --> 00:46:17.580 consider that addition. 00:46:17.580 --> 00:46:22.820 We've already done this, a pi, in deriving P2. 00:46:22.820 --> 00:46:24.810 And that's the nice thing about this derivation. 00:46:24.810 --> 00:46:28.400 It's gonna snowball because as the symmetry gets higher, 00:46:28.400 --> 00:46:32.220 we've already done the work for the subgroup that contains 00:46:32.220 --> 00:46:34.955 the elements that are present in the higher symmetry. 00:46:34.955 --> 00:46:39.550 So we'll get an operation a pi over a pi here. 00:46:39.550 --> 00:46:42.010 We'll get an operation a pi here. 00:46:42.010 --> 00:46:45.950 And if we take the diagonal translation, T1 plus T2, I'll 00:46:45.950 --> 00:46:47.600 get the operation a pi here. 00:46:47.600 --> 00:46:52.260 Or I should probably call them a pi, b pi, and c pi to 00:46:52.260 --> 00:46:58.050 indicate that these are different two-fold axes. 00:46:58.050 --> 00:46:58.510 OK. 00:46:58.510 --> 00:47:03.580 Let me emphasize again that this theorem that we have is a 00:47:03.580 --> 00:47:06.330 theorem in individual operations and 00:47:06.330 --> 00:47:07.760 not symmetry elements. 00:47:07.760 --> 00:47:10.670 If I say a four-fold axis is present, it means that all of 00:47:10.670 --> 00:47:13.560 these operations exist about that locus. 00:47:13.560 --> 00:47:18.260 I cannot say that a four-fold axis sits here. 00:47:18.260 --> 00:47:20.550 I cannot say a four-fold axis sits here. 00:47:20.550 --> 00:47:24.740 I'm gonna have to take each of these combinations in turn and 00:47:24.740 --> 00:47:29.250 then step back and see what operations exist about the 00:47:29.250 --> 00:47:31.720 various loci within the cell. 00:47:31.720 --> 00:47:34.890 So the one that's really different is the combination 00:47:34.890 --> 00:47:40.465 of a translation T with the operation a pi over 2. 00:47:42.980 --> 00:47:52.300 And my theorem says if I put in a motif and rotate it by 90 00:47:52.300 --> 00:47:53.550 degrees to here-- 00:47:53.550 --> 00:47:56.770 so this is number 1 and this is number 2-- 00:47:56.770 --> 00:48:02.780 and then translate it to a location that's related to the 00:48:02.780 --> 00:48:05.010 first by symmetry-- 00:48:05.010 --> 00:48:08.030 by translation to get number 3-- 00:48:08.030 --> 00:48:15.240 they will be related by an operation b pi over 2 that 00:48:15.240 --> 00:48:19.020 sits up along the perpendicular bisector by an 00:48:19.020 --> 00:48:25.100 amount x, which is T over 2 times the cotangent of alpha 00:48:25.100 --> 00:48:36.483 over 2, T over 2 times the cotangent of 1/2 of pi over 2, 00:48:36.483 --> 00:48:37.780 45 degrees. 00:48:37.780 --> 00:48:43.660 The cotangent of 45 degrees is unity because the two edges of 00:48:43.660 --> 00:48:46.220 the triangle are equal in length. 00:48:46.220 --> 00:48:50.860 So this says I should go up along the perpendicular 00:48:50.860 --> 00:48:54.760 bisector by an amount that is half the length of the cell. 00:48:57.810 --> 00:49:01.410 And that puts me right in the middle of the cell. 00:49:01.410 --> 00:49:08.330 So the operation b pi over 2 sits not at the edge of the 00:49:08.330 --> 00:49:12.290 cell, but at the center of the cell. 00:49:12.290 --> 00:49:17.660 And that, I think you will agree, even in this hastily 00:49:17.660 --> 00:49:21.950 sketched diagram, is the way number 1 is related directly 00:49:21.950 --> 00:49:23.860 to number 3 in one shot. 00:49:26.686 --> 00:49:29.970 Now I could do it on the cheap from here. 00:49:29.970 --> 00:49:33.260 But let's show that we have first an 00:49:33.260 --> 00:49:35.340 operation a pi over 2. 00:49:35.340 --> 00:49:38.520 We've already got the operation c pi. 00:49:38.520 --> 00:49:43.070 Let's now combine the operation a minus pi over 2. 00:49:46.810 --> 00:49:54.480 And that says that for c minus pi over 2, I would go up the 00:49:54.480 --> 00:49:58.270 perpendicular bisector a distance T over 2 times the 00:49:58.270 --> 00:50:03.400 cotangent of minus 1/2 of pi over 2. 00:50:03.400 --> 00:50:08.010 And the cotangent of a negative angle is minus that. 00:50:08.010 --> 00:50:11.520 So I go a distance minus T over 2 along the 00:50:11.520 --> 00:50:13.710 perpendicular bisector. 00:50:13.710 --> 00:50:19.870 And what that is gonna do is to bring me to the center of 00:50:19.870 --> 00:50:22.510 the cell that's directly below. 00:50:22.510 --> 00:50:30.000 And this will be the rotation c minus pi over 2. 00:50:30.000 --> 00:50:32.650 But everything has to be translation equivalent at the 00:50:32.650 --> 00:50:38.460 interval T. So I can just move this operation c minus pi over 00:50:38.460 --> 00:50:42.160 2 up to the center of my original cell. 00:50:42.160 --> 00:50:45.780 So now I have at the center of the cell all the operations of 00:50:45.780 --> 00:50:48.700 a four-fold axis. 00:50:48.700 --> 00:50:53.370 So a four-fold axis exists in the middle of the cell. 00:50:53.370 --> 00:50:56.660 And let me write the results in here. 00:50:56.660 --> 00:50:59.420 I've now got all of the operations of a four-fold axis 00:50:59.420 --> 00:51:00.540 sitting here. 00:51:00.540 --> 00:51:03.030 I've got all the operations of a two-fold axis 00:51:03.030 --> 00:51:04.430 sitting here and here. 00:51:04.430 --> 00:51:08.650 So this is the final result, if I translate the two-fold 00:51:08.650 --> 00:51:10.950 axes over to the other edges of the cell. 00:51:17.410 --> 00:51:18.846 AUDIENCE: What is that angle [? by the d project? ?] 00:51:18.846 --> 00:51:21.080 That should be 90 degrees? 00:51:21.080 --> 00:51:22.410 PROFESSOR: That should be 90 degrees, 00:51:22.410 --> 00:51:23.580 if I drew it carefully. 00:51:23.580 --> 00:51:25.090 Yeah. 00:51:25.090 --> 00:51:32.250 So let me give the final result with a pattern in it. 00:51:32.250 --> 00:51:34.350 These are the two translations, equal in length, 00:51:34.350 --> 00:51:37.800 four-fold here, four-fold here, four-fold here. 00:51:37.800 --> 00:51:40.620 They are all the same four-fold axis. 00:51:40.620 --> 00:51:43.180 Four-fold axis in the middle of the cell. 00:51:43.180 --> 00:51:46.970 Two-fold axis in the middle of the cell edges. 00:51:46.970 --> 00:51:51.780 And the pattern of objects is the set of things that are 00:51:51.780 --> 00:51:56.330 rotated by 90 degrees that form a square about the corner 00:51:56.330 --> 00:51:57.580 of the cell. 00:51:59.570 --> 00:52:00.960 Another one here. 00:52:00.960 --> 00:52:04.760 Another one that sits up here. 00:52:04.760 --> 00:52:08.000 And another one that sits like this. 00:52:08.000 --> 00:52:11.060 Four of them on the corners of the square. 00:52:11.060 --> 00:52:14.830 And that's all we're going to get in this pattern. 00:52:14.830 --> 00:52:19.440 Again, it's just a pattern of a four-fold axis that is hung 00:52:19.440 --> 00:52:23.970 at every lattice point of the square net. 00:52:23.970 --> 00:52:34.010 And these other symmetry elements that arise simply are 00:52:34.010 --> 00:52:37.360 things that relate to the squares that are hanging at 00:52:37.360 --> 00:52:40.350 every corner of the square cell. 00:52:47.820 --> 00:52:48.180 OK. 00:52:48.180 --> 00:52:51.970 So the square in the center of the cell, for example, relates 00:52:51.970 --> 00:52:53.520 these four. 00:52:53.520 --> 00:52:55.030 And you can pick any other four and they'd 00:52:55.030 --> 00:52:56.230 be related as well. 00:52:56.230 --> 00:52:58.780 The two-fold axis relates this to this. 00:52:58.780 --> 00:53:02.490 This two-fold axis relates this to this and this to this. 00:53:02.490 --> 00:53:06.910 So the pattern is just the square produced by four 00:53:06.910 --> 00:53:09.620 stamped out at every corner of the square net. 00:53:09.620 --> 00:53:10.870 And it's that simple. 00:53:14.400 --> 00:53:17.164 The name of this thing-- 00:53:17.164 --> 00:53:18.080 what have we done? 00:53:18.080 --> 00:53:20.740 We've taken a four-fold axis, we've put it 00:53:20.740 --> 00:53:22.390 in a primitive net. 00:53:22.390 --> 00:53:26.520 I don't have to tell you it's squared because in your heart 00:53:26.520 --> 00:53:28.455 of hearts know that that is what a 00:53:28.455 --> 00:53:30.480 four-fold axis requires. 00:53:30.480 --> 00:53:32.050 So this describes completely the 00:53:32.050 --> 00:53:33.810 combinations that's been made. 00:53:33.810 --> 00:53:36.010 And this is a representative pattern. 00:53:36.010 --> 00:53:38.680 A pattern of this sort is one that we've seen several times 00:53:38.680 --> 00:53:40.620 already in describing sample symmetries. 00:53:40.620 --> 00:53:44.610 That is the plane group of square floor tiles. 00:53:44.610 --> 00:53:50.900 It's the plane group of the square mesh that's in the 00:53:50.900 --> 00:53:54.540 overhead lighting fixtures. 00:53:54.540 --> 00:53:58.380 It's the pattern that is present in the tiles up above 00:53:58.380 --> 00:54:00.060 the lighting fixtures. 00:54:00.060 --> 00:54:03.160 It's a pattern that is very commonly present in square 00:54:03.160 --> 00:54:06.820 shirts, but not too often. 00:54:06.820 --> 00:54:09.070 It's kind of dull and uninteresting. 00:54:09.070 --> 00:54:09.890 Yes. 00:54:09.890 --> 00:54:13.560 Fellow back there against the wall wearing P4, probably 00:54:13.560 --> 00:54:16.410 didn't know until now. 00:54:16.410 --> 00:54:17.660 Thank you. 00:54:20.030 --> 00:54:21.050 Actually, he's a shield. 00:54:21.050 --> 00:54:21.710 I made him come in. 00:54:21.710 --> 00:54:23.280 I called him up last night and said be sure 00:54:23.280 --> 00:54:24.610 you wear the P4 shirt. 00:54:24.610 --> 00:54:27.530 And he said, OK, I will. 00:54:27.530 --> 00:54:28.090 All right. 00:54:28.090 --> 00:54:28.890 I'm getting silly. 00:54:28.890 --> 00:54:31.190 So that's tells me it's time to quit. 00:54:31.190 --> 00:54:33.550 So let's take our usual 10 minute break, 00:54:33.550 --> 00:54:34.950 And then we'll resume. 00:54:34.950 --> 00:54:38.120 And I think I'll go lickety split through the remaining 00:54:38.120 --> 00:54:41.610 plane groups that consists of combinations of a rotation 00:54:41.610 --> 00:54:42.690 with a lattice. 00:54:42.690 --> 00:54:44.580 OK. 00:54:44.580 --> 00:54:45.830 So do come back.